HMMT 二月 2004 · 冲刺赛 · 第 43 题

英文解析

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 teamthat 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 bestpossible move given the other teams’ choices) are fairly degenerate: every team butone 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 utilityfunctions the same, since the goal is to score higher than other teams, not to maximizeone’s own expected number of points. It will be interesting to see what the submissionsare.