# Existence nash equilibrium - Discontinuous utility function infinite valued

I am new to game theory, I want to know if a Nash equilibrium exists for a game having the following properties:

• Finite number of players
• The strategy space for each player is R^2 (R for real numbers)
• The utility function could have infinite values (notably, the utility function of an individual for a chosen strategy is infinite if a certain constraint depending on the strategies chosen by others is not satisfied)

I know that the problem can be formulated in a generalized game framework, but the difference here is that I accept strategies with infinite utility value if and only if it is the only possible choice for the player.

Do not hesitate to ask for more details if my question is not sufficiently clear.

• In general, no. Take a game with a single player who obtains utility $x$ from playing strategy $(x,y)$. Since there is no largest real number, there is no Nash equilibrium. Jun 13 '17 at 10:01
• Ok thanks! But if I exclude the case of one player, and I suppose that I have at least two players, can we have existence of Nash equilibria in that case?
– F.AR
Jun 14 '17 at 10:12
• You can rewrite the example to make it a two-player game. You will not get a general result without some kind of compactness assumption. Jun 14 '17 at 10:14

• The set of strategies of each players is not a compact (which may be surpassed if you select instead a compact interval on $\mathbb{R}^2$).