Question

Suppose that there are two workers, i = 1, 2, and each can “work” (si = 1) or “shirk” (si = 0). The total output of the team is 4(s1+s2) and is shared equally between the two workers. Each worker incurs private cost 3 while working and 0 while shirking. Write down this as a normal-form game. Check whether there is any dominated strategy or not. Does iterated elimination of strictly dominated strategy give us any solution?

Answer 1

Solution:

The two strategies for two players is to work or to shirk.

If both work, total output = 4(1+1) = 8, and share for each worker is (8/2=) 4. In this case, both incur a private cost of 3 each, thus, each receive a payoff = 4 - 3 = 1

Similarly, if none worls, that is both shirk, total output = 4*(0+0) = 0. Then, share of each = (0/2=) 0, and with shirking, both incur a private cost of 0, ultimately receiving a payoff = 0 - 0 = 0

Finally, for the case where one works and one shirks, total output = 4*(1 + 0) = 4, and share of each = 4/2 = 2. But the one who is working receives payoff = 2 - 3 = -1 (since he/she incurred private cost of 3) and the one shirking receives payoff = 2 - 0 = 2 (as no cost is incurred by him/her).

Accordingly, we can write down the normal form game as follows (payoff (x, y) means payoff of x to worker 1 and of y to worker 2):

		Worker 2
		Work (s1=1)	Shirk (s2=0)
Worker 1	Work (s2=1)	(1, 1)	(-1, 2)
	Shirk (s2=0)	(2, -1)	(0, 0)

Checking for dominated strategy:

If worker 1 choose to work, the best response for worker 2 is to shirk (as that way, a higher payoff is received: 2 > 1). Similarly, if worker 1 choose to shirk, the best response of worker 2 is again to shirk (as 0 > -1). So, whatever worker 1 does, worker 2 will always choose to shirk, thus, shirk is dominant strategy for worker 2.

Similarly, if worker 2 choose to work, best response of worker 1 is to shirk (as 2 > 1; a higher payoff is received), and if worker 2 choose to shirk, worker 1 still prefers to shirk, making shirk a dominant strategy for worker 1 as well.

Thus, working is the dominated strategy for both the workers.

Following the iterated elimination of strictly dominated strategy, if we start by eliminating strategy 'work' for worker 1, we will be left with the last column of above table. Worker 2 then, will clearly choose to shirk (0 > -1), giving (shirk, shirk) as the solution (payoff: (0, 0)). (Same solution will be resulted if we start by eliminating 'work' strategy of worker 2, since unique pure strategy Nash equilibrium).