Question

Alex and Barry have a joint project. Each has first to decide whether to invest 10 or zero (i.e., not to invest) into the project. They make these individual investment choices simultaneously. Once made, these investments are sunk. If no-one invests, the project generates a total revenue of 0. If just one of them invests, then the project generates a (gross) total revenue of 15. If both of them invest, the project generates a (gross) total revenue of 30. Suppose that both Alex and Barry are told how much the project has generated before they make their “share demands”.Alex and Barry then use the following scheme to divide the total project revenue. Each player simultaneously writes down a “share demand” on a piece of paper. The demands can be either 1/5, 1/2 or 4/5. If the two “share demands”add up exactly to one, then each player is given his demand. Otherwise, all the money is thrown away. Thus, for example, if Alex and Barry each invest 10,then the project generates a gross total of 30. If Alex then writes down 4/5 and Barry writes down 1/5, then (since this adds up to one) Alex gets his demand 2 of 24 (= 4/5 x 30) for a net profit of 14( i.e., 24 minus his initial investment of 10), while Barry gets his demand of 6 (= 1/5 x 30) for a net profit of -4(i.e., 6 minus his initial investment of 10). If Barry had demanded 1/2 while Alex was still demanding 4/5, then the project money would have been thrown away and each would simply have lost his initial investment of 10.

a. Consider the subgame that follows Alex choosing to invest 10 and Barry choosing to invest 0. Find all the pure NE of this subgame .

b. Is there a pure SPE in which each player starts by investing 0? If so,explain the equilibrium strategies clearly. If not, explain why not clearly.

c. Is there a pure SPE in which each player starts by investing 10? If so,explain the equilibrium strategies clearly. If not, explain why not clearly.

Answer 1

Now let us first understand the pay off matrix where 0 represents loss and 1 represents gain . However the proportion of gains for each strategy is different.

We can see that there are three optimal strategies.

1) Alex invests , Barry doesn't

2) Barry invests ,Alex doesn't

3) Both invest .

Calculations of sub game :-

There are 3 choices for two players . Either ask for 1/5 , 1/2 or 4/5 share making 6 probability.

ALEX	BARRY	TOTAL
1/5	1/2	0.7
1/5	4/5	1
1/5	1/5	0.4
1/2	4/5	1.3
4/5	4/5	1.6
1/2	1/2	1

THUS EQUILIBRIUM STRATEGY WILL BE :-

Total Revenue generated	Alex share	Barry share	P/L Alex	P/L Barry	Who invests
15	1/5	4/5	3-10= -7	12-0=12	Alex
15	4/5	1/5	12-10=2	3-0=3	Alex
15	1/5	4/5	3-0=3	12-10=2	Barry
15	4/5	1/5	12-0=12	3-10=-7	Barry
15	1/2	1/2	7.5-10=-2.5	7.5-0=7.5	Alex
15	1/2	1/2	7.5-0=7.5	7.5-10=-2.5	Barry
30	1/5	4/5	6-10=-4	24-10=14	Both
30	4/5	1/5	24-10=14	6-10=-4	Both
30	1/2	1/2	15-10=5	15-10=5	Both

Thus most equilibrium strategy is when both invest and ask for equal shares .