This page is to catalogue some of Artem Kaznatcheev's preliminary results for probabilistic strategies in Hammond and Axelrod [1,2] style simulations. The code was written and data collected by AK in May of 2008 when he was working in Thomas R. Shultz's lab. The core of the simulations is the same as the original H&A simulations. They key difference is that instead of having in-group (igs) or out-group (ogs) strategies that are either 0 (defect) or 1 (cooperate), we now allow probabilistic values p in the range of 0 to 1 where an agent cooperated with probability p. Also, we look at both Prisoner's dilemma (PD) and the Hawk-Dove (HD) game.

[1] Hammond, R.A. & Axelrod, R. (2006) "Evolution of contingent altruism when cooperation is expensive," Theoretical Population Biology 69:3, 333-338
[2] Hammond, R.A. & Axelrod, R. (2006) "The evolution of ethnocentrism," Journal of Conflict Resolution 50, 926-936.

Parameter information

The parameters that were constant across simulations, are listed in the table below:

default ptr0.12
death rate0.10
mutation rate0.005
immigration rate1 per epoch
number of tags4

For the game matrix, we considered the standard R,S,T,P parametrization. Where R is the reward for mutual cooperation, S is the suckers payoff for cooperating with a defector, T is the temptation to defect for defection against a cooperator, and P is the punishment for mutual defection. For the HD game we had (R,S,T,P) = (0.02, 0.00, 0.03, x) and for PD we had (0.02, x, 0.03, 0.00) where x is the value listed in the parameters section in the table of results.

A further stressor was added to some of the PD simulations, and is listed in the comments section. Here we varied the number of tags as the simulation went on. A comment of the form 'XT(+/-)YT(at/per)Zep' means that we start the simulation with X tags, and increase (+) or decrease (-) the number of tags available to new immigrants by Y tags at (at) the Z-th epoch or at every multiple (per) of Z epochs. This was to study how sensitive our results were to stress induced by new tags.


The results for each case are averages from 10 worlds. The videos show 2 bar graphs the left one shows the number of agents with an igs strategy in a given range, and the right does the same for ogs strategy. Note that the y-axis varies between videos, so be careful! The red error bars are standard error from averaging the 10 worlds. The horizontal green line corresponds to 1/10th of the total world population, and the dotted green lines are the error bars for the green line. This line is present so that it is easy to notice world saturation. The videos start at epoch 1 and goes through all the epochs in the simulations (the current epoch number is tracked by the number ontop). Each frame shows the strategy distributions from that epoch.

Game Parameters Comments Link
Hawk-Dove -0.01 video
-0.02 video
-0.04 video
Prisoner's dilemma -0.01 video
-0.01 2T+1Tper100ep video
-0.01 4T+1Tat350ep video
-0.01 4T+4Tat350ep video
-0.01 4T-2Tat350ep video
-0.02 video
-0.04 video