The survival of the weakest in a biased donation game

Abstract

Cooperating first then mimicking the partner's act has been proven to be effective in utilizing reciprocity in social dilemmas. However, the extent to which this, called Tit-for-Tat strategy, should be regarded as equivalent to unconditional cooperators remains controversial. Here, we introduce a biased Tit-for-Tat (T) strategy that cooperates differently toward unconditional cooperators (C) and fellow T players through independent bias parameters. The results show that, even under strong dilemmas in the donation game framework, this three-strategy system can exhibit diverse phase diagrams on the parameter plane. In particular, when T-bias is small and C-bias is large, a ``hidden T phase'' emerges, in which the weakest T strategy dominates. The dominance of the weakened T strategy originates from a counterintuitive mechanism characterizing non-transitive ecological systems: T suppresses its relative fitness to C, rapidly eliminates the cyclic dominance clusters, and subsequently expands slowly to take over the entire population. Analysis in well-mixed populations confirms that this phenomenon arises from structured populations. Our study thus reveals the subtle role of bias regulation in cooperative modes by emphasizing the ``survival of the weakest'' effect in a broader context.

Figures (5)

• Figure 1: Schematic illustration of the three-strategy game system. Left: A C-player pays a cost $-r$ to provide $+1$ to the partner. Right: A T-player's behavior depends on the opponent's strategy: when facing a C-player, it pays $-\theta_\text{C} r$ to confer a benefit $+\theta_\text{C}$; when facing a T-player, it pays $-\theta_\text{T} r$ to confer a benefit $+\theta_\text{T}$.
• Figure 2: System behavior under different dilemmas. (a) In the traditional two-strategy donation game, increasing $r$ reduces the level of cooperation hauert2025phase. (b) When cooperation can emerge in the traditional setting ($r = 0.01$), the $\theta_\text{T}$-$\theta_\text{C}$ parameter plane exhibits a C+D phase, a coexistence C+D+T phase, a C+T phase, and a T phase. (c1), (c2) Interestingly, when cooperation in the C+D phase is suppressed in the traditional two-strategy games ($r = 0.1$ and $r = 0.2$), a hidden T phase emerges for small T-bias $\theta_\text{T}$ and large C-bias $\theta_\text{C}$. The red lines indicate the emergence or extinction of the D strategy.
• Figure 3: Spatial dynamics underlying the coexistence phase and the hidden T phase. (a1)--(a3) Under moderate T-bias ($\theta_\text{T} = 1$), the coexistence of all available strategies is sustained by their intransitive relation described as $\text{D} \to \text{C} \to \text{T} \to \text{D}$. (b1)--(b3) Under low T-bias ($\theta_\text{T} = 0.1$), the hidden T phase emerges through a two-step mechanism: cyclic-dominance clusters first die out, after which isolated T individuals slowly expand. (a1), (b1) Typical time evolution of the strategy frequencies in a $600 \times 600$ population for the two phases. (a2), (b2) Snapshots of macroscopic spatial patterns in a $200 \times 200$ population for the two phases. (a3), (b3) Schematic illustrations of the microscopic spatial dynamics for the two phases. The arrows indicate the invasion direction between strategies. Other parameters: $r=0.1$, $\theta_\text{C}=1.5$.
• Figure 4: Extensive results for the stationary strategy frequencies as functions of the bias parameters. (a) For small T-bias ($\theta_\text{T} = 0.1$), moderate C-bias $\theta_\text{C}$ is most detrimental to cooperation. (b) For large T-bias ($\theta_\text{T} = 1$), increasing C-bias $\theta_\text{C}$ reduces cooperation. (c) For small C-bias ($\theta_\text{C} = 1$), moderate T-bias $\theta_\text{T}$ is most detrimental to cooperation. (d) For large C-bias ($\theta_\text{C} = 2$), increasing T-bias $\theta_\text{T}$ promotes cooperation. Other parameter: $r = 0.1$.
• Figure 5: The hidden T phase does not exist in well-mixed populations. In the $\theta_\text{T}$-$\theta_\text{C}$ parameter plane, there are four theoretical phases in well-mixed populations: T (the T strategy dominates), $\text{C}+\text{T}$ (the C and T strategies coexist), $(\text{C}+\text{D}+\text{T})^{**}$ (the three strategies coexist and are stable at the interior equilibrium), and $\text{C}+\text{D}+\text{T}$ (the three strategies coexist in cyclic dominance). The T equilibrium is stable when $\theta_\text{T} (1 - r) > \max\{\theta_\text{C} - r, 0\}$; the $\text{C}+\text{T}$ equilibrium is stable when $\theta_\text{C}(\theta_\text{C} - r)<\theta_\text{T} (1 - r)<\theta_\text{C} - r$; the $(\text{C}+\text{D}+\text{T})^{**}$ equilibrium is stable when $\theta_\text{C}(\theta_\text{C} - r)>\theta_\text{T} (1 - r)$ and $\theta_\text{C}<1$; otherwise, all equilibria are unstable and the phase is $\text{C}+\text{D}+\text{T}$ in cyclic dominance. Numerical parameter: $r = 0.1$.