Game Theory 6: Pareto Optimality


In non-cooperative game theory, the focus is on the agents in the game and strategies that optimize their payoffs, which results in some form of equilibrium.

As we can see in the prisoner’s dilemma game the issue arises in that what turns out to be the equilibrium is suboptimal for all the agents when taken as a whole. One way of defining what we mean by suboptimal for all is the idea of Pareto Optimality.

Named after Vilfredo Pareto, Pareto optimality is a measure of efficiency.

Whereas Nash Equilibrium is a solution concept of non-cooperative games, Pareto optimality in game theory answers a very specific question of whether an outcome can be better than the other?

Pareto optimality is a notion of efficiency or optimality for all the members involved.

An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. That is to say, a Pareto optimal outcome cannot be improved upon without hurting at least one player.

To illustrate this let’s take the game called the stag hunt, wherein two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by themselves, but a hare is worth less than a stag.

In the stag hunt, there is a single outcome that is Pareto efficient, which is that they both hunt stags. With this outcome, both players receive a payoff of three, which is each player’s largest possible payoff for the game. In this case, we cannot switch to any other outcome and make at least one party better off without making anyone worse off. The stag option is here the only Pareto optimal outcome.

One of the features of a Nash equilibrium is that in general, it does not correspond to a socially optimal outcome. That is, for a given game it is possible for all the players to improve their payoffs by collectively agreeing to choose a strategy different from the Nash equilibrium. The reason for this is that some players may choose to deviate from the agreed-upon cooperative strategy after it is made in order to improve their payoffs further at the expense of the group. A Pareto optimal equilibrium describes a social optimum in the sense that no individual player can improve their payoff without making at least one other player worse off. Pareto optimality is not a solution concept, but it can be an important attribute in determining what solution the players should play, or learn to play over time.

This is the interesting thing about the prisoner’s dilemma, that all options are Pareto optimal except for the unique equilibrium, which is for both to defect. This strong contrast between Pareto optimality and Nash equilibrium is what makes the prisoner’s dilemma a central object of study in game theory. The fact that all of the overall efficient outcomes are the ones that do not occur in equilibrium, makes it a classical illustration of the core dynamic between cooperation and competition.

This is a good segway into the next section of the book where we will be talking about the dynamics of cooperation. Where we will be looking specifically at the overall outcomes trying to optimize them instead of just individual payoffs.