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:
- A player can choose to name any of the $N$ cities as their guess.
- Each player hears the other's guess and can use this info to help further decision.
- 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 not answer this problem, it is from the Mathcamp qualifying quiz here: https://mathcamp.org/prospectiveapplicants/quiz/index.php