Malcolm Rivera
The Nash equilibrium is a game theory concept where no player can improve their outcome by changing their strategy while all other players keep theirs the same. It is named after John Nash, an American mathematician, who won the Nobel Prize for his work. The Nash equilibrium is often compared to the dominant strategy, which asserts that a player will choose a strategy that will lead to the best outcome regardless of the strategies that the other players have chosen. In this case, the Nash equilibrium is a superset of the subgame perfect Nash equilibrium, which requires that the strategy is also a Nash equilibrium in every subgame of that game.
| Characteristics | Values |
|---|---|
| Game Theory State | No player can gain by changing their strategy while all other players' strategies remain unchanged. |
| Decision-Making Theorem | A player's best chance at achieving their desired outcome is by not deviating from their initial strategy. |
| Rationality of Players | Players are considered "rational agents" who desire specific outcomes, attempt to choose the most optimal outcome, and incorporate uncertainty in their decisions. |
| Knowledge of Opponents' Strategies | Players know their opponents' strategies and cannot change their own strategy because it remains optimal. |
| Incentive to Deviate | No player has an incentive to deviate from their strategy, even if they know the strategies of other players. |
| Optimal Strategy | Each player's strategy is optimal when considering the decisions of other players. |
| Number of Equilibria | A game can have one or multiple Nash equilibria. |
| Pareto-Dominance | The Nash equilibrium is not always Pareto-dominant with respect to other outcomes. |
| Mixed Strategies | Nash equilibria can be found in games with mixed strategies, where players choose probability distributions over pure strategies or specific strategy profiles. |
| Sequential Equilibrium | Sequential equilibrium models both strategies and beliefs for a set of players, solving the problem of too many irrational equilibria in normal-form models. |
| Non-Cooperative Games | Nash equilibrium is the most commonly used solution concept for non-cooperative games. |
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What You'll Learn
Mixed-strategy equilibrium
A Nash equilibrium is a situation where no player can gain by changing their strategy, assuming that all other players' strategies remain unchanged. This concept is applied in game theory, specifically for non-cooperative games. The Nash equilibrium is named after John Nash, an American mathematician, and is considered one of the most important concepts in game theory.
In the context of mixed-strategy equilibrium, players can randomize their strategies across multiple options. This is particularly relevant when players are indifferent between different actions, allowing them to choose a mixed strategy that combines these actions. For example, if a player is indifferent between two actions, they can assign a probability of 0.5 to each and randomize their choice.
Equilibrium points in mixed-strategy games have been described as unstable because players can deviate from their equilibrium strategy without penalty, even if they expect others to stick to their strategies. However, a model has been proposed to address this issue, suggesting that players' uncertainty about others' exact payoffs can be modelled as a disturbed game. This disturbed game introduces small random fluctuations in payoffs, leading to the stability of mixed-strategy equilibrium points.
The Nash Theorem states that every finite game has at least one Nash equilibrium in mixed strategies. This includes games where players have more than two strategies, and a numerical method can be used to identify Nash equilibria in such cases. By examining the payoff matrix, if the first payoff number in a cell is the maximum of its column and the second number is the maximum of its row, then that cell represents a Nash equilibrium.
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Pure-strategy equilibrium
A pure-strategy equilibrium is a type of Nash equilibrium where players make their choices deterministically. In other words, players are not allowed to randomize their strategies. This is in contrast to mixed-strategy equilibria, where players choose a probability distribution over possible pure strategies.
The Nash equilibrium is a fundamental concept in game theory, where it is the most commonly used solution concept for non-cooperative games. It is named after American mathematician John Forbes Nash Jr. and refers to a situation where no player can improve their payoff by unilaterally changing their strategy, given the strategies of the other players. This can be determined by fixing each player's strategy one by one and checking if the other players are better off changing their strategy. If neither player can improve their payoff by changing their strategy, then the strategy profile corresponds to a Nash equilibrium.
For example, consider a game where two players, Alice and Bob, can each choose between strategies A and B. If Alice chooses strategy A and Bob chooses strategy B, then this strategy profile (A, B) is a Nash equilibrium if Alice has no other strategy available that would increase her payoff more than A in response to Bob choosing B, and vice versa. In this case, neither Alice nor Bob has an incentive to deviate from their chosen strategy, given the other player's strategy.
The coordination game is a classic two-player, two-strategy game that illustrates the concept of pure-strategy Nash equilibria. In this game, there are two pure-strategy equilibria: (A, A) with a payoff of 4 for each player and (B, B) with a payoff of 2 for each player. The combination $(B, B)$ is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will decrease from 2 to 1.
It is important to note that the existence of a Nash equilibrium does not necessarily guarantee the best cumulative payoff for all players involved. For example, in the Prisoners' Dilemma scenario, the Nash equilibrium occurs when one prisoner defects and testifies against the other, resulting in a higher payoff for the betrayer but a worse outcome for the silent accomplice.
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Subgame perfect Nash equilibrium
The Nash equilibrium is a decision-making theorem within game theory that states a player's best chance at achieving their desired outcome is by not deviating from their initial strategy. It is a situation where no player can gain by changing their strategy, assuming that the other players' strategies remain fixed.
The subgame perfect Nash equilibrium is a refinement of the Nash equilibrium concept, specifically designed for dynamic games where players make sequential decisions. A strategy profile is a subgame perfect Nash equilibrium if it represents a Nash equilibrium in every possible subgame of the original game. This means that at any point in the game, the players' behaviour from that point onwards should represent a Nash equilibrium of the continuation game, regardless of what happened before.
For example, in a game where player one chooses left (L) or right (R), followed by player two being called upon to be kind (K) or unkind (U) to player one, player two only stands to gain from being unkind if player one goes left. If player one goes right, the rational player two would be kind to them in that subgame. However, the non-credible threat of being unkind is still part of the Nash equilibrium. In this case, the subgame perfect Nash equilibrium would be for player one to go right, and for player two to be kind.
The subgame perfect Nash equilibrium is deduced by "backward induction" from the various ultimate outcomes of the game, eliminating branches that involve any player making a move that is not credible (because it is not optimal) from that node. This ensures that strategies are credible and rational throughout the entire game, eliminating non-credible threats.
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Pareto-dominant outcomes
Pareto efficiency, named after Italian economist and sociologist Vilfredo Pareto, is a concept in economics and sociology that refers to the ranking of allocations and the distribution of wealth. Pareto-dominant outcomes are those in which the associated payoffs are greater than the payoffs of another outcome.
A Nash equilibrium is a decision-making theorem within game theory that states that a player has the best chance of achieving their desired outcome by not deviating from their initial strategy. It is a stable state where no participant can gain a higher payoff by changing their strategy as long as the other participants remain unchanged.
While the two concepts are distinct, they can be related. A Nash equilibrium may be Pareto efficient, and every correlated strategy on the Pareto frontier is a coalition-proof Nash equilibrium (CPNE). However, a Nash equilibrium is not always Pareto-dominant with respect to other outcomes. For example, in the prisoner's dilemma, there is a dominant strategy equilibrium that is not Pareto-efficient. There is an alternative, the 'cooperative' outcome, that Pareto-dominates the equilibrium allocation, but the players do not choose this strategy.
In some cases, a Nash equilibrium may be the most desirable outcome, but it is not always the case. It is possible for a Pareto-dominant outcome to exist that is a better option for all players, but if the players are not aware of this or are unable to coordinate, they may still end up at the Nash equilibrium.
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Non-cooperative games
The Nash equilibrium is a key concept in game theory, and it is particularly useful for analysing non-cooperative games. The Nash equilibrium describes a situation where no player can benefit by changing their strategy unless the other players do so as well. In other words, each player's strategy is optimal when considering the decisions of the other players. This means that players have no incentive to deviate from their initial strategy, as they cannot increase their payoff by changing decisions unilaterally.
The Nash equilibrium is often compared to the dominant strategy. The dominant strategy is the optimal move for an individual regardless of how other players act. In contrast, the Nash equilibrium occurs when there is no dominant strategy, and each player's optimal move depends on the moves of their opponents.
The Prisoner's Dilemma is a classic example of a non-cooperative game that is often used to illustrate the Nash equilibrium. In this game, two players must decide whether to cooperate with each other or betray the other player and confess to authorities. The Nash equilibrium occurs when both players betray each other, as they are protecting themselves from being punished. However, this outcome is worse for them collectively.
Another example of a non-cooperative game is the stag hunt. In this game, two players must choose to hunt a stag or a rabbit, with the stag providing more meat but requiring cooperation to hunt. The Nash equilibrium occurs when both players choose to hunt the stag, as this provides the highest payoff for each player.
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Frequently asked questions
The Nash Equilibrium is a decision-making theorem within game theory that outlines a situation where no player can improve their outcome by changing their strategy, assuming all other players' strategies remain fixed. It is considered one of the most important concepts in game theory.
There are a few ways to identify a Nash Equilibrium. One method is to reveal each player's strategy to the others. If no player has an incentive to change their strategy, a Nash Equilibrium is proven. Another method is to use a payoff matrix, which is especially useful in two-person games with multiple strategies.
The Dominant Strategy asserts that a player will choose an action that will lead to the best outcome, regardless of the opponent's strategy. In contrast, the Nash Equilibrium states that no player can improve their outcome by changing their strategy when all other players' strategies are known and fixed.
The Prisoner's Dilemma is a classic example used to illustrate the Nash Equilibrium. In this scenario, two criminals are offered a deal to testify against each other. If both stay silent, they serve a minimal sentence. If one testifies and the other doesn't, the betrayer goes free while the other serves a longer sentence. The Nash Equilibrium occurs when both prisoners choose to stay silent, as neither can improve their outcome by changing their strategy. Additionally, in franchising, a Nash Equilibrium can be observed when the franchisor is indifferent between monitoring and not monitoring, and the franchisee is indifferent between shirking and not shirking, resulting in a best response for each player.
