1 Abstract 3
2 Introduction 5
3 Example with N-person Prisoner Dilemma game 8
3.1 Standard 2-person P.D. game 8
3.2 N-person 2-strategy P.D. game 9
3.2.1 Interval-scale payoffs 9
3.2.2 Common scale 10
3.2.3 Symmetric N-person 2-strategy P.D. game 10
4 Introduction to cooperation indexes 14
4.1 K-index 14
4.2 Social preference 15
4.3 beliefs 17
4.4 Trust 19
4.5 Beliefs with trust 20
5 Cooperation indexes applied in N-person P.D. game 23
5.1 K-index applied in N-person P.D. game 23
5.2 A model of trust based cooperation applied in N-person game 25
6 Interpretation of result 31
7 conclusion 34
The Prisoner’s Dilemma is a classical decision problem in which two players must make decisions at the same time about whether to cooperate or defect in their own narrow self-interest. The Prisoner’s Dilemma game has been used to model many naturally occurring interactions where, at the individual, or¬ganizational, and societal levels, there is a tension between local and global interests when more people face the Prisoner’s Dilemma at the same time. In the prisoner’s Dilemma, there are many factors that affect the occurrence of cooperative behavior, which are called cooperation indexes. Some research on cooperation in the prisoner’s Dilemma problem focused on developing simple indexes of a game’s severity- promoting the degree of cooperative and non¬cooperative choice. In the prisoner’s Dilemma of standard symmetric 2 players, the payoff structure of the game affects the cooperation rate, also affects these indexes. However, when more players take the Prisoner’s Dilemma game at the same time, how do decision makers choose cooperation and non-cooperation through these indexes? Whether these indexes are still valid or whether they change is unknown.
This research aims to explore how cooperation indexes such as social value orientation and trust affect the occurrence of cooperative behavior in the N- person Prisoner’s Dilemma game, and what effect about the indexes of more players has on these indexes compared with fewer players. Through the pris¬oner’s Dilemma of a specific payoff structure, this research first considers and expresses the benefits that different numbers of players who choose to cooperate and those who choose to betray will get when N players participate at the same time. Then the K-index for N people is calculated, and social value orienta¬tions and beliefs are represented by a trust-based cooperation model. Finally, by calculating these indexes and comparing them with those of fewer partici¬pants, analysis the change of these indexes and the influence of the number of participants on these indexes.
The final statement provides the central point for this work. In the N- person Prisoner’s Dilemma game,using a simple model of the social preferences of decision makers , and their beliefs about other players’ expected choices, we use both factors to predict when decision makers will choose to trust and therefore cooperate in this game.
Keywords: N-person Prisoner’s Dilemma game; K-index; Trust; Social Value Orientation; belief.
Through the above research,we get how cooperation indexes such as social value orientation and trust affect the occurrence of cooperative behavior in the N-person Prisoner’s Dilemma game, and what effect about the indexes of more players has on these indexes compared with fewer players.Now let’s make the following summary.
First,the K-index is designed to capture the severity of a game as it reflects the intuition that some PD games are ’easier’ than others, and hence cooperation is more likely to emerge in games with higher K values contrasted with games that have relatively low K values.And when we apply this form of the K-index to a N-person Prisoner’s dilemma game,it does not change with the number of players, nor does it change with the number of people who choose to cooperate, it changes only depending on the payoff structure of the game.It is worth noting that the choice of K-index in cooperative games,in particular, if the C frequencies are estimated from the protocols of a whole population of players, maybe it is not legitimate to conclude that the index for the whole population is the same function of the payoffs as the index for an individual player.Since an estimate of the parameters of individuals, and especially of their distributions in the population, is usually extremely difficult, the limitations of such a ’theoretical’ approach are apparent. For this reason one is forced for the time being to choose indices more or less arbitrarily and to justify them on ’untheoretical’ purely pragmatic grounds.
Second,about social preference,in this paper,as the number of players in-creases,the level of support for others will decrease.It means the Q(N) depends not only on the payoff structure, but also on the number of players in the game.In comparing the indexes of the two payoff structures, as people choose to cooperate more frequently, they need less social value orientation for others.It is worth noting that when the k index is high to a certain extent, the social value orientation will be negative(shows in figure 6.2), that is, it will lose its meaning. This suggests that when people have a high desire to cooperate, they will reduce the degree of consideration of others’ payoffs.
Then,In the discussion of positive expectations of others, we get a fixed value: 0.5(shows in figure 6.3), when the number of people who choose to co¬operate is not equal to half of the total number of people. It does not change with any other factors. This result can be understood as follows: In a N-person Prisoner’s dilemma game, when the number of players choosing cooperation and defection is not equal, the player’s positive expectation for others to choose cooperation must be greater than 0.5, then it is possible to choose cooperation.
Finally, by summarizing the above results, it can be concluded that the cooperation model based on trust has certain applicable value in the game of N-person prisoner’s dilemma. But its practical conditions are relatively limited. This is also the shortcoming of this study. We only discuss the cooperation indexes of the N-person prisoner’s dilemma game when the payoff structure satisfies R-S=T-P. And the range of changes in these indices is not particularly clear. Therefore, if we want to better study the application of cooperation indexes in the N-person prisoner’s dilemma game in the future, we not only need to study the N-person Prisoner’s dilemma game with less limited scope, but also need to add some new parameters to the original cooperation indexes to obtain more valuable results.
