Question

5. For the following games, state the range of interest rate required to sustain cooperation using Grim Trigger and using Tit-for-Tat. a AB A 10 , 10 O 18 18, 03, 3 U 5, 51,7 D7 12,2 CLIR U 16 16 4 20 D20, 49, 9 U D -5, -5 -1, -20 -20 -16 , -1 , -16 U D 65, 50, 65 136 136 100 , 50 , 100

Answer 1

A_10; 10 o, 18 0 olcs wo? 5. 5 7 2, ? olon 16, 164, 2.0 20, 4 9 , 9 R -5, -5 | -20:-|| -1, -20 .16 -16 olce olce 65 65 So, 136 136," so 100, 100 ANS:- Player a Akyo, yo 18 Inefficiency still possible (B, B) in the unique NE of the stage game, so playing that in every period is a SPE - But is situation À possible as well 2 CS Scanned with CamScanner

GRIM TRIGGER for sensen I •If anyone has sitation 'B' at any port will be in situation B. otherwise no - If this can't inspire A' mothing can Now Twocconditions to check. 1 at B has been ang gered, is it an equilibrium S to continue B forever? ii) at no han situation B previously, would any me ا ? 15 مه ده • one shot Deviation Principle to the Rescue > No need to look at complex deviations => only need to check whether a player would not w ant to deviate to defection in each situation, period. Scanned with CamScanner

tor Period ca, situation A Condition A 10, 100, 18 B 18,0 3,3 Equilibrium payoff = 10 +108 +108?+ B's pay off = 18 + 38+382 + 35'+..... 9 10+108 +108?.... 18+ 38 + 38° +38+.... ->10111-8) 18 +381(1-6) => 10), 1811-8) +38 =) 10), 18-185+38 => 10-18), -188+38 =) - 8 - 158 =) is 8 ,8 >> Si 0-53 . I This arquement Applies to all periods! [sunk payoffs]+1099 +10.8947108a+2... sunk pay ost] + 18 5*+38**438*** 38**... 2 1089+1080 1087+?), 188 +387 138+2+ 38+3..... = 10 +108 +1082+....,18+ 38+38?+ 38° +.... + -- Same OS2+1 t1082 st 2 cm - t.... comMENTS F 1. A only work if actors are sufficient patient or 6 }, 0-53 - weird but cool that works for infinite - same but not gume arbitrarily long in finite game. Scanned with CamScanner

for period b, sitration ou indir 5, 5 1 V DI 2, 2. {auilibrium pay off = 5 + 5 & + 58t... E BRD's Pay off = 7+28+ 2 82 + 20 5 +58 +582.) 7+25 +28°+28++ =) 5/(1-8) » 7+28/(1-8) =) 5), 7(1-8) +28 =) 5), 7-78+28 =) 5-77-76+28 =) -21-58 I => 583,2 - -> 2/5 = 0·4. " This argument is applics to alun period i.e. e puisd e, dke... [sunk pay offs7+ 557 58nt 5 gat2 > [same sunk pay offs]+ Form + 282 +4280+2+28+7... => 58° +5 80+ 158212 , it 8 + 28 2+1 1282+2 28 +3 =) 5 + 58+5 82.... 7+28+28 2+283e.. rest of the comMENT same as Period a Scanned with CamScanner

11.FOR TAT Thury Punishment Problem som triggen in hardh one misstep ends A freuer can nicer shateng still yield efficient outermes? Tit fo Tat Being cooperating for all futuare periods, copy opment's strategy form the pressus period. Equilibrium? Is it awan equilibrium for two players to play tit for tat ? e what happens ? one the path, coeryone in situation A forever • off the path, something weird 1 1 2 3 4 S S I A B A B A B A BA BA B A B for Period a situation As condition • Equilibrium pay OH = 107 108 11087.... • B's Day off = 180 +0.8+0.8+0.8+... situation A it..... = lotlostiog?>, 18 +0.8+ 0.8² +0.83.... => 10111-8), 181(1-83)+0.81C1-82) -) 10+108 x 18+8 Panned Site Scando &

-)-8), -98 =) 98>,8 -) 6», 8/9 = 0.88 commouth - Sam Porn Shan mohtadele demnatin for all periods cut pont does not always match grim triggers Tit for Tat does very well in tournaments. Усля ки be c cat, » Per, . •, «, «ке. Scanned with CamScanner