How does human generosity affect the iterated equilibrium?
An experiment study of N-Prisoner’s Dilemma on behavioural game theory
Table of contents
Since game theory is a crucial decision-making strategy in microeconomics, the solution of a game should make sense in the real world. Nash equilibrium (1951) and its extensions make the solution powerful. Unfortunately, the game is played by people and their behaviours increasingly affect the solutions of games. Better solutions become feasible given that human beings take fairness into account. Therefore, a more reasonable strategy, behavioral game theory (BGT), is able to explain why Nash equilibrium is not always or even never achieved. BGT could be considered as a mixture of game theory, sociology and psychology.The aim of this study is to examine how human behaviours affect the final equilibrium by means of a study of well-known Prisoner’s Dilemma. We need to extend it to n prisoners and exploit a random probability model to represent human behaviours. Consequently, the behaviour is formulated in form of probability. Although this heuristic representation will not include all uncertainties that matter, it will be a novel way to think about this kind of analysis. Finally, we will give analytical result through our own-designed experiment. The remaining sections are followed by literature review, empirical study, conclusion and further discussions.
2. Literature review
In this section, we briefly review the literatures in two aspects: behavioural game theory and the representative study N-prisoner’s dilemma.
2.1 Review of game theory
Before the development of theory of two-person zero-sum games, Von Neumann and Morenstern (1947) were considered as the first persons who described an exposition and various application of a mathematical theory of games. Equilibrium point is the basic element in Nash’s (1951) theory, and it also closely associated with the theory of a two-person zero-sum game. Nash states that the set of equilibrium points of a two-person zero-sum game is the set of all pairs of opposing ‘good strategies’ and proves that a finite non-cooperative game always has at least one equilibrium point.
Camerer (2003) provided an extensive study on behavioural game theory. Based on the theory of games, the behavioural game theory is highlighted to describe actual human behaviour. The concept of fairness equilibrium is first introduced by Camerer (1997) in terms of three classical examples.
2.2 Review of N-prisoner’s dilemma
The characteristic function of N person’s game is defined by Von Neumann and Morgenstern (1947). Weil (1966) is the first person to describe the N-prisoner’s dilemma problem and it is widely developed and studied. Some of the studies are considered in the evolutionary context. Yao and Darwen (1995) simplified the Axelrod-style representation and made an experimental study of the iterated games. Firstly, they found that cooperation is less likely to happen in large groups than that in a small group. Secondly, they showed the importance of the environments in which each individual is evaluated. However, there is no analysis on the human generosity in their research. Paul et al. (1990) compared the casual attributions from
Cooperators and Defectors for a cooperative and non-cooperative target in an N-prisoner’s dilemma game. Although they include intrapersonal factors, ignorance, concerns, fear, greed and etc., they did not provide a clear model to represent the player’s choice. Goehring and Kahan (1976) studied a uniform prisoner’s dilemma and concluded that a index of competitiveness strongly affects the proportion of cooperation. However, they did not mention the generosity.
3. Empirical analysis
In this section, we illustrate the N-prisoner’s dilemma in details. After that, a mixed strategy probability model is introduced and the corresponding simulation study is implemented. The empirical result is shown.
3.1 Problem description
The N-prisoner’s dilemma problem is a game played by n persons. The final payoff is decided by the individual choices. The individual rationality indicates non-cooperation (D) no matter how the remaining opponents make their decisions. However, the collective cooperation (C) produces more payoffs than that from non-cooperation for each individual. A reasonable assumption is to consider every player is interchangeable, which means the payoff is only affected by the number of Ds and Cs but no matter to the player who selects. Consequently, the binary choice can be identical for each player. An example is given in Table 1 (Yao and Darwen, 1995).
Table 1 N-prisoner’s dilemma payoff
In Table 1, the number in the column is the number of Cs. The restrictions are: for 0in1??-,
DiCi>; for 0in1?<-, Di1Di+>, Ci1Ci+> and CiDiCi12>(+-). Doubtlessly, the Nash
equilibrium is collective D since everyone is rational. The practical world, however, does not always confirm it since players’ intrapersonal behaviours are not always “individual rational”.3.2 Model
We follow Rezaei et al. (2009). They presented a mixed strategy probability model by
considering the population are connected in a network. We have a population ? and players
in ? is randomly selected into games g with size N. The probability of choosing C changes
according to the average link weight. The weight is defined as
where wij indicates individual links in the game. If both player i and j played C, wij is
updated by wij1+, otherwise wij is set to 0. Therefore, the probability is defined as
where β describes the probability when players are not linked, e.g., human generosity. The probability of choosing D is 1Pig-.
We show our finding in terms of a simulated experiment. In our study, we suppose players are linked with initial probability 0.5, ?=120 and α1=. The game size N ranges from 2 to 10.
We repeat each game 2,000 times and observe the proportion of players who chose C of all games. To make the result clear and explicit, we only list the last 200 iterations. Figure 1-2 shows how β affect this proportion of cooperation for two players and ten players.
Figure 1 – Converged percentage of choosing C for two-player game
The symbol "?" represents the β taking value of 0; "•” represents value of 0.3; "*"
represents value of 0.6 and "+" represents the value of 1. It is clearly shown that for two-players game, when β takes 0, the proportion converges to around 0.1. It converges to 0.5 0.8 and 0.9 when β takes 0.3, 0.6 and 1 respectively. But for ten-players, the converged values are 0 for β0= and 1 for the other βs.
Our experiments indicate the percentage of cooperation is affected by both the human
Figure 2 – Converged percentage of choosing C for ten-player game
generosity and the number of players. However, the two-player case is a special case. The converged percentage of cooperation is highly related to the human generosity, β. Even β
takes value 0, the final percentage is a little above 0, which means people will perform cooperatively without generosity. In the ten-player case, that is not true for the converged percentage is 0. For the non-trivial value of β, the converged percentages are approaching 1.
This is an amazing finding since people’s generosity will produce periodical effect between players and this effect arises when the number players grows, even though the value of β is
small. We also investigate the case of three players’ game and five players’ game. The outcomes are similar to ten players’ game.
This study leaves various topics to be investigated further, among which is the choice of the probability of defining the cooperation. The probability described in this study does not thoroughly characterize human behaviours. It needs to be refined. Based on a illustrative probability model, we also need to put more effort on the link adjustment.
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