# How to find best strategy for 3 hunters hunting in 2 regions?

I have come across this problem while trying to understand game theory. The problem presents two players (hunters) that want to hunt in one of two regions available. Region $$A$$ has $$12$$ rabbits and region $$B$$ has $$18$$ rabbits. If the two hunters go to different regions they will acquire all the rabbits in their respective hunting ground, but if they go to the same region they'll acquire half the rabbits each, assuming they have the same skills. I have found the mixed strategy that they need to follow, which will be the same for both: they will go to region $$A$$ with probability $$p=\frac{3}{5}$$ or to region $$B$$ with probability $$1-p=\frac{2}{5}$$.

The problem emerges when I try to further the problem by adding a third hunter in the game. I would assume that an obvious answer would come up, like one region would give the best result when chosen for all hunters. However, what I'm seeing by creating the payoff tables is that if the third hunter chooses region $$A$$ then it's in everyone's benefit if the other two hunters choose region $$B$$, thus having the result array be $$(H_1,H_2,H_3)=(9,9,12)$$. But if hunter 3 chooses region $$B$$, the best outcome is when the other two hunters don't choose the same region between themselves.

I am unsure as to how I can interpret this or if there's another way to find the answer to this more complicated question. Can I apply a method to this problem and find mixed strategies for the three hunters? I am not necessarily looking for the answer directly, but for someone to point out a method or way of thinking for me to find the answer alone.

• Hi. Interesting problem. Could you please explicitly specify whether the hunters are working together, with objective to maximise the expected total rabbits collected, or each for themselves, everyone acting to maximise their own rabbits collected?
– Stef
Dec 11, 2022 at 17:10
• Hint: Step 1. "But if hunter 3 chooses region B, the best outcome is when the other two hunters don't choose the same region between themselves." <<< Find the exact best mixed strategy for the first two hunters in this case.
– Stef
Dec 11, 2022 at 17:14
• Also, the game is completely symmetrical in all three hunters, so you should expect an equilibrium where all hunters have the same mixed strategy. Call p the proba that a given hunter goes to field A under this common mixed strategy, and calculate a hunter's expected payoff as a function of p.
– Stef
Dec 11, 2022 at 17:16
• Yes. That's my first suggestion. Another, different suggestion, is to look for equilibriums where all three hunters are following the same mixed strategy. This strategy has a single parameter (the probability to go to field A), so it should be easy to compute the expected payoff assuming they all follow it.
– Stef
Dec 11, 2022 at 18:27
• For 3 players, a Nash equilibrium means that no one player can improve their expected reward, holding the other players' strategies constant. That means, for each player whose probability is not 0 or 1, the expected payoffs for the two regions are the same. Dec 14, 2022 at 23:24