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 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: and . Threshold: for and , and for and Iterated Public Goods Game (IPG): for and , and for and 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 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 p_t given in (3.2) and (3.3), where the betas are calculated using 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 parameters								moments of distribution of x
n0	m	r0	a	p0	pay-off function		Q	3rd	4th	5th	6th	7th	p1100
80	0.05	1	0.5	0.1	linear	pred.	−0.822	0.00313	0.00101	0.000388	0.000167	7.90 × 10−5	0.0697
						obs.	−0.816	0.00319	0.00104	0.000402	0.000174	8.30 × 10−5	0.0700
					thresh.	pred.	−0.822	0.0439	0.0211	0.0111	0.00621	0.00368	0.283
						obs.	−0.821	0.0444	0.0214	0.0113	0.00641	0.00383	0.283
					IPG	pred.	−0.822	0.00327	0.00106	0.000404	0.000173	8.12 × 10−5	0.0714
						obs.	−0.819	0.00338	0.00111	0.000427	0.000186	8.93 × 10−5	0.0730
320	0.05	1	0.5	0.1	linear	pred.	−0.822	0.00535	7.31 × 10−5	1.17 × 10−5	2.13 × 10−6	4.40 × 10−7	0.0570
						obs.	−0.826	0.00553	7.66 × 10−5	1.24 × 10−5	2.31 × 10−6	4.77 × 10−7	0.0568
					thresh.	pred.	−0.822	0.0145	0.00416	0.00128	0.00417	0.000144	0.221
						obs.	−0.822	0.0146	0.00428	0.00131	0.00428	0.000148	0.221
					IPG	pred.	−0.822	0.000885	0.000132	2.27 × 10−5	4.39 × 10−6	9.44 × 10−7	0.0710
						obs.	−0.822	0.000911	0.000137	2.39 × 10−5	4.71 × 10−6	1.03 × 10−6	0.0717
20	0.05	1	0.5	0.1	linear	pred.	−0.822	0.0345	0.0234	0.0172	0.0133	0.0107	0.133
						obs.	−0.798	0.0352	0.0241	0.0179	0.0140	0.0114	0.131
					thresh.	pred.	−0.822	0.170	0.131	0.107	0.0901	0.0771	0.366
						obs.	−0.801	0.171	0.133	0.109	0.0923	0.0803	0.397
					IPG	pred.	−0.822	0.0180	0.0118	0.00846	0.00641	0.00510	0.0740
						obs.	−0.797	0.0184	0.0121	0.00881	0.00681	0.00543	0.0787
40	0.05	1	1	0.1	linear	pred.	−0.903	0.00784	0.00381	0.00211	0.00127	0.000822	0.0771
						obs.	−0.903	0.00808	0.00396	0.00221	0.00135	0.000885	0.0740
					thresh.	pred.	−0.903	0.0515	0.0300	0.0191	0.0130	0.00926	0.261
						obs.	−0.902	0.0525	0.0308	0.0198	0.0136	0.00978	0.250
40	0.1	1	0.5	0.1	linear	pred.	−0.681	0.00480	0.00170	0.000700	0.000320	0.000160	0.0842
						obs.	−0.664	0.00475	0.00165	0.000670	0.000305	0.000152	0.0861
					thresh.	pred.	−0.681	0.0908	0.0512	0.0308	0.0196	0.0129	0.380
						obs.	−0.673	0.0908	0.0511	0.0308	0.0196	0.0131	0.384
80	0.05	0.5	0.5	0.1	linear	pred.	−0.698	0.00393	0.00132	0.000513	0.000225	0.000108	0.0802
						obs.	−0.686	0.00408	0.00138	0.000546	0.000243	0.000118	0.0805
					thresh.	pred.	−0.698	0.0794	0.0431	0.0249	0.0152	0.00978	0.361
						obs.	−0.700	0.0801	0.0435	0.0254	0.0156	0.0101	0.367
80	0.05	1	0.05	0.1	linear	pred.	−0.316	0.00774	0.00279	0.00116	0.00531	0.000265	0.120
						obs.	−0.315	0.00800	0.00291	0.00122	0.00566	0.000285	0.120
					thresh.	pred.	−0.316	0.172	0.110	0.0740	0.0515	0.0414	0.504
						obs.	−0.318	0.174	0.112	0.0755	0.0529	0.0383	0.507
80	0.05	1	0.5	0.9	linear	pred.	−0.822	0.668	0.600	0.544	0.497	0.458	0.858
						obs.	−0.818	0.669	0.602	0.547	0.501	0.462	0.860
					thresh.	pred.	−0.822	0.856	0.821	0.790	0.762	0.738	0.942
						obs.	−0.780	0.856	0.822	0.791	0.763	0.740	0.943