Migratory Games

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The videos below are for 49x49 grids with periodic boundary conditions and a random update, assuming 40% empty sites, i.e. a density of 0.6.  49x49 grids were chosen for better visibility, while the evaluations in Fig. 2 of the manuscript were done for a 99x99 grid for comparability with Nowak's and May's spatial games [Nature 259, 826 (1992)].  The color code was also chosen as in their paper, i.e. red = defector, blue = cooperator, green = defector who became a cooperator, yellow = cooperator who turned into a defector in the last iteration. White corresponds to an empty site.

The following snapshots complement Figs. 1 and 2, and compare the migration-only case for M = 5 (left) with the imitation only case for M = 0 (center) and the combined imitation-and-migration case for M = 5 (right). 

Clicking on the pictures activates the link to videos animating the representative spatio-temporal dynamics.

Note that the imitation-only case shows a different dynamics in case of a parallel update.

(A) Classical prisoner's dilemma
with P11 = R = 1, P12 = S = -0.2, P21 = T = 1.4, P22 = P = 0.

(A) Classical prisoner's dilemma with P11 = R = 1, P12 = S = -0.2, P21 = T = 1.4, P22 = P = 0. A Row - Migration OnlyA Row - Imitation OnlyA Row - Migration and Imitation A Row - Migration and Imitation

(B) Prisoner's dilemma according to Nowak and May with

P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0.

 (B) Prisoner's dilemma according to Nowak and May with P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0. B Row - Imitation OnlyB Row - Migration OnlyB Row - Migration and ImitationB Row - Migration and Imitation

(C) P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0, as in B,
but the update rule is  reversed, i.e. an individual first imitates, then migrates.

C - Row C Row - Migration OnlyC Row - Imitation OnlyC Row - Migration and ImitationC Row - Migration and Imitation

(D) P11 = P22 = 1, P12 = P21 = -1.

(D) P11 = P22 = 1, P12 = P21 = -1. D Row - Migration OnlyD Row - Imitation OnlyD Row - Migration and ImitationD Row - Migration and Imitation

(E) P11 = P22 = 0, P12 = P21 = 1.

(E)  P11 = P22 = 0, P12 = P21 = 1. E Row - Migration Only E Row - Imitation OnlyE Row - Migration and Imitation

(F) P11=P22=P12=1, P21=-1.

(F)  P11=P22=P12=1, P21=-1. F Row - Migration OnlyF Row - Imitation OnlyF Row - Migration and Imitation

(G) P11=0.59, P12 = 0.18, P21 = 1, P 22= 0, corresponding to a snowdrift game with "cost-to-benefit ratio" r = 0.7.

(G)  P11=0.59, P12 = 0.18, P21 = 1, P 22= 0, corresponding to a snowdrift game with "cost-to-benefit ratio" r = 0.7. G Row - Migration and Imitation G Row - Migration Only G Row - Imitation Only G Row - Migration and Imitation

The following videos (which are activated by clicking on the respective snapshots) show the dynamics of game B with mobilti M = 2 and density = 0.6, if all payoff values are shifted by the same amount of C = 0 (no shift), C = -0.5, C = -1, and C = -1.5, respectively. This shift changes the dynamics and outcome of the game considerably, in contrast to conventional games with no spatial interactions.

Rows H, I, J, K (H) Payoff Matrix R=1, T=1.4, S=P=0.(I)  Payoff Matrix R=0.5, T=0.9, S=P=-0.5.(J)  Payoff Matrix R=0, T=0.4, S=P=-1.(K)  Payoff Matrix R=-0.5, T=-0.1, S=P=-1.5.

The following videos are simulations results for success-driven motion based on wealth-based "neighborhood tagging".

(L) P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0, cf. B.

(L)  P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0, cf. B.

(M) P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0, as in L, but update rule is  reversed, i.e. an individual first imitates, then migrates.

(M) P11 = R = 1, P12 = S = 0, P21 = T = 1.4, P22 = P = 0, as in L,  but update rule is  reversed, i.e. an individual first imitates, then migrates. M Row - Migration OnlyM Row - Imitation OnlyM Row - Migration and Imitation

We have implemented 3 kinds of noises to test the robustness of the amplification of cooperation. In each time step, 10% of players were selected to perform the following operations after the respective migration and imitation steps.

Noise 1: The location was exchanged with a randomly chosen neighboring site.

Noise 2: Cooperation was replaced by defection and vice versa to reflect "mutation".

Noise 3: The selected players were removed from the grid and an equal number of players was created at randomly chosen free sites, in order to mimic birth and death process. The newly born palyers has a 50% chance to be cooperators and a 50% chance to be defectors.

(N) No Noise T = 1.3, R = 1, p = 0, S = 0

(N) No Noise T = 1.3, R = 1, p = 0, S = 0 N Row - Migration Only N Row - Imitation Only N Row - Migration and Imitation

(O) Noise 1 T = 1.3, R = 1, p = 0, S = 0

(O) Noise 1 T = 1.3, R = 1, p = 0, S = 0 O Row - Migration OnlyO Row - Imitation OnlyO Row - Migration and Imitation

(P) Noise 2 T = 1.3, R = 1, p = 0, S = 0

(P) Noise 2 T = 1.3, R = 1, p = 0, S = 0 P Row - Migration OnlyP Row - Imitation OnlyP Row - Migration and Imitation

(Q) Noise 3 T = 1.3, R = 1, p = 0, S =0

(Q) Noise 3 T = 1.3, R = 1, p = 0, S =0 Q Row - Migration OnlyQ Row - Imitation OnlyQ Row - Migration and Imitation

The following figures show the effect of noise on the number of cooperators.

(R) 2% noise, (S) 10% noise

The following videos are simulation results fot the conventional Prisoner's Dilemma with P > S.

(T) No noise T = 1.3, R = 1, P = 0.1, S = 0

(T) No noise T = 1.3, R = 1, P = 0.1, S = 0 T Row - Migration OnlyT Row - Imitation OnlyT Row - Migration and Imitation

(U) Noise 1 T = 1.3, R = 1, P = 0.1, S = 0

(U) Noise 1 T = 1.3, R = 1, P = 0.1, S = 0 U Row - Migration Only U Row - Imitation OnlyU Row - Migration and Imitation

(V) Noise 2 T = 1.3, R = 1, P = 0.1, S = 0

(V) Noise 2 T = 1.3, R = 1, P = 0.1, S = 0 V Row - Migration OnlyV Row - Imitation OnlyV Row - Migration and Imitation

(W) Noise T = 1.3, R = 1, P = 0.1, S = 0

(W) Noise T = 1.3, R = 1, P = 0.1, S = 0 W Row - Migration OnlyW Row - Imitation OnlyW Row - Migration and Imitation

Impact of “globalization” on cooperation in the migratory prisoner’s dilemma with imitation and noise 3 (with q = 0.05).

(X) Gloabl interaction with local migration., (Y) Local interaction with global migration., (Z) Global interaction with global migration. X RowY RowZ Row

Analytical calculations for a simplified model of strategic interaction in space with success-driven migration and diffusion (read "Pattern formation, social forces, and diffusion instability in games with success-driven motion").

 
 
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Sun Jul 23 19:35:11 CEST 2017
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