Pareto Efficiency¶

Definition¶

Let $S$ be the set of pure-strategy profiles of a normal-form game with players $N=\{1,\ldots,n\}$ and payoff functions $v_i$ .

A profile $s\in S$ Pareto dominates $s'\in S$ if

v_i(s)\ge v_i(s')\quad\text{for every } i\in N,\quad\text{and}\quad v_j(s)>v_j(s')\quad\text{for at least one } j\in N.

A profile $s^*\in S$ is Pareto efficient (Pareto optimal) if no profile in $S$ Pareto dominates it. Moving from $s'$ to a profile that Pareto dominates it is a Pareto improvement.

Interpretation¶

Pareto efficiency judges outcomes from the point of view of the players as a group: an outcome is inefficient if someone could be made better off without anyone being made worse off. It is a statement about payoffs, not about how the outcome could be reached.

The four Prisoner's Dilemma profiles plotted as payoff pairs (v_1, v_2); a grey quadrant north-east of the equilibrium (F,F) contains (M,M), and the other three points are efficient.

The figure plots each profile of the Prisoner's Dilemma example as the point $(v_1,v_2)$ . A profile Pareto dominates $(F,F)=(-4,-4)$ exactly when its point lies in the grey quadrant to the north-east, weakly better for both and not equal; $(M,M)=(-2,-2)$ lies there, so the equilibrium is inefficient and the arrow is a Pareto improvement. The three green points have no other point north-east of them, so they are Pareto efficient, and $(M,F)$ and $(F,M)$ cannot be ranked against $(M,M)$ or each other.

This separates two questions. A Nash equilibrium asks whether any single player gains by deviating; Pareto efficiency asks whether the group could do better. Non-cooperative analysis predicts play from individual incentives, so equilibria can be inefficient.

Two misunderstandings are common. Pareto efficiency is not fairness: giving one player everything can be efficient. And an efficient profile need not be better for everybody than an inefficient one; it only needs to be undominated.

Only the players' payoffs count. In a duopoly the players are the firms, so a profile that raises both firms' profits at the expense of consumers is still a Pareto improvement for the game.

Key Results¶

Pareto dominance is transitive and never holds between a profile and itself, but it is not complete: two profiles can be incomparable.
In a finite game a Pareto-efficient profile always exists: any profile that maximizes $\displaystyle \sum_{i\in N}v_i(s)$ is Pareto efficient, because a Pareto improvement would raise the sum.
A Nash equilibrium need not be Pareto efficient, and a Pareto-efficient profile need not be a Nash equilibrium.
A strictly dominant strategy equilibrium can be Pareto dominated (Prisoner's Dilemma).

Examples¶

1. Prisoner's Dilemma with payoffs $(M,M)=(-2,-2)$ $(- 2, - 2)$ , $(M,F)=(-5,-1)$ $(- 5, - 1)$ , $(F,M)=(-1,-5)$ $(- 1, - 5)$ , $(F,F)=(-4,-4)$ $(- 4, - 4)$ .

$(M,M)$ Pareto dominates $(F,F)$ , since $-2>-4$ for both players. So the equilibrium $(F,F)$ is not Pareto efficient.
$(M,F)$ is Pareto efficient: every other profile gives player 2 less than $-1$ .
Likewise $(F,M)$ and $(M,M)$ are Pareto efficient, so three of the four profiles are efficient.

2. Battle of the Sexes with $(O,O)=(2,1)$ $(2, 1)$ , $(F,F)=(1,2)$ and mismatches $(0,0)$ . Both $(O,O)$ and $(F,F)$ are Pareto efficient and incomparable: each is better for one player. Both mismatches are Pareto dominated.

3. Cournot duopoly. In a Cournot duopoly with price $100-q_1-q_2$ and unit cost $10$ , the equilibrium $q_1=q_2=30$ gives price $40$ and profit $30\cdot30=900$ each. If each firm produced $22.5$ , the price would be $55$ and each profit $45\cdot22.5=1012.5$ $1012.5$ , a Pareto improvement for the firms that is not an equilibrium.

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: Pareto optimality, Pareto efficient, Pareto optimal, Pareto optimum Keywords: Pareto dominates, Pareto dominated, Pareto improvement, collective welfare, cooperation versus equilibrium, Prisoner's Dilemma, Battle of the Sexes Symbols: element of, sum

Last updated: 22 October 2026 19:04