## A Simple but Challenging Game, Part I

My 15-year-old son Adam likes game theory. He invented the following simple game, and asked me about it when I got on the phone with him while I was away at a conference last month (I’ve simplified and formalized the set-up slightly):

There are two players, each of whom is given a real number which is chosen randomly from a uniform distribution between 0.0 and 1.0. The players know their own number but not their opponent’s. One player moves first and has the choice of passing or challenging. If he challenges, both players reveal their number, and the player with the higher number receives a payoff of 1, while the other player receives a payoff of 0. If the first player passes, the second player has a choice of challenging or passing. If he challenges, again both players reveal their numbers and the player with the higher number receives a payoff of 1, while the other player receives a payoff of 0. If the second player also passes, both players receive a payoff of 1/2. They play the game one time, and are interested in maximizing their expected payoff.

What is the right strategy? For example, if you received the number 0.17, would you pass or challenge if you were the first player? What about if you were the second player? What would you do if the number you received was 0.0017?

I’ll tell you more in a later post, but for now why don’t you think about it….

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### 2 Responses to “A Simple but Challenging Game, Part I”

1. optionalstopping Says:

My inkling is that Player 1 is in a pretty bad state; that is, by going second, Player 2 has a natural advantage. So if I was Player 1, I’d be tempted to challenge every single time, then I ought to win about half of the money.

2. A Simple but Challenging Game: Part II « Nerd Wisdom Says:

[…] Simple but Challenging Game: Part II Here’s Part I […]