Minimax solution but game has no value

Question

Fix convex sets $\Delta,\Pi$ and let $r: \Pi \times \Delta \in [0,\infty]$ be linear (i.e., concave and convex) in its first parameter for every fixed second parameter.

I'm looking for a situation where 1) the corresponding zero-sum game has no value, i.e., $$ \sup_{\pi \in \Pi} \inf_{\delta \in \Delta} r(\pi,\delta) \neq \inf_{\delta \in \Delta} \sup_{\pi \in \Pi} r(\pi,\delta) $$ but 2) there exists a minimax solution, i.e., there exists $\delta_0 \in \Delta$ such that $$ \sup_{\pi \in \Pi} r(\pi,\delta_0) = \inf_{\delta \in \Delta} \sup_{\pi \in \Pi} r(\pi,\delta). $$ (Note that I've swapped the order of variables from the usual game presentation to match the statistical setting when $r$ is interpreted as (average) risk.)

Clearly, such a situation cannot arise when both $\Pi,\Delta$ are compact and $r : \Pi \times \Delta$ is concave-convex because such games always have values by the minimax theorem. When I've come across counterexamples to a game having a value, this usually also results in there being no minimax solution.

Am I missing an obvious obstruction, or can someone point me to an example, or to where I can read more about existence of minimax solutions when the game has no value?

Answer 1

Actually, it's not hard to construct such a solution. Ferguson (Mathematical Statistics, 1967, pg. 83) gives an example where $\Delta, \Pi$ are the space of probability measures on $\mathbb{N}$ and $r(\pi,\delta)$ is the probability that independent draws $x \sim \pi$ and $y\sim\delta$ are such that $ x > y$. Then every element in $\Delta$ is minimax because $\sup_{\pi} r(\pi,\delta) = 1$ for all $\delta$, but $\sup_{\pi} \inf_{\delta} r(\pi, \delta) = 0$ (but this supremum is not achieved).

I see that it is also easy to produce a game where not every $\delta$ is minimax. I suppose a more interesting example would be one where the minimax was unique, yet the game had no value. I dot't yet see a way to do this. Perhaps this is ruled out logically, but I don't see it yet.