So we turn now to many-to-one matching with money, which is generally seen as a model of firms hiring workers
|
|
- Ambrose Price
- 5 years ago
- Views:
Transcription
1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 20 November So far, we ve considered matching markets in settings where there is no money you can t necessarily pay someone to marry you instead of another guy, and you can t necessarily pay Harvard to admit you instead of another student. In some settings, though, it s natural to assume money is a factor if you re choosing between two jobs, you may have preferences over the jobs themselves (location, hours, work environment), but they may also be offering you very different salaries and they may increase their offers as they compete over you. So we turn now to many-to-one matching with money, which is generally seen as a model of firms hiring workers Crawford and Knoer generalize the deferred acceptance algorithm to a setting with firms, workers, and money, but under two strong assumptions: firms production is additive as a function of the workers it hires, and each firm has an exogenous size cap. Kelso and Crawford ( Job Matching, Coalition Formation, and Gross Substitutes ) eliminate these two assumptions, replacing them with a much less restrictive condition, and show that everything still works. That s the paper we ll look at today. m workers 1,..., m, and n firms 1,..., n Each worker i has preferences over employers and wage, and gets utility u i (j, s) from working for firm j at a salary s this is assumed to be strictly increasing and continuous in s (doesn t have to be separable) Each firm has productivity that s a function of the set of workers it hires: if firm j hires a set of workers C j and pays them each s ij, its payoff is y j (C j ) i C j s ij Kelso and Crawford make two innocuous assumptions and one significant one. First, they define σ ij as the lowest salary at which worker i would work for firm j as opposed to remaining unemployed, so u i (j, σ ij ) = u i (0, 0); they assume that for any j, any C and any i / C, y j (C j {i}) y j (C) σ ij (This seems strong, until they point out that a firm could just pay someone zero to sit at home and do nothing.) 1
2 Second, they assume a firm with no workers doesn t produce y j ( ) = 0 And most importantly, they require that each firm s production function satisfies the gross substitutes condition Formally, for a given vector of prices s j = (s 1j, s 2j,..., s mj ), let M j (s j ) solve the firm s profit maximization problem, that is, { M j (s j ) = arg max y j (C) } s ij C i C Let s j be another vector of prices, weakly higher than s j, that is, s j s j. And for any C j, let T j (C j ) = { i C j : s ij = s ij } Then gross substitutes requires that if C j M j (s j ), there exists C j M j ( s j ) such that T j (C j ) C j. This is the same substitutes condition we saw in multi-unit auctions. Start out at a set of prices s j, and choose your favorite set of workers. Now raise the prices of some workers, leaving others the same. Gross substitutes says that you still want the workers whose prices stayed the same. (The definition is a bit clunkier just because you might be indifferent among more than one possible set of workers it says there has to be an optimal choice that includes all the old workers whose prices didn t go up.) 2
3 Recall Tuesday, we showed the equivalence between the set of stable matchings in the college admissions model and the core defined by weak dominance Here, the same equivalence holds, so Kelso and Crawford talk only about the core Define an allocation as an assignment of workers to firms, f : {1,..., m} {1,..., n}, and a set of salaries s if(i) An allocation is individually rational if u i (f(i), s if(i) ) u i (0, 0) (each worker prefers his deal to unemployment), which is the same as s if(i) σ if(i) ; and no firm makes negative profits, y j (C j ) i C j s ij where C j = {i j = f(i)} Kelso and Crawford use funny language for distinguishing the regular core, defined by strict dominance, from the core defined by weak dominance they call the latter the strict core An allocation is in the strict core if it is individually rational and there is no firm j, and set of workers C, and set of salaries r j that satisfy for each i C and u i (j, r ij ) u i (f(i), s if(i) ) π j (C, r j ) π j (C j, s j ) with at least one of these inequalities holding strictly The allocation is in the core if there is no j, C such that all these equations hold strictly Now, when money is continuous, there is no distinction between the core and the strict core as long as one person is strictly better off, they could give away a little money and still be strictly better off, so you could give everyone else a little bit more money, so everyone ends up strictly better off However, the proofs in Kelso and Crawford involve making money discrete, in which case the two can be different An allocation is in the discrete strict core, or the discrete core, if there is no such deviating coalition when the salaries r are all required to be on the discrete grid (say, integers) (Also note that the reason they name them the way they do is that the strict core is smaller than the regular core, since it s easier for something to not be in the strict core) One other solution concept Kelso and Crawford mention is a competitive equilibrium. This is when there is a price (wage) s ij for every firm-worker pair, and at those prices, the workers choose their favorite firm, the firms choose their favorite sets of workers, and the market clears (each worker demands the firm that demands him). These can be thought of as allocations where every firm and worker who aren t matched together, still know the price at which they could match together, but they re both content not to. 3
4 On, then, to results Like in the other papers we ve seen recently, Kelso and Crawford prove the existence of a core allocation by giving an algorithm to find one But first, they make money discrete that is, transfers can only occur in multiples of some amount (say $1) For any discrete-money model, they give an algorithm to find a core allocation the algorithm is a generalization of the deferred-acceptance algorithm Then they show that if the core was empty when money was continuous, it would also have to be empty when money is discrete but small enough, so there you go So, suppose salaries must be denominated in whole dollars, and let s see how the algorithm works It s actually very similar to the simultaneous-ascending-auction that we saw earlier, except that each firm faces a different set of personalized prices In each round, each firm j has a set of permitted prices (s 1j (t), s 2j (t),..., s mj (t)) at which it s allowed to make a job offer to each worker These prices start out at s ij (0) = σ ij, and go up over the course of the algorithm In the first round, by assumption, firms would be willing to hire everyone at those prices, so they are required to make job offers to everyone In each later round, facing a set of permitted prices (s ij (t)), firm j chooses the set of workers that maximizes profits and makes offers to them. Indifferences can be resolved in any way, except that if a firm made an offer to a worker in the previous round and that worker s price to that firm did not go up, they have to make an offer to him again. (Gross substitutes is exactly the condition that there is at least one profit-maximizing set of workers at the new prices that contains all the old workers whose prices didn t rise.) In each round, workers who get one or more job offer reject all but their favorite one, and tentatively accept their favorite If, in round t, firm j made an offer to worker i at price s ij (t) and that offer was rejected, then that firm s permitted price goes up by one increment that is, s ij (t + 1) = s ij (t) + 1 If not if in round t, firm j didn t make an offer to worker i, or made an offer that was tentatively accepted, the price stays the same s ij (t + 1) = s ij (t) We keep going until a round in which no offers are rejected, then we stop and the offers in place are accepted 4
5 So this is basically the firm-proposing deferred acceptance algorithm, with the twist that proposals come with wage offers, and that the lowest wage a firm can offer a given worker goes up each time an offer to that worker is rejected And it s analogous to the simultaneous ascending auction model with straightforward bidding each bidder (firm) faces a vector of personalized prices, and chooses the package that maximizes their payoff at those prices Theorem 1 from Kelso and Crawford: This process (they call it the salary adjustment process) converges in finite time to an allocation in the discrete core of the discrete model First, note that every worker always has at least one offer, so there is no unemployment This is because every worker gets offers in the first round (at σ ij ), nobody ever rejects all their offers, and any unrejected offer is repeated the next period Second, the process stops in finite time This is because in every round where the process continues, at least one permitted price s ij must increase by 1. We can put an upper bound on the wage any firm would ever pay any worker; so in finite time, if the algorithm is still going, no firm would be willing to hire anyone Third, when the process stops, it s at an individually rational allocation No worker tentatively accepts an unacceptable offer; in the round when the algorithm ends, no offers are rejected, and no firm ever makes a set of offers it wouldn t stand by Finally, the big result: when the algorithm ends, it s at a discrete core allocation Let φ be the matching the process stopped at, and s 1φ(1), s 2φ(2),... be the wages Suppose this is not in the discrete core Since it s individually rational, that means there must be some firm j, group of workers C, and set of salaries r j which block it, that is, such that u i (j, r ij ) > u i ( φ(i), s iφ(i) (t ) ) and ( ) π j (C, r j ) > π j C j φ, sj (t ) where t is the round when the algorithm stopped Now, if u i (j, r ij ) is better than worker i s outcome under φ, then worker i must never have rejected an offer from firm j at wage r ij or higher Which means the highest wage worker i every rejected from firm j was at most r ij 1, so the permitted price for firm j at time t was s ij (t ) r ij 5
6 But this holds for every worker i in C If the set of workers in C, at wages r ij, would have given firm j strictly higher profits than he got in round t of the algorithm, well then, that s who he would have proposed to! So we re done Next, Kelso and Crawford prove that when money becomes continuous, the core is still nonempty The proof is by contradiction: basically, they show that if you have a continuous market with an empty core, you could construct a discrete market with a small enough unit of money and that would have to have an empty core too Take any individually rational allocation ( φ, s 1φ(1),..., s mφ(m) ) in a continuous market that is not in the core Let D(j, C) be the amount that firm j and workers C could collectively gain by deviating by assumption, there is some j and C such that D is strictly positive Define F as the max of all the D, as a function of the allocation ( φ, s 1φ(1),..., s mφ(m) ) (so F is the maximum that any coalition could gain by deviating from the allocation φ) So we know F is strictly positive everywhere F is continuous in the wages s iφ(i), and we can make the space of money compact (by bounding wages above by the max profits of any firm), so the min of F over all possible wages is strictly positive And there are only a finite number of possible matchings φ, so the min over those exists and is strictly positive So the min of F over all possible allocations is bounded away from 0 that is, there is some number ɛ such that for any allocation in this continuous market, there is some coalition that can gain at least ɛ collectively by deviating 1 So make the unit of money smaller than m+1ɛ, and you re guaranteed a deviating coalition in the discrete market too But we already proved discrete markets can t have empty cores, and we re done. Given a core allocation, finding a competitive equilibrium set off-equilibrium wages to make workers indifferent 6
7 For a given discrete grid, indifferences are nongeneric; if we assume them away, we re guaranteed a unique path on the salary adjustment process, which is nice In that case, the algorithm converges to a discrete core allocation that is at least as good for every firm as any other This is the extension of what we already knew: the hospitals-proposing algorithm converges to the hospital-optimal stable matching The proof is similar, we ll skip it They also point out that there can be only be a single firms-optimal core allocation This allows Kelso and Crawford to prove some cool comparative statics when it comes to adding firms or workers to the market Basically: if you add a firm, all the workers are weakly better off, and all the firms are weakly worse off And if you take away a worker, all the other workers are weakly better off, and all the firms are weakly worse off The way they prove these is pretty cool Suppose we first run the salary adjustment process with n firms, and reach the end point Now we drop in one more firm, firm n + 1; allow him to start making offers at s i,n+1 (t + 1) = σ i,n+1 ; and restart the algorithm Similar to the original proof, when the restarted algorithm ends, it must be at a firm-optimal discrete core allocation in the new market with n + 1 firms But since there is a unique firm-optimal discrete core allocation, this is the same one as if we had just started out with n + 1 firms And since restarting the algorithm could only make workers better off they start at time t + 1 with the old job offer they would have accepted before workers are better off after the firm is added As for firms, since permitted prices only rise over time, whatever allocation they end up with at the end now, they could have had at least as cheaply before firm n + 1 entered, so they re weakly worse off 7
8 As for removing a worker... Again, start off with n firms and m workers, and run the algorithm until it stops Now at time t + 1, change every firm s permitted salary for worker m to some huge number K And restart the algorithm Wherever it stops, nobody hires worker m And this is a firm-optimal discrete core allocation for the market with m 1 workers And, since every step of the restarted algorithm makes workers better off and firms worse off, workers 1 through m 1 are all weakly better off than in the market with m workers, and the firms are all weakly worse off WE ENDED HERE WILL CONTINUE TUESDAY WITH EXAMPLE Kelso and Crawford also explore a bit the gross substitutes condition When all the workers are equivalent that is, firms don t care who they hire, just how many gross substitutes is exactly equivalent to decreasing returns the condition that y j (w + 1) y j (w) y j (w) y j (w 1) When the workers are not equivalent but m = 2 there are just two workers gross substitutes is equivalent to the production functions being subadditive that is, y j ({1, 2}) y j ({1}) + y j ({2}) However, gross substitutes and subadditivity are not equivalent when there are more than two workers An example: suppose there are three workers, 1, 2, and 3. Firm j s production technology is y j ( ) = 0 y j ({1}) = 4 y j ({2}) = 4 y j ({3}) = y j ({1, 2}) = y j ({1, 3}) = 7 y j ({2, 3}) = 7 y j ({1, 2, 3}) = 9 Clearly subadditive the first worker is always worth at least 4, second worker is worth more than 2 and less than 4, third worker is worth 2 or less. 8
9 At prices (3, 3, 3), firm j demands workers 1 and 2 At prices (3, 4, 3), firm j demands worker 3 only So an increase in the price of worker 2 causes the firm to abandon worker 1 So gross substitutes is violated Kelso and Crawford use this firm to generate an example where gross substitutes is violated and the core is empty (there are no stable matchings) Add a second firm k with a similar production technology: y k ( ) = 0 y k ({1}) = y k ({2}) = 4 y k ({3}) = 4 y k ({1, 2}) = 7 y k ({1, 3}) = 7 y k ({2, 3}) = y j ({1, 2, 3}) = 9 Assume u i (j, s) = u i (k, s) = s each worker just cares about his salary If there is a core allocation, it must maximize total surplus, otherwise the coalition of both firms and all workers could deviate This means it must either assign 1 to firm j and 2 and 3 to k, or 1 and 2 to j and 3 to k Since these are symmetric, it s enough to show one of them is not in the core consider the first, firm j gets 1 and firm k gets 2 and 3 Roth and Sotomayor (chapter 6) show that gross substitutes is not enough when firms have budget constraints they give an example where gross substitutes is violated, firms are budget-constrained, and the core is empty Tuesday: Hatfield and Milgrom, and Ostrovsky 9
While the story has been different in each case, fundamentally, we ve maintained:
Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 22 November 20 2008 What the Hatfield and Milgrom paper really served to emphasize: everything we ve done so far in matching has really, fundamentally,
More informationUp till now, we ve mostly been analyzing auctions under the following assumptions:
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 7 Sept 27 2007 Tuesday: Amit Gandhi on empirical auction stuff p till now, we ve mostly been analyzing auctions under the following assumptions:
More informationWhen we did independent private values and revenue equivalence, one of the auction types we mentioned was an all-pay auction
Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 15 October 28, 2008 When we did independent private values and revenue equivalence, one of the auction types we mentioned was an all-pay auction
More informationDay 3. Myerson: What s Optimal
Day 3. Myerson: What s Optimal 1 Recap Last time, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] and density f i residual valuation
More informationMultiunit Auctions: Package Bidding October 24, Multiunit Auctions: Package Bidding
Multiunit Auctions: Package Bidding 1 Examples of Multiunit Auctions Spectrum Licenses Bus Routes in London IBM procurements Treasury Bills Note: Heterogenous vs Homogenous Goods 2 Challenges in Multiunit
More informationEcon 711 Homework 1 Solutions
Econ 711 Homework 1 s January 4, 014 1. 1 Symmetric, not complete, not transitive. Not a game tree. Asymmetric, not complete, transitive. Game tree. 1 Asymmetric, not complete, transitive. Not a game tree.
More informationLecture 5: Iterative Combinatorial Auctions
COMS 6998-3: Algorithmic Game Theory October 6, 2008 Lecture 5: Iterative Combinatorial Auctions Lecturer: Sébastien Lahaie Scribe: Sébastien Lahaie In this lecture we examine a procedure that generalizes
More informationMA300.2 Game Theory 2005, LSE
MA300.2 Game Theory 2005, LSE Answers to Problem Set 2 [1] (a) This is standard (we have even done it in class). The one-shot Cournot outputs can be computed to be A/3, while the payoff to each firm can
More informationGame Theory Fall 2006
Game Theory Fall 2006 Answers to Problem Set 3 [1a] Omitted. [1b] Let a k be a sequence of paths that converge in the product topology to a; that is, a k (t) a(t) for each date t, as k. Let M be the maximum
More informationOn Existence of Equilibria. Bayesian Allocation-Mechanisms
On Existence of Equilibria in Bayesian Allocation Mechanisms Northwestern University April 23, 2014 Bayesian Allocation Mechanisms In allocation mechanisms, agents choose messages. The messages determine
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationMS&E 246: Lecture 2 The basics. Ramesh Johari January 16, 2007
MS&E 246: Lecture 2 The basics Ramesh Johari January 16, 2007 Course overview (Mainly) noncooperative game theory. Noncooperative: Focus on individual players incentives (note these might lead to cooperation!)
More informationOctober An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution.
October 13..18.4 An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution. We now assume that the reservation values of the bidders are independently and identically distributed
More informationMarch 30, Why do economists (and increasingly, engineers and computer scientists) study auctions?
March 3, 215 Steven A. Matthews, A Technical Primer on Auction Theory I: Independent Private Values, Northwestern University CMSEMS Discussion Paper No. 196, May, 1995. This paper is posted on the course
More informationProblem 3 Solutions. l 3 r, 1
. Economic Applications of Game Theory Fall 00 TA: Youngjin Hwang Problem 3 Solutions. (a) There are three subgames: [A] the subgame starting from Player s decision node after Player s choice of P; [B]
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2014 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationGame theory and applications: Lecture 1
Game theory and applications: Lecture 1 Adam Szeidl September 20, 2018 Outline for today 1 Some applications of game theory 2 Games in strategic form 3 Dominance 4 Nash equilibrium 1 / 8 1. Some applications
More informationEcon 711 Final Solutions
Econ 711 Final Solutions April 24, 2015 1.1 For all periods, play Cc if history is Cc for all prior periods. If not, play Dd. Payoffs for 2 cooperating on the equilibrium path are optimal for and deviating
More information1 Theory of Auctions. 1.1 Independent Private Value Auctions
1 Theory of Auctions 1.1 Independent Private Value Auctions for the moment consider an environment in which there is a single seller who wants to sell one indivisible unit of output to one of n buyers
More informationThe assignment game: Decentralized dynamics, rate of convergence, and equitable core selection
1 / 29 The assignment game: Decentralized dynamics, rate of convergence, and equitable core selection Bary S. R. Pradelski (with Heinrich H. Nax) ETH Zurich October 19, 2015 2 / 29 3 / 29 Two-sided, one-to-one
More informationSolution to Tutorial 1
Solution to Tutorial 1 011/01 Semester I MA464 Game Theory Tutor: Xiang Sun August 4, 011 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationInternet Trading Mechanisms and Rational Expectations
Internet Trading Mechanisms and Rational Expectations Michael Peters and Sergei Severinov University of Toronto and Duke University First Version -Feb 03 April 1, 2003 Abstract This paper studies an internet
More information(v 50) > v 75 for all v 100. (d) A bid of 0 gets a payoff of 0; a bid of 25 gets a payoff of at least 1 4
Econ 85 Fall 29 Problem Set Solutions Professor: Dan Quint. Discrete Auctions with Continuous Types (a) Revenue equivalence does not hold: since types are continuous but bids are discrete, the bidder with
More informationFrom the Assignment Model to Combinatorial Auctions
From the Assignment Model to Combinatorial Auctions IPAM Workshop, UCLA May 7, 2008 Sushil Bikhchandani & Joseph Ostroy Overview LP formulations of the (package) assignment model Sealed-bid and ascending-price
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationTopics in Contract Theory Lecture 1
Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore
More informationCompeting Mechanisms with Limited Commitment
Competing Mechanisms with Limited Commitment Suehyun Kwon CESIFO WORKING PAPER NO. 6280 CATEGORY 12: EMPIRICAL AND THEORETICAL METHODS DECEMBER 2016 An electronic version of the paper may be downloaded
More informationVirtual Demand and Stable Mechanisms
Virtual Demand and Stable Mechanisms Jan Christoph Schlegel Faculty of Business and Economics, University of Lausanne, Switzerland jschlege@unil.ch Abstract We study conditions for the existence of stable
More informationMatching Markets and Google s Sponsored Search
Matching Markets and Google s Sponsored Search Part III: Dynamics Episode 9 Baochun Li Department of Electrical and Computer Engineering University of Toronto Matching Markets (Required reading: Chapter
More informationm 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6
Non-Zero Sum Games R&N Section 17.6 Matrix Form of Zero-Sum Games m 11 m 12 m 21 m 22 m ij = Player A s payoff if Player A follows pure strategy i and Player B follows pure strategy j 1 Results so far
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
More information15-451/651: Design & Analysis of Algorithms November 9 & 11, 2015 Lecture #19 & #20 last changed: November 10, 2015
15-451/651: Design & Analysis of Algorithms November 9 & 11, 2015 Lecture #19 & #20 last changed: November 10, 2015 Last time we looked at algorithms for finding approximately-optimal solutions for NP-hard
More informationECON Microeconomics II IRYNA DUDNYK. Auctions.
Auctions. What is an auction? When and whhy do we need auctions? Auction is a mechanism of allocating a particular object at a certain price. Allocating part concerns who will get the object and the price
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 07. (40 points) Consider a Cournot duopoly. The market price is given by q q, where q and q are the quantities of output produced
More informationSimon Fraser University Spring 2014
Simon Fraser University Spring 2014 Econ 302 D200 Final Exam Solution This brief solution guide does not have the explanations necessary for full marks. NE = Nash equilibrium, SPE = subgame perfect equilibrium,
More informationTopics in Contract Theory Lecture 3
Leonardo Felli 9 January, 2002 Topics in Contract Theory Lecture 3 Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting
More information6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More information15-451/651: Design & Analysis of Algorithms October 23, 2018 Lecture #16: Online Algorithms last changed: October 22, 2018
15-451/651: Design & Analysis of Algorithms October 23, 2018 Lecture #16: Online Algorithms last changed: October 22, 2018 Today we ll be looking at finding approximately-optimal solutions for problems
More informationRegret Minimization and Security Strategies
Chapter 5 Regret Minimization and Security Strategies Until now we implicitly adopted a view that a Nash equilibrium is a desirable outcome of a strategic game. In this chapter we consider two alternative
More informationNot 0,4 2,1. i. Show there is a perfect Bayesian equilibrium where player A chooses to play, player A chooses L, and player B chooses L.
Econ 400, Final Exam Name: There are three questions taken from the material covered so far in the course. ll questions are equally weighted. If you have a question, please raise your hand and I will come
More informationAlgorithmic Game Theory
Algorithmic Game Theory Lecture 10 06/15/10 1 A combinatorial auction is defined by a set of goods G, G = m, n bidders with valuation functions v i :2 G R + 0. $5 Got $6! More? Example: A single item for
More informationCS364B: Frontiers in Mechanism Design Lecture #18: Multi-Parameter Revenue-Maximization
CS364B: Frontiers in Mechanism Design Lecture #18: Multi-Parameter Revenue-Maximization Tim Roughgarden March 5, 2014 1 Review of Single-Parameter Revenue Maximization With this lecture we commence the
More informationRecalling that private values are a special case of the Milgrom-Weber setup, we ve now found that
Econ 85 Advanced Micro Theory I Dan Quint Fall 27 Lecture 12 Oct 16 27 Last week, we relaxed both private values and independence of types, using the Milgrom- Weber setting of affiliated signals. We found
More informationRepeated Games with Perfect Monitoring
Repeated Games with Perfect Monitoring Mihai Manea MIT Repeated Games normal-form stage game G = (N, A, u) players simultaneously play game G at time t = 0, 1,... at each date t, players observe all past
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games Michael Levet June 23, 2016 1 Introduction Game Theory is a mathematical field that studies how rational agents make decisions in both competitive and cooperative situations.
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 5 [1] Consider an infinitely repeated game with a finite number of actions for each player and a common discount factor δ. Prove that if δ is close enough to zero then
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationNotes on Auctions. Theorem 1 In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy.
Notes on Auctions Second Price Sealed Bid Auctions These are the easiest auctions to analyze. Theorem In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy. Proof
More informationStrategies and Nash Equilibrium. A Whirlwind Tour of Game Theory
Strategies and Nash Equilibrium A Whirlwind Tour of Game Theory (Mostly from Fudenberg & Tirole) Players choose actions, receive rewards based on their own actions and those of the other players. Example,
More informationCS364A: Algorithmic Game Theory Lecture #14: Robust Price-of-Anarchy Bounds in Smooth Games
CS364A: Algorithmic Game Theory Lecture #14: Robust Price-of-Anarchy Bounds in Smooth Games Tim Roughgarden November 6, 013 1 Canonical POA Proofs In Lecture 1 we proved that the price of anarchy (POA)
More informationRecap First-Price Revenue Equivalence Optimal Auctions. Auction Theory II. Lecture 19. Auction Theory II Lecture 19, Slide 1
Auction Theory II Lecture 19 Auction Theory II Lecture 19, Slide 1 Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 2 Motivation
More informationNotes for Section: Week 4
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 2004 Notes for Section: Week 4 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
More informationSolution to Tutorial /2013 Semester I MA4264 Game Theory
Solution to Tutorial 1 01/013 Semester I MA464 Game Theory Tutor: Xiang Sun August 30, 01 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationLecture 5 Leadership and Reputation
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible. One definition of leadership is that
More informationCompetition for goods in buyer-seller networks
Rev. Econ. Design 5, 301 331 (2000) c Springer-Verlag 2000 Competition for goods in buyer-seller networks Rachel E. Kranton 1, Deborah F. Minehart 2 1 Department of Economics, University of Maryland, College
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationLecture B-1: Economic Allocation Mechanisms: An Introduction Warning: These lecture notes are preliminary and contain mistakes!
Ariel Rubinstein. 20/10/2014 These lecture notes are distributed for the exclusive use of students in, Tel Aviv and New York Universities. Lecture B-1: Economic Allocation Mechanisms: An Introduction Warning:
More information16 MAKING SIMPLE DECISIONS
247 16 MAKING SIMPLE DECISIONS Let us associate each state S with a numeric utility U(S), which expresses the desirability of the state A nondeterministic action A will have possible outcome states Result
More informationCS269I: Incentives in Computer Science Lecture #14: More on Auctions
CS69I: Incentives in Computer Science Lecture #14: More on Auctions Tim Roughgarden November 9, 016 1 First-Price Auction Last lecture we ran an experiment demonstrating that first-price auctions are not
More informationTopics in Contract Theory Lecture 6. Separation of Ownership and Control
Leonardo Felli 16 January, 2002 Topics in Contract Theory Lecture 6 Separation of Ownership and Control The definition of ownership considered is limited to an environment in which the whole ownership
More informationLecture Notes on Anticommons T. Bergstrom, April 2010 These notes illustrate the problem of the anticommons for one particular example.
Lecture Notes on Anticommons T Bergstrom, April 2010 These notes illustrate the problem of the anticommons for one particular example Sales with incomplete information Bilateral Monopoly We start with
More informationGAME THEORY. Department of Economics, MIT, Follow Muhamet s slides. We need the following result for future reference.
14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Existence and Continuity of Nash Equilibria Follow Muhamet s slides. We need the following result for future reference. Theorem 1. Suppose
More informationOutline Introduction Game Representations Reductions Solution Concepts. Game Theory. Enrico Franchi. May 19, 2010
May 19, 2010 1 Introduction Scope of Agent preferences Utility Functions 2 Game Representations Example: Game-1 Extended Form Strategic Form Equivalences 3 Reductions Best Response Domination 4 Solution
More information1 Appendix A: Definition of equilibrium
Online Appendix to Partnerships versus Corporations: Moral Hazard, Sorting and Ownership Structure Ayca Kaya and Galina Vereshchagina Appendix A formally defines an equilibrium in our model, Appendix B
More informationApril 29, X ( ) for all. Using to denote a true type and areport,let
April 29, 2015 "A Characterization of Efficient, Bayesian Incentive Compatible Mechanisms," by S. R. Williams. Economic Theory 14, 155-180 (1999). AcommonresultinBayesianmechanismdesignshowsthatexpostefficiency
More informationRepeated Games. Econ 400. University of Notre Dame. Econ 400 (ND) Repeated Games 1 / 48
Repeated Games Econ 400 University of Notre Dame Econ 400 (ND) Repeated Games 1 / 48 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment
More informationCUR 412: Game Theory and its Applications, Lecture 4
CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 22, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions
More informationMicroeconomics of Banking: Lecture 5
Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system
More informationLecture Notes on The Core
Lecture Notes on The Core Economics 501B University of Arizona Fall 2014 The Walrasian Model s Assumptions The following assumptions are implicit rather than explicit in the Walrasian model we ve developed:
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationExtensive-Form Games with Imperfect Information
May 6, 2015 Example 2, 2 A 3, 3 C Player 1 Player 1 Up B Player 2 D 0, 0 1 0, 0 Down C Player 1 D 3, 3 Extensive-Form Games With Imperfect Information Finite No simultaneous moves: each node belongs to
More informationECO 426 (Market Design) - Lecture 8
ECO 426 (Market Design) - Lecture 8 Ettore Damiano November 23, 2015 Revenue equivalence Model: N bidders Bidder i has valuation v i Each v i is drawn independently from the same distribution F (e.g. U[0,
More informationChapter 19: Compensating and Equivalent Variations
Chapter 19: Compensating and Equivalent Variations 19.1: Introduction This chapter is interesting and important. It also helps to answer a question you may well have been asking ever since we studied quasi-linear
More informationCompetitive Outcomes, Endogenous Firm Formation and the Aspiration Core
Competitive Outcomes, Endogenous Firm Formation and the Aspiration Core Camelia Bejan and Juan Camilo Gómez September 2011 Abstract The paper shows that the aspiration core of any TU-game coincides with
More informationHierarchical Exchange Rules and the Core in. Indivisible Objects Allocation
Hierarchical Exchange Rules and the Core in Indivisible Objects Allocation Qianfeng Tang and Yongchao Zhang January 8, 2016 Abstract We study the allocation of indivisible objects under the general endowment
More informationS 2,2-1, x c C x r, 1 0,0
Problem Set 5 1. There are two players facing each other in the following random prisoners dilemma: S C S, -1, x c C x r, 1 0,0 With probability p, x c = y, and with probability 1 p, x c = 0. With probability
More information1 Shapley-Shubik Model
1 Shapley-Shubik Model There is a set of buyers B and a set of sellers S each selling one unit of a good (could be divisible or not). Let v ij 0 be the monetary value that buyer j B assigns to seller i
More informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationProblem Set 3: Suggested Solutions
Microeconomics: Pricing 3E00 Fall 06. True or false: Problem Set 3: Suggested Solutions (a) Since a durable goods monopolist prices at the monopoly price in her last period of operation, the prices must
More informationECON DISCUSSION NOTES ON CONTRACT LAW. Contracts. I.1 Bargain Theory. I.2 Damages Part 1. I.3 Reliance
ECON 522 - DISCUSSION NOTES ON CONTRACT LAW I Contracts When we were studying property law we were looking at situations in which the exchange of goods/services takes place at the time of trade, but sometimes
More informationOptimal selling rules for repeated transactions.
Optimal selling rules for repeated transactions. Ilan Kremer and Andrzej Skrzypacz March 21, 2002 1 Introduction In many papers considering the sale of many objects in a sequence of auctions the seller
More informationDepartment of Economics The Ohio State University Final Exam Answers Econ 8712
Department of Economics The Ohio State University Final Exam Answers Econ 8712 Prof. Peck Fall 2015 1. (5 points) The following economy has two consumers, two firms, and two goods. Good 2 is leisure/labor.
More information16 MAKING SIMPLE DECISIONS
253 16 MAKING SIMPLE DECISIONS Let us associate each state S with a numeric utility U(S), which expresses the desirability of the state A nondeterministic action a will have possible outcome states Result(a)
More informationGame Theory Problem Set 4 Solutions
Game Theory Problem Set 4 Solutions 1. Assuming that in the case of a tie, the object goes to person 1, the best response correspondences for a two person first price auction are: { }, < v1 undefined,
More informationEco504 Spring 2010 C. Sims FINAL EXAM. β t 1 2 φτ2 t subject to (1)
Eco54 Spring 21 C. Sims FINAL EXAM There are three questions that will be equally weighted in grading. Since you may find some questions take longer to answer than others, and partial credit will be given
More information1 Two Period Exchange Economy
University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout # 2 1 Two Period Exchange Economy We shall start our exploration of dynamic economies with
More informationOptimal Auctions. Game Theory Course: Jackson, Leyton-Brown & Shoham
Game Theory Course: Jackson, Leyton-Brown & Shoham So far we have considered efficient auctions What about maximizing the seller s revenue? she may be willing to risk failing to sell the good she may be
More informationOther Regarding Preferences
Other Regarding Preferences Mark Dean Lecture Notes for Spring 015 Behavioral Economics - Brown University 1 Lecture 1 We are now going to introduce two models of other regarding preferences, and think
More informationMaximum Contiguous Subsequences
Chapter 8 Maximum Contiguous Subsequences In this chapter, we consider a well-know problem and apply the algorithm-design techniques that we have learned thus far to this problem. While applying these
More informationFinal Examination December 14, Economics 5010 AF3.0 : Applied Microeconomics. time=2.5 hours
YORK UNIVERSITY Faculty of Graduate Studies Final Examination December 14, 2010 Economics 5010 AF3.0 : Applied Microeconomics S. Bucovetsky time=2.5 hours Do any 6 of the following 10 questions. All count
More informationMicroeconomic Theory August 2013 Applied Economics. Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY. Applied Economics Graduate Program
Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY Applied Economics Graduate Program August 2013 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationRelational Incentive Contracts
Relational Incentive Contracts Jonathan Levin May 2006 These notes consider Levin s (2003) paper on relational incentive contracts, which studies how self-enforcing contracts can provide incentives in
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationPROBLEM SET 7 ANSWERS: Answers to Exercises in Jean Tirole s Theory of Industrial Organization
PROBLEM SET 7 ANSWERS: Answers to Exercises in Jean Tirole s Theory of Industrial Organization 12 December 2006. 0.1 (p. 26), 0.2 (p. 41), 1.2 (p. 67) and 1.3 (p.68) 0.1** (p. 26) In the text, it is assumed
More informationJEFF MACKIE-MASON. x is a random variable with prior distrib known to both principal and agent, and the distribution depends on agent effort e
BASE (SYMMETRIC INFORMATION) MODEL FOR CONTRACT THEORY JEFF MACKIE-MASON 1. Preliminaries Principal and agent enter a relationship. Assume: They have access to the same information (including agent effort)
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 017 1. Sheila moves first and chooses either H or L. Bruce receives a signal, h or l, about Sheila s behavior. The distribution
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
More information