Question

3. Dividing Money There is a $10 available prize, and Players 1,2 must try to agree on a di- vision, via the following game: Player 1 writes down a proposed offer x1 to Player 2 (i.e., proposes that he, P1, keep 10-xi of the prize, giving the remaining xi to Player 2). Player 2 simultaneously writes down a demand for himself, x2: If xi >x2 (Pl's offer weakly exceeds P2's demand), the money s divided according to P1's suggested split (ie. P1 gets 10-xi and P2 gets:Ix x2 (P1 offers less than 1P2 demands), then both players get $0. (a) Ix2 are both restricted to the set 10;5;10, write out the payoff matrix, and identify all NE. Now, for parts (b)-(e), allow each player to choose any integer between 0 and 10. (b) Find PI's best responses, (G) for eachx2 E 10,..,9, and (i) for 2 10. (c) Find P2's best responses, (i) for each Xi є { 1, 2, . . ., 10), and (ii) for x,-0. (d) Using your answers to (b) and (c), find all NE, i.e. all pairs (xi; x2) where both players are best-responding to their opponent. (e) For each of your NE in (d), determine whether either player is using a weakly domi- nated strategy.

Answer 1

Player 1 suggests x1,
If Player 2 suggests x2 such that, x1 $\geq$ x2, P2 gets x1 and P1 gets 10-x1
If Player 2 suggests x2 such that x1 < x2, P2 and P1 both get 0.

(a) Let us create a payoff matrix keeping the above in mind:

	Player 2
Player1		0	5	10
	0	(0,10)	(0,0)	(0,0)
	5	(5,5)	(5,5)	(0,0)
	10	(10,0)	(10,0)	(10,0)

The best strategy is for Player 2 to play 0, while the best strategy for Player 1 is to play 10. Nash equilibrium exists when Player 1 plays 5 and Player 2 plays 5, giving a payoff of (5,5).

(b) If x2 = {0,1...9},
It is best if x1 is 10, maximising his payoff.
If x2 = 10, x1 = 10 will remain the best strategy.

(c) If x1 = {1,2...10}, x2 should be 0 to maximise payoff in each case.
If x1 = 0, x2 = 0 will remain the best strategy.

(d) We can find the best response for each player from the payoff table we constructed. For instance, when x1 = 1, x2 should be 0, this gives a payoff (9,1), which is better than (0,0). Nash equilibrium exists at (5,5) and for all combinations which give payoff of (5,5).

Hope this helped!