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Two persons have 2 uniform sticks with equal length which can be cut at any point. Each person will cut the stick into $n$ parts ($n$ is an odd number). And each person's $n$ parts will be permuted randomly, and be compared with the other person's sticks one by one. When one's stick is longer than the other person's, he will get one point. The person with more points will win the game. How to maximize the probability of winning the game for one of the person. What is the best strategy to cut the stick.

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    Maximizing the probability of winning the game isn't well-defined unless you specify a distribution for the opponent's strategies. The usual question for such a game would be to find the [Nash equilibria](http://en.wikipedia.org/wiki/Nash_equilibrium) (if any). Perhaps that's what you meant?2012-08-09
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    @joriki Each person will try his optimail strategy.2012-08-09
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    @Fan: As I said, that's not well-defined. The optimal strategy depends on the opponent's strategy. Do you know about Nash equilibria? If not, I'd suggest to read about them and then consider updating the question.2012-08-09
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    @joriki Thank you for your advice. I will read about Nash equilibria.2012-08-09
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    By the way, please choose a more descriptive title. Mentioning the overall field, such as game theory, in the title isn't useful because that's what the tags are for and the tags are visible wherever the title is (e.g. on the main page), so this is redundant information. The title should be more specific to the question, e.g. "strategy for a game of breaking sticks". (Also, almost all questions are about problems, so "problem" is even more redundant.)2012-08-09
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    @joriki Thank you, updated.2012-08-09
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    This seems more difficult than I would have expected. Choosing the breakpoints randomly with independent uniform distributions is not a Nash equilibrium.2012-08-09
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    This is sort of a [Colonel Blotto game](http://en.wikipedia.org/wiki/Colonel_Blotto), but in the standard version of that, the payoff is the number of comparisons won, whereas you have a $0$/$1$ payoff depending on who wins more comparisons.2012-08-09
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    @joriki Thank you for the information.2012-08-09
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    Actually, for $n=3$, there's no difference, since the difference in the number of comparisons won is always $\pm1$, so you may want to look into the standard Colonel Blotto game (apparently the Wikipedia article links to a full solution) -- the $n=3$ solution will be the same and perhaps it can give you some ideas for the solution for $n\gt3$.2012-08-09
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    Simulation helps.2012-11-03
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    There's another equivalent question here with sausages rather than sticks. I'm not sure how to link to it...2012-12-06

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