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: http://umich.edu/~axe/vtmovie.htm. 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
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 25x25 017 Viscosity and Tag Model, lattice size 100x100 018 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 -
001 - Null model (no tags or viscosity), standard parameters (ref experiment 01038) 1 1110253921534 15000 100.0% 5.7% 76 54 920 1245 2295 5.7% 2 1110268757407 15000 100.0% 3.6% 32 50 727 1481 2290 3.6% 3 1110271018999 15000 100.0% 6.5% 65 84 764 1380 2293 6.5% 4 1110273266831 15000 100.0% 4.2% 40 57 658 1541 2296 4.2% 5 1110275512951 15000 100.0% 3.6% 25 57 595 1615 2292 3.6% 6 1110277761294 15000 100.0% 5.3% 45 76 719 1452 2292 5.3% 7 1110279992802 15000 100.0% 4.7% 73 35 732 1454 2294 4.7% 8 1110282237871 15000 100.0% 4.7% 56 52 605 1582 2295 4.7% 9 1110284506823 15000 100.0% 5.2% 24 93 628 1549 2294 5.1% 10 1110286753163 15000 100.0% 3.2% 14 58 509 1714 2295 3.1% AVG 100.0% 4.7% 45 62 686 1501 2294 4.6% STDEV 0.0% 1.1% 22 17 114 131 2 1.0% SE 0.0% 0.3% 7 6 36 41 1 0.3%
002 - Viscosity model, standard parameters (ref experiment 0117) 1 1016494344570 2000 100.0% 73.7% 690 591 298 183 1762 72.7% 2 1016494451264 2000 100.0% 75.0% 774 523 222 225 1744 74.4% 3 1016494560461 2000 100.0% 82.0% 472 983 150 181 1786 81.5% 4 1016494665642 2000 100.0% 75.6% 799 526 212 230 1767 75.0% 5 1016494769431 2000 100.0% 78.0% 690 655 208 195 1748 76.9% 6 1016494891537 2000 100.0% 75.2% 649 661 234 223 1767 74.1% 7 1016495019361 2000 100.0% 67.9% 572 531 350 213 1666 66.2% 8 1016495148797 2000 100.0% 76.4% 752 578 144 286 1760 75.6% 9 1016495254318 2000 100.0% 74.1% 516 755 253 211 1735 73.3% 10 1016495361903 2000 100.0% 75.3% 768 533 240 204 1745 74.6% AVG 100.0% 75.3% 668 634 231 215 1748 74.4% STDEV 0.0% 3.5% 114 144 62 30 32 3.8% SE 0.0% 1.1% 36 46 20 10 10 1.2%
003 - Viscosity model, high cost (ref experiment 0119) Run # RndmSeed RunLength % MeetOwnTotal %C pure altr conting altr bias fav oth egoist Total Total % altr 1 1016644241351 2000 100.0% 11.4% 37 109 473 680 1299 11.2% 2 1016644318993 2000 100.0% 19.2% 123 127 692 415 1357 18.4% 3 1016644413228 2000 100.0% 11.8% 71 77 467 657 1272 11.6% 4 1016644500694 2000 100.0% 11.1% 84 55 431 679 1249 11.1% 5 1016644578296 2000 100.0% 20.1% 159 109 420 657 1345 19.9% 6 1016644672671 2000 100.0% 15.8% 136 67 611 495 1309 15.5% 7 1016644745917 2000 100.0% 16.1% 120 77 720 352 1269 15.5% 8 1016644836247 2000 100.0% 13.7% 72 104 417 706 1299 13.5% 9 1016644919967 2000 100.0% 7.1% 57 38 487 732 1314 7.2% 10 1016645012951 2000 100.0% 13.9% 46 123 652 455 1276 13.2% AVG 100.0% 14.0% 91 89 537 583 1299 13.7% STDEV 0.0% 3.9% 41 30 119 138 34 3.8% SE 0.0% 1.2% 13 10 38 44 11 1.2%
004 - Viscosity model with lower viscosity (50% of interactions are at random) (ref experiment 01049) 1 1111076531017 5000 100.0% 51.7% 345 448 381 378 1552 51.1% 2 1111076985450 5000 100.0% 52.0% 390 418 379 384 1571 51.4% 3 1111077422779 5000 100.0% 56.4% 500 385 344 357 1586 55.8% 4 1111077864134 5000 100.0% 40.5% 272 331 451 453 1507 40.0% 5 1111078332687 5000 100.0% 46.0% 312 390 390 445 1537 45.7% 6 1111078776586 5000 100.0% 30.1% 258 163 366 640 1427 29.5% 7 1111079224159 5000 100.0% 44.4% 361 317 382 478 1538 44.1% 8 1111079661999 5000 100.0% 36.1% 219 316 420 555 1510 35.4% 9 1111080099117 5000 100.0% 44.9% 297 360 449 388 1494 44.0% 10 1111080538489 5000 100.0% 42.3% 369 249 447 419 1484 41.6% AVG 100.0% 44.4% 332 338 401 450 1521 43.9% STDEV 0.0% 7.8% 80 84 38 89 47 7.8% SE 0.0% 2.5% 25 27 12 28 15 2.5%
005 - Tags Model, standard parameters (ref experiment 0927) Run # RndmSeed RunLength MeetOwn Total %C # pure altr # conting altr # bias other # egoists Total # Total % altr 1 1106333176034 2000 26.5% 17.1% 75 645 212 1364 2296 31.4% 2 1106333457449 2000 27.4% 13.8% 63 452 180 1600 2295 22.4% 3 1106333733976 2000 28.6% 14.8% 82 536 147 1527 2292 27.0% 4 1106334018506 2000 30.0% 11.2% 44 338 173 1738 2293 16.7% 5 1106334293821 2000 25.1% 12.2% 57 397 166 1674 2294 19.8% 6 1106334581315 2000 27.0% 11.9% 41 243 221 1790 2295 12.4% 7 1106334857632 2000 32.9% 15.1% 84 354 219 1635 2292 19.1% 8 1106335132798 2000 28.0% 15.7% 69 414 239 1570 2292 21.1% 9 1106335408604 2000 31.8% 16.7% 80 585 194 1435 2294 29.0% 10 1106335695407 2000 29.0% 16.6% 58 427 253 1556 2294 21.1% AVG 28.6% 14.5% 65 439 200 1589 2294 22.0% STDEV 2.4% 2.1% 15 121 34 130 1 5.7% SE 0.8% 0.7% 5 38 11 41 0 1.8%
006 - Viscosity+Tags model, standard parameters (ref experiment 0104) 1 1016508794929 2000 79.2% 74.8% 254 1314 42 144 1754 89.4% 2 1016508899159 2000 77.0% 74.3% 324 1251 41 157 1773 88.8% 3 1016509002818 2000 76.8% 74.7% 275 1329 23 124 1751 91.6% 4 1016509128088 2000 78.3% 72.6% 131 1386 45 168 1730 87.7% 5 1016509243664 2000 76.4% 73.2% 201 1401 26 145 1773 90.4% 6 1016509363346 2000 80.9% 77.2% 207 1406 20 138 1771 91.1% 7 1016509480725 2000 81.7% 75.6% 197 1397 44 146 1784 89.3% 8 1016509604062 2000 80.6% 73.5% 193 1316 60 185 1754 86.0% 9 1016509720350 2000 77.1% 73.7% 245 1354 22 147 1768 90.4% 10 1016509843477 2000 78.9% 71.9% 212 1288 40 218 1758 85.3% AVG 78.7% 74.2% 224 1344 36 157 1762 89.0% STDEV 1.9% 1.5% 53 53 13 27 15 2.1% SE 0.6% 0.5% 17 17 4 9 5 0.7%
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 1 1016480916582 2000 77.6% 59.8% 146 902 43 373 1464 71.6% 2 1016481003126 2000 76.5% 53.3% 101 853 59 436 1449 65.8% 3 1016481116199 2000 78.3% 58.1% 115 865 53 412 1445 67.8% 4 1016481226297 2000 78.8% 53.8% 99 856 58 484 1497 63.8% 5 1016481327002 2000 77.3% 56.5% 71 944 72 396 1483 68.4% 6 1016481430210 2000 78.3% 53.2% 86 873 51 488 1498 64.0% 7 1016481531316 2000 74.3% 58.2% 74 1037 56 345 1512 73.5% 8 1016481616248 2000 78.7% 63.7% 63 1136 14 331 1544 77.7% 9 1016481728569 2000 77.3% 52.4% 78 872 36 498 1484 64.0% 10 1016481812260 2000 74.3% 52.1% 106 837 52 485 1480 63.7% AVG 77.2% 56.1% 94 918 49 425 1486 68.0% STDEV 1.6% 3.8% 25 97 16 63 30 4.8% SE 0.5% 1.2% 8 31 5 20 9 1.5%
008 - Viscosity+Tags model, low viscosity (ref experiment 0948) 1 1106927821719 2000 64.6% 61.1% 261 1048 44 228 1581 82.8% 2 1106927988619 2000 65.9% 60.3% 221 1120 29 251 1621 82.7% 3 1106928153626 2000 64.9% 54.7% 117 1056 66 325 1564 75.0% 4 1106928323020 2000 63.6% 52.7% 181 954 61 361 1557 72.9% 5 1106928489839 2000 66.3% 58.5% 214 1040 45 302 1601 78.3% 6 1106928656098 2000 64.2% 58.1% 232 1031 100 263 1626 77.7% 7 1106928825142 2000 63.9% 58.3% 206 1085 54 248 1593 81.0% 8 1106928991981 2000 66.4% 56.9% 143 1085 101 293 1622 75.7% 9 1106929161966 2000 64.9% 60.6% 229 1119 27 237 1612 83.6% 10 1106929330688 2000 63.3% 53.8% 151 1036 66 337 1590 74.7% AVG 64.8% 57.5% 196 1057 59 285 1597 78.4% STDEV 1.1% 2.9% 46 49 26 46 24 3.9% SE 0.3% 0.9% 15 15 8 15 8 1.2%
009 - Viscosity+Tags model, weakend indicator of relatedness (ref experiment 0112) 1 1016487015952 2000 84.2% 77.2% 316 1221 59 178 1774 86.6% 2 1016487119121 2000 86.7% 78.8% 437 1072 76 211 1796 84.0% 3 1016487242288 2000 84.7% 79.0% 328 1260 65 156 1809 87.8% 4 1016487357213 2000 86.0% 77.1% 286 1220 71 186 1763 85.4% 5 1016487475773 2000 83.6% 74.8% 222 1279 69 191 1761 85.2% 6 1016487596126 2000 85.2% 79.4% 317 1254 61 146 1778 88.4% 7 1016487724251 2000 86.5% 78.3% 333 1195 73 186 1787 85.5% 8 1016487851894 2000 84.3% 77.4% 301 1255 56 175 1787 87.1% 9 1016487945309 2000 84.7% 78.7% 229 1352 52 143 1776 89.0% 10 1016488048357 2000 86.6% 80.6% 324 1250 68 123 1765 89.2% AVG 85.2% 78.1% 309 1236 65 170 1780 86.8% STDEV 1.1% 1.6% 60 71 8 27 15 1.8% SE 0.4% 0.5% 19 23 2 8 5 0.6%
010 - Viscosity+Tags model, more misperception (10% noise in tag observation) (ref experiment 01052) 1 1111377624382 2000 74.1% 69.8% 285 1294 35 124 1738 90.9% 2 1111377832221 2000 78.0% 67.5% 189 1317 35 199 1740 86.6% 3 1111378036374 2000 77.9% 68.1% 200 1314 29 192 1735 87.3% 4 1111378236402 2000 77.3% 70.7% 248 1339 38 125 1750 90.7% 5 1111378440065 2000 81.2% 70.3% 262 1230 49 193 1734 86.0% 6 1111378647042 2000 80.3% 68.9% 217 1268 47 192 1724 86.1% 7 1111378845287 2000 75.8% 69.9% 291 1256 63 130 1740 88.9% 8 1111379046767 2000 79.2% 67.2% 142 1316 35 198 1691 86.2% 9 1111379260294 2000 76.2% 71.3% 352 1247 51 131 1781 89.8% 10 1111379468163 2000 78.1% 67.5% 138 1395 21 194 1748 87.7% AVG 77.8% 69.1% 232 1298 40 168 1738 88.0% STDEV 2.1% 1.5% 68 49 12 35 22 1.9% SE 0.7% 0.5% 22 16 4 11 7 0.6%
011 - Viscosity+Tags model, more colors (ref experiment 0116) 1 1016493179535 2000 72.2% 69.1% 161 1372 43 157 1733 88.5% 2 1016493284756 2000 76.1% 72.7% 202 1384 29 140 1755 90.4% 3 1016493384910 2000 72.2% 69.4% 162 1417 25 145 1749 90.3% 4 1016493506004 2000 76.3% 70.8% 154 1345 41 187 1727 86.8% 5 1016493628210 2000 76.4% 73.7% 200 1390 36 137 1763 90.2% 6 1016493755764 2000 75.0% 72.7% 202 1406 37 127 1772 90.7% 7 1016493858351 2000 72.9% 70.3% 135 1495 26 122 1778 91.7% 8 1016493946989 2000 76.7% 73.2% 186 1410 33 141 1770 90.2% 9 1016494078798 2000 75.8% 71.2% 171 1375 25 157 1728 89.5% 10 1016494212220 2000 78.5% 73.4% 262 1289 26 191 1768 87.7% AVG 75.2% 71.7% 184 1388 32 150 1754 89.6% STDEV 2.1% 1.7% 36 53 7 23 19 1.5% SE 0.7% 0.5% 11 17 2 7 6 0.5%
012 - Viscosity+Tags model, low mutation rate (ref experiment 0110) 1 1016483250328 2000 80.9% 77.3% 138 1514 21 117 1790 92.3% 2 1016483360175 2000 86.6% 82.2% 176 1423 44 116 1759 90.9% 3 1016483512274 2000 83.1% 80.8% 98 1593 28 67 1786 94.7% 4 1016483638716 2000 82.6% 79.2% 202 1433 42 105 1782 91.8% 5 1016483794510 2000 84.6% 80.6% 142 1516 20 110 1788 92.7% 6 1016483905409 2000 87.4% 81.8% 186 1419 24 152 1781 90.1% 7 1016484006124 2000 82.5% 80.3% 185 1482 25 83 1775 93.9% 8 1016484113869 2000 81.8% 78.6% 186 1467 35 106 1794 92.1% 9 1016484226581 2000 80.9% 78.2% 114 1563 28 89 1794 93.5% 10 1016484356258 2000 81.3% 79.2% 293 1358 19 115 1785 92.5% AVG 83.2% 79.8% 172 1477 29 106 1783 92.5% STDEV 2.3% 1.6% 55 72 9 23 10 1.4% SE 0.7% 0.5% 17 23 3 7 3 0.4%
013 - Viscosity+Tags model, high mutation (ref experiment 0111) 1 1016484981417 2000 75.2% 69.1% 302 1169 62 223 1756 83.8% 2 1016485083393 2000 73.2% 69.0% 279 1162 48 219 1708 84.4% 3 1016485236584 2000 75.7% 69.1% 281 1178 57 240 1756 83.1% 4 1016485365048 2000 75.9% 70.8% 291 1183 41 214 1729 85.3% 5 1016485491420 2000 74.9% 67.4% 271 1173 61 244 1749 82.6% 6 1016485611973 2000 75.0% 71.5% 344 1136 45 203 1728 85.6% 7 1016485725617 2000 73.1% 69.1% 260 1247 38 198 1743 86.5% 8 1016485849635 2000 73.5% 69.4% 296 1173 65 212 1746 84.1% 9 1016485962107 2000 72.3% 65.1% 303 1053 93 267 1716 79.0% 10 1016486117260 2000 74.2% 69.2% 269 1171 62 229 1731 83.2% AVG 74.3% 69.0% 290 1165 57 225 1736 83.7% STDEV 1.2% 1.8% 24 48 16 21 16 2.1% SE 0.4% 0.6% 8 15 5 7 5 0.7%
014 - Viscosity+Tags model, low immigration (ref experiment 0114) 1 1016489985042 2000 80.8% 78.8% 257 1391 35 105 1788 92.2% 2 1016490088641 2000 79.8% 71.1% 154 1358 24 219 1755 86.2% 3 1016490208743 2000 79.5% 77.7% 245 1388 24 100 1757 92.9% 4 1016490332571 2000 80.8% 77.2% 228 1378 39 113 1758 91.4% 5 1016490441678 2000 78.8% 77.4% 320 1324 28 121 1793 91.7% 6 1016490544085 2000 78.1% 74.3% 225 1362 13 165 1765 89.9% 7 1016490648335 2000 79.7% 74.2% 174 1382 36 161 1753 88.8% 8 1016490775548 2000 81.1% 75.8% 185 1407 14 161 1767 90.1% 9 1016490872588 2000 79.3% 74.4% 181 1367 38 154 1740 89.0% 10 1016490975275 2000 81.1% 74.3% 236 1330 20 201 1787 87.6% AVG 79.9% 75.5% 221 1369 27 150 1766 90.0% STDEV 1.0% 2.3% 49 26 10 40 18 2.1% SE 0.3% 0.7% 15 8 3 13 6 0.7%
015 - Viscosity+Tags model, high immigration (ref experiment 0113) 1 1016488734023 2000 76.1% 72.2% 274 1259 40 200 1773 86.5% 2 1016488839825 2000 76.3% 68.6% 163 1325 70 193 1751 85.0% 3 1016488943945 2000 78.4% 76.7% 342 1252 28 157 1779 89.6% 4 1016489073120 2000 75.3% 72.1% 239 1293 75 162 1769 86.6% 5 1016489177020 2000 75.9% 71.0% 173 1361 68 170 1772 86.6% 6 1016489298354 2000 76.7% 74.6% 248 1336 33 145 1762 89.9% 7 1016489422042 2000 77.5% 71.3% 140 1388 51 184 1763 86.7% 8 1016489551338 2000 74.8% 67.8% 196 1291 65 213 1765 84.2% 9 1016489673063 2000 77.0% 71.7% 210 1290 88 170 1758 85.3% 10 1016489793406 2000 75.1% 67.7% 166 1343 49 206 1764 85.5% AVG 76.3% 71.4% 215 1314 57 180 1766 86.6% STDEV 1.1% 2.9% 62 44 20 23 8 1.8% SE 0.4% 0.9% 19 14 6 7 3 0.6%
016 - Viscosity+Tags model, lattice size 25x25 (ref experiment 0108) 1 1016479148850 2000 71.5% 71.3% 90 292 28 42 452 84.5% 2 1016479186584 2000 76.7% 75.5% 74 328 13 31 446 90.1% 3 1016479231659 2000 71.0% 63.7% 71 259 31 63 424 77.8% 4 1016479274971 2000 69.0% 72.8% 113 297 9 31 450 91.1% 5 1016479314899 2000 68.4% 67.7% 41 355 14 36 446 88.8% 6 1016479348948 2000 75.9% 65.4% 61 271 30 83 445 74.6% 7 1016479387493 2000 75.8% 68.4% 25 339 18 46 428 85.0% 8 1016479426038 2000 76.5% 73.4% 78 301 17 44 440 86.1% 9 1016479467989 2000 77.4% 70.2% 25 332 7 64 428 83.4% 10 1016479510360 2000 69.1% 70.5% 68 325 15 31 439 89.5% AVG 73.1% 69.9% 65 310 18 47 440 85.1% STDEV 3.6% 3.7% 28 31 9 18 10 5.4% SE 1.2% 1.2% 9 10 3 6 3 1.7%
017 - Viscosity+Tags model, lattice size 100x100 (ref experiment 0105) 1 1016469307158 2000 80.9% 75.4% 803 5539 63 663 7068 89.7% 2 1016469697479 2000 79.7% 75.2% 1001 5314 144 613 7072 89.3% 3 1016470062304 2000 79.7% 75.8% 732 5726 92 520 7070 91.3% 4 1016470422262 2000 81.0% 75.7% 812 5517 146 569 7044 89.8% 5 1016470929351 2000 78.2% 75.3% 959 5444 90 536 7029 91.1% 6 1016471284461 2000 81.1% 77.7% 884 5612 87 486 7069 91.9% 7 1016471653412 2000 79.7% 75.5% 998 5356 125 563 7042 90.2% 8 1016472017556 2000 80.3% 76.6% 923 5482 95 550 7050 90.9% 9 1016472521580 2000 82.8% 77.3% 938 5386 137 631 7092 89.2% 10 1016472870442 2000 80.1% 75.7% 588 5862 66 540 7056 91.4% AVG 80.4% 76.0% 864 5524 105 567 7059 90.5% STDEV 1.2% 0.9% 131 171 31 54 18 1.0% SE 0.4% 0.3% 42 54 10 17 6 0.3%
018 - Viscosity+Tags model, run length 1000 (ref experiment 0107) 1 1016478130656 1000 80.2% 69.6% 240 1167 73 263 1743 80.7% 2 1016478175370 1000 78.7% 75.7% 268 1317 47 132 1764 89.9% 3 1016478214657 1000 76.3% 72.1% 245 1293 26 191 1755 87.6% 4 1016478261634 1000 80.3% 75.9% 238 1356 31 144 1769 90.1% 5 1016478305077 1000 77.4% 74.0% 276 1306 35 159 1776 89.1% 6 1016478353346 1000 75.7% 68.1% 168 1289 71 189 1717 84.9% 7 1016478400484 1000 81.5% 79.4% 306 1327 33 106 1772 92.2% 8 1016478446270 1000 75.8% 72.0% 254 1304 38 167 1763 88.4% 9 1016478493197 1000 78.7% 73.8% 273 1248 32 178 1731 87.9% 10 1016478533966 1000 78.3% 73.8% 210 1360 44 147 1761 89.2% AVG 78.3% 73.4% 248 1297 43 168 1755 88.0% STDEV 2.0% 3.2% 38 56 16 43 19 3.2% SE 0.6% 1.0% 12 18 5 13 6 1.0%
019 - Viscosity+Tags model, run length 4000 (ref experiment 0106) 1 1016475647465 4000 76.0% 73.4% 191 1391 34 125 1741 90.9% 2 1016475885237 4000 79.5% 75.9% 235 1345 69 120 1769 89.3% 3 1016476157849 4000 77.2% 74.2% 206 1389 30 136 1761 90.6% 4 1016476394660 4000 77.8% 74.5% 237 1343 48 145 1773 89.1% 5 1016476635896 4000 77.6% 76.0% 314 1272 49 122 1757 90.3% 6 1016476876152 4000 79.2% 76.5% 262 1331 39 133 1765 90.3% 7 1016477122086 4000 77.2% 73.3% 138 1470 34 129 1771 90.8% 8 1016477355231 4000 79.7% 75.2% 254 1313 65 131 1763 88.9% 9 1016477597479 4000 77.8% 74.1% 157 1430 39 138 1764 90.0% 10 1016477834740 4000 77.0% 71.0% 180 1344 48 191 1763 86.4% AVG 77.9% 74.4% 217 1363 46 137 1763 89.6% STDEV 1.2% 1.6% 53 58 13 20 9 1.3% SE 0.4% 0.5% 17 18 4 6 3 0.4%
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, 1982. The Extended Phenotype. Freeman, San Francisco. pp. 146-151) 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 1 484 694 316 44 1766 2 542 648 332 36 1779 3 521 819 225 35 1774 4 601 636 249 52 1782 5 704 716 144 2 1786 6 617 758 201 14 1765 7 481 791 309 30 1780 8 572 675 234 159 1796 9 521 711 224 75 1760 10 375 651 403 135 1776 AVERAGE 541.8 709.9 263.7 58.2 1776.4 AVERAGE % of total population 30.5% 40.0% 14.8% 3.3% Strategy distribution in 5-trait variant
021 - Viscosity+Tags model, all-egoist initial condition and no immigration (ref experiment 01004) 1 1109086302493 4000 81.3% 78.6% 241 1423 15 100 1779 93.5% 2 1109086697892 4000 86.3% 81.2% 290 1323 29 148 1790 90.1% 3 1109087098878 4000 80.5% 73.7% 133 1429 41 142 1745 89.5% 4 1109087497061 4000 81.8% 78.0% 306 1275 10 170 1761 89.8% 5 1109087896275 4000 83.2% 79.2% 246 1355 22 125 1748 91.6% 6 1109088286015 4000 79.8% 76.2% 317 1267 28 145 1757 90.2% 7 1109088686712 4000 81.5% 78.6% 216 1418 18 111 1763 92.7% 8 1109089077463 4000 81.9% 75.8% 198 1406 22 165 1791 89.6% 9 1109089477509 4000 81.2% 74.5% 291 1276 16 197 1780 88.0% 10 1109089869142 4000 79.5% 74.8% 148 1479 12 131 1770 91.9% AVG 81.7% 77.1% 239 1365 21 143 1768 90.7% STDEV 1.9% 2.4% 65 76 9 29 16 1.7% SE 0.6% 0.8% 20 24 3 9 5 0.5%
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 1 1016491629106 2000 77.4% 78.0% 300 1448 18 77 1843 94.8% 2 1016491738243 2000 79.1% 77.2% 216 1479 43 97 1835 92.4% 3 1016491869621 2000 79.7% 77.2% 247 1456 56 111 1870 91.1% 4 1016492024684 2000 77.4% 74.7% 309 1376 53 123 1861 90.5% 5 1016492117658 2000 82.9% 79.3% 254 1432 37 128 1851 91.1% 6 1016492225163 2000 76.0% 74.8% 226 1492 12 114 1844 93.2% 7 1016492332807 2000 80.8% 80.0% 410 1297 42 105 1854 92.1% 8 1016492433472 2000 77.2% 77.3% 278 1462 23 92 1855 93.8% 9 1016492559884 2000 81.0% 82.1% 513 1237 45 81 1876 93.3% 10 1016492669221 2000 78.4% 77.8% 304 1384 23 107 1818 92.8% AVG 79.0% 77.8% 306 1406 35 104 1851 92.5% STDEV 2.1% 2.2% 91 83 15 17 17 1.4% SE 0.7% 0.7% 29 26 5 5 5 0.4%
023 - Viscosity+Tags model, low death rate (ref experiment 0120) 1 1017073596866 2000 71.7% 65.1% 206 1659 49 312 2226 83.8% 2 1017073722136 2000 73.7% 67.3% 187 1721 64 260 2232 85.5% 3 1017073861397 2000 78.8% 72.9% 367 1572 51 251 2241 86.5% 4 1017073973818 2000 74.0% 69.8% 250 1703 78 206 2237 87.3% 5 1017074148119 2000 75.0% 67.9% 199 1717 33 283 2232 85.8% 6 1017074257236 2000 74.8% 67.4% 157 1747 47 284 2235 85.2% 7 1017074364310 2000 74.7% 66.8% 196 1687 63 284 2230 84.4% 8 1017074465405 2000 78.0% 74.5% 237 1787 54 168 2246 90.1% 9 1017074623523 2000 75.7% 68.2% 316 1551 50 321 2238 83.4% 10 1017074731207 2000 72.6% 66.8% 184 1749 46 255 2234 86.5% AVG 74.9% 68.7% 230 1689 54 262 2235 85.9% STDEV 2.2% 2.9% 65 76 12 47 6 1.9% SE 0.7% 0.9% 21 24 4 15 2 0.6%
024 - Viscosity+Tags model, higher dth rate (ref experiment 0121) 1 1017074961228 2000 95.5% 46.5% 4 2 4 3 13 46.2% 2 1017074998021 2000 99.5% 45.4% 4 2 2 5 13 46.2% 3 1017075032150 2000 98.9% 47.0% 4 1 2 5 12 41.7% 4 1017075069564 2000 97.8% 64.5% 5 3 3 2 13 61.5% 5 1017075105926 2000 99.8% 65.5% 1 6 3 1 11 63.6% 6 1017075134988 2000 98.5% 47.3% 2 2 3 2 9 44.4% 7 1017075169628 2000 98.3% 67.3% 4 2 3 1 10 60.0% 8 1017075199901 2000 98.4% 56.5% 1 4 3 1 9 55.6% 9 1017075229654 2000 98.1% 89.3% 6 6 2 1 15 80.0% 10 1017075266107 2000 92.3% 42.1% 6 1 5 2 14 50.0% AVG 97.7% 57.1% 4 3 3 2 12 54.9% STDEV 2.2% 14.7% 2 2 1 2 2 11.7% SE 0.7% 4.6% 1 1 0 0 1 3.7%
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
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.
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 = 1.05365 and 0.0391375. 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.