Follow-up: Mismatched Pennies
A correspondent writes:
Greetings,
I think that in the long run both strategies are equivalent. This game doesn’t favor any player.
Demonstration
Chaim Expected Gains = 3 * 1/4 + 1 * 1/4 = 1
Kyle Expected Gains = 1/4 * 2 + 1/4 * 2 = 1
This is so if both of us pick H half of the time, and pick T half of the time.
But!
If I know Kyle is going to pick H half of the time and T half of the time, I should adjust my strategy. I can do better by always picking H; the payout would then be
C: 3*1/2 = 3/2
K: 2*1/2 = 1
Net 1/2 in my favor!!
Conversely, if I am picking H half of the time and T half of the time, Kyle should adjust his strategy and choose T all of the time; this comes out to
C: 1*1/2 = 1/2
K: 2*1/2 = 1
Net 1/2 in Kyle’s favor– rats!
John von Neumann’s celebrated result is that both players have an optimal strategy, one that cannot be exploited by the other player. If we both play optimally, is the game balanced?
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