H. Fort (2003)
Cooperation with random interactions and without memory or "tags"
Journal of Artificial Societies and Social Simulation
vol. 6, no. 2
<http://jasss.soc.surrey.ac.uk/6/2/4.html>
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Received: 19/12/2002 Accepted: 16-Feb-2003 Published: 31-Mar-2003
T > R > P > S. | (1) |
In case one wants that universal cooperation be Pareto optimal, the additional required condition is:
2R > S + T. | (2) |
(3) |
where the variable p_{k} takes only two values: 1 if k is a C-agent and 0 if k is a D-agent. It turns out that, in general, the value of p is not known by the agents. Thus a simpler estimate of the expected utilities ε_{k} is obtained by substituting p in the above expression for the per-capita-income δ C_{k} (R,S,T,P) by his own probability of cooperation pk^{[2]}. In other words, agent k makes the simplest possible extrapolation and assumes that the average probability of cooperation coincides with his p_{k}; the estimate ε_{k} corresponds to the utilities he would made by playing with himself. That is,
(4) |
(5) |
Figure 1. <p> vs. time, for the canonical payoff matrix. f_{C0} = 0, 0.1, 0.25, 0.75 and 0.9. |
Table 1. Possible outcomes for the different PD game interactions. |
(6) |
for which the proposed strategy produces cooperation (in gray), and the rest for which p_{eq} = 0.0 (in white).
Figure 2. Different region in the (R,P) plane: cooperative (gray) and non-cooperative (white). |
Table 2. Transitions for payoff matrices M[3,0,5,1], M[4,0,5,3] and M[0,3,4,5] |
2(1-p_{eq})^{2} = 2 p_{eq}(1-p_{eq}) | (7) |
one of its roots is p_{eq}=0.5 (the other root is p_{eq}=1); and for M[4,0,5,3] we have
0 = 2 p_{eq}(1-p_{eq}) | (8) |
one of its roots is p_{eq}=0.0 (the other root is p_{eq}=1).
Figure 3. p vs. time, for payoff matrices M[3,0,5,1] with p_{eq}=0.5 and M[4,0,5,3] with p_{eq}=0.0. |
(9) |
(10) |
^{2} One might consider more sophisticated agents which have "good" information (statistics, surveys, etc.) from which they can extract the average probability of cooperation in "real time" p(t) to get a better estimate of their expected utilities. However, the main results do not differ substantially from the ones obtained with these simpler agents.
^{3} max{A,B} stands for the maximum of A and B.
^{4} The choice of M_{av} as a reference point is quite arbitrary, indeed whether or not cooperation becomes a stable strategy depends on the fact that two payoffs are greater and the other two smaller than the reference point.
^{5} For instance "altruists", C-inclined independently of the costs, with payoff matrices with high values of S or "anti-social" individuals, for whom cooperation never pays, with very low values of R.
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FORT, H. (2002) to be published elsewhere.
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