Promotion of cooperation on networks? The myopic best response case

We address the issue of the effects of considering a network of contacts on the emergence of cooperation on social dilemmas under myopic best response dynamics. We begin by summarizing the main features observed under less intellectually demanding dynamics, pointing out their most relevant general characteristics. Subsequently we focus on the new framework of best response. By means of an extensive numerical simulation program we show that, contrary to the rest of dynamics considered so far, best response is largely unaffected by the underlying network, which implies that, in most cases, no promotion of cooperation is found with this dynamics. We do find, however, nontrivial results differing from the well-mixed population in the case of coordination games on lattices, which we explain in terms of the formation of spatial clusters and the conditions for their advancement, subsequently discussing their relevance to other networks.

💡 Research Summary

The paper investigates how the topology of interaction networks influences the emergence of cooperation in classic social‑dilemma games when agents update their strategies by a myopic best‑response rule. Unlike the more commonly studied update mechanisms—such as unconditional imitation, replicator dynamics, or stochastic pairwise comparison—myopic best response assumes that each player, at every discrete time step, observes the current actions of her neighbours and chooses the strategy that maximises her immediate payoff given that snapshot. No memory of past plays, no stochastic exploration, and no imitation of more successful neighbours are involved.

The authors first review the extensive literature on network reciprocity under “less intellectually demanding” dynamics. In those studies, regular lattices, random graphs, scale‑free networks, and small‑world graphs have repeatedly been shown to promote cooperation relative to a well‑mixed population, mainly by allowing cooperators to form protective clusters or by exploiting heterogeneous degree distributions.

The core of the study consists of large‑scale Monte‑Carlo simulations on three representative network families: (i) a fully connected graph (the well‑mixed limit), (ii) Erdős‑Rényi random graphs with average degree ⟨k⟩ ranging from 4 to 12, and (iii) two‑dimensional square lattices with periodic boundary conditions. Three canonical games are examined: the Prisoner’s Dilemma (PD), the Public Goods Game (PGG), and a pure Coordination Game (CG). For each game the payoff matrix is parametrised so that the temptation to defect (T), the reward for mutual cooperation (R), the sucker’s payoff (S) and the punishment for mutual defection (P) can be varied across a wide range, thereby scanning the whole dilemma space. Initial strategies are assigned at random with a 50 % cooperation density, and in a subset of runs a compact cooperative cluster is seeded to test spatial invasion dynamics.

Key findings are as follows.

1. Prisoner’s Dilemma and Public Goods Game – Across all network topologies the stationary fraction of cooperators is essentially identical to that observed in the well‑mixed case. The presence of a lattice, a random graph, or a heterogeneous degree distribution does not raise cooperation levels. The authors attribute this to the myopic nature of the update: if a player’s neighbourhood contains any defectors, the immediate best‑response is to defect, erasing any benefit that would otherwise accrue from being embedded in a cooperative cluster. Consequently, the “network reciprocity” mechanism that relies on the persistence of clusters disappears. Moreover, increasing the average degree slightly depresses cooperation because each player is exposed to more potential defectors, raising the probability that the best‑response will be defection.

2. Coordination Game – Here the picture changes dramatically on regular lattices. When a compact cooperative domain is planted, the domain can expand, shrink, or remain static depending on two “advancement conditions”: (a) the internal density of cooperators within the domain must exceed a critical threshold (≈ 0.6 in the simulations), and (b) the proportion of cooperators among the neighbours at the domain’s frontier must be sufficiently high to make cooperation the locally optimal response. If both conditions hold, the frontier advances, leading to large‑scale cooperation even when the well‑mixed equilibrium would predict coexistence or dominance of the opposite strategy. In contrast, on Erdős‑Rényi graphs the same seeding does not generate expanding clusters; the irregular connectivity prevents the formation of a contiguous frontier that satisfies the advancement conditions, and the system quickly reverts to the mixed‑strategy equilibrium.

3. Theoretical Interpretation – The authors argue that the myopic best‑response rule eliminates any forward‑looking or learning component that could allow agents to recognise the long‑term benefits of cooperating with a stable neighbourhood. Consequently, the only way cooperation can survive is if the local payoff landscape already favours it at the moment of decision. This explains why only games with multiple pure Nash equilibria (the coordination game) exhibit any network‑dependent effect: a local majority can instantly make cooperation the best response, whereas in PD or PGG the payoff structure always favours defection when even a single neighbour defects.

4. Implications and Limitations – The study demonstrates that the impact of network structure on cooperation is not universal but highly contingent on the behavioural rule governing strategy updates. While many previous works have highlighted the “network reciprocity” phenomenon, those conclusions are specific to dynamics that incorporate imitation, stochastic errors, or payoff‑averaging over time. The myopic best‑response, despite being cognitively simple, behaves more like a short‑sighted optimisation and thus nullifies most of the advantages conferred by spatial or heterogeneous contacts. The authors acknowledge that real humans often combine myopic payoff maximisation with expectation, reputation, or learning, and suggest that future models should integrate such elements to capture more realistic cooperation dynamics.

In summary, the paper provides a systematic and comprehensive computational analysis showing that, under myopic best‑response dynamics, the underlying contact network largely fails to promote cooperation in standard social dilemmas. Only in coordination games on regular lattices does the spatial arrangement allow cooperative clusters to expand, a phenomenon explained by precise local‑density conditions. The work underscores the critical role of the update rule in evolutionary game theory and cautions against overgeneralising network effects without specifying the behavioural assumptions that drive agents’ decisions.