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### nash equilibrium vs pareto optimal

Nash equilibrium and Pareto optimality are two concepts used in game theory and economics to evaluate the outcomes of strategic interactions between multiple individuals or entities.

Nash equilibrium is a concept that describes a situation in which each player in a game chooses the best strategy given the strategies chosen by all the other players. In other words, in a Nash equilibrium, no player has an incentive to change their strategy, assuming that all other players’ strategies remain the same. A Nash equilibrium does not necessarily lead to an optimal outcome for all players involved; it only ensures that no player can improve their own outcome by changing their strategy, given the strategies of the other players.

Pareto optimality, on the other hand, refers to a situation in which no individual or entity can be made better off without making someone else worse off. In other words, a Pareto optimal outcome is one where it is impossible to improve the situation of one person without making someone else worse off. This concept is often used to evaluate social welfare or collective utility, rather than individual outcomes.

In summary, while a Nash equilibrium is a situation in which no player has an incentive to change their strategy, a Pareto optimal outcome is one in which no individual or entity can be made better off without making someone else worse off. A Nash equilibrium does not necessarily lead to a Pareto optimal outcome, as it only ensures that no player can improve their own outcome by changing their strategy, given the strategies of the other players. A Pareto optimal outcome, however, is always at least as good as any Nash equilibrium.

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### pareto optimal game theory

In game theory, a Pareto optimal outcome is a state in which no player can be made better off without making another player worse off. In other words, it is a state where all possible improvements have been exhausted, and any further change would result in someone being worse off.

A game is said to be Pareto optimal when there is no other outcome that would benefit at least one player without harming any other player. In other words, any change in the game would result in at least one player being worse off.

Pareto optimality is an important concept in game theory because it helps to identify outcomes that are considered “fair” or “efficient” from a social welfare perspective. In such cases, the goal is to maximize the total benefits for all players while minimizing any harm or loss to any individual player.

A common example of a Pareto optimal outcome is the division of a cake between two people. If the cake is divided such that each person gets an equal share, then this is a Pareto optimal outcome because neither person can be made better off without making the other person worse off. However, if one person gets a larger share of the cake, then this is not a Pareto optimal outcome because the other person could be made better off if they received a larger share.

### every pareto optimum is a nash equilibrium.

This statement is not necessarily true. While it is true that every Nash equilibrium is a Pareto optimum, the reverse is not always true.

A Nash equilibrium is a solution concept in game theory, where each player’s strategy is the best response to the strategies of the other players. In a Nash equilibrium, no player can unilaterally change their strategy and improve their own payoff.

A Pareto optimum, on the other hand, is a solution concept that refers to a situation where it is not possible to make any player better off without making at least one player worse off. In other words, a Pareto optimum is a situation where the allocation of resources is as efficient as possible, given the constraints.

While every Nash equilibrium is also a Pareto optimum, the converse is not always true. It is possible for a solution to be a Pareto optimum but not a Nash equilibrium, if there exists a profitable deviation for at least one player that could make them better off without making any other player worse off.

Therefore, we cannot conclude that every Pareto optimum is a Nash equilibrium.

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