Jijun Zhao, Ferenc Szidarovszky and Miklos N. Szilagyi (2007)
Finite Neighborhood Binary Games: a Structural Study
Journal of Artificial Societies and Social Simulation
vol. 10, no. 3 3
<https://www.jasss.org/10/3/3.html>
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Received: 22-Jun-2006 Accepted: 11-Feb-2007 Published: 30-Jun-2007
Figure 1. Payoff functions |
Figure 2. x(100) as a function of S with varying value of x(0) |
Table 1: The trajectory center of the last 20 iterations for different values of S and x(0) | ||||||||
S | average of x | std | ||||||
x0=0.2031 | x0=0.2999 | x0=0.4 | x0=0.5898 | x0=0.7017 | x0=0.7969 | x0=0.8991 | ||
0.01 | 0.3309 | 0.305 | 0.3171 | 0.3142 | 0.3137 | 0.3183 | 0.3097 | 0.008128 |
0.1 | 0.3309 | 0.305 | 0.3171 | 0.3142 | 0.3137 | 0.3183 | 0.3097 | 0.008128 |
0.2 | 0.3309 | 0.305 | 0.3171 | 0.3142 | 0.3137 | 0.3183 | 0.3097 | 0.008128 |
0.3 | 0.3151 | 0.3195 | 0.3311 | 0.3189 | 0.3215 | 0.3226 | 0.3282 | 0.00554 |
0.4 | 0.3193 | 0.3073 | 0.3205 | 0.3063 | 0.3188 | 0.3119 | 0.3158 | 0.005841 |
0.5 | 0.2993 | 0.3001 | 0.2931 | 0.3116 | 0.2928 | 0.3011 | 0.2918 | 0.006929 |
0.6 | 0.2838 | 0.2846 | 0.2923 | 0.2919 | 0.2791 | 0.3023 | 0.2927 | 0.007663 |
0.7 | 0.3009 | 0.3064 | 0.3064 | 0.293 | 0.2988 | 0.2927 | 0.3013 | 0.005599 |
0.8 | 0.2929 | 0.2905 | 0.2941 | 0.2985 | 0.2905 | 0.2905 | 0.2897 | 0.003122 |
0.9 | 0.293 | 0.2905 | 0.2917 | 0.2985 | 0.294 | 0.2905 | 0.2877 | 0.003411 |
1 | 0.3157 | 0.3147 | 0.3152 | 0.3209 | 0.3137 | 0.3158 | 0.3216 | 0.003126 |
1.1 | 0.3468 | 0.347 | 0.3411 | 0.3457 | 0.3491 | 0.3448 | 0.3438 | 0.002571 |
1.2 | 0.3651 | 0.3531 | 0.3542 | 0.342 | 0.3548 | 0.3497 | 0.3618 | 0.007605 |
1.3 | 0.3612 | 0.3544 | 0.3592 | 0.3597 | 0.3577 | 0.3578 | 0.36 | 0.002213 |
1.4 | 0.3565 | 0.3594 | 0.364 | 0.3602 | 0.3565 | 0.3646 | 0.3626 | 0.003336 |
1.5 | 0.369 | 0.3695 | 0.3741 | 0.3723 | 0.3664 | 0.3697 | 0.3701 | 0.002458 |
1.6 | 0.3575 | 0.3571 | 0.3594 | 0.3609 | 0.3542 | 0.3593 | 0.3624 | 0.002697 |
1.7 | 0.384 | 0.3786 | 0.3867 | 0.3911 | 0.3834 | 0.3826 | 0.3782 | 0.004494 |
1.8 | 0.3909 | 0.3922 | 0.3889 | 0.3818 | 0.3852 | 0.3836 | 0.3899 | 0.003966 |
1.9 | 0.3909 | 0.3922 | 0.3889 | 0.3818 | 0.3852 | 0.3836 | 0.3899 | 0.003966 |
1.99 | 0.3909 | 0.3922 | 0.3889 | 0.3818 | 0.3852 | 0.3836 | 0.3899 | 0.003966 |
std | 0.035286 | 0.037074 | 0.035114 | 0.033596 | 0.035327 | 0.033676 | 0.036484 | |
Figure 3. Time series of x with different values of S (a)S=0.01 (b)S=1 (c)S=1.99 |
Figure 4. Time series of the ratios of cooperators when S<0, S=0 and S=0.01 |
Figure 5. 3D plot of x as function of t and S, 2.3≤S≤6, x(0)=0.1 |
Figure 6. 3D plot of x as function of t and S, 2.3≤S≤6, x(0)=0.5 |
Figure 7. 3D plot of x as function of t and S, 2.3≤S≤6, x(0)=0.9 |
Figure 8. Comparison of time series patterns of x with different S values, x(0)=0.1 |
(1) |
While in the case of greedy agents all trajectories are deterministic, in the case of Pavlovian agents they are random, since the behavior of the agents is determined by the probability values (1). Each simulation can produce different trajectory, therefore we made 16 runs with identical P(0)=0.5 for all agents and computed the average values of the x(t) ratios for all t. If x*(t) denotes the "true" expectation and x_{average}(t) the average value computed from NS simulation runs, then from the Chebyshev inequality we know that with any ε>0,
(2) |
where σ^{2}(t) is the variance of the x(t) values. In our case σ^{2}(t) was approximated by the sample variance, which was very small, less than 10^{-6}, since the different simulation runs were very similar. Notice that we had 10,000 agents, therefore, the relative frequencies were calculated from a very large sample size. The values of x(t) are then accurate for at least 2 decimal figures with 99.9% probability.
Figure 9. 3D plot of x as function of t and S, agents are Pavlovian |
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