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Performance of Zero-determinant Strategies in Repeated Games without Discounting
Masahiko Ueda
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
This paper investigates the performance of Zero-determinant (ZD) strategies in repeated games without discounting. It challenges the previous assumption that the limit of the Cesaro average of the action profile probability distribution must exist. The authors prove that ZD strategies can unilaterally control payoffs and enforce linear relations even without this assumption, providing a stronger theoretical result bounded by specific inequalities involving strategy coefficients.
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Relation Signals (6)
Masahiko Ueda → authored → Performance of Zero-determinant Strategies in Repeated Games without Discounting
confidence 99% · Performance of Zero-determinant Strategies in Repeated Games without Discounting Masahiko Ueda
Zero-determinant strategies → controls → payoffs
confidence 95% · Zero-determinant (ZD) strategies are a class of strategies in repeated games, which unilaterally control payoffs.
Equalizer strategy → isa → Zero-determinant strategies
confidence 92% · The equalizer strategy is a ZD strategy controlling B=s2−r
Zero-determinant strategies → enforces → linear relations between payoffs
confidence 90% · It has been widely believed that the performance of ZD strategies is expressed as 'unilaterally enforce linear relations between payoffs'.
Prisoner's dilemma → isexampleof → social dilemma
confidence 90% · The prisoner’s dilemma is a typical example of social dilemmas
Tit-for-Tat → isa → Zero-determinant strategies
confidence 88% · TFT is known to be a ZD strategy which unilaterally equalizes the payoffs
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Abstract
Abstract:Zero-determinant (ZD) strategies are a class of strategies in repeated games, which unilaterally control payoffs. It has been shown that several ZD strategies promote cooperation in social dilemma games. It has been widely believed that the performance of ZD strategies is expressed as ``unilaterally enforce linear relations between payoffs''. However, in order to interpret properties of ZD strategies in repeated games without discounting, previous studies assumed that the limit of the Cesaro average of the probability distribution of the action profile exists. Here, we explain the performance of ZD strategies without this assumption, which leads to a stronger result than previous ones.
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- Source: https://arxiv.org/abs/2610.02645v1
- Canonical: https://arxiv.org/abs/2610.02645v1
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Performance of Zero-determinant Strategies in Repeated Games without Discounting Masahiko Ueda1 †thanks: ueda@tmu.ac.jp Abstract Zero-determinant (ZD) strategies are a class of strategies in repeated games, which unilaterally control payoffs. It has been shown that several ZD strategies promote cooperation in social dilemma games. It has been widely believed that the performance of ZD strategies is expressed as “unilaterally enforce linear relations between payoffs”. However, in order to interpret properties of ZD strategies in repeated games without discounting, previous studies assumed that the limit of the Cesaro average of the probability distribution of the action profile exists. Here, we explain the performance of ZD strategies without this assumption, which leads to a stronger result than previous ones. 1 Introduction Social dilemma is a situation where rational behavior for individuals conflicts with the optimal outcome for society. The prisoner’s dilemma is a typical example of social dilemmas, where defection is rational for each individual whereas mutual cooperation improves the payoffs of both players [1]. How cooperative behavior of autonomous agents can be realized is a long-standing question [2]. One possible explanation for mutual cooperation is direct reciprocity, where repeated interaction among agents promotes mutual cooperation. Originally, this explanation required rationality of agents [1], which is far from human behavior. The explanation for mutual cooperation based on direct reciprocity was modified by the appearance of zero-determinant (ZD) strategies [3, 4]. ZD strategies are a class of strategies in repeated games, which unilaterally enforce linear relations between payoffs. In the repeated prisoner’s dilemma game, several ZD strategies have been found. The equalizer strategy unilaterally sets the opponent’s payoff [5]. The extortionate strategy obtains payoff no less than that of the opponent [3]. The generous strategy promotes mutual cooperation by adaptive learning of the opponent [6]. Generally, if we can construct a positively-correlated ZD strategy [7], which enforces a linear relation between the payoffs of two players with a positive slope, learning by the opponent increases the payoffs of both players. The performance of ZD strategies in the context of evolution of cooperation has extensively been investigated [8, 9, 6, 10, 11, 12]. Furthermore, the utility of ZD strategies has also been studied in other games, such as the public goods game [6, 13], the continuous donation game [14], and the oligopoly game [15]. Mutual cooperation through ZD strategies is realized not by rationality but by the opponent’s learning, and ZD strategies changed the problem of emergence of cooperation to the problem of control of other agents. Usually, the performance of ZD strategies is expressed as “unilaterally enforce linear relations between payoffs”. Originally, ZD strategies were introduced in games without discounting about future payoffs, and properties of ZD strategies were investigated for the case in which all players use Markov strategies [3]. After that, Akin proved that ZD strategies can control payoffs even if the co-players use non-Markov strategies [16], under the assumption that the limit of the Cesaro average of the probability distribution of the action profile exists. Furthermore, it was also shown that, when there is discounting of future payoffs, this assumption is not necessary [17]. Recently, a necessary and sufficient condition for the existence of ZD strategies was specified for games without discounting [18]. However, the performance of ZD strategies without the assumption about the limit has not been known when there is no discounting about future payoffs. The purpose of this paper is to investigate properties of ZD strategies without the assumption about the existence of the limit of the Cesaro average. We show that ZD strategies have stronger ability of payoff control than “unilaterally enforce linear relations between payoffs”. The performance of ZD strategies is described by some inequality. 2 Preliminaries We consider a repeated version of the normal-form game G:=(,Ajj∈,sjj∈)G:= (N, \A_j \_j , \s_j \_j ), where N is the set of players, AjA_j is the set of actions of player j∈j , and sj:∏k∈Ak→ℝs_j: _k A_k is the payoff function of player j∈j [1]. We introduce the notation :=(ak)k∈ a:= (a_k )_k and a−j:=(ak)k≠ja_-j:= (a_k )_k≠ j. We assume that AjA_j is a finite set for all j∈j . In repeated games, a strategy is a plan for choosing an action at each round when all previous action profiles are given. As a special class of strategies, memory-one strategies are ones using only information about the last round, that is, Markovian. Typically, a memory-one strategy of player j∈j is described by Tj(aj|′)T_j (a_j| a ), which is the conditional probability to take aj∈Aja_j∈ A_j when the previous action profile is ′=(ak′)k∈ a = (a _k )_k , with some initial distribution. Zero-determinant (ZD) strategies are a special class of memory-one strategies. Concretely, a memory-one strategy TjT_j of player j is a ZD strategy controlling the quantity B:∏k∈Ak→ℝB: _k A_k if it satisfies ∑ajcaj[Tj(aj|′)−δaj,aj′] _a_jc_a_j [T_j (a_j| a )- _a_j,a _j ] =B(′) =B ( a ) (1) with some coefficients cajaj∈Aj \c_a_j \_a_j∈ A_j and B is not identically zero [18], where δaj,aj′ _a_j,a _j is the Kronecker delta. When we write the probability distribution of the action profile a at the t-th round by Pt()P_t ( a ), and introduce the the limit of the Cesaro average P∗() P^* ( a ) :=limT→∞1T∑t=1TPt(), := _T→∞ 1T _t=1^TP_t ( a ), (2) a ZD strategy controlling B unilaterally enforces [16, 19] ∑B()P∗() _ aB ( a )P^* ( a ) =0. =0. (3) Therefore, when B is related to the one-shot payoffs skk∈ \s_k \_k , a ZD strategy can control payoffs. If there exists a stationary distribution P(st)()=limt→∞Pt()P^(st) ( a )= _t→∞P_t ( a ), P∗P^* coincides with P(st)P^(st). It should be noted that the payoff of player k in repeated games without discounting is given by ∑sk()P∗() _ as_k ( a )P^* ( a ). Originally, B()=∑k∈αksk()+α0B ( a )= _k _ks_k ( a )+ _0 with some coefficients αkk∈∪0 \ _k \_k ∪\0\ has been studied [3], which corresponds to “unilaterally enforce a linear relation between payoffs”. However, this argument assumes the existence of P∗P^*. 3 Results 3.1 Performance of ZD strategies Here, we investigate the performance of ZD strategies without assuming the existence of P∗P^*, and prove a stronger result than Eq. (3). Below we write the action profile at the t-th round by (t) a^(t). Generally, the joint probability distribution of the action profiles (t′)t′=1t \ a^(t ) \_t =1^t in a repeated game is described as [19] P((t′)t′=1t) P ( \ a^(t ) \_t =1^t ) =[∏t′=2t[∏k∈Tk(t′)(ak(t′)|(t′)t′=1t′−1)]][∏k∈Tk(1)(ak(1))], = [ _t =2^t [ _k T_k^(t ) (a_k^(t )| \ a^(t ) \_t =1^t -1 ) ] ] [ _k T_k^(1) (a_k^(1) ) ], (4) where Tk(t′)(ak(t′)|(t′)t′=1t′−1)T_k^(t ) (a_k^(t )| \ a^(t ) \_t =1^t -1 ) is the player k’s conditional probability of taking ak(t′)a_k^(t ) in the t′t -th round given the history of the action profiles (t′)t′=1t′−1 \ a^(t ) \_t =1^t -1. This joint probability distribution satisfies a recursion relation P((t′)t′=1t+1) P ( \ a^(t ) \_t =1^t+1 ) =[∏k∈Tk(t+1)(ak(t+1)|(t′)t′=1t)]P((t′)t′=1t). = [ _k T_k^(t+1) (a_k^(t+1)| \ a^(t ) \_t =1^t ) ]P ( \ a^(t ) \_t =1^t ). (5) Here, we introduce the marginal distribution of (t) a^(t): Pt((t)) P_t ( a^(t) ) :=∑(t′)t′=1t−1P((t′)t′=1t). := _ \ a^(t ) \_t =1^t-1P ( \ a^(t ) \_t =1^t ). (6) When we calculate ∑a−j(t+1)∑(t′)t′=1t _a_-j^(t+1) _ \ a^(t ) \_t =1^t of Eq. (5), we obtain ∑′δaj′,aj(t+1)Pt+1(′) _ a _a_j ,a_j^(t+1)P_t+1 ( a ) =∑(t′)t′=1tTj(t+1)(aj(t+1)|(t′)t′=1t)P((t′)t′=1t). = _ \ a^(t ) \_t =1^tT_j^(t+1) (a_j^(t+1)| \ a^(t ) \_t =1^t )P ( \ a^(t ) \_t =1^t ). (7) If player j uses a memory-one strategy, this is rewritten as ∑′δaj′,aj(t+1)Pt+1(′) _ a _a_j ,a_j^(t+1)P_t+1 ( a ) =∑′Tj(aj(t+1)|′)Pt(′). = _ a T_j (a_j^(t+1)| a )P_t ( a ). (8) By renaming aj(t+1)→aja_j^(t+1)→ a_j, we obtain ∑′δaj,aj′Pt+1(′) _ a _a_j,a _jP_t+1 ( a ) =∑′Tj(aj|′)Pt(′) = _ a T_j (a_j| a )P_t ( a ) (9) regardless of the co-players’ strategies. It should be noted that the both sides describe the probability to take aja_j at the (t+1)(t+1)-th round. Suppose that TjT_j is a ZD strategy (1) controlling B. By taking ∑ajcaj _a_jc_a_j of the both sides, we obtain 0 0 =∑ajcaj∑′Tj(aj|′)Pt(′)−∑ajcaj∑′δaj,aj′Pt+1(′) = _a_jc_a_j _ a T_j (a_j| a )P_t ( a )- _a_jc_a_j _ a _a_j,a _jP_t+1 ( a ) =∑′[∑ajcajδaj,aj′+B(′)]Pt(′)−∑′caj′Pt+1(′) = _ a [ _a_jc_a_j _a_j,a _j+B ( a ) ]P_t ( a )- _ a c_a _jP_t+1 ( a ) =∑′caj′[Pt(′)−Pt+1(′)]+∑′B(′)Pt(′). = _ a c_a _j [P_t ( a )-P_t+1 ( a ) ]+ _ a B ( a )P_t ( a ). (10) Then, by taking ∑t=1T _t=1^T of the both sides with arbitrary T∈ℕT , we obtain 0 0 =∑t=1T∑′caj′[Pt(′)−Pt+1(′)]+∑t=1T∑′B(′)Pt(′) = _t=1^T _ a c_a _j [P_t ( a )-P_t+1 ( a ) ]+ _t=1^T _ a B ( a )P_t ( a ) =∑′caj′[P1(′)−PT+1(′)]+∑t=1T∑′B(′)Pt(′). = _ a c_a _j [P_1 ( a )-P_T+1 ( a ) ]+ _t=1^T _ a B ( a )P_t ( a ). (11) This implies ∑t=1T∑′B(′)Pt(′) _t=1^T _ a B ( a )P_t ( a ) =∑′caj′[PT+1(′)−P1(′)] = _ a c_a _j [P_T+1 ( a )-P_1 ( a ) ] ≤maxajcaj−minajcaj ≤ _a_jc_a_j- _a_jc_a_j (12) and ∑t=1T∑′B(′)Pt(′) _t=1^T _ a B ( a )P_t ( a ) ≥minajcaj−maxajcaj. ≥ _a_jc_a_j- _a_jc_a_j. (13) Thus, we finally obtain our main result |∑t=1T∑′B(′)Pt(′)| | _t=1^T _ a B ( a )P_t ( a ) | ≤maxajcaj−minajcaj. ≤ _a_jc_a_j- _a_jc_a_j. (14) Therefore, even if P∗P^* does not exist, ZD strategies can still control payoffs in the sense of Eq. (14). Note that the right-hand side does not depend on T, and Eq. (14) means that total expected B is bounded by some constant. Particularly, if we consider limT→∞1T _T→∞ 1T of minajcaj−maxajcaj _a_jc_a_j- _a_jc_a_j ≤∑t=1T∑′B(′)Pt(′)≤maxajcaj−minajcaj, ≤ _t=1^T _ a B ( a )P_t ( a )≤ _a_jc_a_j- _a_jc_a_j, (15) we obtain limT→∞1T∑t=1T∑′B(′)Pt(′) _T→∞ 1T _t=1^T _ a B ( a )P_t ( a ) =0. =0. (16) That is, ZD strategies unilaterally enforce Eq. (16) even if P∗P^* does not exist. When P∗P^* exists, Eq. (16) coincides with Eq. (3). Therefore, Eq. (14) is stronger than Eq. (3). 3.2 Akin’s lemma Usually, properties (3) of ZD strategies (1) are derived by the Akin’s lemma [16, 19]: ∑′[Tj(aj|′)−δaj,aj′]P∗(′) _ a [T_j (a_j| a )- _a_j,a _j ]P^* ( a ) =0(∀aj). =0 (∀ a_j). (17) However, this lemma also requires the existence of P∗P^*. Here, we consider the extension of the Akin’s lemma without assuming the existence of P∗P^*. According to Eq. (9), we obtain ∑′[Tj(aj|′)−δaj,aj′]Pt(′) _ a [T_j (a_j| a )- _a_j,a _j ]P_t ( a ) =∑′δaj,aj′[Pt+1(′)−Pt(′)] = _ a _a_j,a _j [P_t+1 ( a )-P_t ( a ) ] (18) for all aja_j. By taking ∑t=1T _t=1^T of the both sides with arbitrary T∈ℕT , we obtain ∑t=1T∑′[Tj(aj|′)−δaj,aj′]Pt(′) _t=1^T _ a [T_j (a_j| a )- _a_j,a _j ]P_t ( a ) =∑t=1T∑′δaj,aj′[Pt+1(′)−Pt(′)] = _t=1^T _ a _a_j,a _j [P_t+1 ( a )-P_t ( a ) ] =∑′δaj,aj′[PT+1(′)−P1(′)]. = _ a _a_j,a _j [P_T+1 ( a )-P_1 ( a ) ]. (19) Therefore, we obtain |∑t=1T∑′[Tj(aj|′)−δaj,aj′]Pt(′)| | _t=1^T _ a [T_j (a_j| a )- _a_j,a _j ]P_t ( a ) | ≤1 ≤ 1 (20) for arbitrary aja_j. This is a modified version of the Akin’s lemma, which does not assume the existence of P∗P^*. Similarly as Eq. (16), if we consider limT→∞1T _T→∞ 1T of −1 -1 ≤∑t=1T∑′[Tj(aj|′)−δaj,aj′]Pt(′)≤1, ≤ _t=1^T _ a [T_j (a_j| a )- _a_j,a _j ]P_t ( a )≤ 1, (21) we obtain limT→∞1T∑t=1T∑′[Tj(aj|′)−δaj,aj′]Pt(′) _T→∞ 1T _t=1^T _ a [T_j (a_j| a )- _a_j,a _j ]P_t ( a ) =0. =0. (22) This equality holds even if P∗P^* does not exist. When P∗P^* exists, Eq. (22) coincides with Eq. (17). 3.3 Numerical simulation In order to check the validity of our theoretical result (14), we perform a numerical simulation for a simple game. Concretely, we consider the prisoner’s dilemma game [3], where =1,2N=\1,2\ and A1=A2=C,DA_1=A_2=\C,D\. The actions C and D correspond to cooperation and defection, respectively. The payoffs are given by 1 s_1 :=(s1(C,C),s1(C,D),s1(D,C),s1(D,D)) := (s_1(C,C),s_1(C,D),s_1(D,C),s_1(D,D) ) T =(R,S,T,P) = (R,S,T,P ) T 2 s_2 :=(s2(C,C),s2(C,D),s2(D,C),s2(D,D)) := (s_2(C,C),s_2(C,D),s_2(D,C),s_2(D,D) ) T =(R,T,S,P), = (R,T,S,P ) T, (23) where T>R>P>ST>R>P>S and 2R>T+S2R>T+S hold. For this game, we consider the equalizer strategy of player 11 [5, 3]: 1(C) T_1(C) :=(T1(C|C,C),T1(C|C,D),T1(C|D,C),T1(C|D,D)) := (T_1(C|C,C),T_1(C|C,D),T_1(C|D,C),T_1(C|D,D) ) T =(1−R−rW,1−T−rW,−S−rW,−P−rW) = (1- R-rW,1- T-rW,- S-rW,- P-rW ) T (24) with r∈[P,R]r∈[P,R] and W:=max|s2()−r|>0W:= _ a |s_2( a)-r |>0 [18]. Usually, the equalizer strategy is described as a strategy which unilaterally enforces ∑s2()P∗()=r _ as_2 ( a )P^* ( a )=r. In other words, the equalizer strategy unilaterally sets the opponent’s payoff in repeated games. The equalizer strategy is a ZD strategy controlling B=s2−rB=s_2-r, because it satisfies Eq. (1) as 1(C)−(1,1,0,0) T_1(C)-(1,1,0,0) T =(−R−rW,−T−rW,−S−rW,−P−rW) = (- R-rW,- T-rW,- S-rW,- P-rW ) T =−1W[2−r(1,1,1,1)]. =- 1W [ s_2-r(1,1,1,1) T ]. (25) That is, cC=−Wc_C=-W and cD=0c_D=0 in the definition (1). For this situation, our bound (14) is described as |∑t=1T[∑′s2(′)Pt(′)−r]| | _t=1^T [ _ a s_2 ( a )P_t ( a )-r ] | ≤W. ≤ W. (26) Here, we set the parameters as (R,S,T,P)=(3,0,5,1)(R,S,T,P)=(3,0,5,1) and r=(R+P)/2=2r=(R+P)/2=2. For this case, W is calculated as W=3W=3. We consider the situation where player 11 adopts the equalizer strategy (24) and player 22 takes a random action in t≥2t≥ 2. The initial condition is set to (1)=(C,C) a^(1)=(C,C). The numerical result is displayed in Fig. 1. Figure 1: Numerical verification of our theoretical result (26). The horizontal axis is T, and the vertical axis is ∑t=1T∑′B(′)Pt(′) _t=1^T _ a B ( a )P_t ( a ) with B=s2−rB=s_2-r, where the ensemble average is calculated from 10610^6 samples. The parameters are (R,S,T,P)=(3,0,5,1)(R,S,T,P)=(3,0,5,1) and r=(R+P)/2=2r=(R+P)/2=2. W is calculated as W=3W=3. We find that the numerical result is consistent with our theory (26). 4 Concluding remarks In this paper, we investigated the performance of ZD strategies without assuming the existence of P∗P^* in repeated games without discounting. We proved that a ZD strategy controlling B unilaterally enforces the inequality (14). Furthermore, this inequality leads to a “linear relation” (16). Note that Eq. (16) is different from Eq. (3), since the latter assumes the existence of P∗P^*. Therefore, even if P∗P^* does not exist, we can discuss the performance of ZD strategies. For example, in the repeated prisoner’s dilemma game, when player 1 adopts the Tit-for-Tat (TFT) strategy, which imitates the opponent’s last action, and player 2 uses the strategy a2(t) a_2^(t) =C(t∈[32n+1,32n+1])D(t∈[32n+1+1,32n+2]) = \ array[]lC& (t∈ [3^2n+1,3^2n+1 ] )\\ D& (t∈ [3^2n+1+1,3^2n+2 ] ) array . for each n≥0n≥ 0 with some initial condition, the limit P∗P^* does not exist. Therefore, even though TFT is known to be a ZD strategy which unilaterally equalizes the payoffs of two players in the repeated prisoner’s dilemma game [3], one cannot interpret the property if P∗P^* does not exist. In contrast, by using our framework, we can interpret the property of ZD strategies in terms of Eqs. (14) and (16) with B=s1−s2B=s_1-s_2. In general two-player symmetric games, an inequality similar to Eq. (14) was obtained for unbeatable imitation strategies [20, 21]. After that, the unbeatable Tit-for-Tat strategy was proved to be a ZD strategy which unilaterally equalizes the payoffs of two players [22]. Similar results were also obtained in two concurrent repeated games played in parallel [23]. Therefore, we can regard our results as an extension of the results in Refs. [20, 21] from unbeatable imitation to arbitrary ZD strategies. It should be noted that the problem considered in this paper is specific to repeated games without discounting. When there is a discount factor δ∈[0,1)δ∈[0,1), P∗P^* is modified as [17] P∗() P^* ( a ) =(1−δ)∑t=1∞δt−1Pt(), =(1-δ) _t=1^∞δ^t-1P_t ( a ), (29) which always exists. Therefore, the results of this paper complete characterization of the performance of ZD strategies for all δ. We remark on the meaning of cajc_a_j in Eq. (1). In construction of ZD strategies, the coefficients cajaj∈Aj \c_a_j \_a_j∈ A_j have been heuristically obtained [18]. For potential games [24], cajc_a_j is related to potential functions [22]. Recently, a learning algorithm to obtain a ZD strategy together with cajaj∈Aj \c_a_j \_a_j∈ A_j was proposed [25]. Since the coefficients cajaj∈Aj \c_a_j \_a_j∈ A_j determine the bound in the right-hand side of Eq. (14), a ZD strategy with small Δ:=maxajcaj−minajcaj := _a_jc_a_j- _a_jc_a_j is more powerful. How to obtain ZD strategies with small Δ should be investigated in future. acknowledgment This study was supported by JSPS KAKENHI Grant Number JP26K21335, and JST PRESTO, Japan, Grant Number JPMJPR26K1. References [1] D. Fudenberg and J. Tirole: Game Theory (MIT Press, Massachusetts, 1991). [2] M. A. Nowak: Science 314 (2006) 1560. [3] W. H. Press and F. J. Dyson: Proceedings of the National Academy of Sciences 109 (2012) 10409. [4] C. Hilbe, K. Chatterjee, and M. A. Nowak: Nature Human Behaviour 2 (2018) 469. [5] M. C. Boerlijst, M. A. Nowak, and K. Sigmund: The American Mathematical Monthly 104 (1997) 303. [6] A. J. Stewart and J. B. Plotkin: Proceedings of the National Academy of Sciences 110 (2013) 15348. [7] J. Chen and A. Zinger: Journal of Theoretical Biology 357 (2014) 46. [8] C. Hilbe, M. A. Nowak, and K. Sigmund: Proceedings of the National Academy of Sciences 110 (2013) 6913. [9] C. Adami and A. Hintze: Nature Communications 4 (2013) 2193. [10] C. Hilbe, M. A. Nowak, and A. Traulsen: PLOS ONE 8 (2013) e77886. [11] A. Szolnoki and M. Perc: Physical Review E 89 (2014) 022804. [12] X. Chen, L. Wang, and F. Fu: New Journal of Physics 24 (2022) 103001. [13] L. Pan, D. Hao, Z. Rong, and T. Zhou: Scientific Reports 5 (2015) 13096. [14] A. McAvoy and C. Hauert: Proceedings of the National Academy of Sciences 113 (2016) 3573. [15] M. Ueda, S. Yagi, and G. Ichinose: PLOS Complex Systems 2 (2025) e0000081. [16] E. Akin: Ergodic Theory, Advances in Dynamical Systems (2016) 77. [17] C. Hilbe, A. Traulsen, and K. Sigmund: Games and Economic Behavior 92 (2015) 41. [18] M. Ueda: Journal of the Physical Society of Japan 91 (2022) 084801. [19] M. Ueda: Operations Research Forum 3 (2022) 48. [20] P. Duersch, J. Oechssler, and B. C. Schipper: Games and Economic Behavior 76 (2012) 88. [21] P. Duersch, J. Oechssler, and B. C. Schipper: International Journal of Game Theory 43 (2014) 25. [22] M. Ueda: Journal of the Physical Society of Japan 91 (2022) 054804. [23] M. Ueda: arXiv preprint arXiv:2507.16221 (2025). [24] D. Monderer and L. S. Shapley: Games and Economic Behavior 14 (1996) 124. [25] A. McAvoy, U. Madhushani Sehwag, C. Hilbe, K. Chatterjee, W. Barfuss, Q. Su, N. E. Leonard, and J. B. Plotkin: Proceedings of the National Academy of Sciences 122 (2025) e2319927121.