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Ind Organ & Perform

by: Leopoldo Ritchie DDS

Ind Organ & Perform ECON 431

Leopoldo Ritchie DDS
GPA 3.99


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This 7 page Class Notes was uploaded by Leopoldo Ritchie DDS on Thursday October 29, 2015. The Class Notes belongs to ECON 431 at University of Michigan taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/231656/econ-431-university-of-michigan in Economcs at University of Michigan.


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Date Created: 10/29/15
Economics 431 In nitely repeated games Let us compare the pro t incentives to defect from the cartel in the short run when the rm is the only defector versus the long run when the game is repeated The payoffs come from a duopoly forming a cartel on a market with linear demand p 30 7 Q where each rm has a marginal cost 0 6 If each rm stays in the cartel each gets half of monopoly pro t 72 If one rm leaves the leaving rm produces more than its share of cartel output and gets 81 If both rms leave they play a Cournot game with payoffs of 64 to each player This is a prisoner s dilemma whoever leaves rst reaps the pro t from selling more at high price but if the other rm leaves both rms are worse off than they would be in a cartel S L 8 7272 54g L 54 64g If the rms simultaneously decide whether to stay or leave the unique Nash equilib rium is for both to leave Suppose that the cartel exists for multiple periods and each period there are two defectors who simultaneously decide whether to stay or leave We know from our analysis of the chain store paradox what will happen if the cartel exists for a predetermined number of periods say T If the rms agree to stay in the cartel they will necessarily both cheat in the last period it is like your landlord who never returns a security deposit if you move out of town By backwards induction we can say that since staying now never affects future payoff Therefore every rm will choose an action that maximizes current payoff The unique subgame perfect equilibrium is to choose L every period The problem with nitely repeated games is the so called end game problem no matter how far away the end is if the end occurs at a certain time the chain store paradox will work However if we assume that the number of periods is not predetermined we can avoid end game problems If there is some probability that the game continues to morrow then actions today always have at least some rami cations for tomorrow In contrast in a nitely repeated game there is a period with no tomorrow We will thus consider in nitely repeated games They are called so not because it is necessary that the game goes on forever It is more precise to call them inde nitely repeated games there is tomorrow with some probability As we have seen a defector has a short run gain and a long run loss How do we compare the payoffs receive in different periods We use a discount factor The concept of present value We compare streams sequences of payoffs from different periods using a single number the present value of a payoff sequence 1 o The present value is the sum that we are willing to accept today instead of future payoff Let us compute this sum What sum would you accept instead of 7T dollars payable a year from now Will it be more than 7T or less than 7139 If you have today you can put it in the bank for a year and get exactly 7T a year from now 7139 PV 17 What if a year from now you are offered a lottery where you win 7T with probability p and win 0 with probability 1 7 p What sum will you accept today instead of this lottery You will probably accept something that equal the present value of your expected winning Pvp7r1p0 p 1 7 1 7 5 L 1 7 is called discount factor It shows you that if the game continues tomorrow with probability p and the interest rate is 7 then every dollar received in the tomorrows game will be worth 6 dollars payable today Example compute the present value of an in nite sequence of payoffs using the Sum of in nite geometric series 1 1 6 62 63 176 101010 2 10 PV10105106 07271717quotquot PV026162163 62 7 2 2 70266166726176 17271727172739 PV 1266226364265 1 26 162 64 26162 64 17621762 What is a strategy in an in nitely repeated game 2 A strategy should tell you what actions to take in every decision node But how we describe decision nodes when we have an in nitely large tree A history of the game is a record of all past actions that the players took Each history corresponds to a path to a particular decision node or information set Then when we know the history of the game all past actions of all players we know what information set we are in For example suppose that the cartel game S L S 72 72 54 L Q 54 647 Q is repeated twice Consider the second period of the game A history is the record or all past actions taken by players One example of a history would be S1L1 meaning that rm 1 stayed in the rst period S1 and rm 2 left in the rst period The set of all possible histories H1 includes all possible action combinations that could have occurred in the rst period This set not surprisingly has 4 elements H1 J51517 S1L17L1S17L1L1 The strategy for the twice repeated cartel game is a rule that tells a player what to do after every history The strategy must tell a player what to do in the rst period and what to do after each of the 4 histories in H1 Thus a strategy is a veletter word each letter being S or L For example S1L2L2S2S2 is a strategy telling a player to play S in the rst period S1 play L in the second period if the history was S1S1 or S1L1 and play S in the second period if the history was L1S1 or L1L1 Even in this simple game there are 32 25 different strategies for either player Because of the huge number of strategies full description of all strategies is im practical if we have a game repeated many times However it is not necessary to look at all possible strategies We can look at strategies that tell the players to take the same action after a whole lot of somehow similar histories A strategy that tells a player to play S in the rst period and play S in the second period if and only if both players stayed in the rst period can be written as S1S2L2L2L2 Notice that the same action L2 is speci ed after all histories where some player has left in the past What is a subgame perfect equilibrium in an in nitely repeated game A subgame perfect strategy tells a player to play the best response to the oppo nents strategy after every history That is given the opponent s strategy a subgame perfect strategy must always after every history prescribe an action that is a best response to the opponent s strategy The action that is a best response must give a player a higher payoff than my other action available after this particular history Theoretically we need to go history by history and check every time whether a strategy prescribes a best response But since the same action may be prescribed after many similar histories many of these checks are redundant 3 Example The trigger strategy in prisoner s dilemma Let us reconsider the cartel game which will now be in nitely repeated Let us divide all histories in two groups The rst group call it C is all the histories that involved both rms playing S in the past There is only one such history 0 55551 SS The second group call it P is all other histories all histories that involved someone leaving in the past Let the rm s strategy be the following Start with playing S After the history where both rms have played only S in the past ie after a history from group C play S now After all other histories play L This strategy describes the grim trigger agreement 1 will stay in the cartel today as long as everybody stayed in the past If not 1 will leave and never stay again Let us check for which discount factors this strategy is subgame perfect We have just two groups of histories to worry about Is playing S after C a best response to the opponent s trigger strategy If it is not then choosing an alternative action L should give more payoff Leave now LS LL LL Stay now SS SSSS Staying now is better if payoff from leaving is less 64 1 lt 72 1 7 6 1 7 6 V Get 64 forever beginning tomorrow Get 72 forever 8117 5 646 lt 72 817277139137713907 9 81764 Warp 17 After all other histories group P the strategy tells to play L the other player plays L every period after these histories n6 Get this today 64 LLLLLL payoff m If a player plays S instead the current payoff is reduced and the future payoff is unaffected SL LLLL payoff 54 5 64 176 Therefore if 6 gt 1 97 player 1 plays best response after every history The trigger strategy is subgame perfect Player 2 is absolutely symmetric For him the trigger strategy is also subgame perfect if 6 gt 1 97 The trigger strategy imposes the harshest possible punishment it tells the players to stay in the punishment phase forever Note that both rms using this strategy have an incentive to declare that quotbygones are bygonesquot quot forget quot the previous history of play and start afresh 1n this sense strategies that involve permanent punishments are less satisfactory why would anyone follow these strategies if both parties have an incentive to renegotiate the punishment agreement An alternative strategy for enforcing cooperation in the repeated game involves punishing the deviating rm for just one period and then forgiving it and going back to cooperation The strategy works as follows All histories of play are subdivided into three groups or phases C cooperative phase all histories that involved both players choosing the same action in the every previous period either LL or SS P1 punishment phase for player 1 histories where player 1 has cheated in the previous period by either leaving the cartel or not accepting the punishment P2 punishment phase for player 2 histories where player 2 has cheated in some way in the previous period Players choose their actions based only on the phase of the game The candidate equilibrium strategies are Player 1 Player 2 Phase Action Description Phase Action Description C S Cooperate C S Cooperate P1 S Accept punishment P1 L 1mpose punishment P2 L 1mpose punishment P2 S Accept punishment The rule by which the game changes phases depends on the phase it started in and on the actions that players choose in the current period The table below shows the phase that the game goes into depending on the initial phase and the outcome of the current period Current play 1nitial phase SS LS SL LL 0 0 P1 P2 0 P1 0 P1 0 P1 P2 0 0 P2 P2 1n words players cooperate play S as long as the game is in the cooperative phase 1f player i has cheated the game changes to the punishment phase B look at the rst row of the table above 1f the game starts in phase B and player i accepts the punishment ie plays S he is forgiven and the game returns to cooperative phase next period However if player i does not accept the punishment he is not forgiven the game stays in the punishment phase next period That is the game stays in the punishment phase until the cheater player accepts the punishment ie until player 7 plays S To see if the strategy described is a subgame perfect equilibrium we must show that there are no pro table deviations after any history that is there are no pro table deviation in any phase of the game From phase C The strategy tells player 1 to play S Then the game will stay in phase C and player 1 will have the payoff of 2 72 72 726 6 1f player 1 deviates to L he is going to be punished next period and forgiven two periods from now In other words if player 1 chooses action L instead of S the game will go into phase P1 one period from now and back into phase C to periods from now The corresponding payoffs are 81 now 54 next period and 72 in each subsequent period with the present value of 72 81 546 62 1 7 6 Equilibrium action S has a higher payoff if 72 72 72 726 62 81 546 62 1 7 6 gt 17 6 1 186gt96gt From phase P1 The strategy tells player 1 to play S while his opponent will be playing L Then the game will return to phase C next period and player 1 will have the payoff of 72 54 726 62 1 7 6 1f player 1 deviates to L he is going to be punished again next period and forgiven only two periods from now That is he will get 64 now 54 next period and 72 in each subsequent period with the present value of 72 64 546 62 1 7 6 In order for player 1 to accept the punishment doing so must give a higher payoff 72 72 54 726 62 64 546 62 1 i 6 gt 17 6 1 5 186gt106gt6gt


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