Table 1.

Simulation results. Evolution in a finite ILGR population simulation with 106 groups was simulated for 1100 time periods. Initially, frequency of type A was p0 and groups were formed at random. Population regulation was given by the logistic Embedded Image Embedded Image and δ = 0.01. There were three pay-off functions that gave the incremental effect of social behaviour on the expected number of offspring of types A and N in a group in which As had frequency x. Linear: Embedded Image and Embedded Image. Threshold: for Embedded Image Embedded Image and Embedded Image, and for Embedded Image Embedded Image and Embedded Image Iterated Public Goods Game (IPG): for Embedded Image Embedded Image and Embedded Image, and for Embedded Image Embedded Image and Embedded Image The observed value of Q is the average of values in the interval [100,1100] calculated using (33) in the electronic supplementary material. The first 100 time periods were ignored to allow the population to reach demographic quasi-equilibrium. The predicted value of Q is given by (3.4). The observed moments of the distribution of x were calculated using the empirical distribution of x at t = 1100 and the predicted ones were calculated using the beta(lp,lq) distribution for the values of p and Embedded Image calculated from the empirical distribution of x at t = 1100. This is the beta distribution that has the first two moments identical to the observed ones, and for these reason those first two moments are omitted from the table. The observed value of p1100 was calculated from the distribution of x at time t = 1100, and the predicted value was calculated by iterating the recursion for pt given in (3.2) and (3.3), where the betas are calculated using Embedded Image for period t, and Q is given by (3.4), beginning at time 100 with the value of p100 calculated from the simulation at time 100.

population parametersmoments of distribution of x
n0mr0ap0pay-off functionQ3rd4th5th6th7thp1100
800.0510.50.1linearpred.−0.8220.003130.001010.0003880.0001677.90 × 10−50.0697
obs.−0.8160.003190.001040.0004020.0001748.30 × 10−50.0700
thresh.pred.−0.8220.04390.02110.01110.006210.003680.283
obs.−0.8210.04440.02140.01130.006410.003830.283
IPGpred.−0.8220.003270.001060.0004040.0001738.12 × 10−50.0714
obs.−0.8190.003380.001110.0004270.0001868.93 × 10−50.0730
3200.0510.50.1linearpred.−0.8220.005357.31 × 10−51.17 × 10−52.13 × 10−64.40 × 10−70.0570
obs.−0.8260.005537.66 × 10−51.24 × 10−52.31 × 10−64.77 × 10−70.0568
thresh.pred.−0.8220.01450.004160.001280.004170.0001440.221
obs.−0.8220.01460.004280.001310.004280.0001480.221
IPGpred.−0.8220.0008850.0001322.27 × 10−54.39 × 10−69.44 × 10−70.0710
obs.−0.8220.0009110.0001372.39 × 10−54.71 × 10−61.03 × 10−60.0717
200.0510.50.1linearpred.−0.8220.03450.02340.01720.01330.01070.133
obs.−0.7980.03520.02410.01790.01400.01140.131
thresh.pred.−0.8220.1700.1310.1070.09010.07710.366
obs.−0.8010.1710.1330.1090.09230.08030.397
IPGpred.−0.8220.01800.01180.008460.006410.005100.0740
obs.−0.7970.01840.01210.008810.006810.005430.0787
400.05110.1linearpred.−0.9030.007840.003810.002110.001270.0008220.0771
obs.−0.9030.008080.003960.002210.001350.0008850.0740
thresh.pred.−0.9030.05150.03000.01910.01300.009260.261
obs.−0.9020.05250.03080.01980.01360.009780.250
400.110.50.1linearpred.−0.6810.004800.001700.0007000.0003200.0001600.0842
obs.−0.6640.004750.001650.0006700.0003050.0001520.0861
thresh.pred.−0.6810.09080.05120.03080.01960.01290.380
obs.−0.6730.09080.05110.03080.01960.01310.384
800.050.50.50.1linearpred.−0.6980.003930.001320.0005130.0002250.0001080.0802
obs.−0.6860.004080.001380.0005460.0002430.0001180.0805
thresh.pred.−0.6980.07940.04310.02490.01520.009780.361
obs.−0.7000.08010.04350.02540.01560.01010.367
800.0510.050.1linearpred.−0.3160.007740.002790.001160.005310.0002650.120
obs.−0.3150.008000.002910.001220.005660.0002850.120
thresh.pred.−0.3160.1720.1100.07400.05150.04140.504
obs.−0.3180.1740.1120.07550.05290.03830.507
800.0510.50.9linearpred.−0.8220.6680.6000.5440.4970.4580.858
obs.−0.8180.6690.6020.5470.5010.4620.860
thresh.pred.−0.8220.8560.8210.7900.7620.7380.942
obs.−0.7800.8560.8220.7910.7630.7400.943