Conrad Power (2009)
A Spatial Agent-Based Model of N-Person Prisoner's Dilemma Cooperation in a Socio-Geographic Community
Journal of Artificial Societies and Social Simulation
vol. 12, no. 1 8
<http://jasss.soc.surrey.ac.uk/12/1/8.html>
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Received: 28-May-2008 Accepted: 01-Dec-2008 Published: 31-Jan-2009
Table 1: Payoff Matrix for a General Prisoner's Dilemma Game | |||
Player A | |||
Player B | Cooperate | Defect | |
Cooperate | R, R | S, T | |
Defect | T, S | P, P | |
Table 2: Payoff Matrix for an Example Iterated Prisoner's Dilemma Game | |||
Player A | |||
Player B | Cooperate | Defect | |
Cooperate | 3, 3 | 0, 5 | |
Defect | 5, 0 | 1, 1 | |
Figure 1. N-Person Prisoner's Dilemma Payoff Functions for Preferred Defection and Un-Preferred Cooperation, Relative to the Number of Other Players that Decide to Cooperate: (from Akimov and Southchanski 1994) |
Figure 2. Interface of the Spatial Agent-Based Model of NPPD Cooperation |
Figure 3. Analyzed Environment of Central Catalina, Newfoundland and Labrador, Canada |
Figure 4. Configuration of a 50 Meter Neighbourhood Buffer of Agent A |
D = -0.5 + 2x | (1) |
C = -1 + 2x | (2) |
where x represents the ratio of the number of cooperators to the total number of neighbours. The stochastic factor is a parameter that accounts for any uncertainty in the agent interactions and noise in the environment. When dealing with linear payoff functions, stochasticity is applied by thickening the width of each line relative to the y axis to produce a range of payoffs for a cooperation ratio. For example, an agent with previous action C in a neighbourhood with 0.60 cooperation receives a payoff reward of 0.207 ± 0.033. A line drawn from the bottom of the C function intersects the payoff axis at 0.174 while a line from the top of C hits the axis at 0.24. The derived cooperation value will be a random value chosen within the range of 0.174 to 0.24. In a deterministic environment where the stochastic factor is zero, the width of the payoff function would be and the payoff reward would equal 0.207.
Figure 5. Reward/Penalty Payoff Functions for Pavlovian Defectors and Cooperators (from Szilagyi 2003) |
(3) |
,where Wi is a weighting parameter such that all weights sum to one, and Mc_{i} is the history payoff (i.e. Mc_{1} stores the current payoff). Assuming that the effects of memory decrease with time, W_{1} ≥ W_{2} ≥W_{3}.
p( t+1) = p( t) + (1-p( t)) * α_{ i}, if at time t, S(t) = C and RP_{wt} > 0 | (4) |
p( t+1) = (1-α_{ i}) * p( t), if at time t, S(t) = C and RP_{wt} ≤ 0 | (5) |
Note that for every t there must be q( t) = 1 - p( t). The same set of equations are also used for updating the action probabilities when the previous action is D:
q( t+1) = q( t) + (1-q( t)) * α_{ i}, if at time t, S(t) = D and RP_{wt} > 0 | (6) |
q( t+1) = (1-α_{ i}) * q(t), if at time t, S(t) = D, and RP_{ wt }≤ 0 | (7) |
(8) |
where C_{j} is the payoff value for agent j and N is the total number of agents in the neighbourhood. The average neighbourhood function for three memory events is formulated as:
(9) |
Thus, the state of agent i at time t+1 with S( t):
For S(t) = C: S( t+1) = |
(10) |
For S(t) = D: S( t+1) = |
(11) |
,where R_{u}∈ [0,1] is a uniform random value.
Figure 6. Map of Cooperation Pattern for Mobile Agents in a 50 Meter Neighbourhood |
Figure 7. Graph of Cooperation Pattern for Mobile Agents in a 50 Meter Neighbourhood |
Figure 8. Map of Cooperation Pattern for Fixed Agents in a 50 Meter Neighbourhood |
Figure 9. Graph of Cooperation Pattern for Fixed Agents in a 50 Meter Neighbourhood |
Figure 10. Map of Cooperation Pattern for Mobile Agents in a 150 Meter Neighbourhood |
Figure 11. Graph of Cooperation Pattern for Mobile Agents in a 150 Meter Neighbourhood |
Figure 12. Map of Cooperation Pattern for Fixed Agents in a 150 Meter Neighbourhood |
Figure 13. Graph of Cooperation Pattern for Fixed Agents in a 150 Meter Neighbourhood |
Figure 14. Fuzzy Membership Functions of Cooperation and Defection |
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