Topic 1 Game Theory Lecture 2

# Topic 1 Game Theory Lecture 2 - Basic model • Let A and B...

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Unformatted text preview: Basic model • Let A and B be the agents who are bargaining • V is the total value that will be produced if they can agree to a particular solution • If A were to give up on B and pursue other options, then he can assure himself a value a Bargaining as a Game Topic 1 Lecture 2 Week 3 – e.g. the worker obtaining a job at another firm • If B were to pursue other options, he can assure himself a value b – e.g., the firm replacing the worker with a new hire ECON 1401 S1 2011 Lecture 3 1 ECON 1401 S1 2011 Lecture 3 y v • For a meaningful problem: we must have V – a – b > 0 • e.g., the worker does not produce as much in another firm (because he has acquired skills specific to this firm—learnt the “corporate culture”), and a new worker is not as productive in this firm. is the surplus that • Then S = V – a – b is available to bargain over • Let x denote what A gets, and let y denote what B gets in a potential solution. Representation If S is constant independent of how it is distributed between A and B, the situation can be represented like this Any point in the triagle to the NE of P is a potential solution. P b x+y=v a ECON 1401 S1 2011 Lecture 3 2 3 v ECON 1401 S1 2011 Lecture 3 x 4 The standard problem Why will they agree? • So we can think of A and B bargaining over a pie of size S=1 • If they can agree to split the pie with shares a and b, then A gets a and B gets b • Of course, a+b=1 • If they cannot agree then the pie disappears • Both A and B like pie, the more the better • Note that if there is no agreement, then S is just destroyed, neither A nor B gets any part of it by default. • So why will A not hold out for almost all of S, after all B cannot have any without A’s participation? Same for B. • There must be some reason which makes holding out costly, so players are willing to give up some of the pie to reach an agreement. ECON 1401 S1 2011 Lecture 3 5 ECON 1401 S1 2011 Lecture 3 6 Alternating offers • This can happen in one of three ways, all of which depend essentially on how time enters the problem: – agreement must be reached within a certain time, else everybody gives up and goes home – pie gets smaller the longer parties hold out – parties discount the future ECON 1401 S1 2011 Lecture 3 7 • To use the tools we already have, we must define bargaining as a game, i.e. with specific rules and payoffs. • In “alternating offers bargaining”, parties alternately make offers to one another on how to split the pie. e.g.: – – – – A offers to give Y to B and keep (S—Y) If B accepts then game is over If B rejects then B offers to give X to A, and keep (S—X) If A accepts then game is over, else A offers... • The game may also start with B rather than A making the first offer. ECON 1401 S1 2011 Lecture 3 8 Fixed number of rounds Fixed number of rounds • Two rounds • Ultimatum Game: – A offers – if reject then B offers – end – Analysis – A makes offer – B accepts or rejects – if accept, they get shares of S according to A’s offer – if reject then neither gets anything – game is over – (this is the game James and Sachi played last week) ECON 1401 S1 2011 Lecture 3 • Three rounds – A offers – if reject then B offers – if reject then A offers – end – Analysis 9 ECON 1401 S1 2011 Lecture 3 10 Pie shrinks Discounting • Each period the pie shrinks to a fraction of its previous size. There will always be pie, but it can get very very small • If there is no agreement in period 1, then a fraction (1‐q) of the pie disappears • In period 2 there is only q < 1 units of pie left • In period 3 there will only be q2 units of pie left • This is very much like the discounting case (coming up). • Players discount the future • After A offers B can reject, but then he will have to wait at least until next period to get some pie. If he gets Y next period, it is just as good (or bad) as getting Y right now (where is his discount factor). • Note that A suffers as well if B rejects, because he too discounts later payoffs relative to immediate ones. A also has the same discount factor . ECON 1401 S1 2011 Lecture 3 ECON 1401 S1 2011 Lecture 3 11 12 Finite alternating offers with discounting Two rounds • Key: If B rejects A’s initial offer, then in the next round we start a new game which is just like the original game, in which B offers first. • One round‐‐‐ultimatum game. • Two rounds • Three rounds • Note there is an advantage to being the first person to move • A makes offer in the first round, B makes offer in the second round. • If the game goes to the second round, then it becomes an ultimatum game, and B will take (almost) the whole pie. • But if he gets fraction of the pie in period 1, he will be just as happy • So A can offer to give to B in round 1, and B will accept ECON 1401 S1 2011 Lecture 3 13 14 Infinite number of rounds Three rounds • If the game gets to round 3, then A will keep the whole pie (ultimatum game) • So in round 2, B can offer to A, and keep (1‐ ) • But (1‐ ) in round 2 is no better than (1‐ ) in round 1. • So A can offer this to him, and keep 1 ‐ (1‐ ) • (is this better for A than in round 2?) ECON 1401 S1 2011 Lecture 3 ECON 1401 S1 2011 Lecture 3 15 • First, observe that no one ever getting any pie cannot be a solution • Because then A can offer B a small amount in round 1, and both players will prefer this to a solution that never gives anything to either player • So suppose the solution gives a to A and b to B in some future round t (e.g., t=20) • But then in period 1 A can offer B an amount 19b, and B is just as happy to take this as wait for the solution • And A is happier. • SO IN THE EQUILIBRIUM, AN AGREEMENT MUST BE REACHED IN ROUND 1 ECON 1401 S1 2011 Lecture 3 16 Finding the equilibrium • We will use a neat trick • We know that – There is an equilibrium – The equilibrium offer is made and accepted in the first period • Let this equilibrium be an offer of (1‐X) from A to B. B gets (1‐X), A keeps X. ECON 1401 S1 2011 Lecture 3 17 • Suppose B declines the first‐round offer, so game goes to second round. • Now the game looks to B exactly as the original game looked to A • So the following must be an equilibrium: • B offers (1‐X) to A and keeps X • i.e., if (a=X, b=1‐X) is an equilibrium for the original game, then (a=1‐X, b=X) must be an equilibrium once the game gets to the second round. ECON 1401 S1 2011 Lecture 3 18 The role of patience • is a measure of patience • So B can assure himself X in round 2 by declining A’s offer in round 1. • So in round 1 A must offer him at least X. • But A offers 1‐X, which B accepts. • So it must be true that 1‐X > X • A does not need to offer any more – near zero implies high impatience – near unity implies patience • In the equilibrium we have X = 1/(1+ ) • If agents are very impatient then first mover keeps most of the pie – Second mover will take a small cut rather than wait • If agents are very patient then shares are nearly equal. – So 1‐X = X • Which gives X = 1/(1+ ) ECON 1401 S1 2011 Lecture 3 19 ECON 1401 S1 2011 Lecture 3 20 ...
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## This note was uploaded on 06/12/2011 for the course ECONOMICS 3291 taught by Professor Professorsnamespublishedtheyarethesoleowners during the Three '11 term at University of New South Wales.

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