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[EQUATION] Hence [EQUATION] Now the LHS of ( 49 ) is reduced to the following [EQUATION] \qed We derive the energy expression using [MATH] from ( ), |
[EQUATION] Again from the special case for expression [MATH] in Lemma , where [MATH] [MATH] and [MATH] are replaced by [MATH] [MATH] and [MATH] respectively, we have |
[EQUATION] Hence [EQUATION] This energy expression is consistent with ( 50 ) (see Appendix A) for the ground state, which is derived from mean-field analysis. |
Conclusion The continuum limit approximation for calculating the ground-state energy of the [MATH] model, via the Bethe Ansatz solution, was studied. Starting with the closed model, we revisited the formulations of |
which undertake calculations by assuming a form of density function for the Bethe root distribution. It was found that this approach does not provide a consistent solution for particular choices of the momentum density distribution. This was established by a close examination of the case known as the Moore-Read line, w... |
. An alternative approach, which avoids the need to postulate a form for the Bethe root density, was proposed in terms of the coupled equations satisfied by the conserved operator eigenvalues |
. In this case a solution corresponding to the ground state in the continuum limit was found. Curiously, the expression obtained for the ground-state energy coincides with that obtained by the Bethe root distribution. That is to say that although the Bethe root approach involves a flawed methodology, it nonetheless pro... |
The conserved operator eigenvalue approach was then extended to accommodate the open case based on results from . Again, it was found that the result for the ground-state energy is in agreement with mean-field calculations. The open case can be considered as a model allowing for the exchange of particle between the sys... |
where the environment was modelled by a single bosonic degree of freedom in such a way that integrability was preserved. There too, arbitrarily small coupling to the environment was found to annihilate the existence of any quantum phase transition. |
For future work it would be natural to extend this analysis to calculate the leading order finite-size correction to the ground-state energy, for both open and closed models. For the closed [MATH] -wave pairing systems there are results obtained by series expansions |
, which can be extrapolated and shown to be valid more generally Obtaining analogous expressions for the open and closed [MATH] -pairing systems appears to be entirely feasible. |
Acknowledgements This work was supported by the Australian Research Council through Discovery Project DP150101294. Appendix A Mean-field analysis for the open model |
We introduce the following notation adopted in [EQUATION] Then let [MATH] [MATH] and [MATH] . The extended Hamiltonian ( ) can be rewritten as |
[EQUATION] Note that in the mean-field approximation for this extended model, the Lagrange multiplier is [MATH] . The matrix form is derived from the representation of [MATH] acting on [MATH] . Consider the following eigenvalue problem, |
[EQUATION] by minimizing each eigenvalue, we derive the ground-state energy [EQUATION] where [MATH] [MATH] . Then we calculate the following |
[EQUATION] Apply Hellmann-Feynman theorem, since [EQUATION] and [EQUATION] we have [EQUATION] Furthermore from ( 51 ), [EQUATION] |
and [EQUATION] hence [EQUATION] Comparing ( 52 ) and ( 53 ), and assuming [MATH] , we conclude that [EQUATION] i.e. [MATH] . It follows that [MATH] . Hence we immediately have the following results [EQUATION] and the “gap” equation [EQUATION] |
# Source: arxiv 1807.00472 # Title: Linear algebraic structure of zero-determinant strategies in repeated games # Sections: all # Downloaded: 2026-03-02T09:23:13.658343+00:00 |
Linear algebraic structure of zero-determinant strategies in repeated games Masahiko Ueda 1* Toshiyuki Tanaka Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan |
* ueda.masahiko.5r@kyoto-u.ac.jp Abstract Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional... |
Introduction Game theory is a powerful framework explaining rational behaviors of human beings and evolutionary behaviors of biological systems |
In a simple example of prisoner’s dilemma game, mutual defection is realized as a result of rational thought, even if mutual cooperation is more favorable. On the other hand, when the game is repeated infinite times, cooperation can be realized if players are far-sighted, which is confirmed as folk theorem. Axelrod’s f... |
also showed that cooperative but retaliating strategy, called the tit-for-tat strategy, is successful in the setting of infinitely repeated game. |
Recently, in repeated games with perfect monitoring, a novel class of strategies, called zero-determinant (ZD) strategy, was discovered |
Surprisingly, ZD strategy unilaterally enforces a linear relation between average payoffs of players. A strategy which unilaterally sets her opponent’s average payoff (equalizer strategy) is one example. Another example is extortionate strategy in which the player can earn more average payoff than her opponent. ZD stra... |
, and it was found that some kind of ZD strategies, called generous ZD strategies, can stably exist. Performance of ZD strategies has also been studied in human experiments |
Although ZD strategy was originally formulated in two-player two-action (iterated prisoner’s dilemma) games, ZD strategy was extended to multi-player two-action (iterated social dilemma) games |
, two-player multi-action games , and multi-player multi-action games In addition, ZD strategy was extended to two-player two-action noisy games |
, which is one example of the repeated games with imperfect monitoring. Furthermore, besides these fundamental theoretical studies, ZD strategies are also applied to resource sharing in wireless networks |
See Ref. for a review of ZD strategies in the context of direct reciprocity. The contributions of this paper are four-fold. First, we extend ZD strategy for general multi-player multi-action repeated games with public monitoring, where players know the structure of games (players, sets of actions of all players, and pa... |
We remark on discounting. In standard repeated games, discounting of future payoffs is considered by introducing a discounting factor [MATH] |
In the original work on ZD strategy by Press and Dyson, only the case without discounting (i.e., [MATH] ) was investigated After their work, ZD strategy was extended to [MATH] case |
In this paper, we consider only the non-discounting case [MATH] Setup We consider an [MATH] -player multi-action repeated game, in which player [MATH] has [MATH] possible actions, where [MATH] is a positive integer. Let [MATH] denote a state of the game, which is the combination of the actions taken by the [MATH] playe... |
with the conditional probability [MATH] , where [MATH] is some set. We also define the conditional probability that common information [MATH] arises when actions of players in the preceding round are [MATH] by [MATH] (An example of [MATH] is the winner in each round; see Supporting information .) Then the sequence of s... |
[EQUATION] with the transition probability [EQUATION] where [MATH] denotes the state distribution at time [MATH] We assume that all players know the function [MATH] but cannot directly observe [MATH] When [MATH] and [MATH] the above formulation reduces to that of perfect monitoring games. Otherwise, it represents games... |
For each state [MATH] , a payoff of player [MATH] is defined as [MATH] Let [MATH] be the [MATH] -dimensional vector representing the payoffs of player [MATH] , which we call the payoff vector of player [MATH] It should be noted that in the following analysis we do not assume the payoffs to be symmetric, unless otherwis... |
Results Zero-determinant strategies Because a discounting factor [MATH] is one, the payoffs of players are the average payoffs with respect to the stationary distribution of the Markov chain. Let [MATH] denote the stationary distribution, which may depend on the initial condition when the Markov chain is not irreducibl... |
[EQUATION] Taking summation of both sides of Eq. ( with respect to [MATH] with an arbitrary [MATH] , we obtain [EQUATION] where we have defined |
[EQUATION] Regarding [MATH] as representing the strategy “Repeat”, where player [MATH] repeats the previous action with probability one, one can readily see that Eq. ( ) is an extension of Akin’s lemma |
relating a player’s strategy with the stationary distribution, to the multi-player multi-action public-monitoring case. Letting [EQUATION] |
Eq. ( ) means that the average of [MATH] with respect to the stationary distribution is zero for any [MATH] and [MATH] We remark that all players are assumed to know the functional form of [MATH] , and that [MATH] and thus [MATH] as well, are solely under control of player [MATH] Because of the normalization condition ... |
[EQUATION] holds. Let [MATH] which we call the strategy vector of player [MATH] associated with action [MATH] (Another name for [MATH] |
is the Press-Dyson vector .) A strategy of player [MATH] is represented as an [MATH] matrix [MATH] composed of the strategy vectors for her actions [MATH] For a matrix [MATH] let [MATH] be the subspace spanned by the column vectors of [MATH] Let [MATH] and [MATH] denote the [MATH] -dimensional zero vector and the [MATH... |
[EQUATION] for any player [MATH] implying that the dimension of [MATH] is at most [MATH] Let [MATH] be the vector representation of the stationary distribution |
[MATH] When player [MATH] chooses a strategy [MATH] for any vector [MATH] one has [MATH] due to Eq. ( ). In other words, the expectation of [MATH] with respect to the stationary distribution |
[MATH] vanishes. Let [MATH] and [MATH] The following definition is an extension of the notion of the ZD strategy to multi-player multi-action public-monitoring games. |
Definition 1 zero-determinant (ZD) strategy is defined as a strategy [MATH] for which [MATH] holds. To see that this is indeed an extended definition of the ZD strategy, note that any vector [MATH] is represented as |
[MATH] , where [MATH] is the coefficient vector. Let [MATH] be the vector with element [MATH] equal to the expected payoff [MATH] of player [MATH] in the steady state. When player [MATH] employs a ZD strategy, it amounts to enforcing linear relations [MATH] |
on [MATH] with [MATH] satisfying [MATH] Consistency A question naturally arises: When more than one of the players employ ZD strategies, are they “consistent”, that is, do linear payoff relations enforced by the players always have solutions? For example, in a two-player game, when player [MATH] enforces [MATH] by a ZD... |
[MATH] consists of all combinations of the expected payoffs that satisfy the enforced linear relations by the players in [MATH] If [MATH] is empty, then it implies that the set of ZD strategies is inconsistent in the sense that there is no valid solution of the linear relations enforced by the players. |
Definition 2 ZD strategies are said to be consistent when [MATH] is not empty. In the multi-player setting, one may regard [MATH] as a variant of a ZD strategy alliance |
where the players in [MATH] agree to coordinate on the linear relations to be enforced on the expected payoffs. The above question then amounts to asking whether it is possible for a player to serve as a counteracting agent who participates in the ZD strategy alliance with a hidden intention to invalidate it by adoptin... |
The following proposition is the first main result of this paper. Proposition 1 Any set of ZD strategies is consistent. Proof. We first note that the following property holds for strategy vectors, whose proof is given in Methods. |
Lemma 1 Let [MATH] Then [MATH] For any set [MATH] of ZD strategies, let [MATH] be the dimension of [MATH] and let [MATH] be a basis of [MATH] The expected payoff vector [MATH] should be given by a non-zero solution of the linear equation [MATH] |
in [MATH] , where we define [MATH] [MATH] , and [MATH] as [EQUATION] One has [EQUATION] where [MATH] The Rouché-Capelli theorem tells us that [MATH] is a necessary and sufficient condition for the linear equation [MATH] |
in [MATH] to have a solution, that is, for [MATH] to be consistent (because [MATH] is augmented matrix). An equivalent expression of this condition is that there is no vector [MATH] |
such that [MATH] and [MATH] hold (which ensures that there is no elementary operations which make the rank of augmented matrix larger than that of the original matrix). Assume to the contrary that there exist [MATH] |
such that [MATH] and [MATH] hold. One would then have [EQUATION] On the other hand, [MATH] is a linear combination of [MATH] so that Lemma states that it should be zero if it is proportional to [MATH] leading to contradiction. |
Proposition states that it is impossible for any player to serve as a counteracting agent to invalidate ZD strategy alliances. This statement is quite general in that it applies to any instance of repeated games covered by our formulation. |
In Ref. , it was shown that every player can have at most one master player, who can play an equalizer strategy on the given player (that is, controlling the expected payoff of the given player), in multi-player multi-action games. Indeed, our general result on the absence of inconsistent ZD strategies (Proposition imm... |
Since the dimension of [MATH] is at most [MATH] depending on [MATH] it should be possible for player [MATH] with [MATH] to adopt a ZD strategy for which [MATH] holds. The dimension of [MATH] corresponds to the number of independent linear relations to be enforced on the expected payoffs of the players, so that it impli... |
independent linear relations if it is consistent. This in turn implies that if the dimension of [MATH] is equal to [MATH] for a subset [MATH] of players then players not in [MATH] cannot employ independent ZD strategy any more. |
Independence Another naturally-arising question would be regarding independence for a set of ZD strategies, which we define as follows: |
Definition 3 A set [MATH] of ZD strategies is independent if any set [MATH] of non-zero vectors [MATH] in [MATH] is linearly independent. Otherwise, [MATH] is said to be dependent |
If a set of ZD strategies is dependent, then there exists a ZD player whose ZD strategy adds no linear constraints other than those already imposed by other ZD players. One of the simplest example of a dependent set of ZD strategies is the case where two players enforce exactly the same linear relation to the expected ... |
Proposition 2 Let [MATH] be a subset of players. Assume that [MATH] does not have zero elements for any [MATH] and any [MATH] Then, any set [MATH] of ZD strategies of players in [MATH] is independent. |
See Methods for the proof. It should be noted that when [MATH] has zero elements then one might have dependent ZD strategies. A simple example can be found in a two-player two-action perfect-monitoring (iterated prisoner’s dilemma) game: Let the payoff vectors [MATH] and [MATH] for players 1 and 2 be |
[MATH] and [MATH] , with [MATH] If player 1 adopts the strategy [EQUATION] then it enforces the linear payoff relation [MATH] This strategy is a well-known tit-for-tat strategy |
By symmetry, player 2 can also adopt the same strategy [MATH] implying that these two strategies are indeed dependent. Simultaneous multiple linear relations by one player |
As mentioned above, when the number [MATH] of possible actions for player [MATH] is more than two, player [MATH] may be able to employ a ZD strategy with [MATH] |
to simultaneously enforce more than one linear relations. (We note that this is impossible for public goods game because the number of action for each player is two.) Such a possibility has never been reported in the context of ZD strategies. Here, we provide a simple example of such a situation in a two-player three-a... |
We consider the [MATH] symmetric game [EQUATION] We remark that [MATH] [MATH] , and [MATH] are linearly independent when [MATH] and [MATH] We choose strategies of player [MATH] as |
[EQUATION] with [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] Then we obtain [EQUATION] Therefore, player [MATH] can simultaneously control average payoffs of both players, [MATH] and [MATH] , as [MATH] Note that [MATH] with [MATH] is an absorbing state regardless of the strategy of player [MATH] in this case. |
In general, when one player simultaneously enforces two linear relations in two-player multi-action symmetric games, only [MATH] is allowed with some [MATH] This is explained as follows: Assume that player [MATH] can simultaneously enforce [MATH] and [MATH] with [MATH] by one ZD strategy. Because the game is symmetric,... |
The above argument can be extended straightforwardly to the multi-player case. For that purpose, we introduce some notions of symmetric multi-player games. The following definition of a symmetric multi-player game is due to von Neumann and Morgenstern 30 , Section 28] |
Definition 4 A game is symmetric with respect to a permutation [MATH] on [MATH] if [MATH] holds for any [MATH] and if [MATH] preserves the payoff structure of the game, that is, |
[EQUATION] holds for any [MATH] and for any [MATH] where [MATH] The following definition is due to Ref. Definition 5 A game is weakly symmetric if for any pair of players [MATH] |
and [MATH] there exists some permutation [MATH] on [MATH] satisfying [MATH] such that the game is symmetric with respect to [MATH] |
Consider an [MATH] -player weakly symmetric game. Assume that one player simultaneously enforces [MATH] independent linear relations on the average payoffs |
[MATH] of [MATH] players via adopting an [MATH] -dimensional ZD strategy. (Note that for this to be possible the number [MATH] of actions should satisfy [MATH] ). Then, the average payoffs [MATH] |
should be simultaneously controlled, but they should satisfy [MATH] due to the consistency of ZD strategies. The difficulty of construction of a ZD strategy of one player with dimension [MATH] in weakly symmetric [MATH] -player games can be seen in the following two propositions, whose proofs are given in Methods. |
Proposition 3 In a weakly symmetric [MATH] -player game, if the strategy vectors of one player contain no zero element, then a ZD strategy of the player with dimension [MATH] is impossible. |
Proposition 4 In a weakly symmetric [MATH] -player game, if payoffs [MATH] of player [MATH] are different from each other for all [MATH] , then a ZD strategy with dimension [MATH] is impossible. |
Discussion In this paper, we have derived ZD strategies for general multi-player multi-action public-monitoring games, in which players cannot observe actions of other players. By formulating ZD strategy in terms of linear algebra, we have proved that linear payoff relations enforced by ZD players are consistent. Furth... |
Although we have discussed mathematical properties of ZD strategies if exist, we do not know the criterion for whether ZD strategies exist or not when a game is given. For example, we can easily show that ZD strategy does not exist for the rock-paper-scissors game, which is the simplest two-player three-action symmetri... |
In addition, it should be noted that ZD strategies are not always “rational” strategies, which have been a main subject of game theory. Therefore, investigation of ZD strategies in terms of bounded rationality |
may be needed. Specifying the situation where ZD strategies are adopted is another important problem. Another remark is related to memory of strategies. In this work, we considered only memory-one strategies. In Ref. |
, it has been proved that a player with longer memory does not have advantage over a player with short memory in terms of average payoff in two-player games. In Ref. |
, it has been shown that this statement also holds for multi-player games. Therefore, considering only memory-one strategies should be sufficient even in our public-monitoring situation. Longer memory strategies attract much attentions in repeated games with implementation errors |
Extension of ZD strategies to longer memory case may lead to different evolutionary behavior compared to memory-one strategies. We remark on the effect of imperfect monitoring. In perfect monitoring case, the strategy vectors are arbitrary as long as they satisfy the conditions for probability distributions. In contras... |
Methods Proof of Lemma Assume to the contrary that [MATH] with [MATH] Taking the inner product of [MATH] with the stationary distribution [MATH] , one has |
[MATH] since [MATH] is represented as a linear combination of the strategy vectors and since the inner product of a strategy vector and the stationary distribution is zero. On the other hand, |
[MATH] holds because of the normalization of the stationary distribution. Therefore we obtain [MATH] , leading to contradiction. |
Proof of Proposition We first show the following lemma. Lemma 2 Let [MATH] be a subset of players. Assume that [MATH] does not have zero elements for any [MATH] and any [MATH] For [MATH] , let [MATH] be an arbitrary non-zero vector in [MATH] Then [MATH] are linearly independent. |
Proof. We assume to the contrary that [MATH] are linearly dependent. Then there is a set of coefficients [MATH] with which [MATH] holds. Without loss of generality we assume [MATH] for [MATH] |
Since [MATH] , it is expressed as [MATH] with a non-zero vector [MATH] Let [MATH] where ties may be broken arbitrarily, and [MATH] With Eq. ( ), one obtains |
[EQUATION] and thus [EQUATION] We show that the inequality [EQUATION] holds for any [MATH] , any [MATH] and any [MATH] satisfying [MATH] We first note that for any strategy vector [MATH] |
with action [MATH] one has, from Eq. ( ), [EQUATION] Fix any [MATH] satisfying [MATH] for a moment. Then, for [MATH] one has [MATH] |
by definition, making the left-hand side of Eq. ( 20 equal to zero. For [MATH] , on the other hand, one has [MATH] by definition. Also, since [MATH] , from Eq. ( 21 one has [MATH] These imply that the inequality ( 20 ) holds for [MATH] Putting the above arguments together, we have shown that the inequality ( 20 ) holds... |
Fix any [MATH] satisfying [MATH] for all [MATH] The above argument has shown that the inequality ( 20 holds for any [MATH] and any [MATH] On the other hand, at the beginning of the proof we have assumed that |
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