Nash equilibrium

In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy (i.e., by changing unilaterally). If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. In other words, to be a Nash equilibrium, each player must answer negatively to the question: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision. Likewise, many players are in Nash equilibrium if each one is making the best decision that they can, taking into account the decisions of the others. However, Nash equilibrium does not necessarily mean the best cumulative payoff for all the players involved; in many cases all the players might improve their payoffs if they could somehow agree on strategies different from the Nash equilibrium (e.g. competing businessmen forming a cartel in order to increase their profits).


The concept of the Nash equilibrium (NE) in pure strategies was first developed by Antoine Augustin Cournot in his theory of oligopoly (1838). Firms choose a quantity of output to maximize their own profit. However, the best output for one firm depends on the outputs of others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy NE. However, the modern game-theoretic concept of NE is defined in terms of mixed-strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy NE was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior. However, their analysis was restricted to the very special case of zero-sum games. They showed that a mixed-strategy NE will exist for any zero-sum game with a finite set of actions. The contribution of John Forbes Nash in his 1951 article Non-Cooperative Games was to define a mixed strategy NE for any game with a finite set of actions and prove that at least one (mixed strategy) NE must exist.


Informal definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. As a heuristic, one can imagine that each player is told the strategies of the other players. If any player would want to do something different after being informed about the others' strategies, then that set of strategies is not a Nash equilibrium. If, however, the player does not want to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium.


The Nash-equilibrium may also have non-rational consequences in sequential games because players may "threat" each other with non-rational moves. For such games the Subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

Formal definition

Let (S, f) be a game, where Si is the strategy set for player i, S=S1 X S2 ... X Sn is the set of strategy profiles and f=(f1(x), ..., fn(x)) is the payoff function. Let x-i be a strategy profile of all players except for player i. When each player i in {1, ..., n} chooses strategy xi resulting in strategy profile x = (x1, ..., xn) then player i obtains payoff fi(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile x* in S is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is

forall i,x_iin S_i, x_i neq x^*_{i} : f_i(x^*_{i}, x^*_{-i}) geq f_i(x_{i},x^*_{-i}).


When the inequality above holds strictly (with > instead of geq) for all players and all feasible alternative strategies, then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between x^*_i and some other strategy in the set S, then the equilibrium is classified as a weak Nash equilibrium.


Competition game

A competition game

This can be illustrated by a two-player game in which both players simultaneously choose a whole number from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then he/she has to give up two points to the other. This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number. In the table to the left, for example, when starting at the green square it is in player 1's interest to move to the purple square by choosing a smaller number, and it is in player 2's interest to move to the blue square by choosing a smaller number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria (0,0...1,1...2,2...and 3,3).

Coordination game

A coordination game
Player 2 adopts strategy 1 Player 2 adopts strategy 2
Player 1 adopts strategy 1 A, A B, C
Player 1 adopts strategy 2 C, B D, D

The coordination game is a classic (symmetric) two player, two strategy game, with the payoff matrix shown to the right, where the payoffs satisfy A>C and D>B. The players should thus coordinate, either on A or on D, to receive a high payoff. If the players' choices do not coincide, a lower payoff is rewarded. An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game.


The driving game
Drive on the Left Drive on the Right
Drive on the Left 100, 100 0, 0
Drive on the Right 0, 0 100, 100


Prisoner's dilemma

(note differences in the orientation of the payoff matrix) The Prisoner's Dilemma has the same payoff matrix as depicted for the Coordination Game, but now C > A > D > B. Because C > A and D > B, each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. The Prisoner's Dilemma thus has a single Nash Equilibrium: both players choosing strategy #2 ("betraying"). What has long made this an interesting case to study is the fact that D < A (ie., "both betray" is globally inferior to "both remain loyal"). The globally optimal strategy is unstable; it is not an equilibrium.

Nash equilibria in a payoff matrix


We can apply this rule to a 3x3 matrix:

A Payoff Matrix
Option A Option B Option C
Option A 0, 0 25, 40 5, 10
Option B 40, 25 0, 0 5, 15
Option C 10, 5 15, 5 10, 10


This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and NxN pure strategy Nash equilibria.








A Coalition-Proof Nash Equilibrium (CPNE) (similar to a Strong Nash Equilibrium) occurs when players cannot do better even if they are allowed to communicate and collaborate before the game. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core.


If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:

  2. The players are flawless in execution.
  3. The players have sufficient intelligence to deduce the solution.
  4. The players know the planned equilibrium strategy of all of the other players.
  5. The players believe that a deviation in their own strategy will not cause deviations by any other players.

Where the conditions are not met


  2. Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the 4th 'common knowledge' criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the game of 'chicken', ensuring a no-loss no-win scenario).

Where the conditions are met


NE and non-credible threats

The nash equilibrium is a superset of the subgame perfect nash equilibrium. The subgame perfect equilibrium in addition to the Nash Equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change his strategy.

The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise.

Proof of existence


is a best response to sigma_{-i}. Define

b(sigma) = b_1(sigma_{-1}) times b_2(sigma_{-2}) times cdots times b_n(sigma_{-n}).

One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a sigma^* such that sigma^* in b(sigma^*). Since b(sigma^*) represents the best response for all players to sigma^*, the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium.

When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed point theorem." (See Nasar, 1998, p. 94.)

Alternate proof using the Brouwer fixed point theorem

We have a game G=(N,A,u) where N is the number of players and A = A_1 times ldots times A_N is the action set for the players. All of the actions sets A_i are finite. Let Delta = Delta_1 times ldots times Delta_N denote the set of mixed strategies for the players. The finiteness of the A_is insures the compactness of Delta.

We can now define the gain functions. For a mixed strategy sigma in Delta, we let the gain for player i on action a in A_i be

Gain_i(sigma,a) = max {0, u_i(a_i, sigma_{-i}) - u_i(sigma_{i}, sigma_{-i})}

The gain function represents the benefit a player gets by unilaterally changing his strategy. We now define g = (g_1,ldots,g_N) where

g_i(sigma)(a) = sigma_i(a) + Gain_i(sigma,a)

for sigma in Delta, a in A_i. We see that

sum_{a in A_i} g_i(sigma)(a) = sum_{a in A_i} sigma_i(a) + Gain_i(sigma,a)
= 1 + sum_{a in A_i} Gain_i(sigma,a) > 0

We now use g to define f: Delta rightarrow Delta as follows. Let

f_i(sigma)(a) = frac{g_i(sigma)(a)}{sum_{b in A_i} g_i(sigma)(b)}

for a in A_i. It is easy to see that each f_i is a valid mixed strategy in Delta_i. It is also easy to check that each f_i is a continuous function of sigma, and hence f is a continuous function. Now Delta is the cross product of a finite number of compact convex sets, and so we get that Delta is also compact and convex. Therefore we may apply the Brouwer fixed point theorem to f. So f has a fixed point in Delta, call it sigma^*.

I claim that sigma^* is a Nash Equilibrium in G. For this purpose, it suffices to show that

forall 1 leq i leq N, ~ forall a in A_i, ~ Gain_i(sigma^*,a) = 0 text{.} This simply states the each player gains no benefit by unilaterally changing his strategy which is exactly the necessary condition for being a Nash Equilibrium.

Now assume that the gains are not all zero. Therefore, exists i, 1 leq i leq N, and a in A_i such that Gain_i(sigma^*, a) > 0. Note then that

sum_{a in A_i} g_i(sigma^*, a) = 1 + sum_{a in A_i} Gain_i(sigma^*,a) > 1 So let C = sum_{a in A_i} g_i(sigma^*, a). Also we shall denote Gain(i,cdot) as the gain vector indexed by actions in A_i. Since f(sigma^*) = sigma^* we clearly have that f_i(sigma^*) = sigma^*_i. Therefore we see that
sigma^*_i = frac{g_i(sigma^*)}{sum_{a in A_i} g_i(sigma^*)(a)} Rightarrow sigma^*_i = frac{sigma^*_i + Gain_i(sigma^*,cdot)}{C} Rightarrow Csigma^*_i = sigma^*_i + Gain_i(sigma^*,cdot)
left(C-1right)sigma^*_i = Gain_i(sigma^*,cdot) Rightarrow sigma^*_i = left(frac{1}{C-1}right)Gain_i(sigma^*,cdot)

Since C > 1 we have that sigma^*_i is some positive scaling of the vector Gain_i(sigma^*,cdot). Now I claim that

sigma^*_i(a)(u_i(a_i, sigma^*_{-i}) - u_i(sigma^*_i, sigma^*_{-i})) = sigma^*_i(a)Gain_i(sigma^*, a)

forall a in A_i. To see this, we first note that if Gain_i(sigma^*, a) > 0 then this is true by definition of the gain function. Now assume that Gain_i(sigma^*, a) = 0. By our previous statements we have that

sigma^*_i(a) = left(frac{1}{C-1}right)Gain_i(sigma^*, a) = 0 and so the left term is zero, giving us that the entire expression is 0 as needed.

So we finally have that

= left(sum_{a in A_i} sigma^*_i(a)u_i(a_i, sigma^*_{-i})right) - u_i(sigma^*_i, sigma^*_{-i})
= sum_{a in A_i} sigma^*_i(a) Gain_i(sigma^*, a) quad text{ by the previous statements }
= sum_{a in A_i} left(C -1 right) sigma^*_i(a)^2 > 0

where the last inequality follows since sigma^*_i is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore sigma^* is a Nash Equilibrium for G as needed.

See also


Game Theory textbooks

  • . Suitable for undergraduate and business students.
  • . A modern introduction at the graduate level.

Original Papers

  • Nash, John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences 36(1):48-49.

Other References

  • Nasar, Sylvia (1998), "A Beautiful Mind", Simon and Schuster, Inc.


External links

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