Due April 26

This Bonus is actually several bonus assignments rolled into one. Each question can replace one of your previous assignments. Be warned, every question but the first is quite hard.

There are three mystic hermits, Hassan, John, and Faxian, living in a hut on one of the corners of diamond lake. Each morning they get up and find a place along the shores of diamond lake to meditate. Hassan goes out first, then John, then Faxian. The payoff to a mystic increases with their distance from the nearest other mystic. All that chanting, and locust munching of another mystic can really distract from mediation. Payoff decreases with the distance a monk walks, since time spent walking is time not spent meditating. Hermits are mortified at the thought of interrupting another hermits mediation and so they would rather walk the long way around or end up very near another hermit than actually walk past a meditating hermit. Diamond lake is perfectly square with sides of length 1km. The payoff for a hermit is 2D – W. Where D is the shortest distance from the nearest other Hermit, and W is the distance the hermit has walked along the shore of the lake.

Question 1

Suppose that Hassan is sick today and now this is simply a two player game, between John and Faxian. What are the Nash equilibrium strategies for each player. (Figure out Faxian’s best response to what John does. Suppose that John knows that Faxian will play the best response to John. How does John maximize his payoff.)

Question 2

Suppose Hassan is no longer sick and that this is now a three player game. What are the Nash equilibrium strategies for each player. (Figure out Faxian’s best response to what John and Hassan have done. Supposing that John knows that Faxian will play his best response what is John’s best response to Hassan. Supposing that Hassan knows that John and Faxian will play their best responses, what choice of actions should Hassan take to maximize his payoff.

Question 3

Faxian has transcended the material world, and now pays no attention to where he is walking. He is equally likely to go counterclockwise or clockwise around the lake and walks a random distance drawn from a uniform distribution on zero to two. Faxian has transcended payoff functions. The payoff functions of John and Hassan are as before, but now they have the possibility of crazy Faxian walking past them and interrupting their meditation which causes them to lose 2 satisfaction points.

Question 4

Faxian has passed from crazy to “enlightened”. He can now walk on water, although he still needs to sit on the shore of the lake to meditate. (Everyone else still has to walk along the shore.) Everyone’s payoffs are as in part one, what is the Nash equilibrium strategy of each player.

Hi Dan,

I hit a cubic equation for question one. Just want to double check, the shortest distance between the two hermits is the diagonal line connect the two hermits, not the distance along the shore, is that correct? If that is the case, can we leave it at the cubic step and not actually solve it, but to just outline/describe the following steps? Thanks.

You are correct the shortest distance is the diagonal connecting line across the water. Not the distance along the shore. Just outlining the steps would be great, in some ways even better than actually solving the thing.