The Basics of Record Keeping and

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  • Topic: Game theory, Nash equilibrium, Best response
  • Pages : 4 (565 words )
  • Download(s) : 70
  • Published : February 27, 2013
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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|...
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