Complete strategy spaces reveal hidden pathways to cooperation

TL;DR

This work tackles whether restricting to a four-strategy set masks essential cooperative pathways in cheap-talk games by extending to the complete eight-strategy space in a two-stage cheap-talk framework. It analyzes both well-mixed Moran/Fermi dynamics and spatial lattice interactions, introducing a cognitive-cost parameter $\gamma$ and exploring how network reciprocity shapes outcomes when the full strategy space is available. The results reveal novel pathways, including a cyclic dominance driven by suspicious cooperation and a general fragility of strategic defection, with robust cooperation and up to seven-strategy coexistence emerging under certain conditions. The study highlights the importance of comprehensive strategy spaces in evolutionary modelling and points to future work on noisy communication and richer signaling channels to further understand cooperation in realistic settings.

Abstract

Understanding how cooperation emerges and persists is a central challenge in evolutionary game theory. Existing models often rely on restricted, hand-picked strategy sets, which can overlook critical behavioural pathways. A recent four-strategy framework showed that cheap talk can promote cooperation through local interactions, yet it remained unclear whether modelled strategies might alter these conclusions. Here, we extend this framework to the complete set of eight strategies that naturally arise from communication and decision-making rules. We show that incorporating the full strategy space dramatically changes the evolutionary landscape. Cooperation becomes both more robust and more versatile, driven by novel pathways absent in the restricted model. In particular, we uncover a previously overlooked mechanism in which suspicious cooperation catalyses a cyclic dynamic that sustains cooperation. Conversely, the assumed role of strategic defection in the biased model is fragile, acting mainly as a spoiler rather than a genuine evolutionary attractor. The complete model further reveals a rich spectrum of long-term behaviours, including stable coexistence among up to seven strategies and time-varying patterns of partial coexistence. These results demonstrate that the full strategy space unlocks hidden routes to cooperative behaviour and highlight the importance of comprehensive modelling when explaining the emergence of cooperation.

Figures (7)

• Figure 1: Cheap talk alone can not sustain cooperation in finite well-mixed populations. Panels A and B show the stationary distribution of each strategy as a function of selection intensity with different selection intensities; panels C and D show the stationary distribution of each strategy as a function of reasoning cost with different selection intensities. Results are obtained with parameters $r=0.2$, A: $\gamma=0$; B: $\gamma=0.2$; C: $\beta=0.1$, and D: $\beta=1$.
• Figure 2: A full strategy space makes cheap talk more effective at promoting network cooperation. Panel A shows the frequencies of cooperation as a function of dilemma strength and reasoning cost with the complete strategy space. Panel B shows the frequencies of cooperation as a function of the dilemma strength and reasoning cost with the limited strategy space.
• Figure 3: Cheap talk sustains cooperation through deliberative strategies at low reasoning costs, but its effect vanishes at high costs. Panel A shows the frequencies of each strategy and cooperation as a function of the reasoning cost in the weak dilemma strength regime. Panels B-D show the evolution of each strategy with different reasoning costs, respectively. The shadow shows the region where cooperation dominates. Results are obtained with the parameters $r=0.02$, B: $\gamma=0.02$; C: $\gamma=0.06$, and D: $\gamma=0.15$.
• Figure 4: Persistence of cooperation is catalysed by a cyclic dominance involving the suspicious cooperation (NDC). Shown are the snapshots in the evolutionary process at time steps 0, 20, 70, 100, and 1000 in the top row, and at time steps 0, 15, 70, 100, and 1000 in the bottom row, respectively. Results are obtained with parameters $r=0.02$, A-E: $\gamma=0.06$; F-J: $\gamma=0.15$.
• Figure 5: A valley of defection induced by moderate reasoning costs. Panel A shows the frequencies of each strategy and cooperation as a function of reasoning cost in strong dilemma strength. Panels B-D show the evolution of each strategy with different reasoning costs, respectively. The shadow shows the region where cooperation dominates. Results are obtained with the parameters $r=0.2$, B: $\gamma=0.02$; C: $\gamma=0.1$, and D: $\gamma=0.18$.