Game Theory Question: three person Nash Equilibrium?

Question

Suppose players A, B,C play a game. Each writes an integer from 1 to 10. If A writes i, and B writes j, and C writes k, the payoffs to each player are the following:
A's payoff is |i - j|
B's payoff is |j - k|
C's payoff is |k - i|

Find the Nash equilibrium.

I am really having trouble with three person games, so any help on how to find it would be greatly appreciated.

Answer 1

Now, suppose (a, b, c) is a pure-strategy Nash Equilibrium. Then it is also the case that (a,b) is an equilibrium in the two-person game between A and B where c is known, and so on. So let's look at the payoffs there. A's payoff is |a-b|. B's payoff is |b-c|. So B's payoff is unaffected by A's action. If there's an equilibrium, |b-c| has to be maximal given c.

That tells us that b has to be 1 or 10. Specifically, it is 1 if c>5 and 10 if c<6.

OK. Suppose B chooses 1. Then A's best strategy is definitely to choose 10. C's best strategy is definitely to choose 1. B will then want to change his strategy. So there's no pure-strategy Nash Equilibrium with B choosing 1. Similarly there is none with B choosing 10, and hence there is none at all.

However, consider a mixed strategy outcome in which each of A, B, and C chooses either 1 or 10 with 50% probability. If B chooses that strategy, then A's expected payoff is the same no matter WHAT strategy he chooses. Similarly for C and B. So that strategy is indeed a Nash Equilibrium.

I think it's the only one.