耶鲁大学博弈论第一章答案

3
Combination of player 2 and 3
aa
a
bb
cc
ab
a
ac
a a c
bc
a b c
player 1b (a>b>c)
c
a
b
c
b a
Combination of player 1 and 3
aa
a
bb
cc
ab
a
ac
a a or c c
bc
b or c b c
player 2 (c>a>b)
A B C A B C
3,3 2,2 0,0 2,2 1,1 0,0 1,1 0,0 0,0
In the example above, strategy A strictly dominates strategy B and C. Strategy B weakly dominates strategy C. 2. Iterative Deletion of (weakly) Dominated Strategies.
4
20,80 25,85 30,70 50,50 60,40 55,45 50,50 45,55 40,60 35,65
5
25,75 30,70 35,75 40,60 50,50 50,50 45,55 40,60 35,65 30,70
6
30,70 35,75 40,60 45,55 50,50 50,50 40,60 35,65 30,70 25,75
As shown in the matrix, strategies 2 through 7 all strictly dominate strategy 1. (b) Strategy 1 is weakly dominated by strategy 2, e.g. u1 (1, 2, 3) = u1 (2, 2, 3) = 10. Strategy 1 is also weakly dominated by strategy 3, e.g. u1 (1, 3, 4) = u1 (3, 3, 4) = 15. Once we delete strategies 1 and 10, strategy 2 is, by the same token as in “strategy 1 weakly dominated by strategy 2”, weakly dominated by strategy 3 in the reduced game. 4. “Strength can be weakness.” (a) The game results are shown as follows. The columns are the combination of strategies of two players, e.g. ab means either strategy (a,b) or (b,a).
2
l T D
1,1 1,3
r
3,1 2,2
Now in the reduced game, for player 1, D is weakly dominated by T and for player 2, r is weakly dominated by l. The only left strategy is T for player 1 and l for player 2. (c) A seemingly dominating strategy becomes a dominated strategy after reducing the game. 3. Hotelling’s Location Game. (a) Payoff matrix is shown as follows.
Problem Set 1 Solution
Econ 159a/MGT522a, Yale University
M.Chen momotocmx@hotmail.com
1. Strictly and Weakly Dominated Strategies? A strategy si is a strictly dominated strategy if there exists a strategy si such that si always does strictly better than strategy si no matter what others do, that is ui (si , s−i ) > ui (si , s−i ) for all s−i A strategy si is a weakly dominated strategy if there exists a strategy si such that ui (si , s−i ) ≥ ui (si , s−i ) for all s−i ui (si , s−i ) > ui (si , s−i ) for some s−i Example:
player 3 (b>c>a) b c player 2 (c>a>b) c
a a a a c
Now player 2 has strategy a weakly dominated by c and player 3 has b weakly dominated by c. The predicted outcome is that player 1,2 and 3 will choose strategy a,c and c respectively and finally winner is player candidate c.
b c
a
b
c
b a or b
Combination of player 1 and 2
aa
a
bb
cc
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ab
a
ac
a a or c c
bc
b or c b c
player 3 b
c
a
b
c
b a or b
For player 1, both strategies b and c are weakly dominated by strategy 1. For player 2, strategy b is weakly dominated by strategy c. For player 3, strategy a is weakly dominated by strategy b. (b) The reduced game after deleting b,c of player 1, b of player 2 and a of player 3:
l 1 T M D
1,1 1,0 1,3
2 c
0,1 2,2 3,1
r
3,1 1,3 2,2
(a) There is no strictly dominated strategies. For player 1, M is weakly dominated by D. For player 2, c is weakly dominated by r. (b) Deleting dominated strategies give the following matrix: 1
7
35,75 40,60 45,55 50,50 55,45 60,40 50,50 30,70 25,75 20,80
8
9
40,60 45,55 50,50 55,45 60,40 65,35 70,30 50,50 20,80 15,85
10
45,55 50,50 55,45 60,40 65,35 70,30 75,25 80,20 50,50 10,90 50,50 55,45 60,40 65,35 70,30 75,25 80,20 85,15 90,10 50,50
1
1 2 3 4 5 6 7 8 9 10 50,50 90,10 85,15 80,20 75,25 70,30 65,35 60,40 55,45 50,50
2
10,90 50,50 80,20 75,25 70,30 65,35 60,40 55,45 50,50 45,55
3
15,85 20,80 50,50 70,30 65,35 60,40 55,45 50,50 45,55 40,60