Game susceptibility, Correlation and Payoff capacity as a measure of Cooperative behavior in the thermodynamic limit of some Social dilemmas

TL;DR

This work investigates how cooperative behavior arises in the thermodynamic limit for one-shot social dilemmas by comparing three analytical mappings to a 1D Ising chain—Nash equilibrium mapping (NEM), Aggregate selection (AS), and Darwinian selection (DS)—against a numerical Agent Based Method (ABM). It introduces and analyzes five indicators of cooperation: game magnetization, game susceptibility, correlation, payoff capacity, and average payoff, with a focus on Hawk-Dove and Public Goods games. The key finding is that NEM closely matches ABM across both games for all three new indicators, while AS and DS often diverge, especially in the thermodynamic limit; notably, the average payoff and payoff capacity emerge as the most reliable indicators of cooperation in this regime. The results substantiate the superiority of NEM for one-shot, infinite-population analyses and suggest that payoff-based measures provide the most faithful signatures of cooperative behavior in large systems, with potential extensions to other dilemmas and quantum settings.

Abstract

Analytically, finding the origins of cooperative behavior in infinite-player games is an exciting topic of current interest. In this paper, we compare three analytical methods, i.e., Nash equilibrium mapping (NEM), Darwinian selection (DS) and Aggregate selection (AS), with a numerical Agent based method (ABM) via the game susceptibility, correlation, and payoff capacity as indicators of cooperative behaviour. While the analytical NEM model shows excellent agreement with the numerical ABM, the other analytical models, like AS and DS, show notable divergence with ABM in the thermodynamic limit for the indicators in question. Previously, cooperative behavior was studied by considering game magnetization and individual players' average payoff as indicators. This paper shows that game susceptibility, correlation, and payoff capacity can aid in understanding cooperative behavior in social dilemmas in the thermodynamic limit. The results obtained via NEM and ABM are in good agreement for all three indicators in question, for both Hawk-Dove and the Public goods games. After comparing the results obtained for all five indicators, we see that individual players' average payoff and payoff capacity serve as the best indicators to study cooperative behavior among players in the thermodynamic limit.

Figures (16)

• Figure 1: NEM method: Mapping a $1D$-Ising chain consisting of two-spins $(\uparrow, \downarrow)$ to a social dilemma game with available strategies $(\mathcal{S}_1, \mathcal{S}_2)$. The Ising parameters $(\mathcal{J}, \mathfrak{h})$ are expressed in terms of game payoffs $(\mathcal{A, B, C, D})$.
• Figure 2: Reward susceptibility$\chi_\mathfrak{r}$ vs reward$\mathfrak{r}$ for cost$\mathfrak{t}=6.0$ and punishment$\mathfrak{p}=1.0$ via NEM, AS, DS and ABM in PGG. Cooperation (C) becomes the dominant strategy for the condition $2\mathfrak{r} > (\mathfrak{t} - 2\mathfrak{p})$, with NEM, DS and ABM agreeing on the Nash equilibrium at $2\mathfrak{r}=\mathfrak{t}-2\mathfrak{p}$, while AS is an outlier.
• Figure 3: Cost susceptibility$\chi_\mathfrak{t}$ vs cost$\mathfrak{t}$ for reward$\mathfrak{r}=3.0$ and punishment$\mathfrak{p}=1.0$ via NEM, DS and ABM in PG G. For $t < (2r + 2p)$, even though Cooperation remains the dominant strategy, we observe that for increasing cost $t$, the rate of shifting to Defection always exceeds the rate of shifting to Cooperation. Cost susceptibility for AS model vanishes for all cost values, therefore its not shown.
• Figure 4: Punishment susceptibility$\chi_\mathfrak{p}$ vs punishment$\mathfrak{p}$ for cost$\mathfrak{t}=8.0$ and reward$\mathfrak{r}=3.0$ via NEM, AS, DS and ABM in PGG. For $2\mathfrak{p}<(\mathfrak{t}-2\mathfrak{r})$, even though Defection remains dominant, for increasing punishment $\mathfrak{p}$, the rate of shifting to Cooperation always exceeds the rate of shifting to Defection for all models apart from AS.
• Figure 5: NEM: For PGG, in the ZN limit, if $2\mathfrak{r} < (\mathfrak{t}-2\mathfrak{p})$, all players choose to Defect, whereas, if $2\mathfrak{r} > (\mathfrak{t}-2\mathfrak{p})$, then all players choose to Cooperate.