Cooperation in Dynamic Games with Asymmetric Players: The Role of Social Preferences

Abstract

I study a two-player repeated game where payoffs are asymmetric. Attempts to form cooperative agreements in such an environment, for example by pro-rating actions relative to the one-shot Nash equilibrium, have generally struggled in practice. I show that cooperative arrangements when payoff functions are asymmetric tend to disproportionately favor the larger player. Incorporating social concerns, either via envious feelings on the part of the smaller player or altruistic feelings on the part of the larger player, shift quasi-cooperative incentives toward outcomes that are more favorable to the smaller player.

Appendix: Introducing Asymmetry Makes Player 1 Less Cooperative Under Pro-rata Reductions in the Linear-Quadratic Framework

Consider the linear-quadratic framework introduced above, where ${\uppi }_i = ({\upalpha }_i - X)x_i, X = x_1 + x_2$ and ${\upalpha }_1 = 1, {\upalpha }_2 = 1 - {\upalpha }$. The incentive constraint supporting a quasi-cooperative regime when players choose the grim strategy is

$$\begin{aligned} {\uppi }_1^c \ge (1-{\updelta }) {\uppi }_1^d + {\updelta } pi_1^N, \end{aligned}$$

where ${\uppi }_i^N = (x_i^N)^2, \ x_1^N = (1+{\upalpha })/3, \ x_2^N = (1-2{\upalpha })/3, \ X^N = x_1^N + x_2^N = (2-{\upalpha })/3$. Pro-rata reductions implies $x_i^c = \mu x_i^N$, with $\mu < 1.$ In this setting, ${\uppi }_i^c = \mu ({\upalpha }_i - \mu X^N) x_i^N$. Optimal defection, should it occur, would entail $x_i^d = R_i(x_j^c) = ({\upalpha }_i - \mu x_j^N)/2$ and ${\uppi }_i^d = (x_i^d)^2$.

The incentive constraint can be recast as a condition on the discount factor:

$$\begin{aligned} {\updelta } \ge {\hat{{\updelta }}}_i \equiv \frac{{\uppi }_i^d - {\uppi }_i^c}{{\uppi }_i^d - {\uppi }_i^N}. \end{aligned}$$

Define $\Omega = {\hat{{\updelta }}}_1 - {\hat{{\updelta }}}_2$; it is immediate that $\Omega = 0$ if ${\upalpha } = 0$. I claim that $\partial \Omega /\partial {\upalpha } > 0$, from which it follows that $\Omega > 0$, for ${\upalpha } \in (0, .5]$. It is easy to see that $\Omega$ is proportional to ${\upomega } \equiv ({\uppi }_2^d - {\uppi }_2^N)({\uppi }_1^d - {\uppi }_1^c) - ({\uppi }_1^d - {\uppi }_1^N)({\uppi }_2^d - {\uppi }_2^c)$, so it suffices to show that ${\upomega }$ is increasing in ${\upalpha }$. Expanding the definition of ${\upomega }$ yields:

$$\begin{aligned} {\upomega }&= {\uppi }_1^d ({\uppi }_2^c - {\uppi }_2^N) - {\uppi }_2^d ({\uppi }_1^c - {\uppi }_1^N) + {\uppi }_1^c {\uppi }_2^N - {\uppi }_1^N {\uppi }_2^c \\&= {\uppi }_1^d ({\uppi }_2^c - {\uppi }_2^N) - {\uppi }_2^d ({\uppi }_1^c - {\uppi }_1^N) - {\uppi }_1^c ({\uppi }_2^c - {\uppi }_2^N) + {\uppi }_2^c ({\uppi }_1^c - {\uppi }_1^N) \\&= ({\uppi }_1^d - {\uppi }_1^c)({\uppi }_2^c - {\uppi }_2^N) - ({\uppi }_2^d - {\uppi }_2^c)({\uppi }_1^c - {\uppi }_1^N). \end{aligned}$$

Defining $G_i^d = {\uppi }_i^d - {\uppi }_i^c$ and $G_i^c = {\uppi }_i^c - {\uppi }_i^N$ allows ${\upomega }$ to be expressed more compactly as

$$\begin{aligned} {\upomega } = G_1^d G_2^c - G_2^d G_1^c. \end{aligned}$$

(15)

I note that when ${\upalpha } = 0$ one has $G_i^d = G^d \equiv \frac{(1 - \mu )^2}{4}$ and $G_i^c = G^c \equiv -\frac{(1-\mu )(1 + 2\mu )}{9}$, for $i = 1, 2$.

It is straightforward to calculate

$$\begin{aligned} \frac{\partial {G_1^c}}{\partial {{\upalpha }}}&= - \left( \frac{1-2{\upalpha }}{9} \right) (1 - \mu )^2; \\ \frac{\partial {G_2^c}}{\partial {{\upalpha }}}&= (1 - \mu ) \left( 1 + \Bigl (\frac{ 5 - 4{\upalpha }}{9} \Bigr ) (1 - \mu ) \right) ; \\ \frac{\partial {G_1^d}}{\partial {{\upalpha }}}&= \frac{2}{9} (1-2{\upalpha }) \mu ^2; \\ \frac{\partial {G_2^d}}{\partial {{\upalpha }}}&= -\frac{1}{2} \left( 1 - {\upalpha } - (1+{\upalpha }) \frac{\mu }{3} \right) \left( 1 + \frac{\mu }{3} \right) + \frac{\mu }{3} \left( 3 - 4{\upalpha } \right) + \frac{\mu ^2}{9} \left( 5 - 4{\upalpha } \right) . \end{aligned}$$

Using these expressions along with the characterizations of $G_i^c$ and $G_i^d$, evaluating at ${\upalpha } = 0$, and inserting into Eq. (15) yields

$$\begin{aligned} {\upomega }&= \Bigl (2\mu ^2 {-} \bigl (5\mu ^2 {+} 9\mu {+} (9{+}3\mu )(\mu /6 {-}1/2)\bigr )\Bigr ) G^c/9 {+} \Bigl ((1-\mu )(14 {-} 5\mu ) {-} (1-\mu )^2\bigr )G^d \Bigr )/9 \\&= \frac{\Bigl (\bigl (5\mu ^2 + 9\mu + (9+3\mu )(\mu /6 -1/2)\bigr ) - 2\mu ^2\Bigr ) (1-\mu )(1 + 2\mu )}{81} \\&\quad + \frac{\Bigl ((1-\mu )(14 - 5\mu ) - (1-\mu )^2\Bigr )(1 - \mu )^2 }{36} \\&> \Bigl (\bigl (5\mu ^2 + 9\mu + (9+3\mu )(\mu /6 -1/2)\bigr ) - 2\mu ^2\Bigr ) \frac{ (1-\mu )(1 + 2\mu )}{81} \end{aligned}$$

where I used the fact that $\mu < 1$, which implies the second term in the second line is strictly positive. Noting that $\frac{(1-\mu )(1 + 2\mu )}{81} >0$ for $\mu < 1$, and simplifying the first term, I conclude that a sufficient condition for ${\upomega } > 0$ is $7\mu (\mu + 3) - 9 >0,$ which is easily seen to hold for $\mu \ge 1/2.$