Here's a fun game-theory problem; Martin Gairing helped me find a solution during lunch, which I will add later.
There are 2 political parties and N constituencies; each party wants to win as many of them as possible. Both parties have an amount M of money (to spend on election campaigning) which they split amongst the N constituencies; for each constituency, it is won by the party that allocated it the larger amount of money. A party's payoff is the number of constituencies it wins, so it's a zero-sum game. The problem is to find a Nash equilibrium. You can assume that M is infinitely divisible, or if not, you're allowed to find an approximate solution with error proportional to 1/M.
Note that there is no pure equilibrium; if a party fails to randomize, the other one will be able to narrowly defeat it in nearly all constituencies while allocating no money to one(s) that it loses.
Obviously lots of generalizations are possible...
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