Two Different Questions About an Outcome
A strategic outcome can be stable against individual deviations without being the best outcome for the players collectively. Conversely, an outcome may leave no available way to make one player better off without hurting the other, yet still give one or both players an incentive to change their actions alone. This distinction matters whenever a payoff matrix is used to evaluate strategic choices.
A Nash equilibrium tests individual incentives: holding the other player’s action fixed, can either player gain by changing their own action? Pareto efficiency compares outcomes: is there another feasible outcome that makes at least one player better off and nobody worse off? These are different tests, so neither property automatically implies the other.
Formal Definitions
A player’s best response to the other player’s action is an action that gives that player the highest payoff among their available actions, given that the other player does not change theirs. A pure-strategy outcome is a Nash equilibrium when each player’s action is a best response to the other player’s action.
The comparison holds the other player’s action fixed. It does not ask whether the players could both change actions and obtain higher payoffs.
For two payoff pairs \((x_A,x_B)\) and \((y_A,y_B)\), the first Pareto dominates the second precisely when \(x_A \geq y_A\), \(x_B \geq y_B\), and at least one of those inequalities is strict. Pareto comparisons use each player’s own payoff scale; they do not require comparing one player’s payoff directly with the other’s.
Why the Tests Differ
The Nash condition checks deviations one player at a time. Pareto dominance allows a comparison with any feasible outcome, including one reached by changing both players’ actions. That difference explains how a Nash equilibrium can be Pareto dominated.
More formally, at a Nash equilibrium, a unilateral change by A cannot raise A’s payoff while B’s action stays fixed; the analogous statement holds for B. But a different outcome may raise both payoffs if both actions change. The Nash inequalities do not rule out such an outcome. In the other direction, Pareto efficiency only rules out an outcome that improves one player without hurting another. It does not rule out a profitable unilateral deviation that helps the deviating player while reducing the other player’s payoff.
This is the key justification for keeping the concepts separate: their inequalities compare different sets of alternatives. A Nash test compares each player’s current payoff with payoffs from that player’s own deviations. A Pareto test compares the current payoff pair with payoff pairs from all feasible outcomes.
Worked Examples
Worked Example: A Stable Outcome That Both Players Could Improve
Two hypothetical firms simultaneously choose either a high-promotion campaign (P) or a restrained campaign (Q). Their payoffs are in arbitrary profit units. Find the pure-strategy Nash equilibrium, then determine whether it is Pareto efficient.
| Firm A / Firm B | P | Q |
|---|---|---|
| P | (3, 3) | (7, 1) |
| Q | (1, 7) | (5, 5) |
Method: mark each player’s best response in turn. If B chooses P, A compares 3 from P with 1 from Q, so A’s best response is P. If B chooses Q, A compares 7 from P with 5 from Q, so A’s best response is again P.
For B, if A chooses P, B compares 3 from P with 1 from Q, so B’s best response is P. If A chooses Q, B compares 7 from P with 5 from Q, so B’s best response is P. The only cell in which both actions are best responses is (P, P), with payoffs (3, 3). This is the unique pure-strategy Nash equilibrium.
Now compare (P, P) with (Q, Q). The latter gives (5, 5), and \(5>3\) for each firm. Thus (Q, Q) Pareto dominates the Nash equilibrium. The equilibrium is not Pareto efficient.
Check: At (P, P), either firm changing its action alone receives 1 instead of 3, so neither wants to deviate unilaterally. The improvement to (Q, Q) requires both firms to change. The common mistake is to treat an outcome that benefits both firms as proof that the current cell cannot be a Nash equilibrium; Nash stability concerns unilateral changes only.
Worked Example: Testing an Efficient Nash Equilibrium Carefully
Two hypothetical service providers choose between plans U and D for Provider A and plans L and R for Provider B. Find the pure-strategy Nash equilibrium and test it against every other outcome for Pareto dominance.
| A / B | L | R |
|---|---|---|
| U | (8, 6) | (5, 9) |
| D | (7, 4) | (4, 3) |
Method: list best responses by holding the other action fixed. If B chooses L, A prefers U because \(8>7\). If B chooses R, A prefers U because \(5>4\). If A chooses U, B prefers R because \(9>6\). If A chooses D, B prefers L because \(4>3\).
The only cell where both players are choosing best responses is (U, R), with payoffs (5, 9). At (U, L), B can switch to R and raise its payoff from 6 to 9. At (D, L), A can switch to U and raise its payoff from 7 to 8. At (D, R), A can switch to U and raise its payoff from 4 to 5. Therefore (U, R) is the unique pure-strategy Nash equilibrium.
To test efficiency, compare its payoffs (5, 9) with the other three pairs. Outcome (U, L) gives A 8 rather than 5, but B gets 6 rather than 9. Outcome (D, L) gives A 7 rather than 5, but B gets 4 rather than 9. Outcome (D, R) gives (4, 3), which is lower for both players than (5, 9). None of these outcomes gives both players at least as much as (U, R), with a strict gain for someone. Therefore (U, R) is Pareto efficient among the four outcomes.
Check: The row player prefers U to D when B chooses R because \(5>4\), while B prefers R to L when A chooses U because \(9>6\). This confirms the equilibrium. A common error is to call (U, L) an equilibrium after noticing A’s high payoff of 8. But B would switch from L to R, increasing its payoff from 6 to 9.
Worked Example: An Efficient Outcome That Is Not an Equilibrium
Two hypothetical organizations choose actions X or Y for A and C or D for B. Identify the Nash equilibrium and determine whether the outcome (X, C) is Pareto efficient.
| A / B | C | D |
|---|---|---|
| X | (4, 4) | (1, 6) |
| Y | (6, 1) | (3, 3) |
Method: first find best responses, then make a separate Pareto comparison. If B chooses C, A prefers Y because \(6>4\). If B chooses D, A prefers Y because \(3>1\). If A chooses X, B prefers D because \(6>4\). If A chooses Y, B prefers D because \(3>1\). Thus (Y, D) is the unique pure-strategy Nash equilibrium, with payoffs (3, 3).
At (X, C), neither player is playing a best response: A can switch to Y and raise its payoff from 4 to 6, and B can switch to D and raise its payoff from 4 to 6. So (X, C) is not a Nash equilibrium.
It is nevertheless Pareto efficient. The other outcomes pay (1, 6), (6, 1), and (3, 3). The first gives B more but A less than (4, 4); the second gives A more but B less; and the third gives both less. None Pareto dominates (4, 4). In contrast, (X, C) with (4, 4) Pareto dominates the equilibrium (Y, D) with (3, 3).
Check: A non-equilibrium outcome can be efficient because efficiency does not forbid a player from having a profitable unilateral deviation if that deviation hurts the other player. Here each deviation from (X, C) raises the deviator’s payoff but lowers the other player’s payoff. The common mistake is to conclude that a Pareto-efficient outcome must be a Nash equilibrium.
Reading Payoff Matrices Without Confusing the Criteria
A reliable analysis keeps two tasks distinct. First, find the strategic stability of each cell by comparing unilateral deviations. Second, compare payoff pairs to identify Pareto improvements. The process below prevents one type of comparison from being substituted for the other.
Read each cell in the stated player order. A reversed payoff pair can change both best-response and Pareto conclusions.
For each action of the other player, compare the payoffs from the player’s own available actions. Mark every action tied for the highest payoff, if there is a tie.
A cell is a pure-strategy Nash equilibrium only when both players’ actions are best responses there.
For each outcome of interest, check whether another feasible cell has payoffs at least as high for every player and strictly higher for at least one.
The comparison should not be replaced by simply adding the two payoffs. A larger total does not establish Pareto dominance: one player might gain while the other loses. Likewise, an outcome with equal payoffs is not automatically efficient; another outcome may give both players more. Pareto efficiency is about the direction of each player’s payoff change, not the sum or equality of payoffs.
Why the Distinction Matters
For economic analysis, a Nash equilibrium describes incentives under the rules and available actions of a particular game. It does not certify that the resulting allocation is desirable for all players. A Pareto comparison identifies whether a mutually beneficial alternative exists within the outcomes being considered, but it does not explain whether players have incentives to reach that alternative.
This distinction is useful when evaluating strategic interaction between firms, consumers, or other decision makers. If an equilibrium is Pareto dominated, the matrix reveals a potential joint gain, but the matrix alone does not establish that players can coordinate on the better outcome or have incentives to do so. If an efficient outcome is not an equilibrium, the comparison similarly does not show that players will choose it.
The result depends on the stated feasible outcomes and payoffs. If available actions or payoff consequences change, the equilibrium and the set of Pareto-efficient outcomes may change as well. For the calculations here, all listed cells are treated as feasible, and the payoff numbers represent the preferences relevant to the comparison. This distinction between strategic stability and mutual improvement is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, identify best responses before making Pareto comparisons. Payoff pairs list A’s payoff first and B’s second.
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Two hypothetical producers choose C or D. The payoffs are (C, C): (4, 4), (C, D): (8, 1), (D, C): (1, 8), and (D, D): (3, 3). Find the pure-strategy Nash equilibrium and determine whether it is Pareto dominated.
(Answer: The unique equilibrium is (C, C), with (4, 4); it is Pareto efficient among the four outcomes because none of the other payoff pairs gives both players at least 4.) -
In a two-plan game, A chooses M or N and B chooses X or Y. The payoffs are (M, X): (7, 2), (M, Y): (3, 5), (N, X): (6, 4), and (N, Y): (2, 3). Find the equilibrium and test its Pareto efficiency.
(Answer: The unique equilibrium is (M, Y), with (3, 5); it is Pareto efficient. Each other outcome gives either A less than 3 or B less than 5.) -
A game has payoffs (T, L): (2, 7), (T, R): (4, 5), (B, L): (6, 2), and (B, R): (3, 3). Which outcomes are Pareto efficient? Is there a pure-strategy Nash equilibrium?
(Answer: (T, L), (T, R), and (B, L) are Pareto efficient; (B, R) is dominated by (T, R). There is no pure-strategy Nash equilibrium: the best-response cells do not coincide.) -
In another game, the payoffs are (G, C): (5, 2), (G, D): (2, 6), (H, C): (4, 3), and (H, D): (1, 4). Find the equilibrium and explain why comparing total payoffs is not a substitute for the Pareto test.
(Answer: The unique equilibrium is (G, D), with (2, 6), and it is Pareto efficient. (G, C) and (H, C) give A more but B less, while (H, D) gives both less. The sum alone does not test whether every player is at least as well off.) -
For actions R or S by A and L or Q by B, the payoffs are (R, L): (10, 1), (R, Q): (6, 6), (S, L): (5, 5), and (S, Q): (1, 10). Find the pure-strategy equilibrium, identify the Pareto-efficient outcomes, and state whether any other outcome dominates the equilibrium.
(Answer: The unique equilibrium is (R, Q), with (6, 6). The Pareto-efficient outcomes are (R, L), (R, Q), and (S, Q); (S, L), with (5, 5), is dominated by (R, Q). No outcome dominates the equilibrium.