The Stag Hunt game is based on a story told by Jean Jacques Rousseau in his book Discourses on the Origin and Foundation of Inequality Among Men (1754). The story goes something like this: “Two hunters set out to kill a stag. One has agreed to drive the stag through the forest, and the other to post at a place where the stag must pass. If both faithfully perform their assigned stag-hunting tasks, they will surely kill the stag and each will get an equal share of this large animal. During the course of the hunt, each hunter has an opportunity to abandon the stag hunt and to pursue a share. If a hunter pursues the hare instead of the stag he is certain to catch the hare and the stag is certain to escape. Each hunter would rather share half of a stag than have a hare to himself.”
The matrix below shows payoffs in a stag hunt game. If both hunters hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one hunts stag and the other hunts hare, the stag hunter gets 0 and the hare hunter gets 3.
(a) If you are sure that the other hunter will hunt stag, what is the best thing for you to do? Hunt stag.
(b) If you are sure that the other hunter will hunt hare, what is the best thing for you to do? Hunt hare.
(c) Does either hunter have a dominant strategy in this game? No. If so, what is it? If not explain why not. The best action to take depends on what the other player does.
(d) This game has two pure strategy Nash equilibria. What are they? Both hunters hunt stag. Both hunters hunt hare.
(e) Is one Nash equilibrium better for both hunters than the other? Yes If so, which is the better equilibrium? Both hunt stag.
(f) If a hunter believes that with probability 1/2 the other hunter will hunt stag and with probability 1/2 he will hunt hare, what should this hunter do to maximize his expected payoff? Hunt hare.