Zero-determinant strategies in repeated asymmetric games

https://doi.org/10.1016/j.amc.2019.124862Get rights and content

Abstract

Zero-determinant (ZD) strategies are conditional strategies which allow players adopting them to establish a relation between their expected payoffs and those of their opponents. The ZD strategies were investigated in different models including finite and infinite repeated two players’ games, multiplayer games, continuous action spaces and alternating move games. However, all previous studies considered only symmetric games between players, i.e., players have the same strategies and the same associated payoffs, thus the players’ identities are interchangeable without affecting the game's dynamics. In this study, we analytically model and derive the ZD strategies for asymmetric two players’ games, focusing on one-memory strategies and infinite repeated encounters. We derive the analytical bounds of equalizer and extortionate ZD strategies in 2  ×  2 asymmetric games, which differ from the symmetric games case. Furthermore, we derive under what conditions a player using an extortionate ZD strategy will get a higher expected payoff than his/her opponent. Finally using a numerical example, we investigate ZD strategies in 2  ×  2 asymmetric prisoner's dilemma game.

Introduction

Press and Dyson [1] have changed the considerations of game theorists after formalizing a new class of conditional strategies known as “zero-determinant” (ZD) strategies. A player using a ZD strategy can limit his opponent's expected payoff (i.e., utility) to be within a certain desirable range using equalizer ZD strategies or set a ratio between his expected payoff and his opponent's expected payoff using extortionate ZD strategies. The model proposed by Press and Dyson considered symmetric iterated 2  ×  2 games, where players take actions simultaneously and play infinite number of iterations. Since then, several studies have extended the investigation of the ZD strategies to game-theoretic models with different characteristics.

Hilbe et al. [2] have investigated the extortionate ZD strategies in iterated prisoner's dilemma, where such strategies opened new possibilities for studying the evolution of cooperation and the findings can be extended to other iterated 2  ×  2 games. Hao et al. [3] have studied the extortionate ZD strategies under uncertainty caused by noise and deduced that strong extortions did not exist even under low levels of noise. Mcavoy and Hauert [4,5] have investigated the effect of ZD strategies existing in alternating games, which are often more biologically relevant than simultaneous games, and the conditions under which these strategies exist were derived. Ichinose and Masuda [6] studied analytically the ZD strategies in finitely repeated prisoner's dilemma 2  ×  2 game using a general payoff matrix. It was shown in [6] that the only strategies which enforce a linear relation between the two players’ payoffs are either the ZD strategies or unconditional strategies that independently cooperates with a fixed probability in every round of the game. Adami and Hintze [7] showed that ZD strategies are not evolutionarily stable and weakly dominant, while Hilbe et al. [8,9] developed a theory for ZD strategies for iterated multiplayer social dilemmas, with any number of involved players explored their evolutionary performance and existence for these social dilemmas. More examinations and extensions of ZD strategies have been studied in [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] including evolution, multi-players and structured populations. Chen and Zinger [20] showed that ZD strategies are even more powerful than what was pointed out initially by Press and Dyson.

To the best of our knowledge, all ZD strategies related studies have assumed symmetric games, where players have the same set of pure strategies and the same associated payoff with each strategy. The prevalence of these studies is due to the simplified analysis and computations which symmetric games require [21]. The prisoner's dilemma (PD) game is the most famous and extensively studied symmetric game [22]. The PD game captures the social dilemma rising due to the conflict between the self-interest and the group interest, where players end up by defecting and getting low payoff if they play according to their own self-interest while they can get a higher payoff if they mutually cooperated.

While requiring more complex mathematical modeling and analysis, asymmetric games are more advantageous and realistic than symmetric games and more related to experimental studies. Most of strategic confrontations are more probable to be asymmetric (actually, symmetric games are considered a special case of asymmetric games), either in the strategies each player possesses and the associated payoff (i.e., players’ preferences) with each possible outcome of the game [23]. In spite of the fact that many games of interest (e.g., the ultimatum game [24,25]) are asymmetric in nature, asymmetric games constitute the majority of 2  ×  2 game. When classifying 2  ×  2 games using strict linear relations between the four payoff values a player may earn, we have 576 different 2  ×  2 games with only 78 strategically distinct games (i.e., have different strategic properties in term of domination), 66 of which are asymmetric games and 12 are symmetric games [26]. Nevertheless, studies investigating asymmetric games (e.g., [27,28]) are significantly less in number compared to those dealing with symmetric games.

In this study, we extend the investigation of ZD strategies to asymmetric games. We analytically model and derive the ZD strategies for simultaneous iterated 2  ×  2 asymmetric games, focusing on one-memory strategies and infinite repeated encounters. We derive the analytical bounds of equalizer and extortionate ZD strategies in 2  ×  2 asymmetric games, which differ from the symmetric games case. Furthermore, we derive under what conditions a player using an extortionate ZD strategy will get a higher expected payoff than his/her opponent. Finally we illustrate ZD strategies in 2  ×  2 asymmetric prisoner's dilemma game using a numerical example.

The paper is organized as follows. In the next section, we provide a model formulation for ZD strategies in 2  ×  2 asymmetric games. In Section 3, we derive special kinds of ZD strategies, which are ZD strategies for unilateral determination of opponent's expected payoff and extortionate ZD strategies. Section 4 illustrates a numerical example. Finally, in Section 5 we conclude the study and highlight future research directions.

Section snippets

Model formulation: ZD strategies in asymmetric 2  ×  2 games

Consider an infinitely repeated asymmetric 2  ×  2 game [27], where we have two players and each player has two actions (i.e., choices). At the beginning of the game, one of the players is randomly assigned to be the first player (player 1), and the other player will be the second player (player 2). The actions of player 1 are C1 and D1 while the actions of the player 2 are C2 and D2. At each round of the game, we have four possible profiles which are (C1,C2),(C1,D2),(D1,C2) and (D1,D2). We

Special kinds of ZD strategies

There are two special kinds of ZD strategies which are of interest and widely investigated in ZD studies for symmetric 2 × 2 games (e.g., [1]). The first kind of ZD strategies is equalizer ZD strategies which allow a player to unilaterally set the expected payoff of his opponent to a value within a certain interval, while the second kind is the extortionate ZD strategies where a player guarantees a large share of the total expected payoff above the mutual defection value. Since we are

Numerical example of ZD strategies in 2 × 2 asymmetric PD game

Each 2 × 2 game (e.g., Prisoner's Dilemma, Chicken, Stag and Hunt) is defined according to a specific non-strict linear ordering of the payoff values R, T, S, and P for each player. Alternatively as proposed in [30], 2 × 2 games can be categorized based on gamble-intending (GID) and risk-averting (RAD) dilemmas, defined by Dg = T − R and Dr = P − S, respectively, where Dg and Dr can fully determine the game dynamics and equilibria instead of the payoff values R, T, S, and P. For positive Dg and

Conclusion

Asymmetric games are more realistic (and thus applicable) than symmetric games, and they constitute the larger portion of 2 × 2 games. In this study, we extend the investigation and analysis of ZD strategies to asymmetric games. We formalized a model for 2 × 2  asymmetric games where players have different actions, different associated payoffs and use stochastic strategies for infinitely repeated encounters. We illustrated the ZD strategies in such setting. We investigated two special kinds of

References (32)

  • W.H. Press et al.

    Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent

    Proc. Natl. Acad. Sci.

    (2012)
  • C. Hilbe et al.

    Evolution of extortion in Iterated Prisoner's Dilemma games

    Proc. Natl. Acad. Sci.

    (2013)
  • D. Hao et al.

    Extortion under uncertainty: zero-determinant strategies in noisy games

    J. Phys. Rev. E

    (2015)
  • A. McAvoy et al.

    Autocratic strategies for iterated games with arbitrary action spaces

    Proc. Natl. Acad. Sci.

    (2016)
  • C. Adami et al.

    Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything

    J. Nat. Commun.

    (2013)
  • C. Hilbe

    Cooperation and control in multiplayer social dilemmas

    Proc. Natl. Acad. Sci.

    (2014)
  • Cited by (26)

    • Cooperation and control in asymmetric repeated games

      2024, Applied Mathematics and Computation
    • Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

      2022, Journal of Theoretical Biology
      Citation Excerpt :

      In addition to evolutionary games, ZD strategies have been studied from various directions. Examples include games with observation errors (Hao et al., 2015; Mamiya et al., 2019), multiplayer games (Hilbe et al., 2014; Hilbe et al., 2015; Pan et al., 2015; Milinski et al., 2016; Stewart et al., 2016; Ueda et al., 2020), continuous action spaces (McAvoy et al., 2016; Milinski et al., 2016; Stewart et al., 2016; McAvoy et al., 2017), alternating games (McAvoy et al., 2017), asymmetric games (Taha et al., 2020), animal contests (Engel et al., 2018), human reactions to computerized ZD strategies (Hilbe et al., 2014; Wang et al., 2016), and human–human experiments (Hilbe et al., 2016; Milinski et al., 2016; Becks et al., 2019). Among the various ZD strategies, of particular interest are the so-called Equalizer, Extortion (Press and Dyson, 2012), and Generous strategies (Stewart et al., 2013).

    • Evolutionary dynamics of zero-determinant strategies in repeated multiplayer games

      2022, Journal of Theoretical Biology
      Citation Excerpt :

      With the neutral drift as a reference, a strategy is said to be favored by selection if it resists more than N mutants on average. Zero-determinant strategies are of particular interest due to their disproportionate control over opponents’ long-term payoffs (Taha and Ghoneim, 2020; Ichinose and Masuda, 2018; Hilbe et al., 2015; Tan et al., 2021; Wang et al., 2021; Wu and Wang, 2018; Wu et al., 2018; Zhou et al., 2021; Li et al., 2020; Fu et al., 2008; Fu et al., 2009). Following the original work on ZD strategies in iterated prisoner’s dilemmas, several studies have independently shown that ZD strategies also exist in repeated multiplayer games.

    View all citing articles on Scopus
    View full text