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1、Problem Set 9 SolutionEcon 159a/MGT522a, Yale UniversityM.C1. War of Attrition.f(2) (-c-c,-c-c)F(2)q(2)A(-c+v,-c+0)f(1)Bf(2)Q(2)(-c+0,-c+v)q(2)F(1)q(1)(-c+0,-c+0)(v,0)ABf(1)Q(1)(0,v)q(1)(0,0)First we calculate the pure SPEs in this game.Second sub-gamef(2)(-c,-c)f(2)q(2)F(2)q(2)F(2)-c,-cv, 0(v,0)ABf
2、(2)Q(2)0, v0, 0Q(2)(0,v)q(2)(0,0)The NE is(F (2); q(2); (Q(2); f(2)and corresponding payo s (v; 0) and (0; v).First stage12For (Q(2),f(2) in stage 2For (F(2),q(2) in stage 2f(1)q(1)f(1)q(1)F(1)F(1)-c,-c+vv, 0-c+v,-cv, 0Q(1)Q(1)0, v0, 00, v0, 0NE: (F(1),q(1),(Q(1),f(1)NE: (F(1),q(1),(Q(1),f(1)Therefo
3、re, the pure SPEs are(F (1); q(1); (Q(1); f(1) .Next, we will calculate the mixed SPE.SPE of mixed strategy in both stagesIn second stage, we havef(2)q(2)F(2)p-c,-cv, 0-c +Q(2)0, v0, 01-pp1-pcp + v(1 p) = 0 p + 0 (1 p) p =v; 1 p =cc + vc + vMixed NE is (v;c); (v;c), and the payo is (0; 0). Back to r
4、st stage, we havec+vc+vc+vc+vfollowing payo matrixBf(1)q(1)F(1)q-c+0,-c+0v, 0AQ(1)0, v0, 01-pq1-qAfter some simple calculation, we obtain q = c+vv . Finally, we have the SPE with mixed3strategy in both stages2vcvc;vcvc36(c + v; c + v ); (c + v; c + v )(c + v; c + v ); (c + v; c + v )7676745|z |z6pla
5、yer 1splayer 2s7SPE of mixed strategy inrst stage and pure strategy in second stageWe already compute the pure NE in second stage f(F (2); q(2); (Q(2); f(2)g.Back to therst stage:For (F (2); q(2) in second stage, we have the following payo matrixBf(1)q(1)F(1)q-c+v,-c+0v, 0AQ(1)0, v0, 01-qp1-pAfter s
6、ome calculation, we obtain p =vand q =v. The mixed SPE in this case iscc+v2vcvcv36(v + c; v + c); F (2); ( c ;c); q(2)767674|z |z56player 1splayer 2s7For (Q(2); f(2) in second stage, we have the following payo matrixBf(1)q(1)F(1)q-c+0,-c+vv, 0AQ(1)0, v0, 01-qp1-p4After some calculation, we obtain p
7、=vand q =v. The mixed SPE in this case isc+vc23v;cc;vc6( cc); Q(2)(c + v ; c + v ); f(2)7676745|z |z6player 1splayer 2s72. Its a Wonderful Life.(a) The extensive-form (game tree) of the game is shown as follows.W(r, r)WN12(2r-D, D)WW(R, R)(D, 2r-D)NNWN2(2R-D, D)1W(D, 2R-D)NN(R, R)(b)Stage 2: NE = (W
8、; W )WNWR, R2R-D, DN0, 2R-DR, RStage 1WNWNWWr, r2r-D, Dr, r2r-D, DNN0, 2r-DR,R0, 2r-DR, RR+D 2rR+D 2r, then SPEs are (W; W ); (W; W ) and (N; W ); (N; W ).If R + D 2rNFrom the payo matrix, we can conclude that there is still an SPE in which S&L goes bust, that is both players choose W in rst stage.
9、However, we also notice that now strategy W is weakly dominated by N, which means both players will play N in the rst stage and this reduce the likelihood of S&L going bust in period 1.3. Dating in the Burbs.DMovie-BMovie-RMovie-BCMovie-Rk, 10, 00, 01,2(a) The extensive form and normal matrix form:Ck(3/4, 0)oobmBovieDRB(k, 1)DMovie-BMovie-RR(0, 0)Book3/4, 03/4, 0BCMovie-B(0, 0)k, 10, 0Movie-R0, 01, 2R (1, 2)(b) For NE = (k; 1), k 34 . For NE = (1; 2), Cheryl will always choose going to the movie. Therefore, for all k 1 will all the SPE involve C going to the movie.4. Burning Money.