Mixed Strategy Bayesian Nash Equilibrium

Question

There was an exercise question regarding two players with two types each in a game theory class. The two players were assigned to do a team project together. The utility from doing the team project is 10 for both players. The cost of doing the project for player 1 (C1) can be either 5 or 15, and the cost of doing the project for player 2 can also be either 5 or 15. and the probability that the cost will be 15 is 1/3 for both players.

Now the professor said there were three Bayesian Nash Equilibria. Among the three two were quite similar.

One equilibrium was

Player 1 with high cost type - not do project

Player 2 with high cost type - not do project

Player 1 with low cost type - not do project

Player 2 with low cost type - do project

and another Bayesian equilibrium is the same thing as above with the Player 1 with low cost type and Player 2 with low cost type swapped in terms of their action. And thankfully I understood how the two equilibria were derived using expected utility comparison.

The problem I have right now is with the third equilibrium. The teacher mentioned that there is a third equilibrium existing here using 'Mixed Strategy' but that's all he mentioned and he didn't go over it. How can we solve for the Mixed Strategy Bayesian Nash Equilibrium for this kind of question? I've done mixed strategy for perfect information, but I have no idea how to do it for incomplete information Bayesian Nash Equilibrium. I would appreciate some explanations on how to solve for the mixed strategy Bayesian nash equilibrium using this activity. Thanks.

Answer 1

The trick for finding a mixed strategy Nash Equilibrium is that given everyone else's strategies, all players will be indifferent between each of the options their randomizing over (ie. those options will yield the same payoff). So all you need to do is write an expression relating each player's expected payoffs for each strategy, and solve for the frequencies. Letting x represent the probability that player j does the project, low-cost player i's payoff for doing the project is 5, while their expected payoff for not doing the project is 10x. So you know in the Nash Equilibrium, 5=10x. The incomplete information makes the problem slightly more complicated: player i may expect j to complete the project half the time, the task is now finding j's probability of completing the project given that they are low-cost (not too difficult, since high cost players will never do the project). I'll leave the rest to you.