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### finitetimebargainingTEMP

Course: ECON 409, Fall 2011
School: Michigan
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(Finite-Time Handout Bargaining Horizon) Econ 409 Fall 2007 Daisuke Nakajima Consider the following bargaining process: There is a pie with the size of 1, to be divided between players 1 and 2. There are T rounds t = T; T 1; : : : ; 2; 1, where T is the initial round and 1 is the nal round. We index the times backward. In odd rounds, player 1 makes an oer and in even rounds player 2 makes an oer. (so it is player...

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(Finite-Time Handout Bargaining Horizon) Econ 409 Fall 2007 Daisuke Nakajima Consider the following bargaining process: There is a pie with the size of 1, to be divided between players 1 and 2. There are T rounds t = T; T 1; : : : ; 2; 1, where T is the initial round and 1 is the nal round. We index the times backward. In odd rounds, player 1 makes an oer and in even rounds player 2 makes an oer. (so it is player 1 who makes the nal oer, but the initial proposer depends on if T is odd or even.) In each round, if the oer is accepted, they divide the pie accordingly. If it is rejected, then the player who rejects makes a counter oer in the next round. This process continues until they reach an agreement or they cannot even at period 1. Let (xt ; 1 xt ) denote the oer at period t, where xt is the share of player 1 and 1 xt is the share of player 2. If they reach an agreement of (xt ; 1 xt ) Ttt x and player 2 payo is T t (1 s factor representing the cost of the delay. the next round is indierent to getting at period t, player 1 payo is s xt ). 2 (0; 1) is the discount You can see that getting x in x now. Let us solve this game backward. (T = 1) In this game, player 1 makes an oer and if player 2 declines it, they end up with getting nothing. Hence, player 2 accepts any oer. Given that, player 1 oers (1; 0), and player 2 accepts it. Hence x1 = 1. (T = 2) In this case, it is player 2 who makes the initial oer. If player 1 declines player 2 oer, he can get 1 in the next round, so player 1 accepts any oer s greater than 1 = . Given this, player 2 oers ( ; 1 ) and player 1 accepts it. Hence x2 = (T = 3) Now, player 1 makes the initial oer. If player 2 declines the oer, he gets 1 in the next round so he accepts any oer greater than (1 ). Given that, player 1 oers (1 (1 ); (1 )), which is accepted by player 2. Therefore, x3 = 1 (1 ). 1 What happens for general T ? Let us work for the case when T is odd (so T = 2k + 1 where k = 0; 1; : : :) Consider the rst period t = 2k + 1, it is player 1 who makes the initial oer. Player 2 can obtain 1 x2k in the next round if he declines the oer now, so he accepts any oer greater (1 than x2k ). Given that, player 1 should oer (1 (1 x2k ); (1 x2k )). Therefore, x2k+1 = 1 x2k ) (1 (1) Now consider the second period t = 2k , now player 2 makes an oer. Again, player 1 can obtain x2k 1 = x2(k 1)+1 in the next round by rejecting the current oer, so in order to have him accept the oer, player 2 should propose ( x2(k 1)+1 ; 1 x2(k 1)+1 ). Hence, x2k = x2(k 1)+1 (2) Combining (1) and (2), we have x2k+1 = 1 x2(k 1 2 2(k 1)+1 = x 1)+1 + (1 (3) ) To make it simple, dene X k = x2k+1 ; 2 = , and = (1 ) then (3) becomes Xk = Xk 1 + (4) Applying (4) recursively, we have Xk = = Xk 1 + Xk 2 + = = 2 = 3 Xk k X0 + 2 X + k2 + (1 + ) X + + (1 + ) k3 3 + (1 + + 2 ) . . . = 1+ 2 + + k1 Since we have already seen x0 = 1, and X 0 = x1 by denition, it becomes x2k+1 = k + = k + k (1 = 2k = = = 1+ + 2 (1 k1 + k 1 1 ) + (1 1 k ) + (1 ) )1 2k (5) 2 1 2k (1 + ) + 1 1+ 1 + 2k+1 1+ 1 2k (6) (7) Therefore, when T = 2k + 1 (so T is odd), player 1 proposes ! ! 2k 1 + 2k+1 1 + 2k+1 (1 ) 1 + 2k+1 ;1 = ; 1+ 1+ 1+ 1+ and player 2 accepts it2 . How about when T is odd (so T = 2k )? Do we have to make long and messy algebras again? No. Because (2) implies x2k = x2(k 1)+1 and according to (7) x2(k 1)+1 = 1+ 2(k 1)+1 = 1+ 1 + 2k 1+ 1 1 Note for the second equality: How can we nd the value of 1 + + k 1? + S =1+ + Let + k1 k1 + then S= + + k so (1 k )S = 1 Therefore, S= 1 1 k 2 In order to check if the algebras are correct, it is a good idea to substitute the value such as k = 0 or k = 1. If k = 0 (corresponding to T = 1), our formula says player 1 oers (1; 0), which coincide with our previous analysis. If k = 1, (corresponding to T = 3), the 2 formula implies (1 + 2; ), again matching with our previous analysis. Based on my experiences, if the formula is correct for two values of k, it is almost sure that the formula is correct in this kind of algebras. 3 so 2k x 1+ 2k 1 = 1+ Hence, when T = 2k is even, player 2 makes a proposal ! (1 + 2k (1 + 2k 1 ) (1 + 2k 1 ) = ;1 1+ 1+ 1+ 4 1 2k )1 ; 1+ !
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Michigan - ECON - 409
HandoutFirst-Price Auctions and Second-Price Auctions with UniformDistributionsEcon 409 Fall 2007Daisuke NakajimaLet us nd (symmetric) equilibria for the rst-price and the second-priceauctions when all of vi are independently and uniformly distribut
Michigan - ECON - 409
HandoutBargaining (Innite-Time Horizon)Econ 409 Fall 2007Daisuke NakajimaNow, we assume that they can continue bargaining forever, if they do notreach an agreement at all. Here is the setting.There is a pie with the size of 1, to be divided between
Michigan - ECON - 409
HandoutEconomics 409 Fall 2007Daisuke NakajimaCournot Game with an Investment stageThere are two rms 1 and 2, which compete in a market with inverse demandfunction P = a Q in the Cournot manner, where a is big relative to their costs.Firm 1, if it w
Michigan - ECON - 409
HandoutReputationKreps and Wilson (1982): Reputation and Imperfect InformationEco 409 Fall 2008Daisuke NakajimaConsider the chain store game studied in the problem set/class. There isone monopolist serving to N markets. In each market, there is a (p
Michigan - ECON - 409
Revenue Equivalence TheoremEco 409 Fall 2008Daisuke Nakajima1Revenue Equivalence TheoremSuppose there are n buyers, and each player i values the object at xi , whichis drawn from a distribution Fi over [vi ; v i ] with an associated density f where
Michigan - ECON - 409
HandoutStrategic InvestmentEcon 409 Fall 2005Daisuke NakajimaConsider the following dynamic game played by two players i = 1; 2.First, player 1 makes some investment k 2 R.Observing player 1 investment k , both players play some simultaneoussmove g
Michigan - ECON - 409
Problem Set 1Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on January 20 (Th)Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.I
Michigan - ECON - 409
Solutions for Problem Set 1Econ 409Fall 2010Daisuke NakajimaThroughout this problem set, you may ignore mixed strategies.1. Find all Nash equilibria of the following games.Player 2LMHL1; 14; 2 10; 0(a)Player 1 M2; 4 5; 58; 3H 0; 10 3; 87
Michigan - ECON - 409
Problem Set 2Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on January 27 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.
Michigan - ECON - 409
Problem Set 2Econ 409Winter 2011Daisuke Nakajima1. Consider the modied version of Rock-Paper-Scissors with the followingpayo matrix:RPSR0; 02; 21; 1P2; 20; 01; 1S1; 11; 10; 0In this game, when you win with Paper, you get a doubled pay
Michigan - ECON - 409
Problem Set 3Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on February 3 (Th)Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.I
Michigan - ECON - 409
Problem Set 3Econ 409Winter 2011Daisuke NakajimaFirst understand the following denition!Denition 1 A strategy si weakly dominates s0 if and only if:iui (si ; s i )ui (s0 ; s i ) for all siiui (si ; s0 i ) &gt; ui (s0 ; s0 i ) for some s0iiIn ot
Michigan - ECON - 409
Problem Set 4Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on February 10 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.
Michigan - ECON - 409
Problem Set 4Econ 409Winter 2011Daisuke Nakajima1. Remember the model of the price competition with product dierentiations studied in class. There are two ice cream stands on a beach choosingtheir locations. People (consumers) are uniformly located o
Michigan - ECON - 409
Problem Set 5Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on February 17 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.
Michigan - ECON - 409
Problem Set 5Econ 409Winter 2011Daisuke Nakajima1. Remember the model of the price competition with product dierentiations studied in class. There are two ice cream stands on a beach choosingtheir locations. People (consumers) are uniformly located o
Michigan - ECON - 409
Problem Set 7Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on March 17 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.If
Michigan - ECON - 409
Problem Set 7Econ 409Winter 2011Daisuke Nakajima1. Consider the following Cournot game with two rms. They have the sameper unit cost 3 and the market inverse demand function is P = 4 Q.(a) What is the Nash equilibrium output q c ? What is the prot o
Michigan - ECON - 409
Problem Set 8Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on November 18 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.
Michigan - ECON - 409
Problem Set 8Econ 409Fall 2010Daisuke Nakajima1. Consider a Cournot duopoly operating in a market with inverse demandP = a Q, where Q = q1 + q2 is the aggregate quantity on the market.Throughout this question, you can ignore non-negativity constrain
Michigan - ECON - 409
Problem Set 9Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on March 31 (Th)Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.1.
Michigan - ECON - 409
Problem Set 9Econ 409Winter 2011Daisuke Nakajima.1. There are n risk-neutral bidders and a single object is sold by an auction.Each bidder valuations is identically and independently distributed overs[0; 1] with the distribution function F (and the
Michigan - ECON - 409
Problem Set 10Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on December 9 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.
Michigan - ECON - 409
Problem Set 11Econ 409Winter 2011Daisuke NakajimaPlease submit your answer in the lecture on April 14 (Th).Please write neatly and use only one side of the paper. Do not forget tostaple your answer sheets.Be as systematic and logical as possible.1
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2006Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2006Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Winter 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Winter 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2008Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Final ExaminationEcon 409: Game Theory, Fall 2008Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall a
Michigan - ECON - 409
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Fall 2006Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Fall 2006Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Winter 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating sha
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Fall 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Fall 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating shall
Michigan - ECON - 409
Midterm ExaminationEcon 409: Game Theory, Winter 2007Department of EconomicsUniversity of MichiganDaisuke NakajimaDo NOT turn this page until instructed to do so.Any kind of cheating shall result in severe penalties. Anyone who isfound cheating sha
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