The Basics of Record Keeping and

1. | | Player 2 | | | α | β | ſ | Player 1 | a | 12,9 | 11,1 | 10,3 | | b | 8,6 | 7,8 | 12,4 | | c | 5,5 | 11,3 | 10,7 |

a strictly dominates b a strictly dominates c α strictly dominates β α strictly dominates ſ
Nash Equilibrium: (12, 9)
2.
(a) There are no NE
(b) (5.10) and (10.5)
(c) (9,8) and (3,10)

3.
(a) Players {1, 2}; S1 = Rt and S2 = Rt
Payoff function = TR-TC = (100-5Pi +2P-i)(Pi-10) = 100Pi – 1000 – 5Pi^2+50Pi + 2P-iPi – 20P-i
= -5Pi^2 + (150+2P-i)Pi – (20P-i + 1000)

(b) Best response function dU1(PiP-i)/dPi = -10Pi + 150 + 2P-i = 0
BR1: Pi = 15 + 0.2P-i
BR2: P-i = 15 + 0.2Pi

(c) P1 BR1 BR2 15

P2 15 (d) Pi = P-i = 150/8 = 18.75; NE (18.75, 18.75)
(e) Profit for firm 1= -5Pi^2 + (150+2P-i)Pi – (20P-i + 1000) = 382.81
Profit for firm 2 = 382.81

(f) (P1, P2) = (21.67, 21.67); Profit for P1 or P2 = 408.33

4. (a) | | Player 2 | | | R | P | S | Player 1 | R | 0,0 | -1,1 | 1,-1 | | P | 1,-1 | 0,0 | -1,1 | | S | -1,1 | 1,-1 | 0,0 | (b) U2 (R, P) = 1; U1(S, S) = 0

(c) No

(d) No

(e) U2 ([1/4R, 3/4S],P) = 1/4U2(R,P) + 3/4U2(S,P) = 1/ 4 (1) + 3/ 4(-1) = -0.5

(f) U2 to [1/2 R, 1/2S]

| | Player 2 | | | R | P | S | Player 1 | R | 0,0 | -1,1 | 1,-1 | | P | 1,-1 | 0,0 | -1,1 | | S | -1,1 | 1,-1 | 0,0 | | | 1/2 | 0 | 1/2 |

There is no best pure strategy for player 1.

(g)

U1 (R, [1/2 R, 1/2S]) = 1/2U1(R, R) + 1/2U1(R, S) = 1/2
U1 (P, [1/2 R, 1/2S]) = 1/2U1(P, R) + 1/2U1(P, S) = 0
U1 (S, [1/2 R, 1/2S]) = 1/2U1(S, R) + 1/2U1(S, S) = -1/2

The best response for player 2 to [1/2 R, 1/2S] is that player 1 always plays rock

(h) | | Player 2 | | | | R | P | S | | Player 1 | R | 0,0 | -1,1 | 1,-1 | Qr | | P | 1,-1 | 0,0 | -1,1 | 1-Qr-Qs | | S | -1,1 | 1,-1 | 0,0 | Qs |