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I'm having some trouble showing that the following zero-sum game with 2 players has a value (and in consequence, computing it's value).

The strategies of both players are in $\mathbb{N}-\{0\}$ and the utility function is $u(a_1,a_2)=\frac{1}{a_1+a_2}$. Now, for player number one, choosing $a_1=1$ is a strictly dominant strategy. But I don't understand how player 2 should play (My intuition is that it should choose a number as high as he can, but he can choose arbitrarily large numbers, this is what confuses me). Any hints? Thanks in advance

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  • $\begingroup$ If we define the value of the game $V = \inf_{a_2} \sup_{a_1} \frac1{a_1+a_2}$, then we get $V = \inf_{a_2} \frac1{1+a_2}=0$. If we define V to be $V = \sup_{a_1} \inf_{a_2} \frac1{a_1+a_2}$, what do we get then? $\endgroup$ – irchans Mar 26 at 22:45

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