I've been asked to do this by process of game theory and probability (namely Bayesian theory).
Here is the problem: There is a Hunter (H) and a Rabbit (R). They are playing the following game: - There are 20 holes on a line numbered from 1 to 20.
The game begins: - in the first period (t=0) H chooses a hole and R chooses a hole - if H and R choose the same hole, H wins (because he captures the Rabbit) and the game finishes. Otherwise, the game continues
If the game does not finish a t, we enter the next period: t+1
- in t+1:
- R MUST move away from the hole he was in period t and can move only to the holes that are adjacent to the hole he was in at t. i.e: if he was in hole n ( for example 15), he must move either in hole n-1 (i.e., 14) or in hole n+1 (i.e. 16). while if he was on hole 1 (20), he must move in hole 2 (19).
- H can choose a hole with no restrictions (i.e. he can choose any hole in the list 1,2,3...20)
- if H and R choose the same hole, H wins (because he captures the Rabbit) and the game finishes. Otherwise, the game continues
..... if R does not lose in 100 rounds, he wins the game.
Questions:
- Can you tell me if H has a strategy to win the game for SURE? If so, describe this strategy.
Consider the following modification of the game:
In each period, H announces to R the hole he is going to choose before R decides where to move.
Can you tell me if H has a strategy to win the game? If so, describe this strategy.
Thanks!
$\textbf{Hint: }$ If $R$ is at $1$ or $20$ at time $t$, can $H$ win at time $t+1$?