Question

Two workers (A and B) are on a production line. They can either exert effort E or shirk S. It costs each worker 1 > £c > 0 to provide effort and shirking costs them nothing. The workers together produce one unit of output. If the unit passes the quality control test it can be sold for £2 (shared equally between the workers) and it has value zero if it fails the test. The probability of passing the test, p, depends on how much effort is supplied: p = 0 if neither worker exerts effort, p = 1 2 if one worker exerts effort and p = 1 if two workers exert effort.

(a) Write down the strategic form of this game.

(b) For what values of c can you find dominated actions?

(c) Can you find a Nash Equilibrium of this game in the other case?

Answer 1

a) A and B decide simultaneously whether to exert effort or not given the payments that are conditionated by the other´s decision. In the graph we have the payments to A for each action E and S given B´s decision. Payments to B given A´s action is equivalent.

We can express it as a table too:

A/B	E	S
E	2 - C, 2-C	1 - C, 1
S	1, 1-C	0, 0

b) Let´s suppose C = 1. The payments will be:

A/B	E	S
E	1, 1	0, 1
S	1, 0	0, 0

In this situation, if A chooses E, B will choose any E or S because in both cases he is getting 1 payed.

If A chooses S, B will choose E or S because in both cases he is getting paid 0.

If B chooses E, A will choose any E or S because in both cases he is getting 1 payed.

If B chooses S, A will choose E or S because in both cases he is getting paid 0.

So as we can see, for C=1 there is no dominanted actions.

Now let´s set C1. For example C= 0.9

Then

A/B	E	S
E	1,1; 1,1	0,1; 1
S	1; 0,1	0, 0

The numbers in red are A decisions given B´s and in blue are B decisions given A´s.

Now as we can see, no matters what the other worker decides, the best decision is always E (exert effort).

This happens because when 0=C1, the payment for E is higher than shirking even when the other worker shirks. Contrary to what happens with C=1, when the worker is indifferent to choose E or S because the payments are the same no matters what the other worker decides.

So the answer is 0=C1.

c) When C=1:

A/B	E	S
E	1, 1	0, 1
S	1, 0	0, 0

We can see that we have an Nash equilibrium in E,E where both workers maximize their payments given the other´s decision and have no incentives to modify its strategy.