# Value of an infinite game

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

• 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? – irchans Mar 26 at 22:45