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Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 39
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 40516)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 39
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
\noindent Human performance is rarely perfectly consistent over time. Instead, the same workers and their teams can give outstanding performances on some days but be rubbish on others. In striving to predict the future, humans often fixate on salient patterns in historical performance data, such as streaks of success or failure, even if the underlying data-generating process is somewhat driven by white noise and randomness. One such fixation is the concept of the ``hot hand'', according to which there is a belief that serial correlation exists in human performance. This concept has been analysed often in sports. Since the seminal study by \citet{gilovich1985hot}, who did not find support for the hot hand in professional basketball shooting, the evidence on whether such beliefs could be correct has been generally mixed (e.g., \citealp{green2017hot,miller2016surprised,otting2020hot,tversky1989cold,wetzels2016bayesian}).
While many studies investigate a potential hot hand effect in sports, this notion has also been applied to other settings, especially finance; strategies that attempt to identify, follow and profit from market trends are popular among professional investors. In the finance literature, there is abundant evidence that this ``momentum'' explains some of the cross-section of returns in financial markets
\citep{Asness2013,Barroso2015,Carhart1997,ehsani2022factor,Fama2012,Jegadeesh2011,Novy2012},
as well as defining profitable investment strategies in time series frameworks \citep{chan1996momentum,Moskowitz2012,Zakamulin2022}.
However, as typically occurs in finance due to the complexity of markets and the multitude of factors that drive asset prices, there are also some findings in the literature that do not support the existence of momentum \citep{Huang2020,Kim2016}.
When testing behavioural asset pricing theories, sports betting markets constitute a useful real-world laboratory with many advantages over traditional financial markets \citep{bar2020ask,Thaler1988}. The terminal value of sports betting contracts is exogenous to investor behaviour and provides clean identification of mispricing, unlike in traditional financial markets. In particular, if prices deviate from fundamentals due to cognitive biases or erroneous beliefs among the market participants, they will be corrected on average by the sporting outcomes, which are exogenous to these biases or beliefs \citep{MOSKOWITZ2021}. Biases investigated in betting markets include the overvaluation of longshots (e.g., \citealp{Angelini2019,OTTAVIANI2008favlong,Vlastakis2009}) and overreactions to major in-play events such as goals (e.g., \citealp{Angelini2022,Choi2014,Croxson2014}).
In this paper, we have unusual access to high-resolution betting market data provided by a large European bookmaker. We use these data to investigate how bettors respond to perceived momentum. The data cover second-by-second betting odds and volumes staked on all potential outcomes for two seasons of the German Bundesliga, covering 612 football matches, enabling a clean analysis of the responses to momentum within betting markets. We focus on the equalising goal in matches that have a 1-1 scoreline since this event implies some notion of apparent momentum. Although equalisers reset the two teams to their relative position at kick-off, the team that recently scored the equaliser might be considered more likely to score another goal due to the momentum gained by scoring the equaliser. For these 1-1 scorelines, we investigate the betting activity in the minute after the equalisers. We find no evidence that the sequence of goals on average impacts the winning chances of teams or the betting odds provided by the bookmaker. However, we find that bettors have a strong tendency toward betting on the team that has gained momentum. The traded volumes placed are considerably higher on teams that scored the equaliser, compared to teams that conceded at 1-1. In particular, stakes placed on teams that scored the equaliser are about 40\% higher compared to teams that just conceded the equaliser. We further find that a corresponding betting strategy of always following this momentum, or believing it could be profitable, would in fact yield substantially negative returns to bettors.
Correspondingly, in financial markets, trade volumes have been shown to have a fundamental link with price momentum \citep{lee2000price}. Closely related to our study, \cite{Levitt2004}, in his seminal study on the economics of gambling markets, analysed some data akin to real-world betting stakes, but these related to a pre-match season-long prediction competition for American football, with an entry fee and no cost per bet, for a relatively small number of selected participants, and thus did not cover general market activity. \citeauthor{Levitt2004} though used these data to challenge the balanced book hypothesis, whereby bookmakers would just adjust their prices according to the flow of bets and eliminate risk from their position in the market. \citeauthor{Levitt2004}'s results instead suggested that bookmakers may not only be good at forecasting outcomes but also know the likely behavioural biases of their customers, which they can exploit to maximise profits.
Our paper contributes to the literature on behavioural asset pricing in financial markets, by investigating behavioural biases in high-frequency (in-play) betting markets.
There are other contributions in the literature that analyse some sort of momentum effects in betting markets (e.g., \citealp{brown1993does,camerer1989does,Durand2021,Krieger2021,Metz2022,paul2005bettor,wheatcroft2020profiting,woodland2000testing}), but at a much lower frequency level of data, through the long-term patterns in final event outcomes, typically covering weeks or months, and mostly focusing only on posted prices (odds) and not volumes (stakes). \cite{Paul2014hothand}, however, provide the exception, being to the best of our knowledge the only study to use actual data on betting action from bookmakers to study the ``hot hand'' in sports. They found that the volume of pre-game betting activity on National Football League matches followed teams on a hot streak of wins and avoided those on a losing streak. Our study is the first to consider momentum and betting activity at a much higher frequency and immediacy.
The rest of the paper is structured as follows. Section 2 introduces the unique high-frequency betting market dataset. Section 3 empirically
investigates the impact of momentum on match outcomes and betting markets. Section 4 concludes. Several robustness checks are provided in the Appendix.
\section{Data}
\noindent Our dataset was provided by a major European bookmaker that has a large customer base in Germany. It covers second-by-second betting odds and volumes for all 612 German Bundesliga football matches in the 2017/18 and 2018/19 seasons, including in-play information about the timing of major events (such as goals and red cards). The betting stakes (amounts) in the dataset have been multiplied by the same constant for all matches since we do not have the bookmaker's permission to represent the true amount of monetary units. Regardless, our data allow us to compare the volumes of betting stakes across and within matches without providing the actual values or statistics of true stakes.
To investigate how bettors (the bookmaker) respond to teams seemingly having momentum, we investigate their betting behaviour (odds-setting process) after the equalising goal for the scoreline 1-1.\footnote{Football is a low-scoring game, and in the entire history of professional football, 1-1 has been the most likely final outcome of a match \cite{reade2021evaluating}.} Equalisers reset the teams to the position they were in prior to the match beginning. Nevertheless, it is possible that the identity of whichever team scored the most recent goal may affect the subsequent behaviour of all the agents involved. In other words, the sequence of goals up to and including the equaliser may help to predict what will happen in the remainder of the match, conditional on pre-match expectations. Both of these possibilities would imply some notion of momentum in a football match, whereby scoring a goal suggests that a team is more likely to score further goals than beforehand.
As betting volumes vary across teams, we model the relative stakes placed at a moment or over some period of time, where we consider the proportion of the total betting volumes that were placed on a particular team to win the match. In particular, we focus on the stakes placed, winning chances, and betting prices during the next minute after a 1-1 equaliser was scored. For the 612 matches considered, 233 had an intermediate scoreline of 1-1. If a 1-1 equaliser is scored fairly late in a match, such as in the 85th minute or later, much lower absolute stakes tend to be placed in the market due to there being little time left to play, and thus relative stakes across the three match outcomes also become noisy. If the 1-1 is scored during injury time, after the regular 90 minutes of play are complete, then the market does not necessarily reopen again at all (depending on the amount of injury time that the referee has awarded). We thus consider only observations where the equaliser is scored before the 85th minute, resulting in 212 match observations in our sample.
Figure~\ref{fig:example_data} shows an example minute-by-minute time series of relative stakes from our data for a match between Schalke 04 and VfL Wolfsburg, which kicked off at 18:00 CEST on January 20, 2019, and ended 2-1. Before the match began, the bookmaker prices suggested it would be closely fought, with decimal odds of 2.25 for the home team and 3.0 for both the away team and the draw outcome. This example also gives a first glance at how bettors respond to goals and whatever pricing strategy the bookmaker follows in their aftermath. Slightly higher relative stakes were placed on the less-favoured away team, Wolfsburg, than on Schalke early in the match, while the stakes placed on Schalke increased after they scored the first goal of the match. When Wolfsburg scored the equaliser, the relative stakes placed on Wolfsburg to win increased, and are even larger compared to the early stages of the match before the first goal. As introduced above, such betting behaviour may well be driven by bettors believing there is value in momentum for the equalising team. For all matches that featured a 1-1 scoreline, we observe in the minute after, on average, 47\% of the stakes placed on the equalising team, 37.2\% placed on the conceding team, and only 18.6\% on the draw. However, for these matches, the draw is the most likely final outcome (39.3\%). A defeat for the equalising team was the second most likely outcome (33.2\%), followed by a win for the equalising team (27.5\%). These descriptives give a first impression that bettors believe there is value in the apparent momentum, although the win for the equalising team is the least likely outcome.
In the example match, the absolute betting activity increased after the equaliser. For the time from the kick-off, until the first goal was scored, there was an average amount staked of 47 per minute (transformed values). During the three minutes before the 1-1 equaliser, the betting activity was slightly reduced, with an average amount staked of 30. The amount staked climaxed in the minute after the equaliser at an average of 60 per minute. However, the betting activity quickly reduced thereafter, with an average stake per minute of 49 in the 10 minutes following the equaliser.
\begin{figure}[ht!]
\centering
\caption{Schalke 04 vs VfL Wolfsburg, 20 January, 2019: Example time series of in-play relative stakes placed on a win for the home team (Schalke), a draw, and the away team (Wolfsburg).}
\includegraphics[scale=1]{example_ts.pdf}
\caption*{\footnotesize Notes.- The vertical dashed lines denote when goals were scored by Schalke 04 (blue lines) and VfL Wolfsburg (green lines). The grey shaded area indicates half time.}
\label{fig:example_data}
\end{figure}
The main variables of interest in our analysis, which are introduced in the following, are summarised in Table~\ref{tab:descriptives}. As bettors are more likely to wager their money on favourites, we consider a team's odds-implied probability (\textit{impprob}) of winning at the kick-off, which is derived by taking the inverse of the posted decimal odds for them to win, and normalising this such that it sums to one with the inverse of the odds for the other two possible match outcomes. Teams conceding the equaliser generally have a higher implied probability at the kick-off, unsurprisingly indicating that favourites are more likely to first have a lead and then concede an equalising goal in matches that get to 1-1. This is also in line with the finding from above that a defeat is more likely than a win for an equalising team. If an equaliser is scored a few minutes before the final whistle, then bettors may be unlikely to place their money on a team to win but rather on a draw. We thus consider the \textit{minute} of the equaliser, which is, on average, minute 47, but also occurs in our sample as early as the 5th minute and as late as the 84th minute. As undermanned teams have a reduced chance to win a match, we consider red cards received before the 1-1 equaliser.\footnote{A red card, or sending off, is a relatively rare but severe punishment in a football match, with the receiving team having to play with one fewer player for the remainder of the match.} In particular, the explanatory variable \textit{redcard} gives the difference in the number of red cards received between the equalising and conceding teams before a match arrives at 1-1, which in our sample lies strictly between minus and positive one.
\begin{table}[ht!] \centering
\caption{Sample descriptive statistics for 212 matches in German Bundesliga seasons 2017/18 and 2018/19 featuring a 1-1 scoreline}
\label{tab:descriptives}
\begin{tabular}{@{\extracolsep{5pt}}lcccc}
\\[-1.8ex]\hline
\\[-1.8ex]
& \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{St.\ dev.} & \multicolumn{1}{c}{Min.\ } & \multicolumn{1}{c}{Max.\ } \\
\hline \\[-1.8ex]
\textit{relstake} (equalising team) & 0.470 & 0.253 & 0.049 & 0.960 \\
\textit{relstake} (conceding team) & 0.372 & 0.247 & 0.012 & 0.911 \\
\textit{impprob} (equalising team) & 0.357 & 0.188 & 0.037 & 0.909 \\
\textit{impprob} (conceding team) & 0.445 & 0.192 & 0.050 & 0.926 \\
\textit{minute} & 46.87 & 20.30 & 5 & 84 \\
\textit{redcard} & 0 & 0.195 & $-$1 & 1 \\
\hline
\end{tabular}
\caption*{\footnotesize Notes.- \textit{relstake}: the relative betting stakes placed in the minute after the equaliser for the 1-1 scoreline; \textit{impprob}: the inverse of decimal odds at kick-off; \textit{minute}: the minute of the match when the 1-1 equaliser arrived; and \textit{redcard}: the difference in red cards received between the equalising and conceding teams before the 1-1 equaliser.}
\end{table}
\FloatBarrier
\section{The Impact of Momentum on Match Outcomes and Betting Markets}
\noindent While the summary statistics in the previous section suggest that bettors tend to adjust their bets in accordance with the match dynamics, an in-depth analysis is needed to understand whether momentum predicts (i) the final match outcome, (ii) how the bookmaker sets prices, and (iii) how bettors respond. Our approach is in three parts. First, we analyse whether the sequence of goals (momentum) in getting to a 1-1 scoreline impacts the eventual overall outcome of matches. If it does, then betting on that momentum could be a rational action if the bookmaker does not reflect it in their pricing. Second, we determine whether the bookmaker adjusts its pricing in line with what might be perceived as momentum. Third and finally, we study the stakes placed after the 1-1 equaliser to determine whether betting activity is affected by what might either be or only appear to be momentum.
\subsection{Do Equalising Goals Generate Momentum?}
\noindent In the absence of any momentum and everything else equal (i.e., the same balance of winning probabilities and thus team strengths as at kick-off, the minute in the match, and the number of red cards received), the probability of a team winning a football match would be identical after conceding the equalising goal at 1-1 compared with scoring it. On the contrary, a greater probability to go on and subsequently win the match when equalising instead of conceding would indicate a genuine sense of momentum.
We modify our dataset for our empirical approach. Each match appears twice in the estimation samples, once from the perspective of the equalising team and once from the perspective of the conceding team. The response variable in this analysis is $\textit{win}_{i,m}$, where $\textit{win}_{i,m}=1$ if the considered team $i$ actually won match $m$ and is zero otherwise. The main explanatory variable of interest is $\textit{equaliser}_{i,m}$, where $\textit{equaliser}_{i,m}=1$ when the considered team $i$ scored the equalising goal in match $m$ and $\textit{equaliser}_{i,m}=0$ if they conceded. For the control variables, $\textit{impprob}_{i,m}$ covers the winning chances of team $i$ prior to match $m$. The higher is $\textit{impprob}_{i,m}$, the higher we would expect the winning chances to be right after the equalising goal. $\textit{minute}_m$ captures the minute the equalising goal was scored. The later in the match that the equaliser is scored, the more likely the match ends in a draw and the less likely either team is going to win. $\textit{redcard}_{i,m}$ is the difference in the number of red cards between the teams. If $\textit{redcard}_{i,m}>0$, then the team received more red cards than the opponent and hence has less players on the field. We expect the winning chances of a team to decrease as $\textit{redcard}_{i,m}$ increases. As such, we model whether a team won the match using logistic regression:
\begin{align}
\text{logit}\big(\Pr(win_{i,m} = 1) \big) =& \beta_0 + \beta_1 \cdot impprob_{i,m} + \beta_2 \cdot equaliser_{i,m} + \notag \\ & \beta_3 \cdot minute_{m} + \beta_4 \cdot redcard_{i,m} \ . \label{eq:one}
\end{align}
Since each match appears twice in the estimations, we cluster standard errors at the match level.
Table~\ref{tab:momentum} displays the results for the full sample of 212 matches (Column I) and for observations of the first (Column II) and second half (Column III), respectively. We do not find evidence for equalising goals generating momentum in the full sample. All the control variables generate coefficients with expected signs for the likelihood of a team going on to win after a 1-1 equaliser: positive for the pre-match expectations of a win according to odds; negative for a late equaliser; and negative for having received more red cards than the opponent. Checking whether momentum exists in the first or second half separately, we again do not find any evidence that equalising goals generate momentum in any of these cases.
In Appendix~Table~\ref{tab:momentum_robustness}, we show that the main results in column (I) of Table~\ref{tab:momentum} are robust to extensions of the model given by Equation~\eqref{eq:one}, including: a squared term for the minute of the equalising goal; interacting the minute of the equaliser and $redcard_{i,m}$, which is insignificant; interacting $impprob_{i,m}$ and $equaliser_{i,m}$, in case there is momentum only for either equalising favourites or longshots, which there is not; and interacting $equaliser_{i,m}$ and $minute_{m}$, in case there is momentum only for either late or early goals, which there is not.
\begin{table}[ht!]
\caption{Does scoring momentum impact match outcomes?}
\centering
\label{tab:momentum}
\begin{tabular}{@{\extracolsep{5pt}}lccc}
\\[-1.8ex]\hline
\\[-1.8ex]
& \multicolumn{3}{c}{\textit{Timing of equaliser for 1-1}} \\
\cline{2-4}
\\[-1.8ex] & Any time & First half & Second half \\
\\[-1.8ex] & (I) & (II) & (III) \\
\hline \\[-1.8ex]
\textit{impprob} ($\beta_1$) & 0.045$^{***}$ & 0.035$^{***}$ & 0.058$^{***}$ \\
& (0.008) & (0.010) & (0.013) \\
& & & \\
\textit{equaliser} ($\beta_2$) & 0.115 & $-$0.010 & 0.257 \\
& (0.286) & (0.397) & (0.415) \\
& & & \\
\textit{minute} ($\beta_3$) & $-$0.014$^{***}$ & $-$0.019$^{*}$ & $-$0.030$^{**}$ \\
& (0.005) & (0.010) & (0.014) \\
& & & \\
\textit{redcard} ($\beta_4$) & $-$2.268$^{***}$ & (no red cards) & $-$2.411$^{***}$ \\
& (0.646) & & (0.672) \\
& & & \\
Constant ($\beta_0$) & $-$2.101$^{***}$ & $-$1.564$^{***}$ & $-$1.672 \\
& (0.446) & (0.562) & (1.096) \\
& & & \\
\hline
$N$ of matches & 212 & 101 & 111 \\
$N$ of observations & 424 & 202 & 222 \\
McFadden $R^2$ & 0.132 & 0.084 & 0.191 \\
\hline
\end{tabular}
\caption*{\footnotesize Notes.- Logistic regression estimates of Equation~\eqref{eq:one}. \newline $^{***}$, $^{**}$, $^{*}$ indicate significance from zero of the model coefficients at the 1\%, 5\% and 10\% levels, respectively, two-sided tests, with standard errors in parentheses that account for match-level clustering.}
\end{table}
\FloatBarrier
\subsection{Does the Bookmaker `Believe' in Momentum?}
\noindent While we find no evidence in our sample of matches that momentum affects the likelihood of match outcomes, the bookmaker could still systematically alter odds according to the sequence of goals in a 1-1 scoreline for two reasons. First though unlikely, the price setting could be biased because the bookmaker believes in the impact of momentum on the final match outcomes. Second, it could anticipate that bettors believe in momentum and adjust betting odds accordingly to secure profits.
To check whether bookmaker pricing of the win outcomes is affected by the sequence of goals scored up to an equaliser, we consider the bookmaker's posted \textit{odds} for each team $i$ to win in the minute after the 1-1 scoreline in match $m$. We use the same control variables as for the match outcome in Equation~\eqref{eq:one} and consider the following linear regression model:
\begin{align}
odds_{i,m} = & \, \beta_0 + \beta_1 \cdot impprob_{i,m} + \beta_2 \cdot equaliser_{i,m} + \beta_3 \cdot minute_m \notag\\ & + \beta_4 \cdot redcard_{i,m} + u_{i,m} \ . \label{eq:two}
\end{align}
Table~\ref{tab:bookmaker} reports the estimated parameters, again for both the full sample and sub-samples for first and second half equalisers. For all model formulations, the identity of who scored the $\textit{equaliser}$ has an insignificant effect on the bookmaker win odds. The estimated direction of the control variable effects on the post-equaliser betting odds are all in line with our expectations. The pre-match expectations of a win for a team, proxied by $\textit{impprob}_{i,m}$, can significantly and positively explain their odds to win after the equaliser. The later in the match that the equaliser was scored, the lower the team's subsequent odds to secure a win, as the draw is the most probable final outcome. The difference in the number of red cards received prior to the equaliser reduced the winning odds of a team.
In columns (I)-(IV) of Appendix~Table~\ref{tab:bookmaker_robustness}, we show that the main results in column (I) of Table~\ref{tab:bookmaker} are robust to extensions of the model given by Equation~\eqref{eq:two}, including: a squared term for the minute of the equalising goal; interacting the minute of the equaliser and $redcard_{i,m}$, which is insignificant; interacting $impprob_{i,m}$ and $equaliser_{i,m}$, in case there is evidence that the bookmaker only alters odds according to whether it was either the pre-match favourite or longshot that scored the equaliser, which there is not; and interacting $equaliser_{i,m}$ and $minute_{m}$, in case there is evidence that the bookmaker only alters odds according to whether it was a late or early equalising goal, which there is not. In columns (V)-(VIII) of Appendix~Table~\ref{tab:bookmaker_robustness}, comparable results are shown that use the odds-implied probability of a win after the equaliser as the response variable, instead of the decimal odds, thus showing that our results are not sensitive to this choice.
\begin{table}[ht!] \centering
\caption{Do bookmaker odds for the win reflect momentum?
}
\label{tab:bookmaker}
\scalebox{0.9}{
\begin{tabular}{@{\extracolsep{5pt}}lccc}
\\[-1.8ex]\hline
\\[-1.8ex]
& \multicolumn{3}{c}{\textit{Timing of equaliser for 1-1}} \\
\cline{2-4}
\\[-1.8ex] & Any time & First half & Second half \\ \\[-1.8ex] & (I) & (II) & (III) \\
\hline \\[-1.8ex]
\textit{impprob} ($\beta_1$) & $-$0.100$^{***}$ & $-$0.109$^{***}$ & $-$0.090$^{***}$ \\
& (0.004) & (0.007) & (0.004) \\
& & & \\
\textit{equaliser} ($\beta_2$) & $-$0.017 & $-$0.255 & 0.204$^{*}$ \\
& (0.128) & (0.223) & (0.122) \\
& & & \\
\textit{minute} ($\beta_3$) & 0.011$^{**}$ & $-$0.008 & 0.040$^{***}$ \\
& (0.005) & (0.012) & (0.009) \\
& & & \\
\textit{redcard} ($\beta_4$) & 1.931$^{***}$ & (no red cards) & 1.902$^{***}$ \\
& (0.300) & & (0.317) \\
& & & \\
Constant ($\beta_0$) & 7.118$^{***}$ & 8.158$^{***}$ & 4.820$^{***}$ \\
& (0.438) & (0.596) & (0.594) \\
& & & \\
\hline \\[-1.8ex]
$N$ of matches & 212 & 101 & 111 \\
$N$ of observations & 424 & 202 & 222 \\
$R^{2}$ & 0.563 & 0.506 & 0.709 \\
\hline
\end{tabular}}
\caption*{\footnotesize Notes.- Estimates of Equation~\eqref{eq:two}. $^{***}$, $^{**}$, $^{*}$ indicate significance from zero of the model coefficients at the 1\%, 5\% and 10\% levels, respectively, two-sided tests, with standard errors in parentheses that account for match-level clustering.}
\end{table}
\FloatBarrier
\subsection{Do Bettors `Believe' in Momentum?}
\noindent The results from the previous two parts of our analysis provide evidence for neither teams generally gaining momentum after scoring a 1-1 equaliser nor the bookmaker pricing the win according to which team scored last. In the third part of our analysis, we study the relative stakes placed by bettors after 1-1 equalisers, to investigate whether or not bettors believe there is value in momentum and bet more money than they should on the equalising teams.
The descriptives in Table~\ref{tab:descriptives} above already indicate that the relative stakes placed on the equalising team are substantially higher compared to the team conceding the 1-1 equaliser. To extend this exploratory analysis, we split our sample according to money bet on strong favourites (pre-match odds $<$ 1.5), moderate favourites (pre-match odds between 1.5 and 2.7), moderate longshots (pre-match odds between 2.7 and 4) and strong longshots (pre-match odds~$>$~4). As our previous results did not indicate any significant momentum effects for the match outcome nor corresponding pricing response by the bookmaker, the relative stakes for both equalising and conceding teams should be very similar if bettors behaved rationally. However, for the four subsamples considered, the mean relative stakes placed on the team that scored/conceded the equaliser are as follows:
\begin{itemize}
\item[-] strong favourites: 0.779 (scored), 0.591 (conceded)
\item[-] moderate favourites: 0.542 (scored), 0.415 (conceded)
\item[-] moderate longshots: 0.431 (scored), 0.248 (conceded)
\item[-] strong longshots: 0.290 (scored), 0.157 (conceded)
\end{itemize}
These summary statistics indicate substantial differences in betting activity between stakes placed on the equalising and conceding teams. In addition to these means, Appendix~Figure~\ref{fig:hists} shows the corresponding histograms of relative stakes for the four subsamples. For bets on strong pre-match favourites, relative stakes are generally larger if they score rather than concede the 1-1 equaliser. In this case, there is some support for bettors acting sensibly to the return on investment from following a hypothetical betting strategy of always betting on a clear pre-match favourite that just scored the equaliser, although from a small sample. Such a strategy would have yielded a return on investment (ROI) of 0.6\% (20 bets, 13 won), compared with the reverse strategy of always betting the same amount on clear pre-match favourites that just conceded the equaliser, which yields an ROI of -13.9\% (29 bets, 17 won). For context, the average overround (sometimes called the `vig' or `profit margin') implied by the bookmaker odds in our data after the equalising goal is 7.9\%.
Similar patterns can be observed for stakes placed on pre-match moderate favourites, moderate longshots, and strong longshots. Reinforcing the patterns observed in Figure~\ref{fig:example_data} for the example match earlier, there is clear exploratory evidence that the relative stakes for the team that scores at 1-1 are, on average, substantially larger in the aftermath, thus suggesting that bettors believe betting on momentum, conditional on the prices offered, will be profitable for them. They were clearly wrong, at least if they were following the simple strategy of always betting the same amount on the equalising team. The ROIs from always backing equalising pre-match moderate favourites, moderate longshots, and strong longshots in such a way, in our sample, would have been -20.1\% (70 bets, 23 won), -7.4\% (50 bets, 12 won), and -23.3\% (72 bets, 10 won), respectively.
To support what appears to be convincing exploratory findings, i.e., that betting activity tends to follow the team with apparent momentum, we use an econometric approach. To explain $\textit{relstake}_{i,m}$ -- the share of volume bet on a particular team $i$ to win in the minute after the equalising goal in match $m$ -- we estimate the following model:
\begin{align}
relstake_{i,m} = & \beta_0 + \beta_1 \cdot startodds_{i,m} + \beta_2 \cdot equaliser_{i,m} + \beta_3 \cdot minute_m \notag\\ & +
\beta_4 \cdot prerelstake_{i,m} + \beta_5 \cdot redcard_{i,m} + u_{i,m} \ , \label{eq:three}
\end{align}
where control variables once again include $minute_m$ and $redcard_{i,m}$. We also control for the starting odds of a win prior to the match for team $i$, $startodds_{i,m}$, to model whether bettors tend to bet on pre-match favourites after an equaliser. Further, we include the relative stakes in the one minute prior to the equaliser, $prerelstake_{i,m}$ to model the general tendency of the market to favour betting on one team over the other possible match outcomes.
The estimation results in Table~\ref{tab:bettors} show that bettors generally tend to back pre-match favourites over longshots after an equaliser, though only significantly so in the second half after conditioning on the relative stakes prior to the first goal of the match. Again, the later the equaliser is scored, the lower are the chances that one team goes on to win, and hence the lower the relative stakes on a win as the final outcome. As red cards reduce the chances of winning, bettors tend to back wins less for undermanned teams (unless there are 15 or fewer minutes remaining in the match). The relative stakes placed on a team prior to an equalising goal significantly predict the relative stakes after, though the coefficient for this variable in the model is also significantly less than one. Most importantly, column (I) of Table~\ref{tab:bettors} supports the exploratory findings from above and shows that significantly and substantially higher betting volumes tend to back teams that score the equaliser compared to teams that concede. The model estimates show that the relative stakes placed on a team to win are on average 12.7 percentage points higher in the minute after they scored the equaliser compared with if they had instead conceded, remembering also that absolute betting volumes tend to double in our sample after the equaliser compared with the three minutes before.
For the model shown in column (I), fixing all control variables at their respective means, the stakes placed on the equalising team are 35.7\% higher than those placed on the conceding team. The corresponding increase for the model shown in column (II) is substantially higher at 46.5\%. The goodness of fit of our model is also promising, with an $R^2$ of more than 0.6, and our findings hold for sub-samples according to when the equaliser went in, shown in columns (II-III) of Table~\ref{tab:bettors}. In particular, an increase in the relative stakes placed on the equalising team is especially pronounced in the second half of matches, at which times relative stakes on teams that have just equalised to win are 19.4 percentage points greater than on the conceding teams, holding everything else in the models equal.
\begin{table}[ht!] \centering
\caption{Do bettors follow the apparent momentum?}
\label{tab:bettors}
\scalebox{0.9}{
\begin{tabular}{@{\extracolsep{5pt}}lcccc}
\\[-1.8ex]\hline
\\[-1.8ex]
& \multicolumn{4}{c}{\textit{Timing of equaliser for 1-1}} \\
\cline{2-5}
\\[-1.8ex] & Any time & Any time & First half & Second half \\ \\[-1.8ex] & (I) & (II) & (III) & (IV) \\
\hline \\[-1.8ex]
\textit{startodds} ($\beta_1$) & $-$0.037$^{***}$ & $-$0.004 & 0.002 & $-$0.013$^{**}$ \\
& (0.008) & (0.004) & (0.004) & (0.006) \\
& & & & \\
\textit{equaliser} ($\beta_2$) & 0.127$^{***}$ & 0.159$^{***}$ & 0.123$^{***}$ & 0.194$^{***}$ \\
& (0.028) & (0.020) & (0.029) & (0.027) \\
& & & & \\
\textit{minute} ($\beta_3$) & $-$0.002$^{***}$ & $-$0.001$^{***}$ & $-$0.0003 & $-$0.003$^{***}$ \\
& (0.000) & (0.000) & (0.0004) & (0.001) \\
& & & & \\
\textit{prerelstake} ($\beta_4$) & & 0.725$^{***}$ & 0.777$^{***}$ & 0.653$^{***}$ \\
& & (0.040) & (0.052) & (0.061) \\
& & & & \\
\textit{redcard} ($\beta_5$) & $-$0.169$^{***}$ & $-$0.189$^{***}$ & (no red cards) & $-$0.190$^{***}$ \\
& (0.052) & (0.069) & & (0.067) \\
& & & & \\
Constant ($\beta_0$) & 0.562$^{***}$ & 0.087$^{***}$ & 0.031 & 0.262$^{***}$ \\
& (0.032) & (0.033) & (0.038) & (0.064) \\
& & & & \\
\hline \\[-1.8ex]
$N$ of matches & 211 & 211 & 100 & 111 \\
$N$ of observations & 422 & 422 & 200 & 222 \\
$R^{2}$ & 0.229 & 0.621 & 0.644 & 0.623 \\
\hline
\end{tabular}}
\caption*{\footnotesize Notes.- Estimates of Equation~\eqref{eq:three}. $^{***}$, $^{**}$, $^{*}$ indicate significance from zero of the model coefficients at the 1\%, 5\% and 10\% levels, respectively, two-sided tests, with standard errors in parentheses that account for match-level clustering. The sample contains 211 instead of 212 matches since in one match no stakes were placed in the next minute after the 1-1 equaliser.}
\end{table}
In Appendix~Table~\ref{tab:bettors_robustness}, we show that the main results in column (I) of Table~\ref{tab:bettors} are robust to extensions of the model given by Equation~\eqref{eq:three}, including: squared terms for the minute of the equalising goal and the relative stakes in the one minutes prior to the equaliser; interacting the minute of the equaliser and $redcard_{i,m}$, which is insignificant; interacting $startodds_{i,m}$ and $equaliser_{i,m}$, in case there is evidence that bettors tend to back pre-match equalising favourites more greatly than longshots, which there is not; and interacting $equaliser_{i,m}$ and $minute_{m}$. Appendix~Table~\ref{tab:bettors_robustness} further includes a robustness check with relative stakes three minutes after the equaliser as the response variable (shown in the first column). In that model formulation, stakes on the win are also substantially larger for teams that score the equaliser than for teams that concede.
\section{Conclusion}
\noindent This paper tackles the question of how gamblers adjust their risk-taking behaviour to the possibility of perceived momentum in the value of a state-contingent asset, viewed through observing betting activity within football matches. We use a novel and rich dataset from a large and well-known international bookmaker, focusing on betting markets just after 1-1 equalisers are scored during matches in the German Bundesliga. We analyse whether the sequence of scoring impacts the final match outcome, the price setting by the bookmaker, and ultimately the amount and direction of betting activity. On the sequence of scoring, we hypothesise that the equalising team has gained momentum. However, our results suggest that on average the sequence of the goals leading to the 1-1 does not influence the ultimate winning chances of a team or the odds setting by the bookmaker. Still, there is convincing evidence that bettors believe in the value of momentum, as considerably higher stakes are placed on the teams that have just equalised to eventually win, compared with the teams that conceded. Such perceived value in momentum among bettors does not translate into profits, as always betting on the team with momentum on average leads to significant negative returns.
To the best of our knowledge, this paper is the first that can cleanly isolate a singular event (in our case the equaliser in a football match) that creates momentum (for the team that scored the goal) and influences investor behaviour. It demonstrates that while the order of goals is irrelevant to the game outcome, bettors heavily believe in its importance. This indicates that investors on betting markets indeed have difficulty quantifying the relevance of even the most important game situation in a football match. Still, there are some potential limitations to our approach. First, since the data are accumulated over all bettors, it is impossible to connect multiple bets by a single bettor over time. Potentially, some bettors stake money on one particular outcome during the match before later staking on a different outcome, to hedge the first bet. Therefore, betting after the equaliser might in part reflect activity by the very same bettors prior to the goal. The impacts of such strategic and dynamic betting behaviour on our findings should be quite low though, as it would most likely only make sense when a bettor had staked money after but not prior to the first goal. Second, while our analysis covers rich betting data, the detail level of in-play statistics that we could reliably link to them is low. While goals (and red cards) are by far the most meaningful events to impact football match outcomes, others, such as yellow cards, substitutions, shots on target or corner kicks, could impact how bettors perceive the momentum of teams. Last, our match-by-match analysis neglects any cross-match momentum or how teams handle certain in-play situations they have experienced in the past. If one team scored the equaliser in a previous match to later go on and win, then bettors could predict the re-occurrence of such a dynamic in the next match and bet accordingly right after an equaliser.
The above mentioned limitations could and should be tackled in future works, as knowledge of the drivers of betting behaviour is still in its infancy --- almost all the literature is focused only on prices (odds). We believe that betting markets are valuable settings to understand price setting and behavioural biases that affect risk-taking. The availability of both odds and actual staked amounts enables investigation of how bettors respond to prices and make potentially biased investment decisions.
\clearpage
\clearpage
|
{
"timestamp": "2022-11-14T02:07:30",
"yymm": "2211",
"arxiv_id": "2211.06052",
"language": "en",
"url": "https://arxiv.org/abs/2211.06052"
}
|
\section{Introduction}
Front-runner predicts large traders' incoming orders, trades in the same direction in front of them, and then supplies liquidity back when they arrive, hoping to profit from the market impacts caused by large traders. For example, if a front-runner receives the signal that a large trader is going to buy, she will buy in advance and then sell back to the large trader.
In today's market, most front-runners are high-frequency traders (HFTs), who process information faster and send orders with lower latency, compared to normal-speed traders. HFTs' speed advantage enables them to apply and possibly make profits from the front-running strategy.
Front-runners' influence on large traders' actions and profits is an important topic in both predatory trading and high-frequency trading. Conventional wisdom has it that front-runners increase large traders' transaction costs by taking away liquidity that might otherwise have gone to the large traders. However, some studies do believe that under certain conditions, front-runners could benefit large traders by providing liquidity when the large traders come. Empirical work supports both views, a unified conclusion has not been reached.
In this paper, we consider the interactions between an informed large trader and an HFT who can predict, to a certain extent, the large trader's future order, in the extended celebrated Kyle's model. We are interested in whether or not the HFT will conduct the front-running strategy and how the large trader is affected. A comprehensive analysis has been made under various circumstances.
We prove that it is optimal for HFT to front-run and we outline situations where HFT harms or benefits the large trader. Specifically, the front-running HFT is favorable to the large trader when (1) there is sufficient noise trading with HFT's
liquidity-consuming trading, no matter her signal is accurate or not; (2) the noise trading with HFT's liquidity-consuming trading is inadequate but her signal is vague enough.
We also explore the influences of market noise and signal accuracy on investors' behavior and profits. For HFT, she trades more aggressively when the market is noisier and the signal is more accurate. For the large trader, her trading intensity and profit always increase with the size of market noise. Surprisingly, we find that, when the HFT's signal is inaccurate to a certain degree, the large trader can be guaranteed not to be harmed, but her profit tends to decrease as the signal gets noisier, which seems counterintuitive with the common sense that the large trader gets better if her trading intentions are deeper hidden. Hence, for a conservative large trader, if she just aims to protect herself from being harmed by potential HFTs, it is always safer to enter a market
where the order anticipation is more inaccurate. However, if she aims to make a profit, it may be not optimal to enter such a market.
The limit results of the above influences are also investigated. We have drawn two interesting conclusions: (1) when there are few market noises, HFT's trading intensity tends to zero, that is she nearly does nothing, but the large trader is still harmed; (2) when the signal is completely accurate and the size of market noise tends to infinity, HFT reaches the maximum front-running volume, which is only half of the large trader's incoming order.
Besides the case where HFT predicts the large trader's order, we also study the case where HFT can predict the aggregate order flow, i.e., the large trader's order and its accompanying noise traders' orders. From a practical point of view, this kind of HFTs has more advanced technology to estimate the trend of the whole market, rather than only the major market trend. In this case, we find that HFT still front-runs and always benefits the large trader.
\section{Related Literature}
The front-running problem has attracted wide attention since the last century. For example, Harris (1997) \cite{harris1997order} introduces that, ``Large impatient traders often significantly impact prices when they trade. If other traders know their intentions, they may front-run them.''
With the development of high-frequency trading technology, more and more front-runners are HFTs, who
receive and process information faster. But how they make profits and affect other market participants remains questionable. In the empirical work Kirilenko et al. (2017) \cite{2017The}, the authors examine the role of market participants before and during Flash Crash by using transaction-level data for the E-mini. They find that before liquidity demanders arrive, HFTs aggressively remove the last few contracts at the best bid and ask, then provide liquidity at a new price level, offsetting their position, which implies that HFTs behave like front-runners at a high frequency. Other supportive work includes Khan and Lu (2013) \cite{khan2013short}, where the authors find significantly positive short sales in the days leading up to large insider's sales. Manahov (2016) \cite{manahov2016front} shows that HFTs trade in the same direction as the strategic informed traders, however keeping ahead of them. Hirschey (2021) \cite{hirschey2021high} presents evidence that HFTs identify patterns from trading history to predict future selling or buying pressure and trade ahead of other investors' order flow.
Front-running strategy exploits other investors' need to trade, thus belonging to the topic of predatory trading. Brunnermeier and Pedersen (2005) \cite{brunnermeier2005predatory} is a pioneering work to study predatory trading. In their model, when time begins from $t_0$ but the distressed large trader starts selling at $t_1>t_0$, predatory traders are actually front-runners who sell before $t_1$ and buy back after $t_1$. They conclude that front-runners bring inferior prices to the large trader.
In \cite{brunnermeier2005predatory}, the price impact is permanent. When the temporary impact is also considered, Schoenebor and Schied (2009) \cite{schoneborn2009liquidation} and Carmona and Yang (2011) \cite{carmona2011predatory} mention that under particular market conditions, e.g.， low permanent impact and high temporary impact, it is optimal for predatory traders to first buy when the large trader is selling and sell when the price bounces back, the large trader is thus benefited by the liquidity provided by them. Bessembinder et al. (2016) \cite{bessembinder2016liquidity} presents a model where the temporary impact is transient. The authors find that when the market is quite resilient and the permanent impact is not so large, front-runners could benefit the large trader. A resilient market can be regarded as active and liquid, which is consistent with our discovery that HFT may be good for the large trader in a market with abundant noise tradings.
Besides front-running the large or informed traders, Bernhardt and Taub (2008) \cite{bernhardt2008front} models an investor who is not only aware of the true value of the asset but also the current and future noise tradings. Her optimal strategy includes front-running the future noise tradings.
When it comes to high-frequency trading, Li (2013) \cite{li2018high} models front-running HFTs who predict the sum of informed and noise orders. The author concludes that informed trader is harmed and trades less aggressively in the presence of HFTs. Brogaard et al. (2014) \cite{brogaard2014high} studies whether HFT increases the execution costs of institutional investors, based on the data of LSE from 2007 to 2011. The authors find that HFTs are more active after the exchange speed is improved. However, the costs of institutional investors remain unchanged. Hens et al. (2018) \cite{hens2018front} investigates front-running HFTs in a limit order model and shows that they extract rents from investors who need to trade large size quickly.
Yang and Zhu (2020) \cite{yang2020back} models back-runners in an extended two-period Kyle's model, where back-runners use past order flow information to predict the informed traders' future path and trade along with them. For a more thorough review of high-frequency trading literature, readers could refer to Menkveld (2016) \cite{menkveld2016economics}.
\section{The Model and Equilibrium}
We start by introducing the following two-stage market model for front-running, which is an extension of the classic Kyle's model.
\textbf{Assets and Participants.}
In this market, a risky asset is traded whose true value or ex-post liquidation value,
$v$, is normally distributed as
\begin{equation*}
v\sim N(p_0,\sigma_v^2).
\end{equation*}
There is also a risk-free asset with zero interest rate, which provides inter-temporal value accumulations only.
There are four types of market participants: (1) \textit{dealers}, who observe the aggregate order flow and are assumed to be competitive and risk-neutral. The Bertrand competition forces them to make zero expected profit and hence set the transaction
price of the risky asset as the expectation of $v$ conditional on their information; (2) a normal-speed large \textit{informed trader} (IT, for short), who privately knows $v$; (3) a strategic \textit{High-Frequency Trader} (HFT, for short), who is capable to get a signal about IT's future trading and might act as a front-runner; (4) \textit{noise traders}, who trade randomly. We suppose that both IT and HFT are risk-neutral and seek to maximize their expected P\&L.
\textbf{Trading Structure and Prices.} For a two-stage model, we consider three time points \footnote{Note that we mark these time stamps just for convenience. It is not required that the time lengths of $[0,1]$ and $[1,2]$ are equal. }, $t=0,\, 1,\, 2.$ At $t=0,$ IT sends a market order of quantity $i=i(v)$, based on her private knowledge of $v$. However, for some reasons, e.g., the submission delay, as in \cite{li2018high}, \cite{baldauf2020high}, the order is not executed until $t=2$.
Very soon after IT sends her order, HFT recognizes IT's trading intention and gets a noisy signal about the informed order $i$:
\begin{equation*}
\hat{i}=i+z,
\end{equation*}
where the noise $z$ is independent of $i$ and follows $ N(0,\sigma_z^2)$. Hirschey (2021) \cite{hirschey2021high} finds evidence supporting that HFTs can recognize non-HFTs' persistent informed order flow in real time. We will not get to the bottom of how HFT predicts the order flow specifically, which is another challenging
work that lies beyond the scope of this paper.
The noise in HFT's signal may come from: (1) market regulations about information disclosure, which makes it difficult for HFT to filter useful news about large trader's tradings; (2) the limitation of HFT's technology, which brings the prediction error.
The standard deviation of $z$ represents the accuracy of HFT's signal. The smaller the $\sigma_z$, the higher the accuracy. When $\sigma_z=0,$ the information hidden in IT's order is perfectly detected by HFT.
At $t=1$, HFT builds up a position on the risky asset by sending a market order of quantity $x=x(\hat{i})$. We assume that HFT's market orders will always be fulfilled at once. This risky position will be offset when HFT sends a market order of size $-x$ at $t=2$. If $x$ is in the same direction of $\hat{i}$, HFT employs the front-running strategy: takes liquidity in front of IT and when IT's order arrives, trades against it to supply liquidity.
To be closer to reality, we suppose that both time-1 and time-2 trades are accompanied by noise tradings. We denote the aggregate noise order flow in each period by $u_1$ and $u_2$, where
$$u_1\sim N(0,\sigma_1^2),\ u_2\sim N(0,\sigma_2^2),$$are independent of each other and any other random variables.
To sum up, the total order flow $y_1$ and $y_2$ executed at $t=1$ and $t=2$ respectively are
\begin{equation*}
y_1=x+u_1 \quad\text{and}\quad y_2=i+u_2-x.
\end{equation*}
Hence the transaction prices of the risky asset at each time point will be
\begin{equation*}
p_1=\mathbb{E}(v|y_1) \quad\text{and}\quad p_2=\mathbb{E}(v|y_1,y_2).
\end{equation*}
In the following, we assume $\sigma_2>0$ as in Kyle's model and use
\begin{equation*}
\theta_z=\frac{\sigma_z^2}{\sigma_2^2}\quad\text{and}\quad\theta_1=\frac{\sigma_1^2}{\sigma_2^2}
\end{equation*}
to characterize the signal noise and time-1 market noise normalized by $\sigma_2^2$, which are in line with the parameters studied in \cite{yang2020back}.
\textbf{Equilibrium.} The main purpose of the current paper is to study HFT's trading strategy and how IT is affected by HFT in equilibrium, and to compare the results with those in one-period Kyle's model when there are only IT and dealers.
We first give the definition of equilibrium in our model.
\begin{definition}
The equilibrium is defined as a collection of strategies of the dealers, IT and HFT: $\{p_1, p_2,i,x\}$, such that the following market-efficiency condition and two optimization conditions are satisfied.
\begin{enumerate}
\item Given IT's strategy $i$ and HFT's strategy $x$, dealers set price according to the weak-efficiency rule:
\begin{equation*}
\begin{aligned}
&p_1=\mathbb{E}(v|y_1),\\
&p_2=\mathbb{E}(v|y_1,y_2).
\end{aligned}
\end{equation*}
\item Given HFT's strategy $x$ and dealers' pricing rule $p_1,p_2$, IT's strategy $i^*$ maximizes her expected profit over all measurable strategies $i=i(v)$:
\begin{equation*}
i^*=\arg\max_{i=i(v)}\pi^{\text{IT}}(i),
\end{equation*}
where $\pi^{\text{IT}}(i)=\mathbb{E}\big((v-p_2)i\big|v\big).$
\item Given IT's strategy $i$ and dealers' pricing rule $p_1,p_2$, HFT's strategy $x^*$ maximizes her expected profit over all measurable strategies $x=x(\hat{i})$:
\begin{equation*}
x^*=\arg\max_{x=x(\hat{i})}\pi^{\text{HFT}}(x),
\end{equation*}
where $\pi^{\text{HFT}}(x)=\mathbb{E}\big((p_2-p_1)x\big|\hat{i}\big)$.
\end{enumerate}
\end{definition}
Within the Normal-distribution framework, it is natural to conjecture a linear structure of the equilibrium, i.e., in equilibrium, the strategies of IT and HFT as well as liquidation prices are linear functions:
\begin{equation*}
p_1=p_0+\lambda_1y_1,
\end{equation*}
\begin{equation*}
p_2=p_0+\mu_1y_1+\mu_2y_2,
\end{equation*}
\begin{equation*}
i=\alpha (v-p_0),
\end{equation*}
\begin{equation*}
x=\beta\hat{i}.
\end{equation*}
This is consistent with the conjectures and results in Kyle (1985) \cite{kyle1985continuous}, Bernhardt and Miao (2004) \cite{bernhardt2004informed}, Bernhardt and Taud (2008) \cite{bernhardt2008front} and Yang and Zhu (2020) \cite{yang2020back}.
We will prove the existence and uniqueness of a linear equilibrium under different circumstances in Section \ref{sectionmain}. The properties of the optimal strategies will also be verified.
Please note that, till now, we assume HFT predicts the informed order $i$ rather than the aggregate order $i+u_2$. In other words, here we focus on those potential front-running HFTs who are skilled at estimating some major market trends but may not know well the whole market. These HFTs' behavior is consistent with the back-runners' in Yang and Zhu (2020) \cite{yang2020back}. Empirically, both kinds of HFTs may exist, we will discuss the second case in Section \ref{sectionprei+u2}, as a supplement to Li (2018) \cite{li2018high}.
\section{Main Results}
\label{sectionmain}
In this section, we investigate market participants', especially HFT's and IT's behavior in equilibrium. It is usually believed that front-runner increases large traders' transaction costs by trading in the same direction in front of them, see \cite{harris1997order}, \cite{brunnermeier2005predatory}, \cite{li2018high}. But there is another line of research suggests that, both theoretically and empirically, see, e.g., \cite{bessembinder2016liquidity}, the front-running behavior might on the contrary decrease the impact of large traders' order under some circumstances, since it also provides liquidity back. Therefore, under the formulated model, we are concerned about the following questions: (1) whether or not the HFT would do front-running; (2) whether or not the HFT's action would harm the large trader IT; (3) under what circumstances would the HFT's action harm or benefit IT.
In the following, the existence and uniqueness of a linear equilibrium will be given after cautiously analyzing the participants' strategies in different situations, according to the size of time-1 noise trading and the accuracy of HFT's signal. Then we discuss how the HFT's strategy affects IT, especially when will the latter be benefited or harmed. Through comparative static analysis, we explore the influences of $\theta_1$ and $\theta_z$ on investors' actions and the results in limit situations.
For the sake of simplicity, in the following discussions, we assume $p_0=0$, for it does not bring out any essential changes in results.
Since there is no general equilibrium for the two cases when there is noise trading at time 1 or not, we discuss the two cases separately: (1) $\theta_1>0$, that is $\sigma_1>0$, which can be characterized as a case that the market is active in both periods, noise tradings might come from high-speed traders as well as normal-speed traders;
(2) $\theta_1=0,$ that is $\sigma_1=0$, which can be viewed as a case that the noise traders are of slower speed relative to HFT, and thus the HFT's time-1 trading is accompanied by few noises.
\subsection{Equilibrium in the case \texorpdfstring{$\theta_1>0$}{}}
In this case, noise tradings occur in both periods.
We study the strategies of main participants: dealers, IT and HFT respectively, given the others' strategies.
\textbf{Dealers' quotes.} As stated in the former section, risk-neutral and competitive dealers set the transaction prices as the expectation of $v$, conditioned on the order flow information.
At $t=1$, when HFT builds up the position $x$, the price should be
\begin{equation*}
p_{1}=\mathbb{E}(v|y_1)=\lambda_1y_1,
\end{equation*}
where, given by the linear conjecture and projection theorem,
\begin{equation}
\label{lambda1}
\lambda_1=\frac{\sigma_{v}}{\sigma_{y_1}}\rho_{(v,y_1)}=\frac{\alpha\beta\sigma_v^2}{\beta^2(\alpha^2\sigma_v^2+\sigma_z^2)+\sigma_1^2}.
\end{equation}
At $t=2$, when IT's order $i$ and HFT's offsetting order $-x$ are being executed, the price should be
\begin{equation*}
p_2=\mathbb{E}(v|y_1,y_2)=\mu_1y_1+\mu_2y_2,
\end{equation*}
where, still by the projection theorem,
\begin{equation}
\label{mu1}
\mu_1 =\frac{\sigma_{v}}{\sigma_{y_1}} \frac{\rho_{(v,y_1)}-\rho_{(y_1,y_2)} \rho_{(v,y_2)}}{1-\rho^{2}_{(y_1,y_2)}}= \frac{\alpha\sigma_v^2(\beta^2\sigma_z^2+\beta\sigma_2^2)}{\alpha^2\sigma_v^2(\sigma_1^2(1-\beta)^2+\sigma_2^2\beta^2+\sigma_z^2\beta^2)+\beta^2\sigma_z^2(\sigma_1^2+\sigma_2^2)+\sigma_1^2\sigma_2^2},
\end{equation}\begin{equation}
\label{mu2}
\mu_2=\frac{\sigma_{v}}{\sigma_{y_2}} \frac{\rho_{(v,y_2)}-\rho_{(v,y_1)} \rho_{(y_1,y_2)}}{1-\rho_{(y_1,y_2)}^{2}}=\frac{\alpha\sigma_v^2(\beta^2\sigma_z^2+(1-\beta)\sigma_1^2)}{\alpha^2\sigma_v^2(\sigma_1^2(1-\beta)^2+\sigma_2^2\beta^2+\sigma_z^2\beta^2)+\beta^2\sigma_z^2(\sigma_1^2+\sigma_2^2)+\sigma_1^2\sigma_2^2}.
\end{equation}
The above impact coefficients, $\lambda_1,\mu_1,\mu_2,$ also represent the informativeness of the corresponding order flow. And we will see in latter sections
that some heuristic results could be figured out in special cases and in comparative static analysis.
Here we point out that, in the special case when HFT decided to do nothing, that is, when $x=0$,
dealers would confront a market as in Kyle's one-period model, except that there is an additional auction before the IT's order $i$ is completed. However, no information is leaked out in this prior auction. So we find that $p_1=p_0=\mathbb{E}[v]$ while
\begin{equation*}
p_2=\lambda_2y_2, \lambda_2=\frac{\alpha\sigma_v^2}{\alpha^2\sigma_v^2+\sigma_2^2},
\end{equation*}
which is the same as the liquidating price in Kyle's one-period model.
Another point to note is, since HFT partly detects the intention of IT and trades against/ along the informed order $i$,
$i$ has indirect impacts on the prices. In fact, under linear structures, the total impact of $i$ on $p_2$ turns out to be a combination of different-time impacts: $(\mu_1\beta+\mu_2(1-\beta))i$. Let $\lambda_2$ denote this total impact, then
\begin{equation}\label{lambda2}
\begin{aligned}
\lambda_{2} &=\mu_{1} \beta+\mu_{2}(1-\beta) \\
&=\frac{\alpha \sigma_{v}^{2}\left(\sigma_{1}^{2}(1-\beta)^{2}+\sigma_{2}^{2} \beta^{2}+\sigma_{z}^{2} \beta^{2}\right)}{\alpha^{2} \sigma_{v}^{2}\left(\sigma_{1}^{2}(1-\beta)^{2}+\sigma_{2}^{2} \beta^{2}+\sigma_{z}^{2} \beta^{2}\right)+\beta^{2} \sigma_{z}^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)+\sigma_{1}^{2} \sigma_{2}^{2}}.
\end{aligned}
\end{equation}
The part $\mu_1\beta$ comes from HFT's time-1 trading $x$ and the part $\mu_2(1-\beta)$ comes from time-2 trading $i-x$.
\textbf{IT's strategy.} IT decides the order $i$ to maximize her expected
P\&L based on her information. Given dealers' quotes and HFT's strategy, IT's expected P\&L is:
\begin{equation*}
\begin{aligned}
\pi^{\text{IT}}&=\mathbb{E}(i(v-p_2)|v)\\
&=i(v-\lambda_2 i),
\end{aligned}
\end{equation*}
which can be uniquely maximized at
$i^*=\frac{1}{2\lambda_2}v,$ under the second order condition (SOC) $\lambda_2>0.$
The optimal $\alpha$ is
\begin{equation}
\label{alpha}
\alpha^*=\frac{1}{2\lambda_2}.
\end{equation}
\textbf{HFT's strategy.}
Given dealers' quotes and IT's strategy, HFT's expected P\&L is:
\begin{equation*}
\begin{aligned}
\pi^{\text{HFT}}&=\mathbb{E}(x(p_2-p_1)|\hat{i})\\
&=x\left((\mu_1-\mu_2-\lambda_1)x+\mu_2\mathbb{E}(i|\hat{i})\right)\\
&=-(\mu_2+\lambda_1-\mu_1)x^2+\mu_2\frac{\alpha^2\sigma_v^2}{\alpha^2\sigma_v^2+\sigma_z^2}\hat{i}x,
\end{aligned}
\end{equation*}
which can be uniquely maximized at $x^*=\frac{\mu_2}{2(\mu_2+\lambda_1-\mu_1)}\frac{\alpha^2\sigma_v^2}{\alpha^2\sigma_v^2+\sigma_z^2}\hat{i}$, under the SOC $\mu_2+\lambda_1-\mu_1>0$.
The optimal $\beta$ is
\begin{equation}
\label{beta}
\beta^*=\frac{\mu_2}{2(\mu_2+\lambda_1-\mu_1)}\frac{\alpha^2\sigma_v^2}{\alpha^2\sigma_v^2+\sigma_z^2}.
\end{equation}
\textbf{Equilibrium.}
We have analyzed the optimal behavior of different participants and now we give the existence and uniqueness of a linear equilibrium.
Note that according to Equation \eqref{lambda2}, $\lambda_2$ is a function of $\alpha$ and $\beta$:
\begin{equation}
\label{lambda2*}
\lambda_2=\lambda_2(\alpha,\beta)=\frac{B \alpha}{B\alpha^2+C},
\end{equation}
where $B=B(\beta)=\sigma_{v}^{2}\left(\sigma_{1}^{2}(1-\beta)^{2}+\sigma_{2}^{2} \beta^{2}+\sigma_{z}^{2} \beta^{2}\right)$,
$C=C(\beta)=\beta^{2} \sigma_{z}^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)+\sigma_{1}^{2} \sigma_{2}^{2}$.
Substituting it into Equation \eqref{alpha}, we find that $\alpha^*$ could be written as the function of $\beta^*$:
\begin{equation}
\label{alpha*}
\alpha^*=\sqrt{\frac{C(\beta^*)}{B(\beta^*)}}.
\end{equation}
Substituting \eqref{alpha*} into \eqref{lambda1},\eqref{mu1},\eqref{mu2},\eqref{beta}, we get the equation of $\beta^*$ which should be satisfied in equilibrium.
\begin{theorem}
\label{mainthm}
Given $\sigma_1,\sigma_2>0,\sigma_z\geq0,$ $(\theta_1=\frac{\sigma_1^2}{\sigma_2^2}>0,\theta_z=\frac{\sigma_z^2}{\sigma_2^2}\geq0,)$ there exists a unique equilibrium $\{p_1,p_2,i^*,x^*\}$,
where HFT follows the strategy $x^*=\beta^*\hat{i}$ and $\beta^*\in(0,1)$ solves the equation
\begin{equation}
\label{betastar}
\begin{aligned}
0=& \beta^6(4\theta_1\theta_z^2+\theta_1\theta_z^3+2\theta_1^2\theta_z^2+2\theta_z^2+\theta_z^3)+\beta^5(4\theta_1\theta_z+4\theta_1\theta_z^2+2\theta_1\theta_z^3+8\theta_1^2\theta_z+4\theta_1^2\theta_z^2+4\theta_1^3\theta_z)\\
+ & \beta^4(2\theta_1\theta_z+\theta_1\theta_z^2-11\theta_1^2\theta_z-8\theta_1^2\theta_z^2-13\theta_1^3\theta_z)
+\beta^3(2\theta_1^2 + 2\theta_1^3 + 8\theta_1^2\theta_z + 4\theta_1^2\theta_z^2+ 16\theta_1^3\theta_z ) \\
- & \beta^2( \theta_1^2\theta_z +5\theta_1^3 + 9\theta_1^3\theta_z) + \beta(4\theta_1^3 + 2\theta_1^3\theta_z) -\theta_1^3;
\end{aligned}
\end{equation}
IT follows the strategy $i^*=\alpha^* v$ and
\begin{equation}
\label{alphastar}
\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_1+\beta^{*2}\theta_z(\theta_1+1)}{\theta_1(1-\beta^*)^2+\beta^{*2}(1+\theta_z)}};
\end{equation}
the liquidation price at time 1 is
\begin{equation*}
p_1=\lambda_1^* (x^*+u_1),
\end{equation*}
where
\begin{equation*}
\lambda_1^*=\frac{\sigma_v}{2\sigma_2}\frac{2\beta^*\sqrt{(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))}}{\beta^{*2}(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)+(\beta^{*2}\theta_z+\theta_1)(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))};
\end{equation*}
the liquidation price at time 2 is
\begin{equation*}
p_2=\mu_1^*(x^*+u_1)+\mu_2^*(i^*+u_2-x^*),
\end{equation*}
where
\begin{equation*}
\mu_1^*=\frac{\sigma_v}{2\sigma_2}\frac{\beta^{*2}\theta_z+\beta^*}{\sqrt{(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)}},
\end{equation*}
\begin{equation*}
\mu_2^*=\frac{\sigma_v}{2\sigma_2}\frac{\beta^{*2}\theta_z+(1-\beta^*)\theta_1}{\sqrt{(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)}};
\end{equation*}
the impact coefficient of the informed order $i^*$ is
\begin{equation*}
\lambda_2^*=\mu_1^*\beta^*+\mu_2^*(1-\beta^*)=\frac{\sigma_v}{2\sigma_2}\sqrt{\frac{\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1)}{\theta_1+\beta^{*2}\theta_z(\theta_1+1)}}.
\end{equation*}
\end{theorem}
Now we give the maximal expected profit of investors and price discovery variables in equilibrium, representing them in the form of $\beta^*$, to prepare for subsequent analysis.
\begin{corollary}
\label{cor1}
The maximal expected profits of IT and HFT are
\begin{equation*}
\mathbb{E}(\pi^{\text{IT}})=\frac{\sigma_v\sigma_2}{2}\sqrt{\frac{\theta_1+\beta^{*2}\theta_z(\theta_1+1)}{\theta_1(1-\beta^*)^2+(1+\theta_z)\beta^{*2}}},
\end{equation*}
\begin{equation*}
\begin{aligned}
\mathbb{E}(\pi^{\text{HFT}})=&\frac{\sigma_v\sigma_2}{2}\frac{\beta^*\theta_1((1-\beta^*)\theta_1+\beta^{*2}\theta_z)}{\sqrt{(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))}}\\
&\frac{\beta^{*2}\theta_z((1-2\beta^*)-\beta^*\theta_z)+(1-\beta^*)\theta_1(1-\beta^*(1-2\beta^*)\theta_z)}{(1-\beta^*)^2\theta_1^2+\beta^{*4}\theta_z(\theta_z+2)+2\beta^{*2}\theta_1(1+(\beta^{*2}-\beta^*+1)\theta_z)}.
\end{aligned}
\end{equation*}
The ex-ante and ex-post price discovery variables are
\begin{equation*}
\mathbb{E}(v-p_1)^2=\sigma_v^2\frac{(\beta^{*2}\theta_z+\theta_1)(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))}{\beta^{*2}(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)+(\beta^{*2}\theta_z+\theta_1)(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))},
\end{equation*}
\begin{equation*}
\mathbb{E}(v-p_2)^2=\frac{\sigma_v^2}{2}.
\end{equation*}
\end{corollary}
\subsubsection{How Front-running HFT affects IT}
From Theorem \ref{mainthm}, we see that in equilibrium $\beta^*\in(0,1),$ which means that when HFT predicts the future order of IT, she front-runs.
So how IT is affected in the presence of the front-running HFT?
Compared to the classic Kyle's model, where in equilibrium the expected profit of IT is $\frac{\sigma_v\sigma_2}{2},$
when HFT exists, IT's expected profit is $\frac{\sigma_v^2}{4\lambda_2^*}$, so if $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$, it turns out to get larger.
Recalling Equation \eqref{lambda2*} and \eqref{alpha*}, we find that $\lambda_2^*$ is the maximum of $\lambda_2(\alpha;\beta^*),$ attained at $\alpha^*$. So in the next proposition, we specifically analyze how HFT's transactions affect informed order $i$'s total impact $\lambda_2(\alpha;\beta), $ and then influence $\lambda_2^*$ and IT's profit.
\begin{proposition}
\label{effectHFT1}
Given $\theta_1>0,\theta_z\geq0$ and IT's action $\alpha>0$, for any $\beta\in(0,1)$,
\begin{equation*}
\lambda_2(\alpha;\beta)=\mu_1(\alpha;\beta)\beta+\mu_2(\alpha;\beta)(1-\beta),
\end{equation*}
where the first part increases with $\beta$ and the second part decreases with $\beta$.
\end{proposition}
Hence, HFT's time-1 trading increases IT's transaction cost, while, her offsetting trading at time 2 decreases it. All in all, front-running HFT is a double-edged sword for IT. However, we can confirm that under certain conditions, HFT is advantageous to IT.
\begin{proposition}
\label{effectHFT2}
In equilibrium, we have $$\mu_2^*<\frac{\sigma_v}{2\sigma_2},$$
which makes it possible that $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$, i.e. IT benefits from HFT.
\end{proposition}
In fact, in Kyle's model, where there are only IT and dealers, the informativeness of order flow $y=i+u_2$ is $\frac{\sigma_v}{2\sigma_2}$. While, in our model, with the front-running HFT, information about $v$ has been exposed through her time-1 trading $x$, moreover, her time-2 reverse trading $-x$ further reduces the information contained in the order flow $y_2$. Thus, the price is less sensitive to the news in $y_2$ ($y_2-\mathbb{E}(y_2|y_1)$), which makes $\mu_2^*<\frac{\sigma_v}{2\sigma_2}.$
The above two propositions tell us that HFT's front-running behavior has two-sided effects on IT's profit and might be helpful. A natural question is that when does it happen? We answer this in the following theorem.
\begin{theorem}
\label{benefit}
In the following cases,
$\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$, i.e.,
IT makes more profits in the case
with a front-running HFT:
\begin{enumerate}
\item $\theta_1\geq\frac{2\sqrt{3}-3}{3},\theta_z\geq0.$
\item $0<\theta_1<\frac{2\sqrt{3}-3}{3},\theta_z>\overline{\theta}_z=\frac{-(\theta_1+5)+2\sqrt{4\theta_1^2+10\theta_1+5}}{-5\theta_1}.$
\item $\theta_z\geq0,\theta_1>\overline{\theta}_1=\frac{-(5\theta_z+3)+2\sqrt{5\theta_z^2+8\theta_z+3}}{-5\theta_z^2+2\theta_z+3}.$
\end{enumerate}
\end{theorem}
In case 1, when the market noise $\theta_1$ is large enough, HFT cannot harm IT even if her signal is perfectly accurate. In case 2, if noise trading at time 1 is not active enough, IT could be benefited when HFT's signal is relatively vague. What's more,
larger the $\theta_1$, smaller the $\overline{\theta}_z$, that is,
as the size of noise trading gets larger, less signal noise is needed to disturb HFT. In case 3, for any accuracy of signal, the IT could be benefited when the market hides her well. In this case, larger the $\theta_z$, smaller the $\overline{\theta}_1$,
when the size of signal noise grows, less market noise is needed to cover the information disclosed by HFT's time-1 trading.
Similarly, we could consider situations where HFT harms IT.
\begin{theorem}
\label{harm}
In the following cases,
$\lambda_2^*>\frac{\sigma_v}{2\sigma_2}$, i.e.,
IT makes less profits in the case
with a front-running HFT:
\begin{enumerate}
\item $0<\theta_1<\frac{2\sqrt{3}-3}{3},0\leq\theta_z<\overline{\theta}_z.$
\item $\theta_z\geq0,0<\theta_1<\overline{\theta}_1.$
\end{enumerate}
\end{theorem}
\subsubsection{Influences of noises on traders' behavior}
In equilibrium, the optimal intensities of investors, $\alpha^*$ and $\beta^*$, both depend on the relative size of noises: $\theta_1$ and $\theta_z$. Now we investigate how $\alpha^*$ and $\beta^*$ change with them.
\begin{proposition}
\label{HFTaction}
The optimal intensity of HFT, $\beta^*$, increases with $\theta_1$ and decreases with $\theta_z$.
\end{proposition}
A larger $\theta_1$ means more noise tradings at time 1, which enables HFT to employ more of her priority information and bear less impact.
A larger $\theta_z$ means a less accurate signal, which makes HFT less confident about IT's future trading, and hence reduces her trading intensity.
The impact of $\theta$s on IT's behavior is more complicated. However, we can still prove the following results.
\begin{proposition}
\label{ITaction}
\begin{enumerate}
\item When $\theta_1\geq\frac{1}{2}$, $\alpha^*$ increases with $\theta_1$ and decreases with $\theta_z$.
\item
When $0<\theta_1<\frac{1}{2},$
\begin{enumerate}
\item given $\theta_z\geq0$, $\alpha^*$ increases with $\theta_1$ when $\theta_1\geq\Tilde{\theta}_1=\frac{\sqrt{(3\theta_z+1)^2+8}-(3\theta_z+1)}{4}$;
\item given
$\theta_1\geq0.0089$, $\alpha^*$ increases with $\theta_z$ when $0\leq\theta_z<\Tilde{\theta}_z=\frac{1-\theta_1-2\theta_1^2}{3\theta_1}$ and decreases with it when $\theta_z\geq\Tilde{\theta}_z.$
\end{enumerate}
\end{enumerate}
\end{proposition}
For the monotonicity concerned in Proposition \ref{ITaction}, we have done a lot of numerical experiments, and the results can be regarded as a supplement to Proposition \ref{ITaction}.
In fact, we find in general that (see Figure \ref{figtheta1-IT}, Figure \ref{figthetaz-HFT} and Figure \ref{figthetaz-example})
\begin{enumerate}
\item $\alpha^*$ increases with $\theta_1$;
\item given $\theta_1\geq\frac{1}{2},\alpha^*$ decreases with $\theta_z$, $\alpha^*$ is maximized at $\theta_z=0$;
\item given $0<\theta_1<\frac{1}{2},$ $\alpha^*$ increases with $\theta_z$ when $0\leq\theta_z<\Tilde{\theta}_z$ and decreases with it when $\theta_z\geq\Tilde{\theta}_z.$
\end{enumerate}
We will illustrate these in Section \ref{sectionNUM}.
Since $\mathbb{E}(\pi^{\text{IT}})=\frac{\sigma_v^2}{2}\alpha^*,$
when IT trades more aggressively, her expected profit is higher. Thus, combining Proposition \ref{ITaction} and Theorem \ref{benefit}, we surprisingly find that when $\theta_1\geq\frac{1}{2}$, IT is benefited by HFT, however, $\alpha^*$, and consequently, IT's expected profit decrease with the size of signal noise.
When the market provides enough noise shelter for IT to protect herself, adding any noise to HFT's signal is unfavorable to IT. This also answers a controversial question:
is it harmful to other traders if HFT has too precise information? From our observation, it depends on the size of market noise, if the noise trading is active enough, it is not necessary to increase the noise in HFT's signal.
\subsubsection{The limit results}
\begin{proposition}
\label{theta1infty}
Given $\theta_z\geq0$, when $\theta_1\rightarrow\infty,$ $\beta^*$ converges to the root of the following equation:
\begin{equation*}
4\theta_z\beta^5-13\theta_z\beta^4+(2+16\theta_z)\beta^3-(5+9\theta_z)\beta^2+(4+2\theta_z)\beta-1=0.
\end{equation*}
The optimal intensity of IT
\begin{equation*}
\alpha^*\rightarrow\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\beta^{*2}\theta_z+1}{(1-\beta^*)^2}}>\frac{\sigma_2}{\sigma_v}.
\end{equation*}
Limits of other variables could be calculated by just substituting $\beta^*$ into Theorem \ref{mainthm} and Corollary \ref{cor1}.
\end{proposition}
It verifies that given $\theta_z\geq0,$ there exists a $\theta_1$ large enough, such that $\alpha^*>\frac{\sigma_2}{\sigma_v},$ i.e. IT trades and profits more when there is a front-running HFT.
\begin{proposition}
\label{theta10}
Given $\theta_z\geq0,$ when $\theta_1\rightarrow0$,
\begin{equation*}
\begin{aligned}
&\beta^*\rightarrow0,\alpha^*\rightarrow\frac{\sigma_2}{\sigma_v}\sqrt{\frac{y^2\theta_z^{2x}+\theta_z}{y^2\theta_z^{2x}+\theta_z+1}}<\frac{\sigma_2}{\sigma_v},\\
&\mathbb{E}(\pi^{\text{IT}})\rightarrow\frac{\sigma_2\sigma_v}{2}\sqrt{\frac{y^2\theta_z^{2x}+\theta_z}{y^2\theta_z^{2x}+\theta_z+1}},\mathbb{E}(\pi^{\text{HFT}})\rightarrow0,\\
&\lambda_1^*,\mu_1^*\rightarrow\infty,\mu_2^*\rightarrow\frac{\sigma_v}{2\sigma_2}\sqrt{\frac{y^2\theta_z^{2x}+\theta_z}{y^2\theta_z^{2x}+\theta_z+1}},\lambda_2^*\rightarrow\frac{\sigma_v}{2\sigma_2}\sqrt{\frac{y^2\theta_z^{2x}+\theta_z+1}{y^2\theta_z^{2x}+\theta_z}},\\
&\mathbb{E}(v-p_1)^2\rightarrow\sigma_v^2\frac{y^2\theta_z^{2x}+\theta_z+1}{y^2\theta_z^{2x}+\theta_z+2},
\end{aligned}
\end{equation*}
where $x=0.3245,y=1.3845$.
\end{proposition}
Given $\theta_z\geq0,$ when $\theta_1$ is small enough, $\alpha^*<\frac{\sigma_2}{\sigma_v},$ i.e., IT trades less and is harmed by HFT. Interestingly, in the case when nearly all noise traders are normal-speed ($\theta_1\rightarrow0$), although HFT almost does nothing ($\beta^*\rightarrow0$), IT is still worse off.
\begin{proposition}
\label{thetazinfty}
Given $\theta_1>0$, when $\theta_z\rightarrow\infty$,
\begin{equation*}
\begin{aligned}
& \beta^*\rightarrow0,\alpha^*\rightarrow\frac{\sigma_2}{\sigma_v},\mathbb{E}(\pi^{\text{IT}})\rightarrow\frac{\sigma_2\sigma_v}{2},\mathbb{E}(\pi^{\text{HFT}})\rightarrow0,\\
& \lambda_1^*\rightarrow0,\mu_1^*\rightarrow0,\mu_2^*\rightarrow\frac{\sigma_v}{2\sigma_2},\lambda_2^*\rightarrow\frac{\sigma_v}{2\sigma_2},\mathbb{E}(v-p_1)^2\rightarrow\sigma_v^2.
\end{aligned}
\end{equation*}
\end{proposition}
In other words, when HFT gets a signal which is too noisy, the market converges to the one without HFT. When $\theta_z\rightarrow0,$ it is easy to prove that investors and market's behavior converges to the case $\theta_z=0,$ which will be fully discussed in the next subsection.
\subsubsection{A special case: \texorpdfstring{$\theta_z=0$}{}}
To have more intuition about how IT is affected by the front-running HFT, we display the results of a special case that $\theta_1>0,\theta_z=0$, where noise tradings occur in both periods and HFT receives a perfectly accurate signal. From the calculations above Theorem \ref{mainthm}, we have
\begin{equation}
\label{example-mu}
\begin{aligned}
p_2&=\mathbb{E}(v|y_1,y_2)=\mu_1y_1+\mu_2y_2,\\
\mu_1&=\frac{\alpha\theta_v\beta}{\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2)+\theta_1},
\mu_2=\frac{\alpha\theta_v\theta_1(1-\beta)}{\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2)+\theta_1},
\end{aligned}
\end{equation}
where $\theta_v=\frac{\sigma_v^2}{\sigma_2^2}.$ The impact coefficient for informed order, $\lambda_2$, is:
\begin{equation}
\label{example-lam2}
\begin{aligned}
\lambda_2&=\mu_1\beta+\mu_2(1-\beta)=\frac{\alpha\theta_v(\theta_1(1-\beta)^2+\beta^2)}{\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2)+\theta_1}.
\end{aligned}
\end{equation}
Based on \eqref{example-mu} and \eqref{example-lam2}, we could conduct further analysis.
Given $\alpha>0$, $\forall\beta\in(0,1),$
\begin{equation*}
\begin{aligned}
\frac{\partial(\mu_1(\alpha;\beta)\beta)}{\partial\beta}&=\frac{2\alpha\theta_v(\alpha^2\theta_v\theta_1\beta(1-\beta)+\beta\theta_1)}{(\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2)+\theta_1)^2}>0,\\
\frac{\partial(\mu_2(\alpha;\beta)(1-\beta))}{\partial\beta}&=\frac{-2\alpha\theta_v\theta_1(\alpha^2\theta_v\beta(1-\beta)+(1-\beta)\theta_1)}{(\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2)+\theta_1)^2}<0,
\end{aligned}
\end{equation*}
which implies that front-running HFT both increases and decreases the impact of informed order through moving up and down $\lambda_2(\alpha;\beta)$.
In equilibrium,
\begin{equation*}
\begin{aligned}
&\mu_1^*=\frac{\sigma_v}{2\sigma_2}\frac{\beta^*}{\sqrt{\theta_1(\theta_1(1-\beta^*)^2+\beta^{*2})}},\mu_2^*=\frac{\sigma_v}{2\sigma_2}\frac{\theta_1(1-\beta^*)}{\sqrt{\theta_1(\theta_1(1-\beta^*)^2+\beta^{*2})}},\\
&\lambda_2^*=\frac{\sigma_v}{2\sigma_2}\sqrt{\frac{\beta^{*2}+\theta_1(1-\beta^*)^2}{\theta_1}}.
\end{aligned}
\end{equation*}
$\mu_1^*$ could be larger than $\frac{\sigma_v}{2\sigma_2},$ the impact coefficient in the classic Kyle's model. But $\mu_2^*<\frac{\sigma_v}{2\sigma_2}$. Since $\lambda_2^*$ is the linear combination of these two, it is possible that $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$ and IT is favored by HFT. This happens when
\begin{equation}
\label{condition}
0<\beta^*<\frac{2\theta_1}{\theta_1+1}.
\end{equation}
In fact, we could solve $\beta^*$ explicitly through Equation \eqref{betastar}. When $\theta_z=0$, it becomes:
\begin{equation*}
2(\theta_1+1)\beta^3-5\theta_1\beta^2+4\theta_1\beta-\theta_1=0,
\end{equation*}
denote $y=\frac{\beta}{1-\beta}$,
\begin{equation*}
\begin{aligned}
2y^3+\theta_1y-\theta_1=0,
\end{aligned}
\end{equation*}
which we can solve analytically:
\begin{equation*}
\begin{aligned}
&y=(\frac{\theta_1}{4})^{\frac{1}{3}}((\sqrt{1+\frac{2}{27}\theta_1}+1)^{\frac{1}{3}}-(\sqrt{1+\frac{2}{27}\theta_1}-1)^{\frac{1}{3}}),\beta^*=\frac{y}{y+1}.
\end{aligned}
\end{equation*}
So \eqref{condition} holds if and only if $\theta_1>\frac{2\sqrt{3}-3}{3}.$
As for how HFT's optimal intensity $\beta^*$ changes with $\theta_1$,
\begin{equation*}
\begin{aligned}
\frac{\partial\beta^*}{\partial\theta_1}
>&(\frac{\theta_1}{4})^{\frac{1}{3}}\frac{(1-\beta^*)^2}{3}((\sqrt{1+\frac{2}{27}\theta_1}+1)^{\frac{1}{3}}-(\sqrt{1+\frac{2}{27}\theta_1}-1)^{\frac{1}{3}}\\
&-((\sqrt{1+\frac{2}{27}\theta_1}+1)^{-\frac{2}{3}}-(\sqrt{1+\frac{2}{27}\theta_1}-1)^{-\frac{2}{3}}))\\
\geq&0.
\end{aligned}
\end{equation*}
For IT, we could also calculate $\frac{\partial\alpha^*(\theta_1)}{\partial\theta_1}$,
\begin{equation}
\begin{aligned}
\alpha^*
&=\frac{\sigma_2}{\sigma_v}\frac{1}{1-\beta^*}\sqrt{\frac{\theta_1}{\theta_1+y^2}},
\end{aligned}
\end{equation}
where both $\frac{1}{1-\beta^*}$ and $\sqrt{\frac{\theta_1}{\theta_1+y^2}}$ increase strictly with $\theta_1$,
so $\frac{\partial\alpha^*(\theta_1)}{\partial\theta_1}>0$. More noise tradings during period 1 enable IT to trade more intensely.
For limit results, we have
\begin{equation*}
\begin{aligned}
&\lim_{\theta_1\rightarrow0}\beta^*=0,\lim_{\theta_1\rightarrow0}\alpha^*=0;\\
&\lim_{\theta_1\rightarrow\infty}\beta^*=\frac{1}{2},\lim_{\theta_1\rightarrow\infty}\alpha^*=2\frac{\sigma_2}{\sigma_v}.
\end{aligned}
\end{equation*}
When HFT gets precise information and there are few noise tradings in period 1, both HFT and IT do nearly nothing.
Overall parameters $\theta_1>0,\theta_z\geq0,$
\begin{equation*}
\beta^*(\theta_1,\theta_z)\leq\beta^*(\theta_1=\infty,\theta_z=0)=\frac{1}{2},
\end{equation*}
\begin{equation*}
\alpha^*(\theta_1,\theta_z)\leq\alpha^*(\theta_1=\infty,\theta_z=0)=2\frac{\sigma_2}{\sigma_v}.
\end{equation*}
Therefore, in any case, HFT will not front-run more than half of the informed order. IT's intensity $\alpha^*$ and expected profit are at most twice of those without HFT.
\subsection{Equilibrium in the case \texorpdfstring{$\theta_1=0$}{}}
\label{theta1=0}
Now we analyze the equilibrium in the case with normal-speed noise traders. In this case, $y_1=x,y_2=i+u_2-x$.
\textbf{Dealers' quotes.} Given $i=\alpha v,x=\beta\hat{i},$ dealers quote linearly,
\begin{equation*}
\begin{aligned}
&p_1=\mathbb{E}(v|y_1)=\lambda_1y_1,\\
&p_2=\mathbb{E}(v|y_1,y_2)=\mu_1y_1+\mu_2y_2.\\
\end{aligned}
\end{equation*}
When $x\neq0,$ i.e., $\beta\neq0,$
\begin{equation*}
\begin{aligned}
&\lambda_1=\frac{\alpha\theta_v}{\beta(\alpha^2\theta_v+\theta_z)},\mu_1=\frac{\alpha\theta_v(\theta_z+1/\beta)}{\alpha^2\theta_v(\theta_z+1)+\theta_z},\mu_2=\frac{\alpha\theta_v\theta_z}{\alpha^2\theta_v(\theta_z+1)+\theta_z}.\\
\end{aligned}
\end{equation*}
When $x=0,$ i.e., $\beta=0,$
\begin{equation*}
\label{type2quotes}
\begin{aligned}
&\lambda_1=\mu_1=\infty,\mu_2=\frac{\alpha\theta_v}{\alpha^2\theta_v+1}.\\
\end{aligned}
\end{equation*}
\textbf{IT's strategy.} Given $x=\beta\hat{i}$ and dealers' linear quotes, IT's expected P\&L is
\begin{equation*}
\begin{aligned}
&\pi^{\text{IT}} = i(v-\lambda_2 i),\\
\end{aligned}
\end{equation*}
where $\lambda_2=\mu_1\beta+\mu_2(1-\beta).$
It can be uniquely maximized at $i^*=\frac{1}{2\lambda_2}v$ with SOC $\lambda_2>0.$ Then we have
\begin{equation*}
\alpha^*=\frac{1}{2\lambda_2}.
\end{equation*}
\textbf{HFT's strategy.} Given $i=\alpha v$ and dealers' linear quotes, HFT's expected P\&L is
\begin{equation}
\label{piHFTtheta1=0}
\pi^{\text{HFT}}=-(\mu_2+\lambda_1-\mu_1)x^2+\mu_2\frac{\alpha^2\theta_v}{\alpha^2\theta_v+\theta_z}\hat{i}x,
\end{equation}
which can be uniquely maximized at $x^*=\frac{\mu_2}{2(\mu_2+\lambda_1-\mu_1)}\frac{\alpha^2\theta_v}{\alpha^2\theta_v+\theta_z}\hat{i}$ with SOC $\mu_2+\lambda_1-\mu_1>0.$ Then we have
\begin{equation*}
\beta^*=\frac{\mu_2}{2(\mu_2+\lambda_1-\mu_1)}\frac{\alpha^2\theta_v}{\alpha^2\theta_v+\theta_z}.
\end{equation*}
\textbf{Equilibrium.}
In summary, we have the following equations in equilibrium.
$$
\begin{aligned}
&\alpha=\frac{1}{2\lambda_2},\lambda_2>0.\\
&\beta=\frac{\mu_2}{2(\mu_2+\lambda_1-\mu_1)}\frac{\alpha^2\theta_v}{\alpha^2\theta_v+\theta_z},\mu_2+\lambda_1-\mu_1>0.
\end{aligned}
$$
$$
\begin{aligned}
\beta>0,&\lambda_1=\frac{\alpha\theta_v}{\beta(\alpha^2\theta_v+\theta_z)},\lambda_2=\frac{\alpha\theta_v\theta_z}{\alpha^2\theta_v(\theta_z+1)+\theta_z},\\
&\mu_1=\frac{\alpha\theta_v(\theta_z+1/\beta)}{\alpha^2\theta_v(\theta_z+1)+\theta_z},\mu_2=\frac{\alpha\theta_v\theta_z}{\alpha^2\theta_v(\theta_z+1)+\theta_z},\\
\beta=0,&\lambda_1=\mu_1=\infty,\mu_2=\lambda_2=\frac{\alpha\theta_v}{\alpha^2\theta_v+1}.\\
\end{aligned}
$$
Unfortunately, the above equations have no solution. So there is no linear equilibrium when $\theta_1=0$. However, we can investigate the partial equilibrium between IT and dealers, which is a collection of strategies $\{p_1,p_2,i^*\}$, satisfying the market-efficiency and IT's optimization conditions, given HFT's strategy $x=\beta\hat{i}$.
When $\beta\in(0,1),$ i.e., HFT does front-run, $i^*=\alpha^* v$ and
\begin{equation}
\label{alpha*=0}
\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_z}{\theta_z+1}}<\frac{\sigma_2}{\sigma_v},
\end{equation}
IT is hurt by HFT's front-running behavior. Substituting \eqref{alpha*=0} into \eqref{piHFTtheta1=0}, we find that
HFT's expected profit is zero.
Due to the absence of time-1 noise trading, HFT's front-running causes so large a price impact that harms IT without benefiting herself. When $\beta=0,$ undoubtedly, $\alpha^*=\frac{\sigma_2}{\sigma_v}$, which is in line with the classic result.
\section{Numerical Results}\label{sectionNUM}
In this section, we focus on the case $\theta_1>0$, since the case $\theta_1=0$ has been fully discussed in Section \ref{theta1=0}.
The sixth-order polynomial equation \eqref{betastar} hardly has analytical solutions, so we study the equilibrium through numerical analysis. In order to illustrate front-running HFT's impacts on IT and market quality, we use orange lines to represent the corresponding results in Kyle's one-period model, i.e., the situation without HFT.
From results of Theorem \ref{mainthm} and Corollary \ref{cor1}, given $\sigma_2$ and $\sigma_v$, we investigate:
\begin{equation}
\label{index}
\frac{\alpha^*}{\sigma_2/\sigma_v},\beta^*,\frac{\mathbb{E}(\pi^{\text{IT}})}{\sigma_v\sigma_2/2},\frac{\mathbb{E}(\pi^{\text{HFT}})}{\sigma_v\sigma_2/2},\frac{\lambda_1^*}{\sigma_v/2\sigma_2},\frac{\mu_1^*}{\sigma_v/2\sigma_2},\frac{\mu_2^*}{\sigma_v/2\sigma_2},\frac{\mathbb{E}(v-p_1)^2}{\sigma_v^2},
\end{equation}
which are values only decided by $\theta_1,\theta_z$, to see how these two parameters affect investors' actions, profits and market quality. For readability, the x-axis in the figures shows $\sqrt{\theta_1}$ and $\sqrt{\theta_z}$, rather than $\theta_1$ and $\theta_z$.
Note that the ex-post pricing error $\mathbb{E}(v-p_2)^2$ is not included in \eqref{index}, since it is always $\frac{\sigma_v^2}{2}$, which is the same as that in the situation without HFT. In fact, front-running HFT makes the price discovered earlier through $p_1$, but the ex-post pricing error is not reduced. HFT's time-1 trading $x$ injects information into the market before the IT arrives, thus the price discovery is advanced. However, HFT does not produce any extra information: she transfers information into the market through $x$ and reduces its exposure through $-x$. Whatever the IT's intensity $\alpha^*$ is, these two effects offset each other and the ex-ante price discovery is identical.
\subsection{Comparative statics with respect to \texorpdfstring{$\theta_1$}{}}
\label{section5.1}
Now we investigate how investors and market react when $\theta_1$ changes. Let $\theta_z=0.04,$ i.e., $\sigma_z=0.2\sigma_2;$
\begin{equation*}
\begin{aligned}
&\theta_1\in[10^{-6},10^{-4}],\text{ i.e., }\frac{\sigma_1}{\sigma_2}\in[10^{-3},10^{-2}];\\
&\theta_1\in[10^{-4},10^{-2}],\text{ i.e., }\frac{\sigma_1}{\sigma_2}\in[10^{-2},10^{-1}];\\
&\theta_1\in[10^{-2},25],\text{ i.e., }\frac{\sigma_1}{\sigma_2}\in[10^{-1},5].
\end{aligned}
\end{equation*}
Since the general shapes of equilibrium are similar for different $\theta_z,$ we only display results for $\theta_z=0.04$.
For HFT, $\beta^*$ and her profit increase with $\theta_1$, as shown in Figure \ref{figtheta1-HFT}. HFT is favored by more noise traders since $u_1$ plays the role of a shield, enabling HFT to employ the signal $\hat{i}$ more intensely and obtain more profits.
\begin{figure}[!htbp]
\centering
\subfigure[HFT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-small/beta.png}
}
\subfigure[HFT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-medium/beta.png}
}
\subfigure[HFT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/beta.png}
}
\subfigure[HFT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-small/piHFT.png}
}
\subfigure[HFT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-medium/piHFT.png}
}
\subfigure[HFT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/piHFT.png}
}
\caption{How HFTs' action and profits change with $\theta_1$.}
\label{figtheta1-HFT}
\end{figure}
For IT, $\alpha^*$ and her profit also increase with $\theta_1$, as shown in Figure \ref{figtheta1-IT}. On the one hand, although a larger $\theta_1$ brings a larger $\beta^*$, which increases the impact of HFT's time-1 trading, this impact is also decreased by more noise tradings. On the other hand, a larger $\beta^*$ means that HFT provides more liquidity back at time 2 and shares more of IT's impact. On average, IT gets more profits from the greater size of time-1 noise trading.
When $\theta_1$ is large enough ($\theta_1\geq0.15$), the advantages of a large $\theta_1$ outweigh the disadvantages, thus IT is benefited by HFT's front-running.
\begin{figure}[!htbp]
\centering
\subfigure[IT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-small/alpha.png}
}
\subfigure[IT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-medium/alpha.png}
}
\subfigure[IT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/alpha.png}
}
\subfigure[IT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-small/piinf.png}
}
\subfigure[IT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-medium/piinf.png}
}
\subfigure[IT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/piinf.png}
}
\caption{How IT's actions and profits change with $\theta_1$.}
\label{figtheta1-IT}
\end{figure}
For other market variables, results are similar when $\theta_1$ changes in different regions. So we only display the results when $\theta_1\in[10^{-2},25]$ in Figure \ref{figtheta1-market}.
In equilibrium, the intensities of order flow $y_1$'s impact, $\lambda_1^*$ and $\mu_1^*$ decrease with $\theta_1,$ both because when $y_1$ contains more noise, its informativeness becomes lower. Correspondingly, the intensities of order flow $y_2$'s impact, $\mu_2^*$ increases with $\theta_1$: when $y_1$ is noisier, the news component in $y_2$, $y_2-\mathbb{E}(y_2|y_1)$, appears more reliable for the dealers. What's more, we find that $\lambda_1^*>\mu_1^*,$ as shown in (d), which can be seen as the decay of $y_1$'s impact.
For the ex-ante price discovery, although IT and HFT employ their private information more intensely as $\theta_1$ increases, a larger size of noise trading prevents dealers from distinguishing the true value. The latter effect goes beyond the former one, thus the time-1 pricing error increases with $\theta_1$.
\begin{figure}[!htbp]
\centering
\subfigure[Ex-ante liquidity, $\lambda_1^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/lambda1.png}
}
\subfigure[Ex-post liquidity, $\mu_1^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/mu1.png}
}
\subfigure[Ex-post liquidity, $\mu_2^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/mu2.png}
}
\subfigure[Decay of $y_1$'s impact.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/decay.png}
}
\subfigure[Ex-ante pricing error.]{
\includegraphics[width = 0.31\textwidth]{fig/theta1/theta1-large/pd1.png}
}
\caption{How $\theta_1$ affects market quality.}
\label{figtheta1-market}
\end{figure}
\subsection{Comparative statics with respect to \texorpdfstring{$\theta_z$}{}}
\label{section5.2}
Now we investigate how investors and market react when $\theta_z$ changes. We first give how the actions and profits of investors change with $\theta_z\in[0,25]$, when $\theta_1=0.12,0.2\text{ and }1,$ i.e.,
\begin{equation*}
\frac{\sigma_z}{\sigma_2}\in[0,5], \frac{\sigma_1}{\sigma_2}=0.34,0.45\text{ and }1.
\end{equation*}
For HFT, the trading intensity $\beta^*$ decreases with $\theta_z$, as shown in (a)-(c) of Figure \ref{figthetaz-HFT}, since a noisier signal makes her less sure about the future informed trading. Surprising results appear for HFT's expected profit. It is usually considered that HFT's profit should decrease when she receives a less accurate signal. However, as shown in (d) of Figure \ref{figthetaz-HFT}, it increases with $\theta_z$ when both $\theta_1$ and $\theta_z$ are relatively small. It is because IT trades more as $\theta_z$ gets larger ((g) in Figure \ref{figthetaz-IT}), HFT's absolute front-running volume $x^*$ also increases, which brings the growth of profit.
\begin{figure}[!htbp]
\centering
\subfigure[HFT's action, $\theta_1=0.12$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/beta_small.png}
}
\subfigure[HFT's action, $\theta_1=0.2$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/beta_medium.png}
}
\subfigure[HFT's action, $\theta_1=1$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/beta_large.png}
}
\subfigure[HFT's profit, $\theta_1=0.12$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piHFT_small.png}
}
\subfigure[HFT's profit, $\theta_1=0.2$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piHFT_medium.png}
}
\subfigure[HFT's profit, $\theta_1=1$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piHFT_large.png}
}
\caption{How HFT's actions and profits change with $\theta_z$.}
\label{figthetaz-HFT}
\end{figure}
For IT, the results are more complicated. The following Figure \ref{figthetaz-IT} shows the trend of $\theta_z$'s impact on IT when $\theta_1$ is in different value ranges. For $\theta_1\in(0,\frac{2\sqrt{3}-3}{3}),$ we find that IT's action as well as her profit, first increases then decreases with $\theta_z$, and IT is favored by the front-running HFT when $\theta_z$ exceeds the critical value $\overline{\theta}_z$ calculated in Proposition \ref{benefit}. For $\theta_1\in[\frac{2\sqrt{3}-3}{3},\frac{1}{2}),$ IT's action and profit change as the former case. However, she is always benefited by HFT. For $\theta_1\geq\frac{1}{2},$ IT is still always benefited by HFT but her profit keeps decreasing with $\theta_z$.
As the signal becomes noisier, HFT's action $\beta^*$ decreases. On the one hand, HFT's time-1 trading causes less impact. On the other hand, her time-2 trading shares fewer transaction costs for IT. When the first effect exceeds the second one, IT's action and profit increase with $\theta_z$, otherwise, they decrease with it.
\begin{figure}[!htbp]
\centering
\subfigure[IT's action, $\theta_1=0.12\in(0,\frac{2\sqrt{3}-3}{3})$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/alpha_small.png}
}
\subfigure[IT's action, $\theta_1=0.2\in[\frac{2\sqrt{3}-3}{3},\frac{1}{2})$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/alpha_medium.png}
}
\subfigure[IT's action, $\theta_1=1\in[\frac{1}{2},+\infty)$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/alpha_large.png}
}
\subfigure[IT's profit, $\theta_1=0.12\in(0,\frac{2\sqrt{3}-3}{3})$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piinf_small.png}
}
\subfigure[IT's profit, $\theta_1=0.2\in[\frac{2\sqrt{3}-3}{3},\frac{1}{2})$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piinf_medium.png}
}
\subfigure[IT's profit, $\theta_1=1\in[\frac{1}{2},+\infty)$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piinf_large.png}
}
\caption{How IT's actions and profits change with $\theta_z$.}
\label{figthetaz-IT}
\end{figure}
To complement the monotonicity of $\alpha^*$ on $\theta_z$ in Proposition \ref{ITaction}, we have done a lot of numerical experiments for $0<\theta_1<0.0089$ and find that the trend of IT's action and profit is the same as $\theta_1\in[0.0089,\frac{1}{2}):$ first increase then decrease with $\theta_z$, as shown by the example in Figure \ref{figthetaz-example}.
\begin{figure}[!htbp]
\centering
\subfigure[IT's action.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/alpha_special.png}
}
\subfigure[IT's profit.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/piinf_special.png}
}
\caption{How IT's action and profit change with $\theta_z$ when $\theta_1=10^{-4}$.}
\label{figthetaz-example}
\end{figure}
For other market variables, results are similar for different $\theta_1$. Thus we only display numerical results for $\theta_1=0.12$ in Figure \ref{figthetaz-market}. As shown in (a) and (b), $\lambda_1^*$ and $\mu_1^*$ decrease with $\theta_z$, the informational proportion of trading $x^*$ is reduced by the noise in HFT's signal, and so is $y_1$. In contrast, time-2 trading enables dealers to learn more about $v$ and brings larger $\mu_2^*$. $y_1$'s impact also decays, as shown in (d).
As for the ex-ante price discovery, the time-1 pricing error increases with $\theta_z$. Although larger noise might enable the IT to employ her information advantage more fiercely, HFT decreases her intensity $\beta^*$. What's more, HFT's trading $x^*$ contains too much noise, thus dealers learn less from the order flow $y_1$.
\begin{figure}[!htbp]
\centering
\subfigure[Ex-ante liquidity, $\lambda_1^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/lambda1.png}
}
\subfigure[Ex-post liquidity, $\mu_1^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/mu1.png}
}
\subfigure[Ex-post liquidity, $\mu_2^*$.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/mu2.png}
}
\subfigure[Decay of $y_1$'s impact.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/decay.png}
}
\subfigure[Ex-ante pricing error.]{
\includegraphics[width = 0.31\textwidth]{fig/thetaz/pd1.png}
}
\caption{How $\theta_z$ affects market quality, when $\theta_1=0.12$.}
\label{figthetaz-market}
\end{figure}
\section{When HFT predicts \texorpdfstring{$i+u_2$}{}}
\label{sectionprei+u2}
In this section, we discuss an extension of our model that HFT can predict $i+u_2,$ i.e., the signal received by HFT is
\begin{equation*}
\hat{y}=i+u_2+z,
\end{equation*}
where $z\sim N(0,\sigma_z^2)$ is the noise independent of other random variables. The existence and uniqueness of the linear equilibrium are given in the following theorem.
\begin{theorem}
Given $\sigma_1,\sigma_2>0,\sigma_z\geq0,(\theta_1=\frac{\sigma_1^2}{\sigma_2^2}>0,\theta_z=\frac{\sigma_z^2}{\sigma_2^2}\geq0,)$ there exists a unique equilibrium $\{p_1,p_2,i^*,x^*\}$, where HFT follows the strategy $x^*=\beta^*\hat{i}$ and $\beta^*\in(0,1)$ solves the equation
\begin{equation*}
\begin{aligned}
0=& \beta^6(4\theta_1^2+6\theta_1^2\theta_z+8\theta_1\theta_z+8\theta_1\theta_z^2+\theta_1\theta_z^3+2\theta_1^2\theta_z^2+4\theta_z^2+2\theta_z^3)\\
+&\beta^5(4\theta_1^3-16\theta_1^2+4\theta_1^3\theta_z-8\theta_1^2\theta_z-16\theta_1\theta_z+4\theta_1^2\theta_z^2-4\theta_1\theta_z^2+2\theta_z\theta_z^3)\\
+ & \beta^4(-18\theta_1^3+24\theta_1^2-13\theta_1^3\theta_z-4\theta_1^2\theta_z+8\theta_1\theta_z-8\theta_1^2\theta_z^2+2\theta_1\theta_z^2) \\
+&\beta^3(32\theta_1^3-16\theta_1^2+16\theta_1^3\theta_z+8\theta_1^2\theta_z+4\theta_1^2\theta_z^2) \\
+ & \beta^2(-28\theta_1^3+4\theta_1^2-9\theta_1^3\theta_z-2\theta_1^2\theta_z) + \beta(12\theta_1^3 + 2\theta_1^3\theta_z) -2\theta_1^3;
\end{aligned}
\end{equation*}
IT follows the strategy $i^*=\alpha^*v$ and
\begin{equation*}
\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2}}{\theta_1(1-\beta^*)^2+\theta_z\beta^{*2}}}\geq\frac{\sigma_2}{\sigma_v},
\end{equation*}
that is, IT is benefited by HFT;
the liquidation price at time 1 is
\begin{equation*}
p_1=\lambda_1^* (x^*+u_1),
\end{equation*}
where
\begin{equation*}
\lambda_1^*=\frac{\sigma_v}{2\sigma_2}\frac{2\beta^*\sqrt{(\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})(\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})}}{\beta^{*2}(\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})+(\beta^{*2}(\theta_z+1)+\theta_1)(\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})};
\end{equation*}
the liquidation price at time 2 is
\begin{equation*}
p_2=\mu_1^*(x^*+u_1)+\mu_2^*(i^*+u_2-x^*),
\end{equation*}
where
\begin{equation*}
\mu_1^*=\frac{\sigma_v}{2\sigma_2}\frac{\beta^{*2}\theta_z}{\sqrt{(\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})(\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})}},
\end{equation*}
\begin{equation*}
\mu_2^*=\frac{\sigma_v}{2\sigma_2}\frac{\beta^{*2}\theta_z+(1-\beta^*)\theta_1}{\sqrt{(\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})(\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2})}};
\end{equation*}
the impact coefficient of the informed order $i^*$ is
\begin{equation*}
\lambda_2^*=\mu_1^*\beta^*+\mu_2^*(1-\beta^*)=\frac{\sigma_v}{2\sigma_2}\sqrt{\frac{\theta_1(1-\beta^*)^2+\theta_z\beta^{*2}}{\theta_1\theta_z\beta^{*2}+\theta_1(1-\beta^*)^2+\theta_z\beta^{*2}}}\leq\frac{\sigma_v}{2\sigma_2}.
\end{equation*}
When $\theta_1=0,$ the linear equilibrium does not exist.
\end{theorem}
For $\theta_1=0,$ similar to the analysis in Section \ref{theta1=0}, given $x=\beta\hat{y},\beta\in(0,1)$, in the partial-equilibrium, $\alpha^*=\frac{\sigma_2}{\sigma_v}$, which means that whatever $\beta$ is, IT is not affected by HFT.
We can also see the two-sided effects of HFT:
\begin{proposition}
\label{effectHFT-i+u2}
Given $\theta_1>0,\theta_z\geq0$ and IT's activity $\alpha>0$, for any HFT's activity $\beta\in(0,1)$,
\begin{equation*}
\lambda_2(\alpha;\beta)=\mu_1(\alpha;\beta)\beta+\mu_2(\alpha;\beta)(1-\beta),
\end{equation*}
where the first part increases with $\beta$ and the second part decreases with $\beta$. What's more, $\mu_1^*,\mu_2^*\leq\frac{\sigma_v}{2\sigma_2},$ which makes $\lambda_2^*\leq\frac{\sigma_v}{2\sigma_2}.$
\end{proposition}
Recalling that in our original model, HFT predicts $i$, she might harm IT under certain conditions, since $\mu_1^*$ can be greater than $\frac{\sigma_v}{2\sigma_2}$. However, when HFT's signal comes from $i+u_2$, $x$ contains less information but more noise, making the time-1 trading $y_1$ less informative, bringing smaller $\mu_1^*$. Thus $\mu_1^*$ and $\mu_2^*$ are both smaller than $\frac{\sigma_v}{2\sigma_2}$, HFT is always in favor of IT.
Li (2018) \cite{li2018high} models a market with normal-speed noise traders, i.e., $\theta_1=0$, where front-running HFTs predict the aggregate order $i+u_2$. The author concludes that IT is always harmed by HFT. The key differences between this model and ours are: (1) \cite{li2018high} models stale dealers who quote linearly with unchanged impact coefficients at times 1 and 2, while we assume the dealers quote according to the market-efficiency condition; (2) \cite{li2018high} assumes $\theta_1=0$. It is precisely because of these two points that our conclusions are different.
Thus our results and \cite{li2018high}'s results can be seen as mutually complementary, explaining how front-running HFT affects IT when confronted with different liquidity providers and different sizes of market noise.
\section{Conclusion}
We study the influences of a front-running HFT on a large informed trader and market quality in various situations, where the size of market noise and signal accuracy differ. Since HFT takes liquidity away as well as provides liquidity back, she has two-sided effects on the large trader. When the market noise is sufficient, even if HFT's signal is perfectly accurate, she could be in favor of the large trader. Without enough noise shelter from the market, the large trader is benefited only when HFT's signal is relatively vague. The price discovery is advanced through HFT's time-1 trading. However, the ultimate pricing error is not reduced, compared to that without HFT.
\section{Appendix}
\textbf{Proof of Theorem \ref{mainthm}.}
When $\beta\leq0$, we have $\mu_2>0$. If the SOC $\mu_2+\lambda_1-\mu_1$ hold, $\beta$ must be positive, which contradicts with each other. Thus we have $\beta^*>0.$
Combining \eqref{lambda1},\eqref{mu1},\eqref{mu2},\eqref{alpha},\eqref{beta}, we get that $\beta^*$ satisfies Equation \eqref{betastar},
and the optimal intensity for IT is \eqref{alphastar}. Next we need only to prove that the root of Equation \eqref{betastar} must be in $(0,1)$ and unique, satisfies the SOC. If we see the right side of Equation \eqref{betastar} the function of $\beta$: $f(\beta;\theta_1,\theta_z),$ we have
\begin{equation*}
\begin{aligned}
&f(0;\theta_1,\theta_z)=-\theta_1^3<0,\\
&f(1;\theta_1,\theta_z)=2 \theta_1^2 + 6 \theta_1 \theta_z + 4 \theta_1^2\theta_z + 2 \theta_z^2 + 9 \theta_1 \theta_z^2 + 2 \theta_1^2 \theta_z^2 + \theta_z^3 + 3 \theta_1 \theta_z^3>0,
\end{aligned}
\end{equation*}
then there must exist $\beta^*\in(0,1)$, satisfying Equation \eqref{betastar}. For $\beta\geq1,$ it is easy to verify $f'(\beta;\theta_1,\theta_z)$ is also positive. Thus there is no root of Equation \eqref{betastar} on $[1,+\infty)$.
Now we denote $\beta^*$ the smallest root of $f$ on $(0,1)$. The SOC is equivalent to
\begin{equation*}
\alpha\sigma_v^2(\sigma_1^2+\alpha^2\beta\sigma_v^2)((1-\beta)\sigma_1^2+\beta^2\sigma_z^2)>0,
\end{equation*}
which holds for $\beta\in(0,1).$
\begin{equation*}
\begin{aligned}
f^{'}(\beta;\theta_1,\theta_z)=&\theta_1^2\theta_z(40\beta^4-44\beta^3+24\beta^2-2\beta)+\theta_1^2\theta_z^2(12\beta^5+20\beta^4-32\beta^3+12\beta^2)\\
+&\theta_1^3\theta_z(20\beta^4-42\beta^3+48\beta^2-18\beta+2)+\theta_1^3(6\beta^2-10\beta+4)\\
+&\theta_1\theta_z(20\beta^4+8\beta^3)+\theta_1\theta_z^2(24\beta^5+20\beta^4+4\beta^3)\\
+&\theta_1\theta_z^3(6\beta^5+10\beta^4)+6\beta^5(2\theta_z^2+\theta_z^3)+6\beta^2\theta_1^2.
\end{aligned}
\end{equation*}
For $\beta\in(0,0.099),40\beta^4-44\beta^3+24\beta^2-2\beta<0.$ If this leads to $f'(\beta)<0$ for $\beta\in(0,0.099),$ we must have $\beta^*>0.099,$ since $f(0)<0.$ So the negativity would not cause $f'(\beta)<0$ for $\beta\in[\beta^*,1).$
For $\beta\in(0.205,0.325),20\beta^4-42\beta^3+48\beta^2-18\beta+2<0.$ If $\theta_1\leq\theta_z,$
\begin{equation*}
\begin{aligned}
&\theta_1^3\theta_z(20\beta^4-42\beta^3+48\beta^2-18\beta+2)+\theta_1^2\theta_z^2(12\beta^5+20\beta^4-32\beta^3+12\beta^2)\\
\geq&\theta_1^3\theta_z(12\beta^5+40\beta^4-74\beta^3+60\beta^2-18\beta+2)\geq0.
\end{aligned}
\end{equation*}
If $\theta_1>\theta_z$ and we want to have
\begin{equation*}
\theta_1^3\theta_z(20\beta^4-42\beta^3+48\beta^2-18\beta+2)+\theta_1^3(6\beta^2-10\beta+1)\geq0,
\end{equation*}
$\theta_z\leq21.$
For $\beta\in(\frac{2}{3},1),6\beta^2-10\beta+4<0.$ If $\theta_1\leq\theta_z,$
\begin{equation*}
\begin{aligned}
&\theta_1^3(6\beta^2-10\beta+4)+\theta_1^2\theta_z(40\beta^4-44\beta^3+24\beta^2-2\beta)\\
\geq&\theta_1^3(40\beta^4-44\beta^3+30\beta^2-12\beta+4)\geq0.
\end{aligned}
\end{equation*}
If $\theta_1>\theta_z$ and we want to have
\begin{equation*}
\theta_1^3(6\beta^2-10\beta+4)+6\beta^2\theta_1^2\geq0,
\end{equation*}
$\theta_1\leq24.$
Now we have proved that if $\theta_1\leq\theta_z$ or $\theta_z<\theta_1\leq21,f^{'}(\beta)\geq0$ for $\beta\in[\beta^*,1)$. So $f(\beta)>0$ for $\beta\in(\beta^*,1)$, $\beta^*$ is the unique root of $f$ on $(0,1).$ We only need to prove the uniqueness for $\theta_1>21>\theta_z$ and $\theta_1>\theta_z>21$, which is equivalent to prove $f(\beta)>0$ for $\beta\in(\beta^*,1]$.
\begin{equation*}
\begin{aligned}
f(\beta;\theta_1,\theta_z)=&\theta_1^2\theta_z(8\beta^5-11\beta^4+8\beta^3-\beta^2)+\theta_1^2\theta_z^2(2\beta^6+4\beta^5-8\beta^4+4\beta^3)\\
+&\theta_1^3\theta_z(4\beta^5-13\beta^4+16\beta^3-9\beta^2+2\beta)+\theta_1^3(2\beta^3-5\beta^5+4\beta-1)\\
+&\theta_1\theta_z(4\beta^5+2\beta^4)+\theta_1\theta_z^2(4\beta^6+4\beta^5+\beta^4)\\
+&\theta_1\theta_z^3(\beta^6+2\beta^5)+\beta^6(2\theta_z^2+\theta_z^3)+2\beta^3\theta_1^2.
\end{aligned}
\end{equation*}
When $\theta_1>21,$ and for $\beta\in(0,0.154),8\beta^5-11\beta^4+8\beta^3-\beta^2<0,$ but
\begin{equation*}
\begin{aligned}
&\theta_1^2\theta_z(8\beta^5-11\beta^4+8\beta^3-\beta^2)+\theta_1^3\theta_z(4\beta^5-13\beta^4+16\beta^3-9\beta^2+2\beta)\\
\geq&\theta_1^2\theta_z(8\beta^5-11\beta^4+8\beta^3-\beta^2+21(4\beta^5-13\beta^4+16\beta^3-9\beta^2+2\beta))\geq0.
\end{aligned}
\end{equation*}
Then for $\beta\in(0,1),$
\begin{equation*}
f(\beta;\theta_1,\theta_z)\geq\theta_1^3(2\beta^3-5\beta^5+4\beta-1)+2\beta^3\theta_1^2=f(\beta;\theta_1,\theta_z=0),
\end{equation*}
we would analyze the special case $\theta_z=0$ in subsection 4.1.1 , and we employ some conclusions in advance.
We now prove $f(\beta;\theta_1,\theta_z=0)>0$ for $\beta\in(\beta^*(\theta_z=0),1)$. Since
\begin{equation*}
\begin{aligned}
&f(\beta;\theta_1,\theta_z=0)=\theta_1^2g(\beta),\\
&g(\beta)=2\beta^3(\theta_1+1)-5\theta_1\beta^2+4\theta_1\beta-\theta_1,
\end{aligned}
\end{equation*}
it is equivalent to investigate $g(\beta).$
$g(\beta)$ is (1) increasing when $21<\theta_1<24$ or (2) has a unique minimum point $\Tilde{\beta}=\frac{5\theta_1+\sqrt{\theta_1^2-24\theta_1}}{6(\theta_1+1)}$ on $(\beta^*(\theta_z=0),1]$, when $\theta_1\geq24.$ Correspondingly, for $\beta\in(\beta^*(\theta_z=0),1]$, (1) $g(\beta)>g(\beta^*(\theta_z=0))=0$, (2)$g(\beta)\geq g(\Tilde{\beta})>0.$ Since $\beta^*(\theta_z=0)\leq\frac{1}{2},$ we have $f(\beta;\theta_1,\theta_z)>0$ for $\beta\in(\frac{1}{2},1)$ when $\theta_1>21$.
When $\theta_z<21,$ $f(\beta;\theta_1,\theta_z)$ is increasing for $\beta\in[\beta^*,\frac{2}{3}],$ then it is also increasing for $\beta\in[\beta^*,\frac{1}{2}],$ which implies $f(\beta;\theta_1,\theta_z)>0$ for $\beta\in(\beta^*,1)$.
When $\theta_z>21,f(\beta;\theta_1,\theta_z)$ may decrease for $\beta\in(0.206,0.324).$ The only possible negative part in $f$ is
$\theta_1^3(2\beta^3-5\beta^5+4\beta-1)$. However,
\begin{equation*}
\begin{aligned}
&\theta_1^3\theta_z(4\beta^5-13\beta^4+16\beta^3-9\beta^2+2\beta)+\theta_1^3(2\beta^3-5\beta^5+4\beta-1)\\
\geq&\theta_1^3(21(4\beta^5-13\beta^4+16\beta^3-9\beta^2+2\beta)+2\beta^3-5\beta^5+4\beta-1)>0,
\end{aligned}
\end{equation*}
so we must have $f(\beta;\theta_1,\theta_z)>0$ for $\beta\in(\beta^*,1]$ no matter $\beta^*<0.206,$ $\beta^*>0.324$ or $\beta^*\in(0.206,0.324).$
\noindent\textbf{Proof of Proposition \ref{effectHFT1}.}
Denote $\theta_v=\frac{\sigma_v^2}{\sigma_2^2},$
\begin{equation*}
\begin{aligned}
\frac{\partial(\mu_1(\alpha;\beta)\beta)}{\partial\beta}
=&\frac{\alpha\theta_v}{(\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2+\theta_z\beta^2)+\beta^2\theta_z(\theta_1+1)+\theta_1)^2}(\alpha^2\theta_v(\theta_1\theta_z\beta^2(1-\beta)(3-\beta)\\
+&\beta^4(\theta_z+\theta_z^2)+2\theta_1\beta(1-\beta))+\beta^4\theta_z^2(\theta_1+1)+\theta_1(3\beta^2\theta_z+2\beta))>0,
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\frac{\partial(\mu_2(\alpha;\beta)(1-\beta))}{\partial\beta}
=&\frac{\alpha\theta_v}{(\alpha^2\theta_v(\theta_1(1-\beta)^2+\beta^2+\theta_z\beta^2)+\beta^2\theta_z(\theta_1+1)+\theta_1)^2}(-\alpha^2\theta_v(\beta^4(\theta_z+\theta_z^2))\\
+&\theta_1\theta_z\beta^2(1-\beta)(3-\beta)+2\theta_1\beta(1-\beta))-\beta^4\theta_z^2(\theta_1+1)-\beta^2\theta_1\theta_z\\
-&2\theta_1^2\theta_z\beta(1-\beta)-2\theta_1^2(1-\beta))<0.
\end{aligned}
\end{equation*}
\noindent\textbf{Proof of Proposition \ref{effectHFT2}.}
\begin{equation*}
(\frac{\mu_2^{*}}{\sigma_v/2\sigma_2})^2-1=\frac{\beta^{*2}(\theta_1+\beta^{*2}\theta_z+(1-\beta^*)^2\theta_1^2\theta_z+\beta^{*2}\theta_1\theta_z(2+\theta_z))}{(\theta_1(1-\beta^*)^2+\beta^{*2}(\theta_z+1))(\beta^{*2}\theta_z(\theta_1+1)+\theta_1)}
>0.
\end{equation*}
\noindent\textbf{Proof of Theorem \ref{benefit}.}
$\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$ is equivalent to
$\beta^*(1+\theta_1-\theta_1\theta_z)-2\theta_1<0,$ which holds for $\theta_1\theta_z\geq1.$
If $\theta_1\theta_z<1,$ we need $\beta^*<\frac{2\theta_1}{1+\theta_1-\theta_z},$ i.e.
\begin{equation*}
f(\frac{2\theta_1}{1+\theta_1-\theta_z})>0\Longleftrightarrow-5\theta_1^2\theta_z^2+2\theta_1^2\theta_z+3\theta_1^2+10\theta_1\theta_z+6\theta_1-1>0.
\end{equation*}
We could rearrange it as
\begin{equation*}
h(\theta_z)=-5\theta_1^2\theta_z^2+\theta_z(2\theta_1^2+10\theta_1)+3\theta_1^2+6\theta_1-1>0.
\end{equation*}
If $3\theta_1^2+6\theta_1-1\geq0,\theta_1\geq\frac{2\sqrt{3}-3}{3},h(\theta_z)>0$ when $0\leq\theta_z<x_2,$ where
\begin{equation*}
x_2=\frac{(\theta_1+5)+2\sqrt{4\theta_1^2+10\theta_1+5}}{5\theta_1}.
\end{equation*}
Since $x_2>\frac{1}{\theta_1},$ $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}.$
If $3\theta_1^2+6\theta_1-1<0,0<\theta_1<\frac{2\sqrt{3}-3}{3},h(\theta_z)>0$ when $x_1<\theta_z<x_2$, where
$x_1=\overline{\theta}_z.$ Since $x_1<\frac{1}{\theta_1}<x_2,$ we have $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$ when $\theta_z>\overline{\theta}_z.$
As for the last conclusion, we investigate
\begin{equation*}
h(\theta_1)=\theta_1(-5\theta_z^2+2\theta_z+3)+\theta_1(10\theta_z+6)-1>0.
\end{equation*}
If $-5\theta_z^2+2\theta_z+3>0,0\leq\theta_z<1,h(\theta_1)>0$ when $\theta_1>x_2=\overline{\theta}_1.$
If $-5\theta_z^2+2\theta_z+3\leq0,\theta_z\geq1,h(\theta_1)>0$ when $x_1<\theta_1<x_2,$ where
\begin{equation*}
\begin{aligned}
x_1&=\overline{\theta}_1,\\
x_2&=\frac{-(5\theta_z+3)-2\sqrt{5\theta_z^2+8\theta_z+3}}{-5\theta_z^2+2\theta_z+3}.
\end{aligned}
\end{equation*}
Since $x_1<\frac{1}{\theta_z}<x_2,$ we have $\lambda_2^*<\frac{\sigma_v}{2\sigma_2}$ when $\theta_1>\overline{\theta}_1.$
\noindent\textbf{Proof of Proposition \ref{HFTaction}.}
Denote the right side of Equation \eqref{betastar} $f$, to investigate $\theta_1$,\begin{equation*}
\begin{aligned}
f'(\theta_1;\beta)&=\theta_1^2(\theta_1(12\beta^5-39\beta^4+48\beta^3-27\beta^2+6\beta)+6\beta^3-15\beta^2+12\beta-3)\\
&+\theta_1(\theta_z^2(4\beta^6+8\beta^5-16\beta^4+8\beta^3)+\theta_z(16\beta^5-22\beta^4+16\beta^3-4\beta^2)+4\beta^3)\\
&+\theta_z^3(\beta^6+2\beta^5)+\theta_z^2(4\beta^6+4\beta^5+\beta^4)+\theta_z(4\beta^5+2\beta^4).
\end{aligned}
\end{equation*}
It is first positive then negative or first positive then negative finally positive on $\theta_1\in(0,\infty]$.
If $f(\beta;\theta_1)=0,f(\theta_1=0;\beta)=\beta^6(2\theta_z^2+\theta_z^3)>0,$ then for $\theta_1'\in[0,\theta_1],$ we have $f(\beta;\theta_1')=f(\theta_1';\beta)>0.$ If $f(\beta';\theta_1')=0,$ we have $\beta>\beta'$, i.e. $\beta^*$ increases with $\theta_1$.
To investigate $\theta_z,$
\begin{equation*}
\begin{aligned}
f'(\theta_z;\beta)=&\theta_z^2(3\beta^6+3\beta^6\theta_1+6\beta^5\theta_1)\\
+&\theta_z(\theta_1^2(4\beta^6+8\beta^5-16\beta^4+8\beta^3)+\theta_1(8\beta^6+8\beta^5+2\beta^4)+4\beta^6)\\
+&\theta_1^3(4\beta^5-13\beta^4+16\beta^3-9\beta^3+2\beta)\\
+&\theta_1^2(8\beta^5-11\beta^4+8\beta^3-\beta^2)+\theta_1(4\beta^5+2\beta^4).
\end{aligned}
\end{equation*}
It is first negative then positive or positive on $\theta_z\in[0,+\infty),$ which implies $f(\theta_z;\beta)$ first decreases than increases or increases on $\theta_z\in[0,+\infty),$
If $f(\beta;\theta_z)=0,$ then
\begin{equation*}
\begin{aligned}
f(\beta;\theta_z=0)=&\beta^3(2\theta_1^2+\theta_1^3)-5\beta^2\theta_1^3+4\beta\theta_1^3-\theta_1^3\\
= &-\beta^6(4\theta_1\theta_z^2+\theta_1\theta_z^3+2\theta_z^2+\theta_z^3)-\beta^5(4\theta_1\theta_z+4\theta_1\theta_z^2+2\theta_1\theta_z^3)\\
&-\beta^4(2\theta_1\theta_z+\theta_1\theta_z^2)+\theta_1\theta_z^2(-8\beta^5+8\beta^4-8\beta^3+\beta^2)\\
&+\theta_1^2\theta_z^2(-4\beta^5+8\beta^4-4\beta^3)+\theta_1^3\theta_z(-4\beta^5+13\beta^4-16\beta^3+9\beta^2-2\beta)\\
&<0.
\end{aligned}
\end{equation*}
So for $\theta_z'\in[0,\theta_z],f(\beta;\theta_z')=f(\theta_z';\beta)<0.$ If $f(\beta';\theta_z')=0,$ we have $\beta'>\beta$. Thus $\beta^*$ decreases with $\theta_z$.
\noindent\textbf{Proof of Proposition \ref{ITaction}.} $\alpha^*(\theta_1,\theta_z)=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_1+\beta^{*2}\theta_z(\theta_z+1)}{\theta_1(1-\beta^*)^2+(1+\theta_z)\beta^{*2}}}.$
\begin{equation*}
\begin{aligned}
\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_1}&=\frac{\partial\alpha^*(\beta^*,\theta_1,\theta_z)}{\partial\beta^*}\frac{\partial\beta^*}{\partial\theta_1}+\frac{\partial\alpha^*(\beta^*,\theta_1,\theta_z)}{\partial\theta_1}.
\end{aligned}
\end{equation*}
Then $\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_1}\geq0$ if and only if
\begin{equation*}
2\theta_1(\theta_1-(1+\theta_1)\beta^*)\frac{\partial\beta^*}{\partial\theta_1}+\beta^{*2}(1+\beta^*\theta_z)\geq0.
\end{equation*}
Since $\frac{\partial\beta^*}{\partial\theta_1}>0,$ the above inequality holds if $\theta_1-(1+\theta_1)\beta^*\geq0,$ which is equivalent to
\begin{equation*}
f(\frac{\theta_1}{1+\theta_1})=\frac{\theta_1^3(1+\theta_1+\theta_1\theta_z)^2(2\theta_1^2+\theta_1(3\theta_z+1)-1)}{(1+\theta_1^5)}\geq0,
\end{equation*}
in other words,
\begin{equation*}
\begin{aligned}
&2\theta_1^2+\theta_1(3\theta_z+1)-1\geq0,\\
&\theta_1\geq\frac{\sqrt{(3\theta_z+1)^2+8}-(3\theta_z+1)}{4}=\frac{2}{\sqrt{(3\theta_z+1)^2+8}+(3\theta_z+1)}|_{max}=\frac{1}{2}.
\end{aligned}
\end{equation*}
Actually we specify two sufficient conditions: (1) $\theta_1\geq\frac{1}{2}$, (2) $\theta_z\geq0,\frac{\sqrt{(3\theta_z+1)^2+8}-(3\theta_z+1)}{4}\leq\theta_1<\frac{1}{2}.$
$\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_z}$ has the same sign as
\begin{equation*}
2\theta_1(\theta_1-(1+\theta_1)\beta^*)(1+\beta^*\theta_z)\frac{\partial\beta^*}{\partial\theta_z}+\beta^{*2}(\theta_1-(1+\theta_1)\beta^*)^2.
\end{equation*}
If $\theta_1-(1+\theta_1)\beta^*>0,\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_z}\leq0$ is equivalent to
\begin{equation*}
g(\beta^*;\theta_1,\theta_z)=\beta^{*2}(\theta_1-(1+\theta_1)\beta^*)\frac{\partial f}{\partial\beta^*}-2\theta_1(1+\beta^*\theta_z)\frac{\partial f}{\partial\theta_z}\leq0.
\end{equation*}
For simplicity, we still use $\beta$ to represent $\beta^*$.
\begin{equation*}
\begin{aligned}
g&=\theta_1^2(-4\beta^4-14\beta^5)+\theta_1^3y_1+\theta_1^4(-4\beta+22\beta^2-46\beta^3+42\beta^4-14\beta^5)\\
&+\theta_1\theta_z(-16\beta^6-20\beta^7)+\theta_1^2\theta_z(-2\beta^4-36\beta^5+32\beta^6-60\beta^3)\\
&+\theta_1^3\theta_z(-18\beta^3+60\beta^4-110\beta^5+112\beta^6-60\beta^7)+\theta_1^4\theta_z(-2\beta^2-2\beta^3+34\beta^4-74\beta^5+64\beta^6-20\beta^7)\\
&-12\beta^8\theta_z^2+\theta_1\theta_z^2(-10\beta^6-16\beta^7-36\beta^8)+\theta_1^2\theta_z^2(-24\beta^5+26\beta^6-32\beta^7-36\beta^8)\\
&+\theta_1^3\theta_z^2(-4\beta^4-12\beta^5+36\beta^6-16\beta^7-12\beta^8)-6\beta^8\theta_z^3\\
&+\theta_1\theta_z^3(-10\beta^7-12\beta^8)+\theta_1^2\theta_z^3(-2\beta^6-10\beta^7-6\beta^8),\\
y_1&=2\beta^2-20\beta^3+38\beta^4-28\beta^5.
\end{aligned}
\end{equation*}
Except $y_1,$ other coefficients are both non-positive for $\beta\in[0,\frac{1}{2}].$ (We would see in following paragraphs that $\beta^*\leq\frac{1}{2}$.) $y_1\geq0$ when $\beta\in[0,0.1286].$ Let
\begin{equation*}
\begin{aligned}
p&=-4\beta^4-14\beta^5,\\
q&=-4\beta+22\beta^2-46\beta^3+42\beta^4-14\beta^5.
\end{aligned}
\end{equation*}
$\theta_1^2p+\theta_1^3y_1+\theta_1^4q\leq0$ is equivalent to $p+\theta_1y_1+\theta_1^2q\leq0,\Delta=y_1^2-4pq\geq0$ when $\beta\in[0,0.0328].$ Thus if
\begin{equation*}
\theta_1\geq\frac{-y_1-\sqrt{\Delta}}{2q}|_{max}=0.0089,
\end{equation*}
and $\theta_z>\frac{1-\theta_1-2\theta_1^2}{3\theta_1},$ we have $\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_z}\leq0$. In a word, we specify two conditions, (1) $\theta_1\geq\frac{1}{2},$ (2) $0.0089\leq\theta_1<\frac{1}{2},\theta_z>\frac{1-\theta_1-2\theta_1^2}{3\theta_1}.$
When $\theta_1-(1+\theta_1)\beta^*\leq0,\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_z}\geq0,$ which implies $\theta_z<\frac{1-\theta_1-2\theta_1^2}{3\theta_1}.$ Thus when $0<\theta_1<\frac{1}{2}$ and $\theta_z<\frac{1-\theta_1-2\theta_1^2}{3\theta_1},\frac{\partial\alpha^*(\theta_1,\theta_z)}{\partial\theta_z}\geq0.$
\noindent\textbf{Proof of Proposition \ref{theta1infty}.} Divide both sides of Equation \eqref{betastar} $\theta_1^3$ and calculate the limit when $\theta_1\rightarrow\infty$, we have the equation for the limit of $\beta^*$. Then by L's Hospital rule,
\begin{equation*}
\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_1(\beta^{*2}\theta_z+1)+\beta^{*2}\theta_z}{\theta_1(1-\beta^*)^2+(1+\theta_z)\beta^{*2}}}\rightarrow\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\beta^{*2}\theta_z+1}{(1-\beta^*)^2}}.
\end{equation*}
\noindent\textbf{Proof of Proposition \ref{theta10}.}
Calculate the limit of both sides of Equation \eqref{betastar} when $\theta_1\rightarrow0$, we have $\beta^*\rightarrow0$. However, in order to calculate the limit of $\alpha^*$, we need a more precise estimate of $\beta^*$. We assume $\beta^*=O(\theta_1^a)$. If $a>\frac{1}{2},\lim_{\theta_1\rightarrow0}\alpha^*=\frac{\sigma_2}{\sigma_v},$ which contradicts that $\lim_{\theta_1\rightarrow0}\alpha^*=0$ when $\theta_z=0$. If $a<\frac{1}{2},\lim_{\theta_1\rightarrow0}\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_z}{\theta_z+1}},$ which is also incorrect (let $\theta_1=10^{-20}$, compare $\alpha^*$ and $\frac{\sigma_2}{\sigma_v}\sqrt{\frac{\theta_z}{\theta_z+1}}$ under different $\theta_z$). So $a=\frac{1}{2}$ and we further assume $\beta^*\sim\frac{\sqrt{\theta_1}}{y\theta_z^x}$ for $\theta_z>0$ and $\theta_1\rightarrow0$, then
\begin{equation*}
\lim_{\theta_1\rightarrow0}\alpha^*=\frac{\sigma_2}{\sigma_v}\sqrt{\frac{y^2\theta_z^{2x}+\theta_z}{y^2\theta_z^{2x}+\theta_z+1}}.
\end{equation*}
In order to get the value of $x=y$, we calculate $\alpha^*$ when $\theta_1=10^{-20}$ for different $\theta_z$, get $10^4$ pairs of $(\theta_z,\alpha^*)$ for $\theta_z\in(0,25].$ Then we do regression for the sample $(\theta_z,\alpha^*)$ and the result is shown in Figure \ref{goodnessoffit}.
\begin{figure}[!htbp]
\centering
\includegraphics[width = 0.31\textwidth]{fig/theta1/limit.png}
\caption{The goodness of fit.}
\label{goodnessoffit}
\end{figure}
Other limits could be calculated by substitute $\beta^*\sim\frac{\sqrt{\theta_1}}{y\theta_z^x}$ to Corollary \ref{cor1}. The case that $\theta_z=0$ could also be included.
\noindent\textbf{Proof of Proposition \ref{thetazinfty}.}
Recall the equation of $\beta^*$ \eqref{betastar}, we have
\begin{equation*}
\begin{aligned}
&\beta^6(2\theta_z^2 + 4\theta_1\theta_z^2 + 2\theta_1^2\theta_z^2 + \theta_z^3 +
\theta_1\theta_z^3)+\beta^5(4\theta_1\theta_z + 8\theta_1^2\theta_z + 4\theta_1^3\theta_z + 4\theta_1\theta_z^2 + 4\theta_1^2\theta_z^2 + 2\theta_1\theta_z^3)\\
+ & \beta^4(2\theta_1\theta_z + \theta_1\theta_z^2 )
+\beta^3(2\theta_1^2 + 2\theta_1^3 + 8\theta_1^2\theta_z + 16\theta_1^3\theta_z + 4\theta_1^2\theta_z^2) +\beta(4\theta_1^3 + 2\theta_1^3\theta_z)\\
=&\beta^4( 11\theta_1^2\theta_z + 13\theta_1^3\theta_z + 8\theta_1^2\theta_z^2)+ \beta^2(5\theta_1^3 + \theta_1^2\theta_z + 9\theta_1^3\theta_z) +\theta_1^3.
\end{aligned}
\end{equation*}
Thus,
\begin{equation*}
\begin{aligned}
&\beta^6\theta_z^3\leq \beta^4( 11\theta_1^2\theta_z + 13\theta_1^3\theta_z + 8\theta_1^2\theta_z^2)+ \beta^2(5\theta_1^3 + \theta_1^2\theta_z + 9\theta_1^3\theta_z) +\theta_1^3\\
&\beta^6\leq\beta^4( \frac{11\theta_1^2+13\theta_1^3}{\theta_z^2} + \frac{8\theta_1^2}{\theta_z})+ \beta^2(\frac{5\theta_1^3}{\theta_z^3} + \frac{\theta_1^2+9\theta_1^3}{\theta_z^2}) +\frac{\theta_1^3}{\theta_z^3}\rightarrow0.
\end{aligned}
\end{equation*}
In order to calculate $\lim_{\theta_z\rightarrow\infty}\alpha^*$, we need know the convergence speed of $\beta^*$. We prove $\beta^*<\frac{1}{\theta_z},$ equivalently, we only need to prove $f(\frac{1}{\theta_z};\theta_z)>0.$ Denote $x=\frac{1}{\theta_z}$,
\begin{equation*}
\begin{aligned}
f(x;\theta_z)&\geq\theta_1^2\theta_z(8x^5-11x^4+8x^3-x^2)+\theta_1^2\theta_z^2(4x^5-8x^4+4x^3)\\
&+\theta_1^3\theta_z(4x^5-13x^4+16x^3-9x^2+2x)+\theta_1^3(2x^3-5x^2+4x-1)\\
&=\theta_1^2(8x^4-11x^3+8x^2-x)+\theta_1^3(4x^5-13x^4+18x^3-14x^2+6x-1)\\
&\geq0.
\end{aligned}
\end{equation*}
Then $\beta^{*2}\theta_z\rightarrow0,\alpha^*\rightarrow\frac{\sigma_2}{\sigma_v}.$ Other limits are direct results of Theorem \ref{mainthm} and Corollary \ref{cor1}.
\noindent\textbf{Proof of Proposition \ref{effectHFT-i+u2}.}
\begin{equation*}
\begin{aligned}
&\mu_1(\alpha;\beta)=\frac{\alpha\theta_v(\beta^2\theta_z)}{(\alpha^2\theta_v+1)(\theta_1(1-\beta)^2+\theta_z\beta^2)+\beta^2\theta_1\theta_z},\\
&\frac{\partial(\mu_1(\alpha;\beta)\beta)}{\partial\beta}=\frac{\alpha\theta_v(2\alpha^2\theta_v+1)\theta_1\theta_z\beta(1-\beta)}{((\alpha^2\theta_v+1)(\theta_1(1-\beta)^2+\theta_z\beta^2)+\beta^2\theta_1\theta_z)^2}\geq0.
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
&\mu_2(\alpha;\beta)=\frac{\alpha\theta_v(\beta^2\theta_z+(1-\beta)\theta_1)}{(\alpha^2\theta_v+1)(\theta_1(1-\beta)^2+\theta_z\beta^2)+\beta^2\theta_1\theta_z},\\
&\frac{\partial(\mu_2(\alpha;\beta)(1-\beta))}{\partial\beta}=\frac{-\alpha\theta_v((\alpha^2\theta_v+1)\theta_z\beta^2(2\theta_1(1-\beta)+3\theta_z\beta^2)+\beta^2\theta_1\theta_z(3\theta_z\beta^3+2\theta_1(1-\beta^2)))}{((\alpha^2\theta_v+1)(\theta_1(1-\beta)^2+\theta_z\beta^2)+\beta^2\theta_1\theta_z)^2}\leq0.
\end{aligned}
\end{equation*}
As for the second conclusion,
\begin{equation*}
\mu_1(\alpha;\beta),\mu_2(\alpha;\beta)\leq\frac{\alpha\theta_v}{\alpha^2\theta_v+1}\leq\frac{\sigma_v}{2\sigma_2}.
\end{equation*}
Since $\mu_1^*$ and $\mu_2^*$ are maximum of $\mu_1(\alpha;\beta^*)$ and $\mu_2(\alpha;\beta^*)$ respectively, it holds.
\newpage
\bibliographystyle{unsrt}
|
{
"timestamp": "2022-11-14T02:07:21",
"yymm": "2211",
"arxiv_id": "2211.06046",
"language": "en",
"url": "https://arxiv.org/abs/2211.06046"
}
|
"\\section{Introduction}\n The project titled Measuring at Intermediate metallicity\n Neutron-Cap(...TRUNCATED)
| {"timestamp":"2022-11-14T02:08:26","yymm":"2211","arxiv_id":"2211.06086","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nHumans have long wondered whether other life exists elsewhere in the uni(...TRUNCATED)
| {"timestamp":"2022-11-14T02:07:29","yymm":"2211","arxiv_id":"2211.06050","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\n\\noindent \\textcolor{black}{Social media has been identified as a cent(...TRUNCATED)
| {"timestamp":"2022-11-14T02:09:19","yymm":"2211","arxiv_id":"2211.06116","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\\label{sec:intro}\nReducing inequality among regions is one of the 17 Soc(...TRUNCATED)
| {"timestamp":"2022-11-14T02:08:45","yymm":"2211","arxiv_id":"2211.06095","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nThe `baryon cycle' --- the cycle of gas condensation, the resulting star(...TRUNCATED)
| {"timestamp":"2022-11-14T02:06:09","yymm":"2211","arxiv_id":"2211.06005","language":"en","url":"http(...TRUNCATED)
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"\n\\section{Introduction}\nIn this section the concept of ion beam therapy is introduced followed b(...TRUNCATED)
| {"timestamp":"2022-11-14T02:09:11","yymm":"2211","arxiv_id":"2211.06113","language":"en","url":"http(...TRUNCATED)
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"\\section{Conclusion}\n\nIn this work, we propose a one-stage weakly semi-supervised method that jo(...TRUNCATED)
| {"timestamp":"2022-11-14T02:08:57","yymm":"2211","arxiv_id":"2211.06104","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\nAccurate dust and star-formation scaling relations are essential in studie(...TRUNCATED)
| {"timestamp":"2022-11-14T02:08:55","yymm":"2211","arxiv_id":"2211.06099","language":"en","url":"http(...TRUNCATED)
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