HMMT 二月 2004 · 冲刺赛 · 第 43 题

HMMT February 2004 — Guts Round — Problem 43

Write down an integer from 0 to 20 inclusive. This problem will be scored as follows: if N is the second-largest number from among the responses submitted, then each team that submits N gets N points, and everyone else gets zero. (If every team picks the same number then nobody gets any points.)

解析

Write down an integer from 0 to 20 inclusive. This problem will be scored as follows: if N is the second-largest number from among the responses submitted, then each team that submits N gets N points, and everyone else gets zero. (If every team picks the same number then nobody gets any points.) Solution: The only Nash equilibria of this game (where each team plays its best possible move given the other teams’ choices) are fairly degenerate: every team but one plays 1, and the remaining team is more likely to choose 2 than any higher number. Of course, we cannot assume perfectly rational play in reality — nor are the utility functions the same, since the goal is to score higher than other teams, not to maximize one’s own expected number of points. It will be interesting to see what the submissions are.