Mixed Equilibrium of ISR & ALLD Dynamics

• The paper presents a robust quadratic model detailing how ISR and ALLD coexist, with explicit formulations for helping and punishing probabilities.
• It shows that productive costly punishment can enhance collective welfare by elevating ISR's frequency under precise parameter thresholds.
• The analysis reveals that ALLD acts as an evolutionary shield against ALLC, ensuring the stability of the mixed equilibrium even when facing complexity costs.

Mixed equilibrium of integrated strong reciprocity (ISR) and unconditional defection (ALLD) refers to a robust, stable coexistence between two strategies—ISR, which combines upstream and downstream reciprocity with costly punishment, and ALLD, which unconditionally defects—in evolutionary game dynamics. This equilibrium emerges under parameter regimes where the cost and efficacy of punishment, as well as modest cognitive or complexity burdens, permit both strategies to persist within a population. Critically, under ISR, costly punishment not only enforces cooperation but, when sufficiently efficient, increases collective welfare over indirect reciprocity alone. The presence of ALLD serves a distinctive protective function, precluding the invasion of unconditional cooperation (ALLC) and stabilizing the mixed polymorphism against second-order free-riding and complexity penalties (Sasaki et al., 7 Jan 2026).

1. Mathematical Formulation of the ISR–ALLD System

In the restricted two-strategy subspace, the population frequency of ISR is denoted by $x$ and ALLD by $1-x$. Key model parameters include the benefit ($b$) and cost ($c$) of helping acts ($b>c>0$), the fine levied against defectors ($K>0$), the private cost incurred by punishers ($k>0$), and a complexity cost specific to ISR ($d\geq 0$).

The probability that an ISR individual helps or punishes depends on the population composition:

• The helping probability: $h(x) = x(2-x)$,
• The punishing probability: $p(x) = x(1-x)$.

Payoff functions for ISR and ALLD are: $$\begin{align*} \pi_{\text{ISR}}(x) &= b x - c h(x) - k p(x) - d, \ \pi_{\text{ALLD}}(x) &= b h(x) - K p(x). \end{align*}$$ Selection dynamics are governed by the replicator equation: $$\dot{x} = x(1-x)[\pi_{\text{ISR}}(x) - \pi_{\text{ALLD}}(x)],$$ with the selection-difference function $G(x) = \pi_{\text{ISR}}(x) - \pi_{\text{ALLD}}(x)$. Simplification yields a quadratic form: $G(x) = -A x^2 + B x - C,$ where $A = b-c+K+k$, $B = b-2c+K+2k$, and $C = d+k$. The stable mixed equilibrium $x^*$ exists at the larger root: $x^* = \frac{B + \sqrt{B^2 - 4AC}}{2A},$ provided $B^2 > 4AC$ and $x^*\in (0,1)$.

2. Stability and Dynamics of the Mixed Equilibrium

Linearization of the replicator equation at $x^*$ gives the local stability criterion. The sign of $G'(x^*) = -2A x^* + B$ determines stability; negativity guarantees local asymptotic stability of $x^*$. Algebraic analysis confirms that, when $B^2 > 4AC$, $x^*$ is stable, and the smaller root $x_1$ serves as an invasion threshold; $x$ below $x_1$ leads to extinction of ISR, while $x > x_1$ evolutionarily attracts to $x^*$. No interior equilibrium exists in the full simplex including ALLC, as ALLC is strictly dominated by ALLD on this edge; global bistability thus holds between ALLD fixation and the ISR–ALLD mixed equilibrium (Sasaki et al., 7 Jan 2026).

3. Collective Welfare and Productive Punishment

Collective welfare at equilibrium is quantified as the average payoff in the population. At coexistence, $\pi_{\text{ISR}}(x^*) = \pi_{\text{ALLD}}(x^*)$, so welfare is given by: $W(x^*) = b[2x^* - (x^*)^2] - K x^*(1-x^*).$ Without punishment or complexity costs ($K = k = d = 0$), the baseline integrated indirect reciprocity (IIR) equilibrium is $x_0 = \frac{b-2c}{b-c}$ for $b>2c$, and its welfare is $W_0 = b[2x_0 - x_0^2]$. Punishment is termed productive if $W(x^*) > W_0$. Analytic and implicit criteria identify parameter ranges (notably sufficiently large $K$) where this welfare increase occurs. Under $b \geq 2c$, increasing $K$ (with $k$ and $d$ fixed) both increases the ISR frequency $x^*$ and raises welfare, securing the productivity of costly punishment beyond a critical threshold.

4. ALLD as Evolutionary Shield and Barrier to ALLC

ALLD persists in the population even when ISR is stable, serving as an "evolutionary shield." In ISR–ALLD mixtures, ALLC individuals are always classified as "Bad" for failing to punish ALLD, and consequently become targets of ISR punishment; this prevents invasion or advantage by ALLC or other second-order free-riding strategies. On the ISR–ALLC edge, ISR and ALLC are neutral at $d=0$, but any $d>0$ or $k>0$ makes ISR strictly dominated. However, the presence of ALLD pins ALLC at a vanishing frequency and safeguards ISR from exploitation by ALLC. This structural role of ALLD is essential for the maintenance of the polymorphic equilibrium.

5. Impact of Complexity Costs

Complexity costs ($d>0$) incurred by ISR strategies reduce but do not destroy the stable coexistence with ALLD. The quadratic discriminant $B^2 - 4A(d+k)$ shrinks as $d$ increases, lowering the equilibrium frequency $x^*$,

$$\frac{\partial x^*}{\partial d} = -\frac{x^*}{G'(x^*)} < 0,$$

yet $x^*$ remains positive and stable as long as $B^2 > 4A(d+k)$. Importantly, ISR is robust to modest complexity costs: defectors continue to function as evolutionary shields, preserving the key invasion threshold $x_1$ and the stable equilibrium $x^*$. This resilience sharply contrasts with standard strong reciprocity (SR, combining downstream reciprocity and punishment without upstream reciprocity), where any $d>0$ eliminates the mixed equilibrium by making SR strictly dominated by ALLC along their edge.

6. Summary and Significance

The mixed equilibrium of ISR and ALLD represents a robust, stable coexistence under evolutionary game theory. Its defining features include:

• Stability arises from selection dynamics on the ISR–ALLD edge, with explicit quadratic forms for both selection-difference and equilibrium frequency.
• Increased punishment efficiency can yield "productive punishment," where population welfare exceeds that of baseline indirect reciprocity.
• ALLD serves both as an evolutionary shield against ALLC invasion and as a structural necessity for dynamical stability.
• ISR's resilience to complexity (cognitive or reputational) costs highlights a qualitative distinction from traditional downstream reciprocity and punishment schemes.
• No interior three-strategy equilibrium with ALLC exists in the relevant parameter regimes, leading to global bistability between ALLD fixation and the ISR–ALLD polymorphism.

This framework delineates the parameter landscape and evolutionary logic ensuring the coexistence and evolutionary viability of costly cooperation enforcement mechanisms, conditional on the continued presence of defectors and underlines the broader relevance of integrated strong reciprocity models to the evolution of large-scale cooperation (Sasaki et al., 7 Jan 2026).