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I have this game theory like problem

Tom and Kathy play a guessing game played in N cities located on a large ring around the earth. Two cities that are adjacent are chosen at random. Tom is sent to one and Kathy the other. Each knows his/her own location and the fact that they are adjacent, but not exact location.

Starting with Tom, they take turns guessing where the other is. More precisely:

  1. A player can choose to name any of the $N$ cities as their guess.
  2. Each player hears the other's guess and can use this info to help further decision.
  3. A player's guessing strategy can be probabilistic: they can decide to guess city 1 with probability $p_1$, city 2 with $p_2$ and so on.

Whoever guess correct wins.

I need help with the following to questions:

1.If $N=3$, find a strategy for Tom that wins with at least probability $\frac{2}{3}$

2.What are Tom and Kathy's optimal strategy for the general $N$.

Thanks for the help!

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  • $\begingroup$ Do Tom and Kathy know that they are adjacent to one another? $\endgroup$
    – Nathan H.
    Mar 8 '17 at 0:13
  • $\begingroup$ @NathanH. Yes, they do. I will edit the question to make this clear. Thanks $\endgroup$
    – Rust Z
    Mar 8 '17 at 0:16
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    $\begingroup$ I like this problem. Where did you get it? $\endgroup$
    – Nathan H.
    Mar 8 '17 at 0:26
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    $\begingroup$ Do not answer this problem. It is from the Mathcamp qualifying quiz here: mathcamp.org/prospectiveapplicants/quiz/index.php $\endgroup$
    – Bob Jones
    Mar 8 '17 at 0:53
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    $\begingroup$ I'm voting to close this question as off-topic because it is from a live contest. $\endgroup$
    – lulu
    Mar 8 '17 at 0:58
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Do not answer this problem, it is from the Mathcamp qualifying quiz here: https://mathcamp.org/prospectiveapplicants/quiz/index.php

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