Question

Construct your own coordination game, by selecting an appropriate game matrix (choose your own payoffs). Find the Nash equilibrium (or equilibria) for this game. What do you need to change in your payoff matrix to obtain a potential game? Find the potential function for your proposed potential game.

Answer 1

An example of a coordination game would be :

There are two pure nash equilibria for the given game : (A,A) = 3,3 and (B,B) = 1,1

To obtain a potential game, we need a slight modification in just two payoffs of the matrix. We will try to make the above coordination game into a prisoner's dilemmna(as prisoner's dillemma is a potential game). The required matrix would be :

So, we changed strategy (B,A) and strategy (A,B) by adding 5 to player 1's payoff and player 2's payoff respectively. This is now an example of prisoner's dilemma, where (B,B) is a pure strategy nash equilibrium.
The potential function for this prisoner's dilemma game is given by :