Game theory and the willingness to look bad
Here’s a fun game of Jeopardy!. Let's think about it.
Vickie Taviola, an apparently excellent player who had defeated a four-game champion the day before, went into Final Jeopardy! with a small lead: $25,800 to $25,600 to $7,600.
Vickie bet very small ($150).
Everyone got the question right.
The second-place player bet a non-zero amount ($10,000) and won. Without attempting a full analysis of this Final Jeopardy! scenario, I’ll note that this basic structure, where the leader has more than the second-place player but less than 1.5x as much as the second-place player, comes up a lot. Ignoring, for now, the third-place player:
- If the leader bets to “cover”–that is, to guarantee a win if everyone is correct, no matter how much the second-place player bets–then the second-place player is better off betting small or zero. (That way, they beat the leader if the leader gets the question wrong, and they can’t possibly win if the leader gets the question right.)
- But if the leader knows that the second-place player is going to do that, then they can also bet small or zero, and guarantee the win no matter who gets what right.
That is not a full game-theoretic analysis (which depends on the exact ratio, each player’s true and perceived likelihood of getting it right, and what’s going on with the third-place player, among other things). But when you have a situation with this structure (if I do A, you should do B, but if you do B, I should do C, but if I do C, you should do D…), you’re probably going to wind up needing a mixed strategy.
Mixed strategies have many important formal properties. Here's an important informal one: they're likely to make the user look really bad sometimes. This comes up a lot in poker, where (at equilibruim) you often need to make near-hopeless bluffs, near-hopeless bluff-catches, occasionally-catastrophic folds, and so on. One thing that separates great players is their willingness to look very bad, and this is one facet of the game where that comes up.
So: do not think it’s a coincidence that the player willing to risk the “feel-bad” and the egg on one’s face is a quant with a Princeton math background.