A Note on Altruism in Asymmetric Games: An Indirect Evolutionary Approach

Abstract

This article studies the evolution of altruism. We consider a model in which a population of agents are assortatively matched to play some asymmetric two-player game, and evolution operates at the level of behavior rules. We find that the relationship between the evolutionarily stable level of altruism and the index of assortativity of matching is determined by two novel features: (1) whether the total payoff function of the game exhibits complementarity or substitutability; (2) whether the two players’ strategies affect each other’s fitness in the same direction or the opposite. These two features combined generalize the stability conditions related to Hamilton’s rule to a class of asymmetric games.

Appendix

Proof of Lemma 1

By the first-order necessary conditions (14) and (15) for interior Nash equilibrium of $\Gamma _{mn}$, we can write

$$\begin{aligned} V_{{m1}} (\alpha ,\beta ) & = \pi _{{m1}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times x_{1}^{*} (\alpha ,\beta ) + \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times y_{2}^{*} (\beta ,\alpha ) \\ & = - \alpha \pi _{{n2}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{1}^{*} (\alpha ,\beta ) + \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times y_{2}^{*} (\beta ,\alpha ); \\ \end{aligned}$$

(24)

$$\begin{aligned} V_{{n1}} (\beta ,\alpha ) & = \pi _{{n1}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times y_{1}^{*} (\beta ,\alpha ) + \pi _{{n2}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{2}^{*} (\alpha ,\beta ) \\ & = - \beta \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times y_{1}^{*} (\beta ,\alpha ) + \pi _{{n2}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{2}^{*} (\alpha ,\beta ); \\ \end{aligned}$$

(25)

$$\begin{aligned} V_{{m2}} (\alpha ,\beta ) & = \pi _{{m1}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times x_{2}^{*} (\alpha ,\beta ) + \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha ) \times y_{1}^{*} (\beta ,\alpha ) \\ & = - \alpha \pi _{{n2}} (y^{*} \beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{2}^{*} (\alpha ,\beta ) + \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times y_{1}^{*} (\beta ,\alpha ); \\ \end{aligned}$$

(26)

$$\begin{aligned} V_{{n2}} (\beta ,\alpha ) & = \pi _{{n1}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times y_{2}^{*} (\beta ,\alpha ) + \pi _{{n2}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{1}^{*} (\alpha ,\beta ) \\ & = - \beta \pi _{{m2}} (x^{*} (\alpha ,\beta ),y^{*} (\beta ,\alpha )) \times y_{2}^{*} (\beta ,\alpha ) + \pi _{{n2}} (y^{*} (\beta ,\alpha ),x^{*} (\alpha ,\beta )) \times x_{1}^{*} (\alpha ,\beta ). \\ \end{aligned}$$

(27)

Recall that the necessary condition for $\alpha$ to be evolutionarily stable in a monomorphic population is that the evolutionary drift function $D(\alpha , \beta )$ (as a function of $\beta$) takes value of zero at the point where $\beta =\alpha$. Hence, we substitute all $\beta$ in (24) to (27) with $\alpha$ and plug them into the evolutionarily stationary condition $D(\alpha , \alpha )=0$, we have:

$$\begin{aligned} (\sigma _0-\alpha )(\pi _{n2}x^*_1(\alpha , \alpha )+\pi _{m2}y^*_1(\alpha , \alpha ))+(1-\sigma _0\alpha )(\pi _{n2}x^*_2(\alpha , \alpha )+\pi _{m2}y^*_2(\alpha , \alpha ))=0. \end{aligned}$$

(28)

Q.E.D. $\square$

Proof of Theorem 1

Let us first differentiate the first-order necessary conditions (14) and (15) for interior Nash equilibrium of $\Gamma _{mn}$ with respect to $\alpha$:

$$\begin{aligned}&\pi _{m11}x^*_1 + \pi _{m12}y^*_2+\alpha (\pi _{n12}y_2^*+\pi _{n22}x^*_1)+\pi _{n2} = 0; \end{aligned}$$

(29)

$$\begin{aligned}&\pi _{n11}y^*_2 +\pi _{n12}x^*_1+\beta (\pi _{m12}x^*_1+\pi _{m22}y^*_2) = 0. \end{aligned}$$

(30)

Equations (29) and (30) give us

$$\begin{aligned}&x^*_1(\alpha , \beta )=\frac{-(\pi _{n11}+\beta \pi _{m22}) \pi _{n2}}{(\pi _{m11}+\alpha \pi _{n22})(\pi _{n11}+\beta \pi _{m22})-(\pi _{n12}+\beta \pi _{m12})(\pi _{m12}+\alpha \pi _{n12})}, \end{aligned}$$

(31)

$$\begin{aligned}&y^*_2(\beta , \alpha )=\frac{(\pi _{n12}+\beta \pi _{m12}) \pi _{n2}}{(\pi _{m11}+\alpha \pi _{n22})(\pi _{n11}+\beta \pi _{m22})-(\pi _{n12}+\beta \pi _{m12})(\pi _{m12}+\alpha \pi _{n12})}. \end{aligned}$$

(32)

Next, let us differentiate (14) and (15) with respect to $\beta$:

$$\begin{aligned}&\pi _{m11}x^*_2 + \pi _{m12}y^*_1+\alpha (\pi _{n12}y_1^*+\pi _{n22}x^*_2) = 0;\end{aligned}$$

(33)

$$\begin{aligned}&\pi _{n11}y^*_1 +\pi _{n12}x^*_2+\beta (\pi _{m12}x^*_2+\pi _{m22}y^*_1)+\pi _{m2} = 0. \end{aligned}$$

(34)

Equations (33) and (34) give us

$$\begin{aligned}&x^*_2(\alpha , \beta )=\frac{(\pi _{m12}+\alpha \pi _{n12}) \pi _{m2}}{(\pi _{m11}+\alpha \pi _{n22})(\pi _{n11}+\beta \pi _{m22})-(\pi _{n12}+\beta \pi _{m12})(\pi _{m12}+\alpha \pi _{n12})}, \end{aligned}$$

(35)

$$\begin{aligned}&y^*_1(\beta , \alpha )=\frac{-(\pi _{m11}+\alpha \pi _{n22}) \pi _{m2}}{(\pi _{m11}+\alpha \pi _{n22})(\pi _{n11}+\beta \pi _{m22})-(\pi _{n12}+\beta \pi _{m12})(\pi _{m12}+\alpha \pi _{n12})}. \end{aligned}$$

(36)

Recall that the necessary condition for $\alpha$ to be evolutionarily stable in a monomorphic population is that the evolutionary drift function $D(\alpha , \beta )$ (as a function of $\beta$) takes the value of zero at the point where $\beta =\alpha$. Hence, we replace $\beta$ in (31), (32), (35), and (36) with $\alpha$ and plug them into the evolutionarily stationary condition given by (28), we have

$$\begin{aligned} (\sigma _{0} - \alpha ) ( - \pi _{{n2}}^{2} (\pi _{{n11}} + \alpha \pi _{{m22}} ) - \pi _{{m2}}^{2} (\pi _{{m11}} + \alpha \pi _{{n22}} )) \\ + (1 - \sigma _{0} \alpha )(1 + \alpha )\pi _{{n2}} \pi _{{m2}} (\pi _{{m12}} + \pi _{{n12}} ) = 0. \\ \end{aligned}$$

(37)

The second-order necessary conditions for an interior Nash equilibrium of $\Gamma _{mn}$ tell us that $\pi _{n11}+\alpha \pi _{m22}<0$, and $\pi _{m11}+\alpha \pi _{n22}<0$, and we know $T_{12}=\pi _{m12}+\pi _{n12}$. Hence, $\alpha>\sigma _0$ requires that $\pi _{n2}\pi _{m2}T_{12}>0$, $\alpha <\sigma _0$ requires that $\pi _{n2}\pi _{m2}T_{12}<0$, and $\alpha =\sigma _0$ requires that $\pi _{n2}\pi _{m2}T_{12}=0$. These give us the desired results as stated in the theorem. Q.E.D. $\square$