耶鲁大学博弈论第一章答案

1、3Problem Set 1 SolutionEcon 159a/MGT522a. Yale UniversityM.ClienmomotocinxhotniaiLcoin1. Strictly and Weakly Dominated Strategies?A strategy sj is a strictly doininated strategy if there exists a strategy s： such that Si always does strictly better than strategy s no matter what others do. that isfo

2、r all sA strategy is a weakly dominated strategy if there exists a strategy si such that 血for all s_i %(s,St) for some s_jExample:ABCA3,3B2.21J0.0c0,00,00,0In the example above, strategy A strictly dominates strategy B and C. Stratcgr B weakly dominates strategy C.2. Iterative Deletion of (weakly) D

3、ominated Strategies.(a) There is no strictly doininated strategies. For player 1. M is weakly dominated by D For player 2. c is weakly dominated by r.(b) Deleting doininated strategies give the following matrix:Now in the reduced game, for player L D is weakly dominated by T and for player 2. r is w

4、eakly dominated by 1. The only left strategy is T for player 1 and 1 for player 2.(c) A seemingly dominating strategy becomes a dominated strategy after reducing the game.3. Hotellings Location Game.(a) Payoff matrix is shown as follows.12345678910150.5010,9015,8520.80257530,70357540.6045,5550?50290

5、.1050.5020,8025,85307035.7540.6045.5550.5055,45385.15802050,503070357540,6045,5550,5055,456040480.2075,2570,3050,5040.6045.5550,5055.4560.4065,355752570?3065,3560.4050.5050,5055,4560.4065,3570;30670.3065,3560,4055.4550.5050,5060,4065.3570,3075,25765.3560.4055.4550.5045.5540.6050.5070.3075.2580.20860

6、.4055,4550,5045.5540.6035,6530,7050.5080,2085,15955.4550,5045,5540.6035.6530,70257520.8050.5090J01050.5045,5540,6035,6530,7025,7520,8015.8510,9050,50As shown in the matrix strategies 2 through 7 all strictly dominate strategy 1(b) Strategy 1 is weakly doiiiiiiated by strategy 2. e.g. ui(1.2.3)=血(2,2

7、,3) = 10. Strategy 1 is also weakly dominated by strategy 3, e.g ui(l,3,4)= 血(3,3,4) = 15. Once we delete strategies 1 and 10. strategy 2 is. by the same token as in strategy 1 weakly doiiiiiiated by strategy 2. weakly dominated by strategy 3 in the reduced game4. Strength can be weakness.(a) The ga

8、me results are shown as follows. The columns are the combination of strategies of two players, e.g. ab means cither strategy (a.b) or (b.a).player(abc)player 2(cab)aaaborebabcba orcbca orbccplayer 3Combination of player 1 and 2aa bb cc ab ac beFor player 1. both strategies b and c are weakly cloinii

9、iated by strategy 1. For player 2, strategy b is weakly dominated by stmtegy c. For player 3. str a tegy a is weakly dominated by strategy b(b) Tlie reduced game after deleting b,c of player 1, b of player 2 and a of player 3:player 3 (bca)Now player 2 has strategy a weakly dominated by c and player 3 has b weakly domin