Question

Suppose n people are choosing between two activities, hiking or fishing, where n is an even number. The payoff from going to hike is 1 if more than half of the people go hiking, and 0 otherwise. The payoff from going to fish is 1 if more than half of the people go fishing, and 0 otherwise.

A.Find all the Nash Equilibria of this game, if any.

B.Suppose instead that the payoffs are such that the payoff to hiking is 1 if less than half of the people go hiking, and 0 otherwise, and the payoff to going fishing is 1 if less than half of the people go fishing, and 0 otherwise (remember that n is an even number). The goal will be to find all Nash Equilibria. First consider the case where h<(n/2)-1, where h=the number of people hiking. Are there Nash Equilibria where h<(n/2)-1?

C.Now consider the case where h=(n/2)-1.Are there Nash Equilibria where h=(n/2)-1?

D.Now Consider the case where h=(n/2). Are there Nash Equilibria where h=(n/2)?

E.Now consider the case where h=(n/2)+1. Are there Nash Equilibria where h=(n/2)+1?

F.Now Consider the case where h>(n/2)+1. Are there Nash Equilibria where h>(n/2)+1?

Answer 1

A. Two Nash Equilibria

(n, 0) and (0, n) i. e., either everyone go hiking or everyone go fishing.

Nash Equilibrium is a situation where not a single person has any inventive to deviate from the strategy he/she has chosen.

Let's start by taking a case where the number of people going to hiking is not equal to the number of people going to fishing. It doesn't matter which is higher. If number of people going to fishing is lower then everyone who is going to fishing has an incentive to move away because the number of people going to hiking is more than half of the total people and moving away will earn a payoff of 1 whereas staying in the group of fishers will earn a payoff of 0. This will continue till everyone changes their decision from fishing to hiking. After everyone go to hiking there is no incentive to anyone to deviate. So everyone going to hiking is a Nash equilibrium.

Similar analysis works when the number of people going to hiking is lower. Then the result will be everyone going to fishing.

Now let's assume that number of people going to fishing and hiking is same i. e., h=f=(n/2). Here everyone earns a payoff of 0.Here at least one person in either group has incentive to move away because the person that moves away will make his/her destination group higher in number and earns a payoff of 1.

So {(n/2) (n/2) } is not a Nash Equilibrium.

Hence Either everyone go fishing or everyone go hiking are the two Nash Equilibria.

B. h< (n/2) -1

This is not a Nash Equilibrium. Because one person in the other group (fishing) has an incentive to join the group of hiking. If one person moves from fishing to hiking, 'h' will still be lesser than (n/2) and this person will get a payoff off of 1. This deviation will stop only when the h=(n/2) -1.

When h=(n/2) -1, no one has an incentive to deviate i. e., deviating from fishing to hiking will make h=f=n/2 and everyone gets pay off 0. So the person who was fishing earlier hasn't got any better. People who are hiking are already getting a pay off of 1. So they don't have any incentive to deviate.

So h=(n/2) -1 is Nash equilibrium. Then f=(n/2)+1

Similar analysis for fishing will result in f=(n/2) -1 being Nash equilibrium. Then h=(n/2)+1

So {(n/2) -1, (n/2) +1} and {(n/2) +1, (n/2)-1} are both Nash Equilibria.

Now h=f=(n/2)

Now no one has any incentive to deviate because deviation in any direction will not improve the playoff of any person.

Therefore, h=f=n/2 i. e. , [(n/2) (n/2)] is a Nash equilibrium.

Half people going to fishing and half going to hiking is a Nash equilibrium.

Hence {(n/2), (n/2)}; {(n/2) -1, {(n/2) +1}; {(n/2) +1, (n/2) -1} are the three Nash Equilibria.

C. Yes. h=(n/2)-1 is a Nash Equilibrium

See part B

D. Yes. h=(n/2) is a Nash equilibrium.

See part B

E. Yes. h=(n/2) +1 is a Nash equilibrium.

f=(n/2)-1 is equivalent of h=(n/2) +1

See part B

F. No.

When h>(n/2)+1, f<(n/2) -1

This means people who are fishing are getting a payoff of 1 and hiking people are getting 0. In this case at least one person in hiking group has incentive to move to fishing because doing so the number of fishing people will still be lesser than (n/2) and this person will get an improvement in his/her payoff from 0 to 1.

So this person has an incentive to deviate.

Hence h>!(n/2) +1 is not a Nash Equilibrium.