Question

1. Each of two firms has one job opening. The firms offer different wages: firm 1 offers the wage $10 per hour, and firm 2 offers the wage $12 per hour. There are two workers, each of whom can apply to only one firm. The workers simultaneously decide whether to apply to firm 1 or to firm 2. If only one worker applies to a given firm, that worker gets the job; if both workers apply to one firm, the firm hires one worker at random (each worker is hired with probability 1/2) and the other worker is unemployed (which has a payoff of zero). (a) Write the payoff matrix of this game. (b) Find the best-response function of player 1. (c) Find all NE in this game.

Answer 1

Firm1 pays W1=$10 per hour

Firm2 pays W2=$12 per hour

Probability of each worker hired when applying to 1 firm = 0.5 = p = q

(a) The pay-off matrix of the game is:

		Worker 2
		Firm1	Firm2
Worker 1	Firm1	5,5	10,12
	Firm2	12,10	6,6

(b)

It shall be noted that in Nash equilibrium, every player plays a best response against the other players simultaneously.

Player 1's best response function function is choosing Firm1 to Player 2's choice of Firm2

(c) It is a simultaneous game. Simultaneous games are those where decisions are simultaneous: both we and the other ‘player’ choose at the same time.

The Nash equilibrium for (Worker1, Worker2) is {Choose Firm2 with pay-off of $12, Choose Firm1 with pay-off of $10} and {Choose Firm1 with pay-off of $10, Choose Firm2 with pay-off of $12}