Best Strategy for a guessing game

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!

• Do Tom and Kathy know that they are adjacent to one another? Mar 8, 2017 at 0:13
• @NathanH. Yes, they do. I will edit the question to make this clear. Thanks Mar 8, 2017 at 0:16
• I like this problem. Where did you get it? Mar 8, 2017 at 0:26
• Do not answer this problem. It is from the Mathcamp qualifying quiz here: mathcamp.org/prospectiveapplicants/quiz/index.php Mar 8, 2017 at 0:53
• I'm voting to close this question as off-topic because it is from a live contest.
– lulu
Mar 8, 2017 at 0:58