Question

4 Game Theory II (40 points) Using the following infromation about normal-form game payoff matrix to answer the questions from (a) to (). Tony Confess Silent (4,-1) (3,3) (11) (-14) Confess Jane Silent (a) Identify pure strategy NE. What is the name for this type of game? What is the main issue of this game? (4 points) (b) Suppose this game is repeated infinitely and each time the probability of game end in that game is 1 -g where 0<8<1 is probability. Calculate expected payoff for cooperation that last forever. (4 points) (c) Continue from (b), calculate expected payoff for Jane when she deviates from cooperation and Tony penalizes her forever once Jane deviates from cooperation (this is called grime strategy). (2 points) (d) Continue from (b), calculate expected payoff for Jane when she deviates from cooperation and Tony penalizes only once (e.g., Jane applies silent first and Tony applies confess only once in the next game but he will go back to silent after that. This is called tit-for-tat strategy). (2 points) (e) What is the probability g that induces cooperation between Jane and Tony when Tony applies grime strategy. (4 points) () What is the probability g that induces cooperation between Jane and Tony when Tony applies tit-for-tat strategy. (4 points)

Answer 1

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Enpected payoft Cinder gaim enatrgy =Cis)+ CE C)+ CcCiC)+ - - olviotes -4th+a + -- 1 e'g 1-7 Expetd payoft Jane 0enfols (CiS)+(CIC)+ C, s)+ Ceic) +CCis)+CCiC)-- Cuder tet-for-tat st rategy y1.4. +4.81.4 3 -3+ [7t E =4+ 41+a+-+ g [1 +g+84+--1 4 + 1- 2 4+9 ET As Jame diviates she Rnon that shu sfu b pufshed after pinishment ane knouos Tony hou Cooprate Cas Tit - Pr-Tat sutey blayd by Tony) so if She cfesses sere gets highee fay ff of 4-indtead of Coofuretfing uitth Tomy & remain silent So, afteru pnishmenls &ue cloosi bo Confers7 Probalafuty&udueiny cooperatfen woen Exteeted Payoff Prom Deviation DE21 Expeeted bays ff pion coopeiat m 3 1-2 1-9

3-4>4 1-8 3-3 4-49 3g 1 > 1 3 fox E11 woiu fndue te coopatin hetoesn ane&Tony(athdes geim rategy). frobnsai bity ndueiy cooporatim Epeeted payolt Inom Reviation Te Entenered ayoft of coo beration TE1 4+2 FAR-8 = 4+B) (A-8)] Cotnce-) 4+9 3 3CH8) 3+ 39 2 2. leiu induce cooperetion betoeen Jome tTony Cappiies tit-fr-tat strategy