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CSC5350-06 - Page 1 of 4 75 5 “l” i 3 $3...

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Unformatted text preview: Page 1 of 4 75: 5% “l” i 3: $3 fiéfiffifi game: The Chinese UanCI'Sl of Hon Kon ° ' “a” r Ex minati n ls Term 2 -2 7 Course Code 8: Title ; ....................... C SC5350GameThe0rymComputerSCIence .............................. Time allowed : 2 hours. .............................................. minutes Student ID. No. ................................................ Answer all four (4) questions. John is the father of Tom and Mary. One day, John gives Tom $3, and asks Tom to share it with his sister Mary. If Tom gives x dollars to Mary, then Tom will keep the remaining 3 — x dollars for himself (for simplicity, we assume that x is an inte- ger and 0 S x S 3). However, John wants to know whether Tom is a good boy, so he tells Tom to do the following. First, Tom should decide the amount of x, then put at dollars in a wooden box, and pass the box to Mary. Therefore, Mary does not know how much money there is in the box. However, Mary must decide Whether to accept the amount before opening the box, (i.e., Without knowing how much there is in the box). If Mary accepts it, then Tom will have 3 — x dollars and Mary will have x dollars. Otherwise, if Mary does not accept it, then John will get back the $3 and neither Tom nor Mary will receive any money. (a) Assume that the utilities of Tom and Mary are the amount of money they re- ceive. Model the scenario as a strategic game T = <N,(Ai),(ui)). i. (1 mark) Write down N in the game T . ii. (1 mark) Write down (A,) in the game T . iii. (2 marks) Write down the set A of outcomes in the game T . iv. (2 marks) Write down (ui) in the game T . (3 marks) What are the pure strategy Nash equilibria in the game T ? Justify your answer. (2 marks) If Tom is self-interested, how much money should Tom put in the box? Justify your answer. Now suppose the scenario is repeated every day. That is, suppose every day John gives $3 to Tom and Tom has to determine how much to give Mary. We further as- sume that both Tom and Mary Wish to maximise the average amount of money they re- ceive every day. Course Code CSC5350 ............ Page 2 0f 4 (d) Consider the hypothetical scenario that the game T is repeated for infinitely many days, which can be modelled as an infinitely repeated game T°° = (N,H,P,(u;)) of T = (N,(A,),(ui)). i. (1 mark) Write down P in the game T°°. ii. (3 marks) Write down (u: ) in the game T‘”. iii. (5 marks) Give gne feasible enforceable payoff profile in the game T°°. Justify your answer. iv. (2 marks) Give the enforceable outcome that corresponds to the feasible enforceable payoff profile in your answer to question iii above. v. (6 marks) In general, player i's minmax payoff v, is caused by the collec- tion p_, of actions of all other players, which is intuitively the most se- vere punishment that other players can inflict upon player i. What are Tom's and Mary’s minmax payoffs and the corresponding ’most severe punishing?’ Vi. (4 marks) Describe precisely a trigger strategy, such that if it is used by all players, a Nash equilibrium will result, which yields the feasible en- forceable payoff profile in your answer to question iii above. One day a little boy and a little girl share 11 identical pieces of chocolates, where n is even. They agree that they should use the following method to allocate the 11 pieces of chocolate. First, the little boy divides the n pieces into two shares. The little girl then has two options. The first option is that the little girl picks the smaller share, and let the little boy have the larger share. (Note that the little boy and the little girl agree that the little girl cannot pick the larger share if she chooses this option.) The second option is that the little girl moves k pieces of chocolate from the larger share to the smaller share, and then let the little boy choose. If the little girl chooses the second option, then the little girl can have the remaining share after the little boy chooses. Both the little boy and the little girl prefer more pieces of chocolate to less. (a) This chocolate allocation procedure can be formulated as an extensive game with perfect information G = (N, H ,P,(,>;,)) . i. (1 mark) Write down N in the game G. ii. (2 marks) Write down H in the game G. iii. (2 marks) Write down P in the game G. iv. (4 marks) Write down (:i) in the game G. (b) (6 marks) Are there any strategies that can be reduced? For each strategy that can be reduced, write down the original strategy and the reduced strat- egy. (6 marks) Find 93 subgame perfect equilibrium of the reduced game. (d) Now suppose the little girl is in a bad mood on that day, and the little boy knows that unless he divides the chocolate into two equal shares, the little girl will be upset and will simply randomly picks some (including none, and all) chocolate from the larger share, throw them into the rubbish bin, and grab the remaining larger share (not necessarily the originally larger share). Course Code CSC5350 ........... Page 3 of 4 The rubbish bin is so dirty that nobody will eat the chocolate that has been put into it. If the two shares are equal, then the little girl will be happy to pick one of them. i. (6 marks) Formulate the new situation as an extensive game with perfect information and chance moves. You may present your solution using a game tree. ii. (4 marks) Find one Nash equilibrium of the game. Justify your answer. 3. Consider the following game that is played by two players with a red box and a green box. First, player 1 puts a ball into either the red box (action R) or the green box (action G). Player 2 is not allowed to see when player 1 puts the ball into the box, so player 2 does not know which box contains the ball and which is empty. Then, player 1 tells player 2 Whether the ball is in the red box (action r) or the green box (action g). However, player 1 can choose to tell the truth or tell a lie, and then player 2 can choose to believe (action y) or not to believe (action n). The rule is as follows. 0 Case 1: player 1 puts the ball into the red box. In this case, player 1’s utility is ten (10) and player 2’s is zero (0) if player 1 successfully cheats player 2, or player 1’s utility is minus ten (-10) and player 2’s is one (1) if player 1 does not succeed in cheating player 2. ° Case 2: player 1 puts the ball into the green box. In this case, player 1’s utility is five (5) and player 2’s is zero (0) if player 1 successfully cheats player 2, or player 1’s utility is minus five (—5) and player 2's is one (1) if player 1 does not succeed in cheating player 2. ° In both cases, if player 1 decides not to cheat player 2, but to tell the truth to player 2, then both players’ utilities are three (3) if player 2 believes the truth, but both players’ utilities are minus one (—1) if player 2 does not believe the truth. (a) This game can be modelled as an extensive game with incomplete informa- tion G = (NIH/Plfuaoxzay i. (1 mark) Write down N in the game G. ii. (2 marks) Write down P in the game G. iii. (2 marks) Write down I] in the game G. iv. (2 marks) Write down 12 in the game G. (b) (2 marks) Is this game a game with perfect recall? Justify your answer. (c) (2 marks) How many pure strategies does player 1 have? What are they? ((1) (2 marks) How many pure strategies does player 2 have? What are they? (e) If player 2 thinks that it is more likely that player 1 puts the ball into the red box, discuss how player 2 should respond. i. (2 marks) Discuss under what condition player 2 should choose to be- lieve. Course Code : CSC5350 Page 4 of 4 ii. (2 marks) Discuss under what condition player 2 should choose not to believe. (2 marks) If player 1 thinks that player 2 will use the behavioural strategy fiz , which is to simply purely randomly choose to believe or not, what should be player 1’s best behavioural strategy '81 in response to [32 ? (2 marks) Describe a consistent assessment that contains the strategy profile 0311.32) in question (0- (2 marks) Is the assessment in question (g) a sequential equilibrium? Justify your answer. (2 marks) Suppose you, as an observer, know that player 1 puts the ball into the red box and tells a lie. Assume that the probability that player 2 believes is 0.8, then What is the outcome of the game? Consider a group of seven (7) players, each having one card. Players 1 and 2 each has a red card, and the other players each has a green card. There is a rule saying that players can form groups, such that a group can receive $1 if the group mem- bers have one red card gig one green card, and, in general, $11 if they have 11 red cards M 11 green cards. (a) This scenario can be modelled as a conational game with transferable payoff (N, v) . i. (1 mark) Write down N in the game. ii. (1 mark) Write down 12 in the game. (4 marks) Consider the following set of imputations Y1 = {(x,x,y,y,y,y,y):2x+5y= 2,x 2 0,y20} That is, the utilities of players 1 and 2 are x, and the utilities of other players are y. Discuss whether or not Y1 is a stable set. Prove your answer. (2 marks) Give a stable set different from Y1 (if Y1 is a stable set). Prove your answer. (2 marks) What is the ’standard of behaviour’ in your answer to question (c)? (2 marks) Identify one imputation that is in the core of the game. - End of Paper - ...
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