A theory of cooperation in games with an application to market socialism

Abstract

Economic theory has focused almost exclusively on how humans compete with each other in their economic activity, culminating in general equilibrium (Walras–Arrow–Debreu) and game theory (Cournot–Nash). Cooperation in economic activity is, however, important, and is virtually ignored. Because our models influence our view of the world, this theoretical lacuna biases economists’ interpretation of economic behavior. Here, I propose models that provide micro-foundations for how cooperation is decentralized by economic agents. It is incorrect, in particular, to view competition as decentralized and cooperation as organized only by central diktat. My approach is not to alter preferences, which is the strategy behavioral economists have adopted to model cooperation, but rather to alter the way that agents optimize. Whereas Nash optimizers view other players in the game as part of the environment (parameters), Kantian optimizers view them as part of action. When formalized, this approach resolves the two major failures of Nash optimization from a welfare viewpoint – the Pareto inefficiency of equilibria in common-pool resource problems (the tragedy of the commons) and the inefficiency of equilibria in public-good games (the free rider problem). An application to market socialism shows that the problems of efficiency and distribution can be completely separated: the dead-weight loss of taxation disappears.

Keywords:

• Kantian equilibrium
• cooperation
• tragedy of the commons
• free rider problem
• market socialism

JEL codes:

• D70
• D50
• D60
• D70

Acknowledgement

The author grateful to Roberto Veneziani for organizing the symposium that led to this issue, and to the authors of the contributions herein for stimulating my thinking on Kantian cooperation.

Notes on contributor

John Roemer is a professor of political science and economics at Yale University. He works on problems at the intersection of economic theory, political theory and political philosophy. Along with cooperation, his recent work has focussed on climate change and intergenerational ethics, and the political consequences of income and wealth inequality.

Notes

1 Unravelling will not occur at the $q^{*}$ equilibrium in Figure 1, which is a stable equilibrium. But it will occur if the equilibrium is at $q=1$ .

2 The set of strategy profiles where all players play the same strategy is called an isopraxis, by J. Silvestre, in his contribution to this issue.

3 Simple Kantian equilibria exist for a broader class of games than symmetric ones (see Roemer in press, chapter 2).

4 Indeed, one can prove that if the payoff functions are differentiable, any interior Nash equilibrium in a monotone game is Pareto inefficient (Roemer in press, Proposition 3.3).

5 Let the payoff functions of the game be $\left\{V^i\right\}$ where the strategy space for each player is an interval of real numbers. Then $(E^1,\dots ,E^n)$ is a $K^{\times }$ equilibrium of the game when

$(\forall i)(\underset{r\ge 0}{\arg \max }{V}^i(r\mathbf{E})=1)$ .

6 For general games, $(E^1,\dots ,E^n)$ is an additive Kantian equilibrium if:

$(\forall i)\left(\underset{r}{\arg \max }{V}^i(E^1+r,\dots ,E^n+r)=0\right).$

7 There are two caveats. If $\text{β}<0$, the allocation rules can also be efficiently implemented; however, Kantian allocations may not exist. And for $\text{β=}0$ we must insist that the allocation be strictly positive. (The zero vector is a multiplicative Kantian equilibrium but it is not Pareto efficient.)

8 The ‘standard conditions’ on G are the Inada conditions and homotheticity; a consumption good is normal if an increase in income generates an increase in the good’s consumption. These conditions are sufficient for equilibrium; they may not be necessary.

9 The results exposited in this section, in a somewhat more complicated model, are given in Roemer (in press).

10 See Sonnenschein (1972), Debreu (1974), Mantel (1974).

11 In like manner, a sequence of iterated best responses also often converges to a Kantian equilibrium. See Roemer (in press, chapter 7).