After discussing last week’s Mismatched Pennies Game, Kyle and Chaim are hauled off to jail!
First, here are some of the details of the calculation for mismatched pennies.
The expected payouts (from Kyle’s point of view) are
HH -> -3
HT or TH -> 2
TT -> -1
1) Is it really true, if Kyle steadily plays 3/8ths Heads and 5/8ths Tails, that Chaim can’t adjust his own strategy to take advantage of Kyle? Suppose Chaim plays some fraction p Heads and then (1-p) Tails. Then we expect to see
HH exactly 3/8 * p of the time, for an expected payout of 3/8 * p * (-3)
TH exactly 5/8 * p of the time, for exp. payout of 5/8 *p *2
HT exactly 3/8 * (1-p) of the time, for exp. payout of 3/8 * (1-p) * 2
TT exactly 5/8 * (1-p) of the time for an expected payout of 5/8 * (1-p)*(-1)
The total expected payout is thus
3/8 * p * (-3) + 5/8 *p *2+3/8 * (1-p) * 2+5/8 * (1-p)*(-1) =
-9/8 p + 10/8 p – 6/8 p + 5/8 p – 5/8 + 6/8 = 1/8
No matter what p is chosen, Kyle will come out 1/8th of a penny ahead, per game, on average, in the long run.
Similarly, if Chaim chooses to play 3/8ths H and 5/8ths T, Kyle cannot exploit this and do any better.
As discussed in the podcast, there is a quick trick for calculating this, at least for simple games like this one.
2) What’s really going on? Why is von Neumann’s theorem true, for more complicated games?
Really, one can think of the expected payout as a function of the various probabilities assigned to each outcome. If you think about it, this function is quadratic– each term will have degree at most two– and the von Neumann minimax is a search for a saddle point. The existence of such saddle points– unless the game is always tilted to one inevitable outcome– is straightforward enough, but is beyond what we can discuss here.