Three-Strategy Evolutionary Game Dynamics

• Three-strategy evolutionary games are models where individuals choose among three options (e.g., cooperator, defector, and a third role) with dynamics driven by replicator equations and update rules.
• The framework reveals rich behaviors such as cyclic dominance, dominant absorption, and stable coexistence, using analytical methods, simulations, and mutation-selection dynamics.
• Applications include modeling ecological and social systems where environmental feedback and spatial structure stabilize cooperation and maintain biodiversity.

A three-strategy evolutionary game is a class of models in evolutionary game theory where individuals in a population choose among three distinct strategies, and the frequencies and abundances of these strategies evolve through differential success in strategic interactions. This framework generalizes the classical two-strategy case, enabling the modeling of richer ecological, social, and biological phenomena such as cyclic dominance, bistability, cooperation under environmental feedback, and the effects of neutral or buffer states. The interplay among three strategies can generate complex dynamical regimes, including oscillatory coexistence, extinction, and phase transitions that are not possible in two-strategy systems.

1. Fundamental Model Structure

A population comprises agents, each adopting one of three possible strategies. In typical models, the strategies represent distinct behavioral or biological options such as:

• Cooperator (C): Pays a cost $c$ to provide benefit $b$ to another.
• Defector (D): Pays no cost but seeks benefits from cooperators.
• Third strategy ($L$, $I$, or $E$): This may be a loner who opts out ($L$) and receives a fixed payoff, an individualist ($I$), or an empty site ($E$) representing a neutral or vacant state.

Let $x_1, x_2, x_3$ (or, where named, $x_C, x_D, x_L$) denote the frequencies of each strategy, always satisfying $x_1 + x_2 + x_3 = 1$. Interactions may be pairwise (standard matrix games) or involve larger groups, and payoffs are encoded by a $3 \times 3$ payoff matrix $A = (a_{ij})$ or higher-order tensors in multiplayer settings.

The temporal evolution of strategy abundances can be governed by continuous-time replicator equations, discrete-time update rules, or stochastic simulation protocols. In well-mixed populations, deterministic replicator dynamics typically read: $\dot{x}_i = x_i \left( \pi_i - \bar\pi \right),$ where $\pi_i$ is the expected payoff of strategy $i$, and $\bar\pi$ is the population mean. Alternatively, purely competitive or mutation-selection models can be used (Veller et al., 2012, Gokhale et al., 2011, Boccabella et al., 2011).

2. Dynamical Regimes and Phase Structure

Three-strategy games generate a diverse landscape of dynamical regimes absent in binary systems, as supported by both theoretical analyses and simulations:

• Dominance and Absorption: If one strategy strictly outperforms both competitors in all pairwise matchups, the population converges to a pure state with only that strategy present (Veller et al., 2012).
• Cyclic Dominance (Rock-Paper-Scissors cycles): When no strategy is globally dominant—e.g., $C$ beats $D$, $D$ beats $L$, $L$ beats $C$—the population exhibits cycling in the simplex, often realized as limit cycles or persistent oscillations (Veller et al., 2012, Jiang et al., 2018, Takesue, 8 Nov 2024).
• Stable Coexistence: Under certain payoff and feedback parameters, all three strategies coexist at a fixed interior point, with convergent or softly oscillating abundances. Stability conditions are analytically derived in eco-evolutionary models (Li et al., 2021).
• Noise-Induced Resonance and Phase Transitions: In the presence of mutation, small but nonzero mutation rates can stabilize previously unstable mixed equilibria, inducing transitions via Hopf bifurcations. This leads to noise-sustained oscillations or to the stabilization of coexistence states at mutation rates typically $10^{-3}$ to $10^{-2}$ (Veller et al., 2012, Chen et al., 9 Oct 2025).
• Buffer-mediated and Individual Solution Dominance: Introduction of empty or neutral states (e.g., $E$ or $I$) allows for phases where these states dominate, suppressing both cooperation and defection. On structured populations, coexistence is often sustained by cyclic domain invasion (Chen et al., 9 Oct 2025, Takesue, 10 Feb 2025, Takesue, 8 Nov 2024).

Table: Example Phase Boundaries in Three-Strategy Dynamics

Model	Coexistence Condition	Absorption/Extinction Condition
Replicator (well-mixed, RPS)	Cyclic dominance ($p^*$ unstable, limit-cycle)	Dominant type: pure strategy fixed point
Coevo. PDG + loner + environment (Li et al., 2021)	$(1+\theta)+c<\delta<1$ (see full formula)	$\delta<1/(1+\theta)$: extinction of $L$
PDG + Individual solution (Takesue, 10 Feb 2025)	Structured: $c_I \in (1.2c,1.8c)$	Well-mixed: only $I$ survives

These boundaries are parameterized by payoff cost-to-benefit ratios, feedback strengths, and mutation rates.

3. Model Variants and Representative Formalisms

3.1. Purely Competitive 3-Strategy Dynamics

In infinite populations, each period involves random pairing and offspring inherit strategy with probability proportional to pairwise contest outcomes: $$x_i' = x_i^2 + 2 x_i \sum_{j \neq i} x_j \tilde u_{ij},$$ with normalization $\tilde u_{ij} = a_{ij}/(a_{ij} + a_{ji})$. Mutations are incorporated as: $$x_i' = (1-\tfrac{3\mu}{2})\left[x_i^2 + 2x_i \sum_{j \neq i} x_j \tilde u_{ij}\right] + \frac{\mu}{2}.$$ A supercritical Hopf bifurcation occurs at a critical mutation rate $\mu_c$, stabilizing the interior fixed point, with oscillatory cycling below $\mu_c$ (Veller et al., 2012).

3.2. Coevolutionary Games with Environmental Feedback

Incorporating a dynamically-coupled environment $n(t)$, strategy dynamics become intertwined with environmental state: $$\dot x_C = x_C\left[\omega\pi_C - \Phi\right] - d_C(n)x_C + b_C(n)x_E,\ \dot n = \epsilon\left(\theta x_C - x_D\right)$$ Clustering of cooperators amplifies survival and environment quality; the system admits tragedy-of-commons, dynamic equilibrium, and resonance regimes as parameters $r$, $\epsilon$, and mutation $\mu$ vary (Chen et al., 9 Oct 2025, Li et al., 2021).

3.3. Individual Solution and Buffer Strategies

Adding a “safe harbor” strategy (such as individual solution $I$ or empty node $E$) produces dominant absorber states in well-mixed models, with only network structure rescuing cooperation through spatial reciprocity and domain-level cyclic dominance (Takesue, 10 Feb 2025, Takesue, 8 Nov 2024).

3.4. Hypergame and Strategy-Set Heterogeneity

Hypergames admit individuals with restricted access to the full strategy set. Dynamical phases include single-strategy absorption, coexistence via cyclic dominance, and uncertain phases with stochastic revival. Mean-field and pair-approximate equations predict phase transitions controlled by internal adoption probability $\rho$ (Zhang et al., 29 Sep 2025, Jiang et al., 2018).

4. Spatial Structure, Mutation, and Environmental Feedback

Spatial and network structure fundamentally alter the attractor landscape. For example:

• Structured lattices promote domain-based cyclic dominance, sustaining coexistence regimes unachievable in well-mixed populations. Domain walls become mobility fronts where strategies invade and retreat following rock–paper–scissors cycles (Takesue, 10 Feb 2025, Takesue, 8 Nov 2024, Jiang et al., 2018).
• Mutation suppresses pure-strategy domains, fills low-density phases, and can render interior fixed points stable even at arbitrarily small rates. In hypergames and eco-evolutionary contexts, this produces stochastic resonance phenomena and stabilizes oscillatory or stationary coexistence (Chen et al., 9 Oct 2025, Veller et al., 2012).
• Environmental feedback enables the emergence of new dynamical regimes: cooperators can stabilize by healing environmental degradation, while defectors accelerate collapse. The loop between strategy abundance and environment quality generates feedback-driven phase transitions and nontrivial stationary states (Chen et al., 9 Oct 2025, Li et al., 2021).

5. Analytical and Numerical Methods

Analyses of three-strategy evolutionary games employ a spectrum of methods:

• Replicator ODEs: Analytical stability conditions, phase diagrams on the simplex, and eigenvalue analysis for fixed points (Takesue, 10 Feb 2025, Takesue, 8 Nov 2024, Li et al., 2021).
• Discrete-time, Pairwise Competitive Maps: Nonlinear updates per generation, computation of limit cycles and Hopf bifurcations (Veller et al., 2012).
• Multiplayer and Mutation-Selection Equilibria: Coalescence theory for $d$-player, $n$-strategy systems; equilibrium frequencies as rational functions of mutation rate $\mu = N u$ (Gokhale et al., 2011).
• Continuum Mixed-Strategy PDEs: Integro-differential equations for distributions $f(t,p)$ over the simplex with numerical schemes to resolve continuous and Dirac-type stationary solutions (Boccabella et al., 2011).
• Agent-Based and Monte Carlo Simulation: Square lattice models, Fermi imitation rules, asynchronous updates, measurement of statistical survival probabilities, power spectra, and spatiotemporal patterning (Chen et al., 9 Oct 2025, Takesue, 8 Nov 2024).
• Hypergame Mean-Field and Pair Approximations: ODE and pair motif-based analyses for phase boundaries and transition thresholds in heterogeneous strategy-set models (Zhang et al., 29 Sep 2025, Jiang et al., 2018).

6. Broader Implications and Applications

The inclusion of a third strategy—whether as a buffer, absorber, or via restricted access (hypergame)—profoundly alters evolutionary outcomes compared to two-strategy models:

• Biodiversity and Strategy Coexistence: Stable three-way coexistence emerges in eco-evolutionary models, supporting ecological and social diversity, especially with resource feedback or strategic opt-outs (Li et al., 2021).
• Stabilization of Cooperation: Buffer/empty states, environmental coupling, and mutation can stabilize cooperation regimes that would otherwise collapse in classical models (Chen et al., 9 Oct 2025).
• Cyclic Dominance and Dynamic Patterns: Three-strategy games explain the emergence of cyclic dominance in biological/ecological systems (e.g., rock–paper–scissors), domain wall dynamics, and strategy revival phenomena (Takesue, 8 Nov 2024, Jiang et al., 2018).
• Fragility and Hysteresis: Introduction of even rare mutations can destroy full-cooperation attractors, and phase transitions may exhibit significant hysteresis as parameters are varied (Takesue, 10 Feb 2025).
• Realistic Modeling: Mechanisms such as empty sites or individual solutions reflect real-world extinction, recolonization, and fallback behaviors, enhancing the fidelity of socio-ecological game models (Chen et al., 9 Oct 2025, Takesue, 10 Feb 2025).

A plausible implication is that even minimal enrichment of the strategic repertoire, particularly when coupled to feedback from a slow environmental variable or incorporated through agent heterogeneity, is sufficient to generate a phase structure qualitatively richer than the classical two-strategy replicator, and to produce mechanisms for cooperation stabilization and biodiversity maintenance found in natural and social systems.

• Chen, Wu, & Guan, “Three-state coevolutionary game dynamics with environmental feedback” (Chen et al., 9 Oct 2025).
• Veller & Rajpaul, “Purely competitive evolutionary dynamics for games” (Veller et al., 2012).
• Li et al., “Game-environment feedback dynamics for voluntary prisoner's dilemma games” (Li et al., 2021).
• Jiang et al., “Evolutionary hypergame dynamics” (Jiang et al., 2018), and extended hypergame analysis (Zhang et al., 29 Sep 2025).
• Takesue, “Hinderance of cooperation by individual solutions: Evolutionary dynamics of three-strategy games combining the prisoner's dilemma and stag hunt” (Takesue, 10 Feb 2025).
• Takesue, “Evolution of cooperation in a three-strategy game combining snowdrift and stag hunt games” (Takesue, 8 Nov 2024).
• Gokhale & Traulsen, “Strategy abundance in evolutionary many-player games with multiple strategies” (Gokhale et al., 2011).
• Burger, “On a continuous mixed strategies model for evolutionary game theory” (Boccabella et al., 2011).