Supporting Online Material for Evolution of Contingent Altruism when Cooperation is Expensive June, 2005

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1 Supporting Online Material for Evolution of Contingent Altruism when Cooperation is Expensive June, 2005 This file contains supporting online material to accompany the paper Evolution of Contingent Altruism when Cooperation is Expensive by Ross A. Hammond and Robert Axelrod. The first section below contains full data for all of the simulations discussed in the paper. The second section contains the details of the analytic model discussed in the paper, and discusses convergence between the analytic and simulation models. Complete code for the simulation models (in JAVA) is available from the authors on request. In addition to the information presented here, a movie showing the dynamics of a typical simulation run of the Viscosity and Tag model is available on the web at: Questions regarding any of the supporting material can be addressed to rahammon@umich.edu or axe@umich.edu. I. Data used in the paper The data is presented below by experiment. Each experiment consists of 10 separate simulation runs (each with a unique random seed) with the same model specification and parameters. Data presented in the paper represents averages of the 10 runs for a given experiment. For each experiment, data is organized into columns defined as follows: Data column definitions Column A: run number (out of 10 total for the experiment) Column B: the random seed used for the run Column C: the length of the run (periods) Column D: % of interactions that occur between like-type agents, averaged over the last 100 iterations of each run Column E: % of interactions that are cooperative (e.g. donations), averaged over the last 100 iterations of each run Column F: average number of pure altruists in the population during the last 100 iterations Column G: average number of contingent altruists in the population during the last 100 iterations Column H: average number of agents with bias toward non-similar others in the population during the last 100 iterations

2 Column I: average number of egoists in the population during the last 100 iterations Column J: average total population (sum of F-I) during the last 100 iterations Column K: total % altruists [(F+G)/J] Description of data sets in order of appearance 001 Null Model 002 Viscosity Model 003 Viscosity Model, high cost (c/b=2/3) 004 Viscosity Model, with low viscosity (50% of interactions at random) 005 Tags Model 006 Viscosity and Tag Model 007 Viscosity and Tag Model, high cost (c/b=2/3) 008 Viscosity and Tag Model, with low viscosity (50% of interactions at random) 009 Viscosity and Tag Model, weakened indicator of relatedness (2 color types only) 010 Viscosity and Tag Model, misperception (10% noise in observation of other s tags) 011 Viscosity and Tag Model, more colors (8 color types) 012 Viscosity and Tag Model, low mutation rate (0.25% per trait) 013 Viscosity and Tag Model, high mutation rate (1% per trait) 014 Viscosity and Tag Model, low immigration rate (average of one immigrant every 2 rounds) 015 Viscosity and Tag Model, high immigration rate (2 immigrants / day) 016 Viscosity and Tag Model, lattice size 25x Viscosity and Tag Model, lattice size 100x Viscosity and Tag Model, run length 1000 periods 019 Viscosity and Tag Model, run length 4000 periods 020 Viscosity and Tag Model, 5-trait variant (see data set for description) 021 Viscosity and Tag Model, all-egoist start, no immigration 022 Viscosity and Tag Model, low cost (c/b=1/6) 023 Viscosity and Tag Model, low mortality rate (5% per period) 024 Viscosity and Tag Model, higher mortality rate (20% per period) 025 Data for Figure 1 (comparative sensitivity of Viscosity and Viscosity & Tag models to c/b ratio) - 2 -

3 001 - Null model (no tags or viscosity), standard parameters (ref experiment 01038) % 5.7% % % 3.6% % % 6.5% % % 4.2% % % 3.6% % % 5.3% % % 4.7% % % 4.7% % % 5.2% % % 3.2% % AVG 100.0% 4.7% % STDEV 0.0% 1.1% % SE 0.0% 0.3% %

4 002 - Viscosity model, standard parameters (ref experiment 0117) % 73.7% % % 75.0% % % 82.0% % % 75.6% % % 78.0% % % 75.2% % % 67.9% % % 76.4% % % 74.1% % % 75.3% % AVG 100.0% 75.3% % STDEV 0.0% 3.5% % SE 0.0% 1.1% %

5 003 - Viscosity model, high cost (ref experiment 0119) Run # RndmSeed RunLength % MeetOwnTotal %C pure altr conting altr bias fav oth egoist Total Total % altr % 11.4% % % 19.2% % % 11.8% % % 11.1% % % 20.1% % % 15.8% % % 16.1% % % 13.7% % % 7.1% % % 13.9% % AVG 100.0% 14.0% % STDEV 0.0% 3.9% % SE 0.0% 1.2% %

6 004 - Viscosity model with lower viscosity (50% of interactions are at random) (ref experiment 01049) % 51.7% % % 52.0% % % 56.4% % % 40.5% % % 46.0% % % 30.1% % % 44.4% % % 36.1% % % 44.9% % % 42.3% % AVG 100.0% 44.4% % STDEV 0.0% 7.8% % SE 0.0% 2.5% %

7 005 - Tags Model, standard parameters (ref experiment 0927) Run # RndmSeed RunLength MeetOwn Total %C # pure altr # conting altr # bias other # egoists Total # Total % altr % 17.1% % % 13.8% % % 14.8% % % 11.2% % % 12.2% % % 11.9% % % 15.1% % % 15.7% % % 16.7% % % 16.6% % AVG 28.6% 14.5% % STDEV 2.4% 2.1% % SE 0.8% 0.7% %

8 006 - Viscosity+Tags model, standard parameters (ref experiment 0104) % 74.8% % % 74.3% % % 74.7% % % 72.6% % % 73.2% % % 77.2% % % 75.6% % % 73.5% % % 73.7% % % 71.9% % AVG 78.7% 74.2% % STDEV 1.9% 1.5% % SE 0.6% 0.5% %

9 007 - Viscosity+Tags model, high cost (ref experiment 0109) Run # RndmSeed RunLength % MeetOwnTotal %C pure altr conting altr bias fav oth egoist Total Total % altr % 59.8% % % 53.3% % % 58.1% % % 53.8% % % 56.5% % % 53.2% % % 58.2% % % 63.7% % % 52.4% % % 52.1% % AVG 77.2% 56.1% % STDEV 1.6% 3.8% % SE 0.5% 1.2% %

10 008 - Viscosity+Tags model, low viscosity (ref experiment 0948) % 61.1% % % 60.3% % % 54.7% % % 52.7% % % 58.5% % % 58.1% % % 58.3% % % 56.9% % % 60.6% % % 53.8% % AVG 64.8% 57.5% % STDEV 1.1% 2.9% % SE 0.3% 0.9% %

11 009 - Viscosity+Tags model, weakend indicator of relatedness (ref experiment 0112) % 77.2% % % 78.8% % % 79.0% % % 77.1% % % 74.8% % % 79.4% % % 78.3% % % 77.4% % % 78.7% % % 80.6% % AVG 85.2% 78.1% % STDEV 1.1% 1.6% % SE 0.4% 0.5% %

12 010 - Viscosity+Tags model, more misperception (10% noise in tag observation) (ref experiment 01052) % 69.8% % % 67.5% % % 68.1% % % 70.7% % % 70.3% % % 68.9% % % 69.9% % % 67.2% % % 71.3% % % 67.5% % AVG 77.8% 69.1% % STDEV 2.1% 1.5% % SE 0.7% 0.5% %

13 011 - Viscosity+Tags model, more colors (ref experiment 0116) % 69.1% % % 72.7% % % 69.4% % % 70.8% % % 73.7% % % 72.7% % % 70.3% % % 73.2% % % 71.2% % % 73.4% % AVG 75.2% 71.7% % STDEV 2.1% 1.7% % SE 0.7% 0.5% %

14 012 - Viscosity+Tags model, low mutation rate (ref experiment 0110) % 77.3% % % 82.2% % % 80.8% % % 79.2% % % 80.6% % % 81.8% % % 80.3% % % 78.6% % % 78.2% % % 79.2% % AVG 83.2% 79.8% % STDEV 2.3% 1.6% % SE 0.7% 0.5% %

15 013 - Viscosity+Tags model, high mutation (ref experiment 0111) % 69.1% % % 69.0% % % 69.1% % % 70.8% % % 67.4% % % 71.5% % % 69.1% % % 69.4% % % 65.1% % % 69.2% % AVG 74.3% 69.0% % STDEV 1.2% 1.8% % SE 0.4% 0.6% %

16 014 - Viscosity+Tags model, low immigration (ref experiment 0114) % 78.8% % % 71.1% % % 77.7% % % 77.2% % % 77.4% % % 74.3% % % 74.2% % % 75.8% % % 74.4% % % 74.3% % AVG 79.9% 75.5% % STDEV 1.0% 2.3% % SE 0.3% 0.7% %

17 015 - Viscosity+Tags model, high immigration (ref experiment 0113) % 72.2% % % 68.6% % % 76.7% % % 72.1% % % 71.0% % % 74.6% % % 71.3% % % 67.8% % % 71.7% % % 67.7% % AVG 76.3% 71.4% % STDEV 1.1% 2.9% % SE 0.4% 0.9% %

18 016 - Viscosity+Tags model, lattice size 25x25 (ref experiment 0108) % 71.3% % % 75.5% % % 63.7% % % 72.8% % % 67.7% % % 65.4% % % 68.4% % % 73.4% % % 70.2% % % 70.5% % AVG 73.1% 69.9% % STDEV 3.6% 3.7% % SE 1.2% 1.2% %

19 017 - Viscosity+Tags model, lattice size 100x100 (ref experiment 0105) % 75.4% % % 75.2% % % 75.8% % % 75.7% % % 75.3% % % 77.7% % % 75.5% % % 76.6% % % 77.3% % % 75.7% % AVG 80.4% 76.0% % STDEV 1.2% 0.9% % SE 0.4% 0.3% %

20 018 - Viscosity+Tags model, run length 1000 (ref experiment 0107) % 69.6% % % 75.7% % % 72.1% % % 75.9% % % 74.0% % % 68.1% % % 79.4% % % 72.0% % % 73.8% % % 73.8% % AVG 78.3% 73.4% % STDEV 2.0% 3.2% % SE 0.6% 1.0% %

21 019 - Viscosity+Tags model, run length 4000 (ref experiment 0106) % 73.4% % % 75.9% % % 74.2% % % 74.5% % % 76.0% % % 76.5% % % 73.3% % % 75.2% % % 74.1% % % 71.0% % AVG 77.9% 74.4% % STDEV 1.2% 1.6% % SE 0.4% 0.5% %

22 020-5-trait variant In this variant of the model, agents can distinguish all 4 colors individually (e.g. they now have 5 effective traits a tag, and one strategy for each color). As in the original model, an agent can distinguish its own color, but in this variant it can also distinguish among the others. When an agent s color changes by mutation, its new color becomes its own color, and its strategy for the old color is given a random value (donate or not). This is the appropriate way to deal with mutation because technically our model is an armpit model, rather than a greenbeard model. (See Richard Dawkins, The Extended Phenotype. Freeman, San Francisco. pp ) Note that the central model results are not vulnerable to such a change. In the 5-trait variant, 89% of agents cooperate with their own color, and 71% are contingent altruists, where contingent now means that the agent cooperates with its own color and defects with at least two of the other three colors. RUN # Fully Contingent Contingent 2/3 Contingent 1/3 Pure Altruists TOTPOP AVERAGE AVERAGE % of total population 30.5% 40.0% 14.8% 3.3% Strategy distribution in 5-trait variant

23 021 - Viscosity+Tags model, all-egoist initial condition and no immigration (ref experiment 01004) % 78.6% % % 81.2% % % 73.7% % % 78.0% % % 79.2% % % 76.2% % % 78.6% % % 75.8% % % 74.5% % % 74.8% % AVG 81.7% 77.1% % STDEV 1.9% 2.4% % SE 0.6% 0.8% %

24 022 - Viscosity+Tags model, low cost (ref experiment 0115) Run # RndmSeed RunLength % MeetOwnTotal %C pure altr conting altr bias fav oth egoist Total Total % altr % 78.0% % % 77.2% % % 77.2% % % 74.7% % % 79.3% % % 74.8% % % 80.0% % % 77.3% % % 82.1% % % 77.8% % AVG 79.0% 77.8% % STDEV 2.1% 2.2% % SE 0.7% 0.7% %

25 023 - Viscosity+Tags model, low death rate (ref experiment 0120) % 65.1% % % 67.3% % % 72.9% % % 69.8% % % 67.9% % % 67.4% % % 66.8% % % 74.5% % % 68.2% % % 66.8% % AVG 74.9% 68.7% % STDEV 2.2% 2.9% % SE 0.7% 0.9% %

26 024 - Viscosity+Tags model, higher dth rate (ref experiment 0121) % 46.5% % % 45.4% % % 47.0% % % 64.5% % % 65.5% % % 47.3% % % 67.3% % % 56.5% % % 89.3% % % 42.1% % AVG 97.7% 57.1% % STDEV 2.2% 14.7% % SE 0.7% 4.6% %

27 025 - Source data for Figure 1 (Effect of Cost-Benefit Ratio on Viscosity and Viscosity+Tag Models) Viscosity Model cost of altruism (c=) total % altruist strategies (%) AVG STDEV SE 0.0% 92.4% 0.5% 0.2% 0.3% 89.6% 1.4% 0.4% 0.6% 85.5% 2.4% 0.7% 0.9% 78.7% 4.3% 1.4% 1.2% 65.5% 4.8% 1.5% 1.5% 43.4% 6.5% 2.0% 1.8% 22.1% 4.1% 1.3% 2.1% 11.5% 3.4% 1.1% 2.4% 7.0% 1.9% 0.6% 2.7% 4.8% 0.7% 0.2% 3.0% 4.5% 1.1% 0.3% 10-run averages, last 100 iterations of each run Viscosity and Tags Model cost of altruism (c=) total % altruist strategies (%) AVG STDEV SE 0.0% 94.3% 1.0% 0.3% 0.3% 93.1% 1.1% 0.3% 0.6% 92.2% 1.5% 0.5% 0.9% 89.9% 1.7% 0.5% 1.2% 87.7% 1.4% 0.4% 1.5% 83.0% 4.5% 1.4% 1.8% 76.6% 4.0% 1.3% 2.1% 61.4% 5.4% 1.7% 2.4% 47.2% 9.1% 2.9% 2.7% 27.0% 4.7% 1.5% 3.0% 13.2% 3.4% 1.1% 10-run averages, last 100 iterations of each run

28 II. Analytic version of the Null Model The equations and equilibrium derivation for the analytic continuous version of the Null model referred to in the paper are shown below. Convergence with the simulations of the Null model is also discussed. Parameters, with standard values c = cost of donating ( 0.01) b = benefit from getting a donation (0.03) g = natural growth rate, the initial Propensity to Reproduce (0.12) i = immigration rate (1/2500) m = mutation rate (0.05) d = death or mortality rate (0.10) Variable p = proportion of population that is altruist, rest are egoists, while 1-p is the proportion that is egoists. The Null Model: Each agent is paired with another agent at random to play a one move Prisoner s Dilemma in which each player has an opportunity to give a costly benefit to the other. On average, an agent will play two such games each period, once due its own pairing, and once by another agent s pairing. See text for details.

29 The analytic approximation of the Null model assumes that time is continuous, and that population size is effectively unlimited. The later assumption is justified by the fact that the mortality rate is equal and constant for altruists and egoists. The expected number of new offspring per altruist is just its Potential to Reproduce, namely R A = g +2( (b-c)p - c(1-p) ). The expected number of offspring per egoist is R E = g + 2 b p. The expected number of new altruists (N A ) is composed of new immigration (a 1/2 chance the immigrant is an altruist), unmutated offspring of altruists, and mutated offspring of egoists. This is: i N A = + RA p(1 m) + RE (1 p) m 2 i N E = + RE (1 p)(1 m) + R 2 A pm At equilibrium, the altruists and egoists grow at the same rate. This requires that the difference between the number of altruists and egoists remains the same proportion of the population from one period to the next, i.e. (N A - N E ) / (N A + N E ) = p - (1-p). Solving the resulting quadratic equation for p, and using the standard values of the parameters shown above gives p = and Since the first value is not feasible, the prediction of the continuous Null Model is that at equilibrium Altruists are 3.9% of population. In other words, Altruists are reduced to a very small proportion of the population, a proportion which is maintained by immigration and mutation. The agent-based simulation of the Null Model finds p = 6.1% after 5000 periods, 5.1% after 10,000 periods, and p = 4.6% ± 0.3% after 15,000 periods. This suggests an asymptotic value of p of 4.1 % ± 0.3% for the simulation. This is consistent with the value of p = 3.9% from the results of the continuous model.