Hugo Fort and Nicolás Pérez (2005)
The Fate of Spatial Dilemmas with Different Fuzzy Measures of Success
Journal of Artificial Societies and Social Simulation
vol. 8, no. 3
<http://jasss.soc.surrey.ac.uk/8/3/1.html>
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Received: 26-Jul-2004 Accepted: 16-Mar-2005 Published: 30-Jun-2005
T > R > P > S | (1a) |
2R > S + T | (1b) |
i.e. R =1, S = -2, T = 2 and P = -1. Since the four payoffs sum to zero, if all players would update their behavioural variable at random they would get, on average, a null score. The state of players are updated cell by cell so we have an asynchronous cellular automata (ACA)^{[4]}. The initial state at t=0 is taken as C and D chosen at random for each cell i.e. the fraction of cooperators c is equal to 0.5 (we checked that the equilibrium states are independent from the initial configurations. The number of agents in our simulations varied from N=2500 (a 50 × 50 grid) to N = 1,000,000 (a 1000 × 1000 grid). The typical number of lattice sweeps is N_{s}=400 (i.e. from 1,000,000 to 400,000,000 time steps). The results we present here do not change appreciably with the size of the grid.
μ_{IU}(T) = 1
μ_{IU}(R) = 2/3
μ_{IU}(P) = 1/3
μ_{IU}(S) = 0.
μ(X) = 1 if (W_{max} + <w>)/2 < X
μ(X) = 2/3 if <w> < X d (W_{max} + <w>)/2
μ(X) = 1/3 if (W_{min} + <w>)/2 < X ≤ <w>
μ(X) = 0 if X ≤ (W_{min} + <w>)/2
where X ≡ W(x,y) (X ≡ W_{N} ) for measures of success IC_{L} and IC_{G} (for measure of success NW) and <w> ≡ W^{av}_{N } (<w> ≡ W^{av}) for measure of success IC_{L} (IC_{G} and NW).
J_{CD} = 1/3 c^{2} + c(1-c). | (2) |
J_{DC} = 2/3 (1-c)^{2}. | (3) |
4c_{eq}^{2}+ -7c_{eq}+2 = 0. (MS IU ) | (4) |
Figure 1. The fraction of C agents for different measure of success MSs. Thin lines, from below to above: IC_{L}, IU, IC_{G} and NW. Thick upper line corresponds to the discriminating or two-level strategy. The lowest c_{eq} corresponds to IC_{L}. |
U_{eq} _{ }E U(c_{eq}) = (R-S-T+P) c_{eq}^{2} +(S+T-2P) c_{eq} +P = 2 c_{eq} -1. | (5) |
Hence, U_{eq} is greater (smaller) than zero if c_{eq} is greater (smaller) than 1/2. We found that for MS IU and IC_{L} we get respectively U_{eq} ≅ -0.28 and U_{eq} ≅ -0.4 in complete agreement with equation (4). On the other hand, for measures IC_{G} and NW, U_{eq} ≅ 0 in consonance with the fact that c_{eq} ≅ 0.5 for both. Let us denote by ¢ the normalised temporal average of the number of times a player played C. In Figure 2 histograms of ¢and W are depicted for the von Neumann neighbourhood. For measures IC_{L} and IC_{G} both are multi-peaked. The ¢ histograms exhibit two peaks: One centred around c=0.5 and the other at c=0 (indicated by an arrow), which is large for IC_{L} (Figure 2c) and very small in the case of IC_{G} (Figure 2e ), which corresponds to D-agents. The peaks of the respective W histograms can be explained in terms of different local spatial patterns for c. For instance, the right peak at W=400 (large in Figure 2d and small indicated by an arrow in Figure 2f) corresponds to a D player surrounded by four players with c=0.5. This configuration gives to the central player average utilities U=(T+P)/2=1/2 per game. On average each agent plays two times per lattice sweep (one for sure plus another one with each of his z neighbours with probability 1/z). Therefore the average capital accumulated during N_{s} lattice sweeps is given by U × 2 × N_{s} = N_{s} = 400, in accordance with Figure 2. In an analogous way all the peaks can be explained^{[6]}.
Figure 2. Histograms for temporal average ¢ (right column) and average capital (left column). (a) & (b) IU , (c) & (d) IC_{L}, (e) & (f) IC_{G} and (g) & (h) NW. The ¢ and W histograms for measures IC_{L} and IC_{G} are multi-peaked. The cooperation histograms exhibit two peaks, one of them at ¢=0 -large (very small) for IC_{L} (IC_{G}) (see arrows in Figure 2c and 2e)-, corresponding to D-agents, and the other centred around ¢=0.5. The peaks of the respective capital histograms can be explained in terms of different local spatial patterns for c (see text). |
Figure 3. Asymptotic capital maps for different MSs (50 × 50 subsets of 500 × 500 lattices): (a) IU, (b) IC_{L}, (c) IC_{G} and (d) NW. IC measures, produce spatial patterns that are in consonance with their multi-modal histograms, whilst measures IU and NW produce random spatial structure. |
Table 1: Summary of equilibrium properties for the 4 elementary measures of success (von Neumann's neighbourhood) | ||||||
MS | c_{eq} | U_{eq} | W Histogram | C Histogram | W fluctuations | W Spatial patterns |
IU | ≅0.36 | ≅-0.28 | Gaussian like | Gaussian like | ξ = 0.5 | Random |
IC_{L} | ≅0.3 | ≅-0.4 | 6 peaks histogram | 2 peaks histogram | ξ ≅ 2 | "Chess board" patches |
IC_{G} | 0.5- | 0^{-} | 3 peaks histogram | 2 peaks histogram | ξ ≅ 1 | "Flowers" |
NW | 0.5+ | 0^{+} | Gaussian like | Gaussian like | ξ = 0.5 | Random |
μ_{CU}(2R)=1
μ_{CU}(0)=1/2
μ_{CU}(2P)=0.
This strategy gives rise to a more cooperative (and more efficient) society with a c_{eq} higher than all the straightforward applications of the 4 elementary MSs (see Figure 1). It turns out that the interesting "segregation" patterns depicted in Figure 4 emerge: "islands" of rich C-agents (red) and of poor D-agents (yellow) in a sea of medium class (blue).
Figure 4. Asymptotic capital map for the two level strategy (50 × 50 subset of 500 × 500 lattices). (a): von Neumann's neighbourhood; (b): Moore's neighbourhood. A segregation pattern with "islands" of rich C-agents (red) and "islands" of poor D-agents (yellow) in a sea of medium class (blue) is clear. |
^{2}In what follows we will call these steady states by the usual terminology of "equilibrium" states, however one should bear in mind that indeed are dynamical equilibriums (i.e on average the fraction of C's agents is fixed but, there are C agents that transform into D and vice versa).
^{3}We checked that for sufficiently large lattices the results are independent of the boundary conditions.
^{4}We also considered the synchronous dynamics in which all the agents update their states simultaneously at the end of each lattice sweep (that correspond to ordinary CA) to check the robustness of results.
^{5}Indeed, the determination of the global minimum and maximum of W is not a trivial matter for players and the informational assumptions involved in using W_{max} and W_{min} are rather implausible. However, we checked that changing W_{max} and W_{min} whith rough estimates doesn't change results qualitatively.
^{6}This identification of peaks of the histogram of capital with local patterns of c works also for z=2 and z=8 neighbours.
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