Pareto Optimality

Relevant Courses: Game Theory

Often, payoffs between players are not relatively measurable: for example, the payoff for Player A might be in lives taken, while for Player B it could be in money stolen from the bank. There isn’t any strict “value” you can put to a life in terms of money stolen from the bank. So these two payoffs are like “currencies” that don’t have a conversion rate. With these differences in payoff units, how do we determine what is “universally” best? Well, we can’t determine what’s universally best, but we can come up with a list of options that could be universally best, depending on how much each player’s interests are valued. These outcomes are called Pareto optimal.

Pareto Domination

Let $o$ and $o^{\prime }$ be arbitrary outcomes of a game. We say that $o$ Pareto-dominates $o^{\prime }$ if outcomes $o$ is at least as “good” for every agent as $o^{\prime }$, AND there is at least one agent who strictly prefers $o$ to $o^{\prime }$.

Summary: Pareto Domination

Let $o$ and $o^{\prime }$ be arbitrary outcomes of a game $G=⟨{\mathbb{N}}_{\le n},A,u⟩$ with $n$ players. Let’s say $o$ and $o^{\prime }$ are defined by a set of actions $o=(a_1,\dots ,a_n)$ for each player, and $o^{\prime }=(a_1^{\prime },\dots ,a_n^{\prime })$.

$o$ is said to Pareto-dominate $o^{\prime }$ if, $\forall i$, $u_i(a_i)\ge u_i(a_i^{\prime })$ and $\exists k$ for which $u_k(a_k)>u_k(a_k^{\prime })$.

Pareto Optimal

Definition: Pareto Optimal

An outcome $o^{\ast }$ is Pareto-optimal if there is no other outcome that Pareto-dominates it.

Note that every game has at least one Pareto-optimal outcome.