Topology-dependent rationality and quantal response equilibria in structured populations

Given that the assumption of perfect rationality is rarely met in the real world, we explore a graded notion of rationality in socioecological systems of networked actors. We parametrize an actors' rationality via their place in a social network and quantify system rationality via the average Jensen-Shannon divergence between the games Nash and logit quantal response equilibria. Previous work has argued that scale-free topologies maximize a system's overall rationality in this setup. Here we show that while, for certain games, it is true that increasing degree heterogeneity of complex networks enhances rationality, rationality-optimal configurations are not scale-free. For the Prisoner's Dilemma and Stag Hunt games, we provide analytic arguments complemented by numerical optimization experiments to demonstrate that core-periphery networks composed of a few dominant hub nodes surrounded by a periphery of very low degree nodes give strikingly smaller overall deviations from rationality than scale-free networks. Similarly, for the Battle of the Sexes and the Matching Pennies games, we find that the optimal network structure is also a core-periphery graph but with a smaller difference in the average degrees of the core and the periphery. These results provide insight on the interplay between the topological structure of socioecological systems and their collective cognitive behavior, with potential applications to understanding wealth inequality and the structural features of the network of global corporate control.

Figure 1

Illustrative example of a core-periphery topology that minimizes the average divergence (while avoiding isolated nodes) in the case of the (a) Prisoner's Dilemma and Stag Hunt games, where the hubs form a complete subgraph and each has a high degree and (b) Battle of the Sexes, where the hubs for a regular subgraph. For the Matching Pennies the optimal topology is similar to (b), but the periphery consists of isolated $K_2$ graphs.

Figure 2

(a) The evolution of $-\rho$, the average Jensen-Shannon divergence between Nash and LQR equilibria, for the Prisoner's Dilemma ($\beta =1.33$) with linear rationality, of an initially random network ($N=1000$ nodes and $M=2000$ edges) as a function of attempted edge rewirings, while not allowing isolated nodes. The optimization gets within $2\%$ of the estimated minimum (dashed line). (b) The divergence of a core-periphery graph with $N=1000,M=2000$ as a function of the number of hubs according to numerical calculations (dots) and analytic estimates (dashed line) using equation (12) for core-periphery graphs.

Figure 3

(a) The evolution of $-\rho$, the average Jensen-Shannon divergence between Nash and LQR equilibria, for the Prisoner's Dilemma ($\beta =1.33$) with linear rationality, starting with an initially random network ($N=1000$ nodes and $M=2000$ edges) as a function of the number of attempted edge rewirings, allowing isolated nodes. The optimization reaches the theoretical minimum (dashed line), which is achieved for a complete graph with 63 nodes. (b) Numerical (dots) and analytical (dashed line) results for the divergence of regular graphs with $M=2000$ edges in the case of the Prisoner's Dilemma. Analytic values calculated according to Eq. (8).

Figure 4

(a) The evolution of $-\rho$, the average Jensen-Shannon divergence between Nash and LQR equilibria, for the Battles of the Sexes (thin line) and Matching Pennies (thick line) ($\beta =1.33$) with linear rationality, in the range of topologies between an initially random network ($N=1000$ nodes and $M=2000$ edges) as a function of attempted edge rewirings, while not allowing isolated nodes. The optimizations reaches the estimated minima (dashed lines). (b) The divergence for regular graphs with $M=2000$ edges in the case of the Battle of the Sexes (triangular dots) and Matching Pennies (square dots), with analytic results shown by dashed lines.