Marco Janssen (2006)
Evolution of Cooperation when Feedback to Reputation Scores is Voluntary
Journal of Artificial Societies and Social Simulation
vol. 9, no. 1
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Received: 13-Mar-2005 Accepted: 24-Aug-2005 Published: 31-Jan-2006
|Table 1: Pay-off table of the Prisoner's Dilemma with the option to withdraw from the game|
|Table 2: Utility pay-off table of the Prisoner's Dilemma with the option to withdraw from the game|
|Cooperate||R,R||S+ β A,T- α B||E,E|
|Defect||T- α A,S+ β B||P,P||E,E|
where α max is used to scale the symbol value between 0 and 1. In order to represent the assumed nonlinear relationship between the preferences of the agent and the symbols, the equation is squared.
where w0 is the bias, wi is the weight of the ith input, and xi is the ith input. Initially, all weights are zero, but during the simulation the network is trained, when new information is derived, by updating the weights as described below in equation (4).
where Δwi is the adjustment to the ith weight, λ is the learning rate, F is the feedback, F-Pr[Tr] is the difference between the agent's level of trust in the other agent and the observed trustworthiness of the other agent, and xi is the other agent's ith symbol. In effect, if the other agent displays the ith symbol, the corresponding weight is updated by an amount proportional to the difference between the observed trustworthiness of an agent and the trust placed in that agent. The weights of symbols associated with positive experiences increase, while the weights of those associated with negative experiences decrease, reducing discrepancies between the amount of trust placed in an agent and that agent's trustworthiness.
Given the two estimates of expected utility, the player is confronted with a discrete choice problem which addressed with a logit function. The probability to cooperate, Pr[C], depends on the expected utilities and the parameter γ, which represents how sensitive the player is to differences in the estimates. The higher the value of γ, the more sensitive the probability to cooperate is to differences between the estimated utilities.
Logit models are used by other scholars to explain observed behavior in one-shot games such as the logit equilibrium approach by Anderson et al. (forthcoming). Although the functional relation is similar, their approach differs from the one used here since they assume an equilibrium and perfect information of the actions and motivations of other players. Moreover, in the games in this paper agents do not play anonymously but can observe symbols of others in order to estimate the behavior of the opponent.
|Table 3: List of parameters and their default values|
|Population size (n)|
Number of symbols (s)
Conditional cooperation parameter α
Conditional cooperation parameter β
Learning rate (λ)
Number of round
Length of memory (lm)
Probability of getting feedback when cooperate.
Probability of getting feedback when defect.
[ β ,3]
[0, α ]
|Figure 1. Average payoff per agent per game for different length of history. The dots represent the average payoffs over 105 interactions over 25 simulations|
|Figure 2. The average payoff per agent per game for different levels of pC and pD, when agents express only reputation score. The historical memory length is 100 interactions|
|Figure 3. The average payoff per agent per game for different levels of pC and pD, when agents express only reputation score and retaliate. The historical memory length is 100 interactions|
|Figure 4. The average payoff per agent per game for different levels of pC and pD, when agents express reputation scores and 10 additional symbols. The historical memory length is 100 interactions|
|Figure 5. The average payoff per agent per game when pC =1 and pD =0, for different numbers of additional symbols. The historical memory length is 100 interactions|
|Figure 6. The average payoff per agents per game for different levels of pC and pD, when agents express reputation scores and act strategically. The historical memory length is 100 interactions|
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