From rationality to cooperativeness: The totally mixed Nash equilibrium in Markov strategies in the iterated Prisoner’s Dilemma

Abstract

In this research, the social behavior of the participants in a Prisoner's Dilemma laboratory game is explained on the basis of the quantal response equilibrium concept and the representation of the game in Markov strategies. In previous research, we demonstrated that social interaction during the experiment has a positive influence on cooperation, trust, and gratefulness. This research shows that the quantal response equilibrium concept agrees only with the results of experiments on cooperation in Prisoner’s Dilemma prior to social interaction. However, quantal response equilibrium does not explain of participants’ behavior after social interaction. As an alternative theoretical approach, an examination was conducted of iterated Prisoner's Dilemma game in Markov strategies. We built a totally mixed Nash equilibrium in this game; the equilibrium agrees with the results of the experiments both before and after social interaction.

Mutual Cooperation and Tolerance to Defection in the Context of Socialization: The Theoretical Model and Experimental Evidence

“…PD has one Nash equilibrium: it is a mutual choice of Defect strategy which gives the payoff of 1 for two players. However, laboratory experiments show that people in some conditions avoid Nash equilibrium 9,26 . For example, under social framing, individuals may start to choose more frequently the Cooperate strategy, a sort of behavior that could be considered irrational.…”

“…These two variables imply that individuals' strategies at round 𝑡 − 1 determine completely their behavior at round 𝑡. This model will be referred as PD in Markov strategies 26,30 , and for brevity we will refer to subsequent 𝛾 and 𝛼 as Markov strategies.…”

“…(3) The paper 26 found a symmetric (whereby 𝛾 1 = 𝛾 2 = 𝛾 and 𝛼 1 = 𝛼 2 = 𝛼) totally mixed Nash equilibrium for Prisoner's Dilemma in Markov strategies in explicit form. This equilibrium can be represented as the points (𝛼, 𝛾) that meet equation 5𝛼 2 + 9𝛾 2 − 14𝛼𝛾 − 10𝛾 + 1 = 0 (4) and located in the unit square (see Figure 1).…”

“…Proper equilibrium 24 Previous studies demonstrate that social interaction significantly increases the cooperation level in iterated Prisoner's Dilemma games, from a 20% cooperation rate prior to socialization to 53% after socialization 9,[25][26][27] . To model such a high level of cooperative strategy choice we require a specific approach.…”

“…To model such a high level of cooperative strategy choice we require a specific approach. In the paper 26 it was proposed to consider Prisoner's Dilemma in Markov strategies. For this game, a symmetric totally mixed Nash equilibrium was found.…”