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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.

Can anyone help please?

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

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  • $\begingroup$ 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. $\endgroup$ Jun 13 '17 at 10:01
  • $\begingroup$ 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? $\endgroup$
    – F.AR
    Jun 14 '17 at 10:12
  • $\begingroup$ 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. $\endgroup$ Jun 14 '17 at 10:14
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No. For two reasons.

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

  • Second, you cannot ensure the existence of a maximum, since by the Weierstrass theorem, a function reaches a maximum iff is continuous and quasi-concave. Your example violates this condition. Now this may be surpassed using a different tools, and fixed points with more relaxed conditions. However, in a standard and classical way, the maximum is not attainable.

Therefore, is not that you don't have a Nash equilibrium in such a game. You may have. What I cannot guarantee is that it will exists always in a game under such framework.

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