This is the story of how I came to write a little paper called The Coinflipper’s Dilemma.
When I was in high school, my English teacher must have had a free period at the time when my math class met, because every day he would march into the math class and empty his pockets on the table, whereupon my math teacher did the same. Then whoever had put down the most money scooped up everything on the table.
I am ashamed to admit that it took me until this summer to think about computing the equilibrium strategy is in that game.
The problem turns out to be more interesting if we impose a minimum bet of, say, $1. Given that, I briefly convinced myself that there’s an equilibrium where each player chooses a bet between $1 and infinity, subject to the probability density (1/2)x-3/2. (In layman’s terms, this means roughly that the probability you’ll bet some amount x is proportional to (1/2)x-3/2).
It turns out that if the math teacher plays this strategy, then the English teacher, no matter how much he bets, earns an expected return (that is, a return on the average day) of $-1. Therefore, for the English teacher (assuming he’s required to play), one strategy is as good as another, and he might as well play this one. Ditto for the math teacher. That, I briefly thought, sufficed (essentially by definition) to make this an equilibrium outcome.
It turns out that for technical reasons, that’s not quite right. And it turns out also that I was not the first person to stumble in this way, or the first to recover from it. Just a few years earlier, Michael Baye, Dan Kovenock and Casper de Vries had run into exactly the same issue in the context of a slightly more complicated game. Their paper cleared up whatever remaining confusion I had about the equilibrium issue.
But I was still left to wonder what advice I should have given to my old English teacher. Should he stick to the strategy described above? It’s true, as I said, that as long as the math teacher sticks to this strategy, the English teacher can bet any amount he wants and expect to lose $1, so this strategy seems as good as any other — though it would be better still to avoid the game altogether. But at the same time, if the English teacher sticks to this strategy, then no matter what amouont the math teacher bets, the English teacher can expect to win $1 — so maybe it’s a good idea to play after all.
Multiple conversations with some excellent economists convinced me that the resolution of this paradox is not completely obvious, even to (at least some) experts in the field, and that it might be useful to write up an explanation of what’s really going on. In the course of writing, I discovered that it’s a lot easier to highlight the key issues with a different game, which I called the Coinflipper’s Dilemma and I wrote a little paper about it.
My math and English teachers have vanished entirely from the final draft, but their dilemma is not fundamentally different from the coinflipper’s, and if you understand one, it’s not hard to understand the other.
When all is said and done, does this sound like a game you’d like to play?
Edited to add: An earlier version of this post was missing a crucial minus sign; it’s now edited to get it right. Thanks to our diligent reader nivedita for catching this.
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