Games Strategies and Decision Making Second Edition Chapter

Games, Strategies, and Decision Making, Second Edition

Chapter 4, problem 15

Two companies are deciding at what point to enter a market. The market lasts for four periods and companies simultaneously decide whether to enter in period 1,2,3,4, or not enter at all. Thus, the strategy set of a company is (1,2,3,4, do not enter). The market is growing over time, which is reflected in growing profit from being in the market. Assume that the profit received by monopolist in period t (where a monopoly means that only one company has entered) is 10*t -15, whereas each duopolist (so both have entered) would earn 4*t -15. A company earns zero profit for any period that it is not in the market. For example, if company 1 entered in period 2 and company 2 entered in period 3, then company 1 earns zero profit in period 1; 5(=10*2-15) in period 2; -3(=4*3-15) in period 3; and 1(4*4-15) in period 4, for a total payoff of 3. Company 2 earns zero profit in periods 1 and 2, -3 in period 3, and 1 in period 4, for a total payoff of -2.

a. Derive the payoff matrix.
b. Derive a company\'s best reply for each strategy of the other company.
c. Find the strategies that survive the IDSDS.
d. Find the Nash equilibria.

Solution

a.

Strategies for both the companies are:

{1, 2, 3, 4, don\'t enter} i.e. both companies have five strategies in hand.

If both chose to enter in the period t, then payoff for both would be 4*t - 15 for that and all the remaining periods and 0 for the earlier periods.

If both chose not to enter then their payoffs would be 0

so, the pay off matrix would be:

Company 2

1

2

3

4

Don’t enter

1

-20,-20

-14,-9

-2,-2

16,1

40,0

2

-9,-14

-9,-9

3,-2

21,1

45,0

Company 1

3

-2,-2

-2,3

-2,-2

16,1

40,0

4

1,16

1,21

1,16

1,1

25,0

Don’t enter

0,40

0,45

0,40

0,25

0,0

b.

If other company is choosing strategy 1 and 2 i.e. entering in first or second stage, then the best reply is to choose strategy 4 i.e. enter in the fourth period. however, if the other company is choosing strategy 3, 4 i.e. enter in third, fourth period or don\'t enter then the best reply is to choose strategy 2 i.e. enter in the second period.

c.

Only strategies 2 and 4 survive the IDSDS other strategies can be itterated. So, entering in second and fourth period will survive IDSDS.

d.

There are two nash euiliibria, if company 1 chooses 2 then best strategy for company 2 is to choose strategy 4 and if company 1 chooses 4 then the best strategy for company 2 is to choose strategy 2

			Company 2
		1	2	3	4	Don’t enter
	1	-20,-20	-14,-9	-2,-2	16,1	40,0
	2	-9,-14	-9,-9	3,-2	21,1	45,0
Company 1	3	-2,-2	-2,3	-2,-2	16,1	40,0
	4	1,16	1,21	1,16	1,1	25,0
	Don’t enter	0,40	0,45	0,40	0,25	0,0