Groucho Marxism

Game theory is the study of mathematical models of conflict. Such models generally involve a finite set of players labelled 1,2,…,n; a finite set of actions A_i available to each player i; and a reward or utility function r_i for each player i. The reward function maps the set A = ∏_j A_j to the real numbers, with the interpretation that r_i(a) represents the reward or payoff to player i when the players take the action profile a ∈ A. A mixed strategy p_i for player i is a probability distribution over the set A_i, with the interpretation that p_i(a_i) represents the probability that player i will take action a_i ∈ A_i. It is generally assumed that each player will play a mixed strategy. It is also usually assumed that these strategies are statistically independent of each other.

Statistical independence means that the action taken by each player tells us nothing about the actions the other players will play. This is expressed mathematically by saying that probability the players take the action profile a ∈ A is given by the product p(a) = ∏_j p_j(a_j). Then the expected payoff for player i is given by ∑_A r_i(a)p(a). A key concept in game theory is the Nash equilibrium, named after the American mathematician John Nash. The mixed strategy profile p is said to be a Nash equilibrium if for all players i and actions a_i,b_i ∈ A_i, we have ∑_A r_i(a)p(a) ≥ ∑_A r_i([a,b_i])p(a), where [a,b_i] ∈ A is defined by [a,b_i]_i = b_i and [a,b_i]_j = a_j for i ≠ j. In other words, in a Nash equilibrium no player can do better by unilaterally changing their strategy.

Recently, the American Marxist economist John Roemer has devised a novel solution concept for game theory he refers to as a Kantian equilibrium, named after the 18th century German philosopher Emmanuel Kant. It formalizes Kant’s categorical imperative – acting according to a maxim you would wish to see universalized – into mathematical optimization. Instead of asking “What is my best response if I hold others’ strategies fixed?” (the Nash approach), a player in a Kantian equilibrium asks: “What action would I want everyone to take if everyone were to do the same thing as me?” This solution concept only makes sense when the players’ action sets are all the same, and the players only have two actions available to them, so we can set A₁ = A₂ = … = A_n = {0,1}.

In this model a mixed strategy for player i can be represented as a number q_i in the interval [0,1] representing the probability that player i takes action 1, and a mixed strategy profile can then be represented as a vector q = (q_j). For a ∈ A let J(a,0) and J(a,1) denote the set of indices j where a_j = 0 and a_j = 1 respectively. The probability that the players take action profile a ∈ A is then given by the product p(a,q) = ∏_J(a,1) q_ja_j ∏_J(a,0) (1-q_ja_j). The mixed strategy profile represented by  q is said to be a (multiplicative) Kantian equilibrium if for all players i and real numbers c with 0 ≤ cq_j ≤ 1 for all players j,  the following inequality holds: ∑_A r_i(a)p(a,q) ≥ ∑_A r_i(a)p(a,cq). In other words, no player can do better if all players scale their mixed strategies by the same amount.

A correlated equilibrium is a game theory solution concept which relaxes the statistical independence assumption and is therefore more general than the well-known Nash equilibrium. If p is a probability distribution over A then p is said to be a correlated equilibrium if for all players i and actions a_i,b_i ∈ A_i, we have ∑_A r_i(a)p(a) ≥ ∑_A r_i([a,b_i])p(a), where again [a,b_i]_i = b_i and [a,b_i]_j = a_j for i ≠ j. Thus, a Nash equilibrium is a special case of a correlated equilibrium where the actions taken by the players are statistically independent; that is, where p(a) = ∏_j p_j(a_j). On the other hand, a Kantian equilibrium can be seen as a special case of a correlated equilibrium where A₁ = A₂ = … = A_n = {0,1} and the actions taken by the players are linearly dependent. I will leave the demonstration of this to a future blog post.