A Nash equilibrium is a strategy profile from which no player has incentive to unilaterally deviate. Each player correctly anticipates the strategy choice of all other players, and chooses his optimal strategy in response to what he anticipates everyone else will do. If the Nash equilibrium were revealed to each player individually, no player would want to switch from the strategy “assigned” to him by the Nash equilibrium.

What Is a Nash Equilibrium?

A Nash equilibrium is a strategy profile in game theory in which no player has a dominant strategy. Each player correctly anticipates the strategic choice of all other players, and thus has no incentive to unilaterally deviate from their own optimal strategy. The Nash existence theorem dictates that every game has at least one Nash equilibrium. 

Games play a key role in artificial intelligence (AI), and game environments are a popular training mechanism in areas such as reinforcement learning or imitation learning. In theory, any multi-agent AI system can be subjected to gamified interactions between its participants.

The branch of mathematics that formulates the principles of games is game theory. Game theory enables some key capabilities required in multi-agent environments in which different AI programs interact or compete in order to accomplish a goal. The famous Nash equilibrium is the cornerstone of many AI interactions in modern systems.

To best understand this definition of the Nash equilibrium, we will look at a game in which other game theory methods like the dominated strategy equilibrium and iteratively deleting strictly dominated strategies don’t work. 

 

Nash Equilibrium Situation Example

Two bars, Bar A and Bar B, are located near each other in the city center. Each bar seeks to maximize revenue, and chooses which price to set for a beer: $3, $4 or $5. Each bar has 60 potential customers, of which, 20 are locals and 40 are tourists. Locals will buy from the bar setting the lowest price. Tourists will choose a bar randomly.

 

Creating a Payoff Matrix 

The only difference with respect to the first version is the strategy $2 has been switched out for the strategy $3. This is the payoff matrix of the game:

payoff matrix for bar a vs bar b
Payoff matrix for two bars. | Image: Michael Kingston

In this version of the game, neither player has dominant or dominated strategies. In other words, all three prices are rationalizable strategies for each player. Let’s see the matrix again, underlining the associated payoffs for the best responses:

associated best responses underlined
Associated best responses in the payoff matrix underlined. | Image: Michael Kingston
import nashpy as nash
import numpy as np

row = np.array([[90, 120, 120], [80, 120, 160], [100, 100, 150]])
column = np.array([[90, 80, 100], [120, 120, 100], [120, 160, 150]])
bars_v2 = nash.Game(row, column)
print(bars_v2)
equilibria = bars_v2.support_enumeration()
for eq in equilibria:
    print(f"The unique Nash equilibrium is {eq}")
screenshot of code for nash equilibrium
Screenshot of code to find the Nash equilibrium. | Image: Michael Kingston

 What can we say about the strategy profile for which neither payoff is underlined, such as for the strategy Profile ($3, $3)? This means that neither strategy is the best response to the other. If Bar A expected Bar B to price at $3, it would also price at $3. And the same goes for Bar B. If both players were to suspect that ($3, $3) is how the game would end up, then the game wouldn’t end up that way. Both would have incentive to deviate to a different strategy.

What can we say about a strategy profile for which only one payoff is underlined, such as for the strategy profile ($3, $4)? The player whose payoff is underlined is the best result. In this example, that is Bar A. If Bar A expected Bar B to price their beer at $4, it would be in Bar A’s best interest to price theirs at $3. There isn’t another strategy that gives a higher payoff.

But the player whose payoff is not underlined is not best responding. In this example, that is Bar B. If Bar B expects Bar A to price at $3, it should not price at $4 but rather at $5. If both players were to suspect that ($3, $4) is how the game would end up, then the game wouldn’t end up that way. One player would have incentive to deviate to a different strategy.

Finally, what can we say about a strategy profile in which all players’ payoffs are underlined? In this game, there is only one such strategy profile: ($4, $4). If both bars price at $4, each is best responding to the other. If both players were to suspect that ($4, $4) is how the game would end up, then the game would end up exactly that way. Given that Bar B is pricing at $4, pricing at $4 is Bar A’s best response. And given that Bar A is pricing at $4, pricing at $4 is Bar B’s best response.

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How to Find the Nash Equilibrium

This game is useful in illustrating the reasoning behind the Nash equilibrium. Player 1 and Player 2 are indifferent between being alone at the neighborhood cafe or at their respective homes. However, they would like to meet and that can only happen if they both turn up at the cafe. This is the payoff matrix for this game:

payoff matrix for cafe vs home example
The meeting game payoff matrix. | Image: Michael Kingston

What are the Nash equilibria for this game?

import nashpy as nash
import numpy as np

row = np.array([[10, 0], [0, 0]])
column = np.array([[10, 0], [0, 0]])
meeting_game = nash.Game(row, column)
print(meeting_game)
equilibria = meeting_game.support_enumeration()
for eq in equilibria:
    print(f"The unique Nash equilibrium is {eq}")
code to find the nash equilibrium for the meeting game
Meeting game output code. | Image: Michael Kingston

We can identify two Nash equilibria: (cafe, cafe) and (home, home).

nash equilibrium underlined in the meeting game payoff matrix
Nash equilibrium underlined in the meeting game payoff matrix. | Image: Michael Kingston

The result (home, home) may not be as convincing as (cafe, cafe) as an outcome of this game. After all, if you’re indifferent between being alone in one place or the other, and the cafe is the only place you have a remote chance of meeting the other player, why would you ever stay at home? However, (home, home) is a Nash equilibrium of this game. Next, we’ll check that it satisfies the three properties listed below. If either player were told that his counterpart was staying at home, that player may as well stay at home, too. Thus, home is a best response to home for both players, which makes (home, home) a Nash equilibrium.

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How Nash Equilibrium Relates to Other Game Theory Concepts

The below image depicts the relationship of Nash equilibrium with other equilibria.

nash equilibrium compared to other solution strategies illustration
Relationship of Nash equilibrium to other solution concepts. | Image: Michael Kingston

Key points to remember:

  • All Nash equilibria are rationalizable strategy profiles. Iterated deletion of strictly dominated strategies will never “delete” a strategy used in a Nash equilibrium. However, not all rationalizable strategy profiles are a Nash equilibrium
  • All equilibria found by iterated deletion are Nash equilibria, but not all Nash equilibria can be found by iterated deletion (e.g. the second version of the “Two Bars Game.”)
  • All equilibria in dominant strategies can be found by iterated deletion (every other strategy is dominated by that one dominant strategy.) But not all equilibria found by iterated deletion are equilibria in dominant strategies
An outline of the prisoners’ dilemma and the Nash equilibrium. | Video: Khan Academy

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Why Nash Equilibrium Matters

The Nash equilibrium is a powerful solution concept in large part because of the following:

  • Nash’s existence theorem: In all finite games (i.e. games with a finite number of players and strategies per player), there exists at least one Nash equilibrium. We will almost always be able to find at least one Nash equilibrium.
  • The Nash equilibrium is computationally straightforward and elegant.
  • Ability to find a large set of Nash equilibria. This is not necessarily a good thing. This solution concept still has the limitation that it may find an intractably large set of Nash equilibria.
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