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[EQUATION] holds, implying that the summand [MATH] is equal to zero for any [MATH] and any [MATH] By assumption, [MATH] and [MATH] so that one has [MATH] , and consequently, |
[MATH] leading to contradiction. The proof of Proposition is straightforward by taking [MATH] as belonging to [MATH] in Lemma Proof of Proposition |
We first show the following lemma. Lemma 3 Consider an [MATH] -player game which is symmetric with respect to a permutation [MATH] on [MATH] Assume that the column vectors of [MATH] are linearly independent. For any pair of players [MATH] and [MATH] satisfying [MATH] if the strategy vectors of these players contain no ... |
[MATH] with [MATH] and where player [MATH] enforces [MATH] where [MATH] Proof. We assume to the contrary that there exists [MATH] |
satisfying the properties stated in Lemma By assumption, [MATH] and [MATH] There then exist [MATH] and [MATH] satisfying [MATH] and [MATH] One has |
[EQUATION] where the second equality is due to the assumed symmetry of the game with respect to [MATH] Letting [MATH] [MATH] and [MATH] one has |
[EQUATION] implying that [MATH] holds. Let [MATH] Let [MATH] and [MATH] where ties may be broken arbitrarily, and [MATH] and [MATH] One then has |
[EQUATION] Recalling that we have assumed [MATH] let [MATH] be an arbitrary state satisfying [MATH] and [MATH] Then, in view of Eq. ( 21 ), one has |
[EQUATION] implying that [MATH] holds. Since [MATH] for all [MATH] , they are all equal to zero. Since [MATH] is assumed non-zero, one has [MATH] for all [MATH] |
and consequently [MATH] One similarly has [MATH] Therefore, from Eq. ( ) one has [MATH] Due to the assumption of linear independence of the columns of [MATH] it in turn implies that [MATH] holds, leading to contradiction. |
It should be noted that Lemma holds even if one takes [MATH] , in which case the Lemma implies that, if the game is symmetric with respect to [MATH] player [MATH] with [MATH] cannot enforce linear relations [MATH] simultaneously. It should also be noted that Lemma furthermore implies that it is impossible for that play... |
satisfying [MATH] In other words, in a symmetric game no player to whom the game is symmetric can enforce a linear relation with the same symmetry as the game itself. |
Proposition is a direct consequence of Lemma in weakly symmetric multi-player games. Proof of Proposition Without loss of generality, we assume that player [MATH] takes an [MATH] -dimensional ZD strategy determining the average payoffs [MATH] for [MATH] Due to the above discussion, only [MATH] is allowed. Letting [MATH... |
[EQUATION] By the assumption of weak symmetry, for any player [MATH] there exists a permutation [MATH] satisfying [MATH] such that the game is symmetric with respect to [MATH] Noting that [MATH] from Eq. ( Proof. ) one has |
[EQUATION] For [MATH] define [MATH] and [MATH] where ties may be broken arbitrarily provided that [MATH] holds, and [MATH] and [MATH] From Eq. ( ), one has |
[EQUATION] Then, from Eq. ( 28 ) and Eq. ( 21 ), we obtain for an arbitrary [MATH] satisfying [MATH] and [MATH] [EQUATION] implying [MATH] On the other hand, we also obtain for an arbitrary [MATH] satisfying [MATH] and [MATH] |
[EQUATION] implying [MATH] Then, because we have assumed that all elements of the payoff vector [MATH] are different from each other, we have arrived at a contradiction. |
Supporting information ZD strategy in zero-sum games 1.1 Absence of ZD strategies in the rock-paper-scissors game We consider the rock-paper-scissors game: |
[EQUATION] It should be noted that this game is two-player three-action symmetric zero-sum game. Because it is zero-sum game, the payoff vectors are linearly dependent [MATH] , and |
[EQUATION] always holds. When player [MATH] can employ ZD strategy, her strategy takes the form [EQUATION] and she enforces the linear relation |
[EQUATION] Since the game is symmetric, player [MATH] can independently employ ZD strategy which enforces [EQUATION] Then, because of the consistency of ZD strategies, [MATH] must hold. On the other hand, by using [MATH] , Eq. ( S3 ) can be written as |
[EQUATION] where we have defined [EQUATION] Then we find that [EQUATION] and [EQUATION] Because [MATH] and [MATH] , Eq. ( S12 ) and Eq. ( S13 ) are inconsistent with the definition of the payoff ( S1 ). Therefore, we conclude that ZD strategy does not exist in the rock-paper-scissors game. |
1.2 Example of ZD strategy in two-player three-action symmetric zero-sum game We next consider the following two-player three-action symmetric zero-sum game, which is the slightly modified version of the game in the main text: |
[EQUATION] We remark that [MATH] and [MATH] are linearly dependent [MATH] We choose strategies of player [MATH] as [EQUATION] with [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] Then we obtain |
[EQUATION] Therefore, this strategy is ZD strategy which control the payoffs of both players as [MATH] It should be noted that Eq. ( S12 ) and Eq. ( S13 ) are satisfied for this game. |
ZD strategy in game with public monitoring 2.1 Two-player two-action game As an example of ZD strategy for a repeated imperfect-monitoring game, we consider a two-player two-action symmetric game |
We assume [MATH] and the probability [MATH] is given by [EQUATION] This model is different from the noisy games studied by Hao et al. |
in that they consider [MATH] as noisy states, taking four values [MATH] corresponding to the four states in the iterated prisoner’s dilemma game, whereas ours considers [MATH] as taking only two values, representing winning/losing of player 1. The payoff vectors are given by [MATH] and [MATH] We consider equalizer stra... |
[EQUATION] This strategy unilaterally sets the average payoff of player [MATH] in the steady state: [EQUATION] By solving Eq. ( S21 ) with respect to [MATH] , we obtain |
[EQUATION] Concretely, we consider [MATH] and [MATH] By setting [MATH] and [MATH] , we obtain [EQUATION] and [EQUATION] In Fig. S1 , we display the result of numerical simulation of one sample. |
Time-averaged payoffs of two players [MATH] are displayed when the strategy of player [MATH] is all- [MATH] [EQUATION] The initial condition is set to [MATH] and [MATH] The numerical result for player [MATH] well matches with the theoretical prediction Eq. ( S31 ). We can see that the expected payoff of player [MATH] i... |
2.2 Two-player three-action game We consider the same two-player three-action symmetric game in the main text: [EQUATION] However, we assume that players can observe only common information [MATH] and the probability [MATH] is given by |
[EQUATION] The common information [MATH] represents whether payoffs of both players are non-zero or not. We consider the situation that player [MATH] employs the following strategy: |
[EQUATION] with [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] Then, from the definition [EQUATION] we obtain [EQUATION] These strategy vectors are the same as those in the main text, and give |
[EQUATION] which enforce the linear relations [MATH] and [MATH] Therefore, player [MATH] can enforce the same linear relations as those in the perfect monitoring case, even though players can know only [MATH] This means that the space of states [MATH] is successfully reduced to the smaller space [MATH] in terms of ZD s... |
In Fig. S2 , we display the result of numerical simulation of one sample for [MATH] and [MATH] Time-averaged payoffs of two players [MATH] are displayed when [MATH] [MATH] [MATH] [MATH] [MATH] and the strategy of player [MATH] is |
[EQUATION] The initial condition is given by the probability distribution [MATH] for both players. The numerical result is consistent with the theoretical prediction [MATH] |
Acknowledgments We thank Ryosuke Kobayashi for valuable discussions. This study was supported by JSPS KAKENHI Grant Numbers JP18H06476 and JP19K21542. |
# Source: arxiv 1807.00573 # Title: Strategic behaviour and indicative price diffusion in Paris Stock Exchange auctions # Sections: all # Downloaded: 2026-03-03T04:47:29.962437+00:00 |
Strategic behaviour and indicative price diffusion in Paris Stock Exchange auctions Abstract We report statistical regularities of the opening and closing auctions of French equities, focusing on the diffusive properties of the indicative auction price. Two mechanisms are at play as the auction end time nears: the typi... |
Research in market micro-structure has focused on the dynamical properties of open markets (O’Hara, 1997 ; Bouchaud et al., 2009 . However, main stock exchanges have been using auction phases when they open and close for a long time . Auctions are known to have many advantages, provided that there are enough participan... |
Only a handful of papers are devoted to the dynamics of auction phases, i.e., periods during which market participants may send limit or market orders specifically for the auction. Boussetta et al. ( 2016 investigate when fast and slow traders send their orders during the opening auction phase of the Paris Stock Exchan... |
show how and when low-latency traders (identified as high frequency traders) add or remove liquidity in the pre-opening auction of the Tokyo Stock Exchange. Accordingly, Yergeau ( 2018 finds typical patterns of high-frequency algorithmic trading in the auctions of XTRA. Challet and Gourianov ( 2018 analyze anonymous da... |
Auctions, data and notations The opening auction phase of Paris Stock Exchange starts at 7:15 and ends at 9:00 while the closing auction phase is limited to the period 17:30 to 17:35. The auction price maximises the matched volume. |
From the Thomson Reuters Tick History, we extract auction phase data for the 2013-04-16 components of the CAC40 index. This database contains all the updates to either the indicative match price or the indicative matched volume in the 2010-08-02 to 2017-04-12 period, which amounts to 8,095,524 data points for the openi... |
For each asset [MATH] , we denote the indicative price of auction [MATH] of day [MATH] at time [MATH] by [MATH] , the time of auction [MATH] by [MATH] and the auction price by [MATH] . Dropping the index [MATH] since this paper focuses on a single asset at a time, the [MATH] -th indicative price change occurs at physic... |
Similarly, the indicative matched volume is written as [MATH] , while the final volume is [MATH] . Finally, when computing averages over days, since updates occur at random times, we will use time coarsening by [MATH] seconds, i.e. compute quantity averages over days within time slices of [MATH] seconds. |
Figure illustrates why auctions deserve attention: the relative importance of the closing auction volume has more than doubled in the last 10 years. Note that the relative opening auction volume of French equities is quite small (typically around 1%) and has stayed remarkably constant. |
From collisions in event time to diffusion in physical time It is useful to consider the price as the position of a uni-dimensional random walker and assume that each price change is caused by a collision: if collision [MATH] shifts the price [MATH] by [MATH] , after [MATH] collisions the mean square displacement equal... |
[EQUATION] if the increments [MATH] are i.i.d, a straightforward consequence of the central limit theorem. This corresponds to standard diffusion. In addition, if the collisions occur at a constant rate [MATH] , then time is homogeneous and [MATH] . As we shall see, none of these assumptions is true during auctions, wh... |
2.1 Event rates In the case of indicative auction prices, the event rate is not constant: the activity usually increases just before the auction time. This finding is a generic feature of auctions with fixed end time (Borle et al., 2006 , and more generally of human procrastinating nature when faced with a deadline, be... |
Let us denote by [MATH] the number of price events (changes) having occurred up to time [MATH] on day [MATH] for auction [MATH] . The activity pattern of day [MATH] can be measured by the ratio between the number of events up to time [MATH] on day [MATH] and the total number of events which happened that day, defined a... |
There are clear peaks of changes for both [MATH] and [MATH] at unimaginative physical times such as 7:30, 8:30, etc., and at round minutes and multiples of 30 seconds during the closing auction. This of course denotes a regular behavior of some investors. If each peak is systematically caused by a single trader, there ... |
The global pattern of price changes and total volume matched clearly differs between both types of auctions. During opening auctions, the price change rate increases much, starting from a low baseline. During closing auctions, the opposite happens: price change activity is first large, slows down during the first 2-3 m... |
2.2 Activity acceleration The acceleration pattern of price change rate follows some regularity. To characterize it in a simpler way, it is useful to work in Time-To-Auction [MATH] frame. Since the latter reverts the time arrow, the activity decelerates as a function of [MATH] . Let us denote the average event rate [MA... |
Assuming that [MATH] , we perform a robust linear fit of [MATH] for [MATH] seconds and only keep the fits whose t-statistics associated with [MATH] is larger than 5. This particular choice of interval for [MATH] corresponds to a typical period during which the autocorrelation of [MATH] at one lag is roughly constant (s... |
If the typical absolute value of price change [MATH] does not depend on [MATH] and is still i.i.d., Eq. ( ) becomes [EQUATION] hence the Hurst exponent in [MATH] time, denoted by [MATH] , equals [MATH] : the price change rate influences the diffusive pattern in a simple way, given the above approximations. It is worth ... |
2.3 Typical price change When the indicative price changes, it jumps to the next non-empty tick of the auction order book. Thus, the typical indicative price change reflects the density of the latter, which increases as the auction time nears. As a consequence, the typical price change magnitude [MATH] is not constant ... |
2.4 Diffusive properties of indicative prices It is easy to see why the increase of activity and decrease of the typical magnitude of price changes have antagonistic and purely mechanistic effects on the diffusive properties of the indicative auction price in the simplest case: neglecting the autocorrelations and cross... |
[EQUATION] The first approximation assumes that all [MATH] within a time slice are i.i.d, while the second one assumes no correlation between [MATH] and [MATH] . The relative merits of both approximations can be assessed in Fig. . The first approximation corresponds to the continuous black line and the second one to th... |
Let us now compare the TTA Hurst exponents of the above quantities, plotted in Fig. for the 6 stocks whose fits of both [MATH] and [MATH] are deemed significant. Two features stand out. First, [MATH] overestimates [MATH] , even when accounting for the fairly large error bars. This implies that the dynamics caused by th... |
Indeed, in practice, even linear autocorrelation of both [MATH] and [MATH] and the cross-correlation between them are not negligible. Let us focus on the autocorrelation of [MATH] , denoted by [MATH] . For each time slice [MATH] , we average [MATH] over all the days for a given asset. Figure plots this quantity versus ... |
When [MATH] does not depend on [MATH] , it only modifies the prefactor of [MATH] in Eq. ( ) by a factor of the order [MATH] , not the Hurst exponent, and thus explains in part the discrepancy between [MATH] and [MATH] . The dependence of [MATH] on [MATH] modifies the apparent Hurst exponent in a nontrivial way. This is... |
Discussion Indicative auction prices display non-trivial properties due in part to the antagonistic effects of both the acceleration of activity and the reduction of the typical price change magnitude. However, the indicative price is much less over-diffusive than what these two effects alone imply. In other words, the... |
So far, we have used a basic data type, which nevertheless has a rich behavior. More detailed data, such as data from the auction book, will allow us to characterize order strategic placement, the evolution of the average auction book density and the price impact of new orders and order cancellations much before the au... |
# Source: arxiv 1807.00690 # Title: First order logic without equality on relativized semantics # Sections: all # Downloaded: 2026-03-03T02:37:24.224063+00:00 |
First order logic without equality on relativized semantics Abstract. Let [MATH] be any ordinal. We consider the class [MATH] of relativized diagonal free set algebras of dimension [MATH] . With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free ... |
Key words and phrases: free algebras, atoms, zero-dimensional elements 2010 Mathematics Subject Classification: Primary 03G15, 03B45, 03C95. Secondary 03G25, 03C05, 08B20 |
1. Introduction In the middle of the twentieth century, A. Tarski introduced and initiated the investigation of cylindric algebras and relation algebras. These algebras are Boolean algebras with extra additive, closure and complemented operators. The theories of these algebras are directly related to the development of... |
and An important notion in the theories of these algebras is the notion of representable algebras. These algebras can be conceived as expansions of Boolean set algebras whose elements are unary relations to algebras whose elements are relations of higher ranks. The question whether every abstract algebra is isomorphic ... |
One can find well motivated appropriate notions of representable structures by first locating them while giving up classical semantical prejudices. It is hard to give a precise mathematical underpinning to such intuitions. What really counts at the end is a completeness theorem stating a natural fit between chosen intu... |
The classical concrete algebras are cylindric set algebras defined by A. Tarski, these are algebras of sets of sequences in which the top element is a square of the form [MATH] , where [MATH] is a non-empty set and [MATH] is the dimension. Other concrete algebras can be the relativized versions of cylindric set algebra... |
The notion of a relativized algebra has been introduced in algebraic logic by L. Henkin and I. Németi. Relativization was proved extremely potent in obtaining positive results in both algebraic and modal logic, the slogan being relativization turns negative results positive . For instance, I. Németi proved, in a semina... |
The important connections between relativized cylindric set algebras and guarded fragments are discussed in and . More liberal versions of guarded fragments are the so-called loosely guarded clique guarded and packed fragments of first order logic , see |
and 31 , Definitions 19.1, 19.2, 19.3, pp. 586-589] . Relativized algebras and their related logics attracted many logicians and were shown to have several desirable properties, especially concerning decidability and complexity issues. They are widely applied in various areas of computer science and linguistics (e.g.,,... |
and The structures of free cylindric algebras are quite rich since they are able to capture the whole of first order logic, in a sense. One of the first things to investigate about these free algebras is whether they are atomic or not, i.e.,, whether their Boolean reducts are atomic or not. By an atomic Boolean algebra... |
and For a class [MATH] of algebras, and a cardinal [MATH] [MATH] stands for the [MATH] -generated free [MATH] algebra. In particular, for any ordinal [MATH] , the class of all cylindric algebras of dimension [MATH] is denoted by [MATH] , thus [MATH] denotes the [MATH] -generated free cylindric algebra of dimension [MAT... |
If [MATH] , then [MATH] is atomless (has no atoms). This result is due to D. Pigozzi , 2.5.13] and it can be generalized easily to any class of Boolean algebras with operators. So, from now on, let us assume that [MATH] |
If [MATH] then the free algebra [MATH] is finite, hence atomic , 2.5.3 (i)] . Moreover, the free algebra [MATH] is infinite but still atomic , 2.5.3(ii), 2.5.7(ii)] |
If [MATH] , then [MATH] has infinitely many atoms , 2.5.9] , and it was posed as an open question, cf , Problem 4.14] , whether it is atomic or not. |
In , it was shown that [MATH] is not atomic for [MATH] . This was proven by an involved metalogical machinery, namely, Gödel’s incompleteness Theorem. Then the problem of finding purely algebraic proof of this fact was raised in , Problem 4.14] . Such a proof, for [MATH] , was found by I. Németi |
. The problem of finding algebraic proof for the case [MATH] is still open. Similar results concerning representable cylindric algebras are also obtained, c.f. |
. The question whether the finitely generated free relativized cylindric set algebras are atomic was a difficult problem that remained open for three decades. See 10 , Remark 18 (i)] 13 , Problem 38] and 34 , Problem 1.3.3] . However, recently, it was shown that the free relativized cylindric set algebras are not atomi... |
. Investigating the non-atomicity of free algebras in algebraic logic is an ongoing research project punctuated by many deep results and challenges. See section for more details about the current status of this project. |
In this paper, we consider diagonal-free versions of relativized cylindric set algebras [MATH] . We prove that [MATH] is atomless whenever [MATH] and [MATH] . Considering this in line with the results in |
gives us some information about the differences between guarded logics with identity and guarded logics without identity as a privileged logical symbol. The methods we use here are similar to the ones in |
, but applied in new directions. As a strength sign of our methods, we collect other important results too, e.g.,, the decidability of the equational theory of [MATH] , which is proved by these methods. |
Diagonal-free relativized cylindric set algebras correspond to first order logic without equality on general assignment models. We will discuss this correspondence and the applications of our results in section . The interest for the study of languages without equality has its origin in the works of W. Blok and D. Pigo... |
and . Several developments and interesting results in this direction have been made, c.f., e.g.,, and 2. Preliminaries and main results |
Recall the basic concepts of universal algebra from the literature, see, e.g.,, . Let [MATH] be any class of algebras of the same similarity type, then [MATH] [MATH] [MATH] and [MATH] are the classes that consist of the isomorphic copies, subalgebras, (isomorphic copies of) direct products and homomorphic images, respe... |
[MATH] . Throughout this paper, we fix an ordinal [MATH] We start with the following basic notions. For every [MATH] and every two sequences [MATH] of length [MATH] , we write [MATH] if and only if [MATH] for some [MATH] , where [MATH] is the sequence which is like [MATH] except that it’s value at [MATH] equals [MATH] ... |
[EQUATION] This is called the [MATH] -cylindrification in the direction [MATH] . When no confusion is likely, we merely omit the superscript [MATH] from the above defined object. |
Definition 2.1 The class of all representable relativized diagonal free algebras of dimension [MATH] , denoted by [MATH] , is defined to be the class that consists of all isomorphic copies of the subalgebras of the (full) algebras of the form, |
[EQUATION] where [MATH] is a non-empty set of sequences of length [MATH] and [MATH] is the family of all subsets of [MATH] . In other words, [MATH] . For every [MATH] , the set [MATH] is called the unit of [MATH] , while the smallest set [MATH] that satisfies [MATH] is called the base of [MATH] |
Proposition 2.2 The class [MATH] is a variety. Sketch of the proof. We need to show that [MATH] is closed under [MATH] [MATH] and [MATH] . By definition, it is clear that [MATH] is closed under forming subalgebras [MATH] |
[MATH] is closed under [MATH] Let [MATH] and [MATH] be two algebras in [MATH] . We show that their direct product [MATH] . The same method can be applied to show that the direct product of any set of algebras in [MATH] is an element of [MATH] . By definition, there are two non-empty sets (of sequences of length [MATH] ... |
Define the map [MATH] as follows. For each [MATH] and each [MATH] , let [MATH] . It is not hard to see that [MATH] is a homomorphism because [MATH] It remains to prove that [MATH] is an injection. It is enough to show that the kernel of [MATH] is [MATH] , which is clear by the definition of [MATH] . Therefore, [MATH] i... |
[MATH] is closed under [MATH] By the first homomorphism theorem, we know that every homomorphic image of an algebra is isomorphic to a quotient of this algebra. Thus, it is enough to prove that every quotient algebra of a member of [MATH] is a member of [MATH] . Suppose that [MATH] is a subalgebra of [MATH] for some no... |
[EQUATION] Let [MATH] . We prove that [MATH] is embeddable into the full algebra [MATH] . Define [MATH] as follows. For each [MATH] , let [MATH] . The fact that [MATH] is a congruence on [MATH] implies that [MATH] is an injective homomorphism. Therefore, [MATH] |
We assume familiarity with the basic notions of the theory of cylindric algebras, e.g., atoms, zero-dimensional elements, etc. The definitions of such notions can be found in |
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