Single equalizer strategy with no information transfer for conflict escalation

Engel, Assaf; Feigel, Alexander

doi:10.1103/physreve.98.012415

“…In addition to evolutionary games, ZD strategies have been studied from various directions. Examples include games with observation errors [19,20], multiplayer games [12,[21][22][23][24][25], continuous action spaces [23,24,26,27], alternating games [27], asymmetric games [28], animal contests [29], human reactions to computerized ZD strategies [30,31], and human-human experiments [23,32,33].…”

Section: Introductionmentioning

confidence: 99%

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

Miyagawa,

Mamiya,

Ichinose

2021

Preprint

0

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Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

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“…After Stewart and Plotkin raised a question [10], evolution or emergence of ZD strategies became one of the main targets in subsequent studies [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Then, this research spread in many directions including multiplayer games [19,[26][27][28][29], continuous action spaces [28][29][30][31], alternating games [31], asymmetric games [32], animal contests [33], human reactions to computerized ZD strategies [34,35], and human-human experiments [28,36,37], which promote an understanding of the nature of human cooperation. For further understanding, see the recent elegant classification of strategies, partners (called "good strategies" in Ref.…”

Section: Introductionmentioning

confidence: 99%

Zero-determinant strategies under observation errors in repeated games

Mamiya

¹

,

Ichinose

²

2020

Preprint

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Zero-determinant (ZD) strategies are a novel class of strategies in the Repeated Prisoner's Dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, a discount factor and observation errors are both important because they often happen in society. However, they were not considered in the original discovery of ZD strategies. In some preceding studies, each of them were considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first revealed that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we searched for all possible strategies that enforce linear payoff relationships and found that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. Moreover, we numerically derived minimum discount rates for the one subset of the ZD strategies in which the extortion factor approaches to infinity. For the ZD strategies whose extortion factor is finite, we numerically derived the minimum extortion factors above which such strategies exist. These results contribute to a deep understanding of ZD strategies in society. Author summaryRepeated games where two players independently select cooperative or non-cooperative behavior have been used to model interactions of biological organisms. In a real situation, people sometimes cannot observe the direct behaviors that other people select. Instead, they receive signals that reflect other people's behaviors. Those signals are influenced by the environment. Therefore, people sometimes receive the wrong signals. As a result, people mistake other people's behaviors. Hence, in repeated games, assuming such observation errors is important to model biological phenomena close to reality. We mathematically derived that, in the repeated games with observation errors, there are only two types of strategies which enforce a linear payoff relationship to the opponent irrespective of the opponent's strategy. The subsets of the strategies can manipulate the opponent's payoff or enforce an unequal payoff relationship to the opponent. We further numerically revealed the conditions of error rates and a discount factor above which this strategy can exist. January 13, 2020 1/18 49 In this study, we search for ZD strategies under the situations that observation 50 errors and a discount factor are both incorporated. We search for the other possible 51 strategies, not just ZD strategies, that enforce a linear payoff relationship between the 52 January 13, 2020 2/18 two players. By formalizing the determinants for the expected payoffs in the RPD 53 game, we mathematically found that only ZD strategies [9] and unconditional 54 strategies [14,55] are the two types which enforce a linear payoff rela...

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“…After Stewart and Plotkin raised a question [10], evolution or emergence of ZD strategies became one of the main targets in subsequent studies [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Then, this research spread in many directions including multiplayer games [19,[26][27][28][29], continuous action spaces [28][29][30][31], alternating games [31], asymmetric games [32], animal contests [33], human reactions to computerized ZD strategies [34,35], and human-human experiments [28,36,37], which promote an understanding of the nature of human cooperation. For further understanding, see the recent elegant classification of strategies, partners (called "good strategies" in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…Besides evolutionary games, ZD strategies have been studied from various directions. Examples are games with a discount factor [26][27][28][29][30], games with observation errors [31,32], multiplayer games [7,19,[33][34][35][36], continuous action spaces [27,28,35,36], alternating games [28], asymmetric games [37], animal contests [38], human reactions to computerized ZD strategies [39,40], and human-human experiments [35,41,42].…”

Section: Introductionmentioning

confidence: 99%

Zero-determinant strategies under observation errors in repeated games

Mamiya

¹

,

Ichinose

²

2020

View full text Add to dashboard Cite

Zero-determinant (ZD) strategies are a novel class of strategies in the Repeated Prisoner's Dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, a discount factor and observation errors are both important because they often happen in society. However, they were not considered in the original discovery of ZD strategies. In some preceding studies, each of them were considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first revealed that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we searched for all possible strategies that enforce linear payoff relationships and found that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. Moreover, we numerically derived minimum discount rates for the one subset of the ZD strategies in which the extortion factor approaches to infinity. For the ZD strategies whose extortion factor is finite, we numerically derived the minimum extortion factors above which such strategies exist. These results contribute to a deep understanding of ZD strategies in society.

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Single equalizer strategy with no information transfer for conflict escalation

Cited by 10 publications

References 79 publications

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

Zero-determinant strategies under observation errors in repeated games

Zero-determinant strategies under observation errors in repeated games

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