Aspiration dynamics generate robust predictions in heterogeneous populations

Abstract

Update rules, which describe how individuals adjust their behavior over time, affect the outcome of social interactions. Theoretical studies have shown that evolutionary outcomes are sensitive to model details when update rules are imitation-based but are robust when update rules are self-evaluation based. However, studies of self-evaluation based rules have focused on homogeneous population structures where each individual has the same number of neighbors. Here, we consider heterogeneous population structures represented by weighted networks. Under weak selection, we analytically derive the condition for strategy success, which coincides with the classical condition of risk-dominance. This condition holds for all weighted networks and distributions of aspiration levels, and for individualized ways of self-evaluation. Our findings recover previous results as special cases and demonstrate the universality of the robustness property under self-evaluation based rules. Our work thus sheds light on the intrinsic difference between evolutionary dynamics under self-evaluation based and imitation-based update rules.

Fig. 1. Aspiration dynamics on weighted graphs.

a An undirected weighted graph with edge weight w_ij ≥ 0. b Individuals occupy vertices of the graph and each individual l has an imaginary payoff value α_l they aspire, called aspiration level. c For aspiration dynamics, at each time step, an individual is randomly selected (here, the sixth individual, marked by the black circle). It garners an edge-weighted average payoff (π₆) by playing games with its nearest neighbors. Then it self-evaluates the performance of the strategy in use by calculating the aspiration-payoff difference (α₆ − π₆), which is later used by the update function $g:\mathbb{R}\to \left[0,1\right]$ to determine its switching probability. If the payoff exceeds the aspiration, it feels satisfied and is more likely to keep its current strategy; otherwise, it is prone to switch. As illustrated, α₆ − π₆ > 0 (i.e., π₆ < α₆) and the corresponding individual switches from strategy B to A.

Fig. 2. Robust predictions generated by aspiration dynamics on weighted networks.

For the game, we set payoff value b = 0, c = 5, d = 1 and leave a as a tunable parameter. We plot the average frequency of strategy A, 〈x_A〉, as a function of a. Symbols represent simulation results while solid lines are analytical ones. We construct weighted graphs by first generating an undirected graph with average degree $\overline{k}$ and then assigning weights to edges. The undirected graphs considered are random graph (left), regular graph (middle), and scale-free network (right). For each type of network, we test three edge weight distributions: homogeneous—every edge has weight one (i.e., unweighted network); uniform—edge weights are uniformly selected from the integer set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; power-law—edge weights are randomly selected from the discrete power-law distribution (Zipf distribution with the value of the exponent equal to 3). Each data point is obtained by averaging 〈x_A〉 in 200 independent runs. For each run, we calculate 〈x_A〉 by averaging the frequency of strategy A in the last 1 × 10⁷ time steps after a transient time of 1 × 10⁷ time steps. Other parameters: N = 1000, $\overline{k}=6$, α_l = 2.0 (l = 1, 2, ⋯ , N), and β = 0.01.

Fig. 3. The structure coefficients σ for aspiration-based and imitation-based update rules.

The two imitation-based rules shown here are pairwise comparison^, and death-birth update rule. Here, the parameter N represents the population size and k the degree of the regular graph (i.e., the number of neighbors each individual has). For strategy A to be favored over strategy B, the structure coefficient σ can be interpreted as the required degree of assortment among individuals who use the same strategy. All the σs in the table are obtained under the limit of weak selection. In addition, the limit of rare mutation is assumed under imitation-based rules. We also derive a general formula under pairwise comparison update rules for any weighted graphs in Supplementary Note 3.3.

Fig. 4. Robustness of aspiration dynamics vs. sensitivity of imitative dynamics to heterogeneity of decision-making functions for all networks of size six (N = 6).

There are 112 connected and unweighted networks of size six. For each network, we calculate the critical value of a, a^*, above which 〈x_A〉 > 〈x_B〉 and below which 〈x_A〉 < 〈x_B〉. The critical value a^* under two populations are calculated: (i) $a_{\mathrm{Homo}}^{*}$, a homogeneous population where everyone shares the same decision-making function g(u); (ii) $a_{\mathrm{Heter}}^{*}$, a heterogeneous population where each individual has its own decision-making functions, i.e., individual i uses g_i(u). In panels (a) and (b), each symbol represents a pair of critical values $\left(a_{\mathrm{Homo}}^{*},a_{\mathrm{Heter}}^{*}\right)$ calculated under one of the networks. The size of the symbols indicates how much the heterogeneity of decision-making functions affects the evolutionary outcomes, which is proportional to the absolute value of $a_{\mathrm{Heter}}^{*}-a_{\mathrm{Homo}}^{*}$. The larger the size of the symbol, the more sensitive the evolutionary outcome to individual heterogeneity. In panel a, we consider maximum heterogeneity of decision-making functions (i.e., for any i ≠ j, g_i ≠ g_j) under both aspiration-based (red squares) and imitation-based (blue circles) rules. In panel b, we consider the minimum heterogeneity under imitation-based rules, where all the individuals share the same decision-making function g(u) except one. Here, the blue circles are the results obtained when g₆ ≠ g, and the red triangles are the results when g₅ ≠ g. In all the calculations, we set $g\left(u\right)=g_1\left(u\right)=1/\left(1+\exp \left(-u\right)\right)$, $g_2\left(u\right)=\left(1+\mathrm{erf}\left(u\right)\right)/2$, $g_3\left(u\right)=\left(1+\tanh \left(u\right)\right)/2$, $g_4\left(u\right)=1/\left(1+\exp \left(-u/2\right)\right)$, $g_5\left(u\right)=1/\left(1+10\exp \left(-u\right)\right)$, and $g_6\left(u\right)=10/\left(10+\exp \left(-u\right)\right)$. Other parameters: b = 0, c = 5, d = 1, α_l = 1.0 (l = 1, 2, ⋯ , N), β = 0.01, and μ → 0 (see Supplementary Note 3.3 for definitions).