Continuous game

A continuous game is a mathematical generalization, used in game theory. It extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite.

In general, a game with uncountably infinite strategy sets will not necessarily have a Nash equilibrium solution. If, however, the strategy sets are required to be compact and the utility functions continuous, then a Nash equilibrium will be guaranteed; this is by Glicksberg's generalization of the Kakutani fixed point theorem. The class of continuous games is for this reason usually defined and studied as a subset of the larger class of infinite games (i.e. games with infinite strategy sets) in which the strategy sets are compact and the utility functions continuous.

Define the n-player continuous game ${\displaystyle G=(P,\mathbf {C} ,\mathbf {U} )}$ where

Let ${\displaystyle {\boldsymbol {\sigma }}_{-i}}$ be a strategy profile of all players except for player ${\displaystyle i}$. As with discrete games, we can define a best response correspondence for player ${\displaystyle i\,}$, ${\displaystyle b_{i}\ }$. ${\displaystyle b_{i}\,}$ is a relation from the set of all probability distributions over opponent player profiles to a set of player ${\displaystyle i}$'s strategies, such that each element of

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