Online Appendix for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam
|
|
- Ashlynn Teresa Page
- 7 years ago
- Views:
Transcription
1 Online Appendi for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam Christiaan van Bochove, Lars Boerner, Daniel Quint January, 06 The first part of Theorem states that for any, ρ > 0, and > 0, an equilibrium eists. The proof is constructive: for 4, equilibrium strategies are given in the appendi of the paper. Here, we show that these strategies do indeed make up an equilibrium, and deal separately with the cases 3 and. First Round Play First-round strategies and beliefs for player are Do not drop out at P 0 At any history where P > 0 and there is one other active bidder, believe that he has signal s i, and If P < P, remain active If P P, drop out and for player i are Drop out at P 0 At any history where P > 0 and bidder i is still active along with bidder, If P P, believe s U0, ; remain active if s i and drop out if s i < If P < P < P, believe s ; remain active if s i and drop out if s i < If P P, drop out immediately where and c + P c ρ if ρ + ρ if ρ < c + + P + ρ if ρ c + + ρ if ρ < Since all bidders other than bidder are epected to drop out at P 0, any history where P > 0 and more than one of bidders, 3,..., are still active could only be reached following
2 simultaneous deviations by more than one player. Strategies at such histories therefore does not affect equilibrium play, and we do not specify what strategies are played following such deviations. et, we will describe second-round beliefs and strategies, both on and off the equilibrium path, and show that these are equilibrium strategies. After that, we will show that these firstround strategies are best-responses given that continuation play. Second Round Play On the Equilibrium Path Bidder won the first round at a price of 0. Common beliefs are that s i U0, for all i. Second-round bidding functions are bs i c + s i + s i To see that these bid functions are an equilibrium, first note that bids above b are strictly dominated by bidding b, since this will still win with probability ; and bids below b0 earn epected payoff of 0. For 0,, a bidder with signal s i who plans to bid b will win whenever s j < for every j i, which occurs with probability ; in that event, the security is worth, on average, c + si +, so his epected payoff is c + si + c + + s i Differentiating with respect to gives si s i, which is positive when < s i and negative when > s i, so epected payoff is maimized by setting s i. ote also plugging this into the payoff function that this gives nonnegative epected payoff, so bidding less than b0 and losing for sure is not a profitable deviation. Should bidder choose to drop out at P 0 along with the other bidders so that the first-round winner is chosen by a random tiebreaker, second-round beliefs and strategies are unchanged. Second Round Play After a First-Round Deviation By Bidder i We refer to this continuation game as CG. As in the tet, assume without loss of generality that bidder is the one who deviated. Since bidder s equilibrium strategy following such a deviation is to bid up to price P, assume that either the deviating bidder has dropped out at a price P 0, P or has won the first round at the price P. All bidders commonly believe that s ; beliefs about
3 the other bidders signals including bidder match the prior s i U0,. Strategies are: Bidder does not bid regardless of his actual type If P c + +, then nobody bids If P c +, c + +, then let s solve P c + + s. Bidder i does not bid when s i s ; when s i > s, he bids bs i c + If P c +, then bidders i each bid s i + s i s s i + s i + bs i c + s i + s i + In the first case, P c + +, each bidder i epects nobody to bid; winning the object would require bidding strictly more than P, therefore strictly more than c + +, for an object that will be worth, on average, c + si + + c + +, so not bidding is a best-response. Bidder epects the object to be worth c + s + < c + +, so not bidding is a best-response for him as well. In the second case, bs i is strictly increasing, so bids above b are again strictly dominated. By construction, bs P. For s,, a bidder with signal s i who bids b epects to win when the other bidders ecluding himself and bidder have signals below ; he therefore earns epected payoff c + si + + c + s i s s + + s i s Differentiating with respect to gives 3 s i, which is positive for < s i and negative for > s i, so epected payoff is maimized at bs i. Bidding P bs or a tiny bit above that wins only if nobody else bids, giving epected payoff s c + s si + + s s i s c s s + s s s + s + s s i s making it worse than not bidding when s i < s. So for s i < s, any bid equal or greater than P would earn negative epected profits; and for s i > s, bidding bs i earns greater payoff than bs, which is itself positive, so bs i is a best-response. 3
4 ote that if bidder dropped out at a price P < P but above c +, then bidder is the first-round winner. The fact that he will win the object if nobody bids in the second round means that by not bidding, he pays bs for the object whenever all other bidders aside from have signals s i < s. This is equivalent to bidder having to bid bs if he does not bid higher than that; his optimal move is therefore still to bid bs when s > s, and to not bid in the second round when s s. As for bidder, bidding b would give epected payoff c + so not bidding is a best-response. s + c s + + s s + + < 0 In the third case, bids above b are again dominated, and a bid of b for 0, earns epected payoff c + si + + c s i + + The derivative with respect to is again n 3 s i, so epected payoff is again maimized at s i, which gives positive epected payoff so bidding less than b0 is not a profitable deviation. For bidder, bidding b would give c + s + c + + s + < 0 so not bidding is again a best-response. Bidder s Payoff From Deviating In the First Round By deviating in the first round and dropping out before P, bidder guarantees entry into CG discussed above, in which he does not bid and earns epected payoff 0; so remaining active past price 0, but planning to drop out before P, is not a profitable deviation. On the other hand, if bidder deviates in the first round and does not drop out before P, he epects bidder to drop out at price P. We net show that this would earn epected profit of 0 or less for bidder. 4
5 If ρ, then P c ρ c + +, so if the first round reaches price P, nobody bids in the second round. In that case, by winning the first round at price P and not updating beliefs about s, bidder earns second-round payoff c + s + c ρ s ρ Combined with winning the premium, then, bidder s epected payoff would be nonpositive. If ρ <, then if P P c + + ρ, s solves and so s c + + ρ c + + s ρ. After winning the first round at price P, bidder epects to be stuck with the security with probability s, for second-round epected payoff s c + s + s c + + s s s s s s s s s ρ s s ρ So including winning the premium in the first round, winning the first round at price P gives epected payoff 0 to bidder if s, and negative epected payoff otherwise. ote therefore that if s and bidder has already deviated by failing to drop out immediately, he epects payoff 0 whether he drops out before P or waits for bidder drop out at P. This means that once he has already deviated, it is a weak best response not to drop out before P which makes bidder willing to punish him by staying in until P ; but also that deviating in the first place cannot be profitable. Second Round Play After Deviations By Bidders i and Again, assume i is the first deviating bidder. If bidder does not drop out immediately, and bidder either drops out at a price P < P, or fails to drop out at price P so that either bidder or wins the first round at a price P > P, we refer to the resulting continuation game as CG3. Beliefs and strategies in the second round are as follows: Everyone believes s s Bidders and regardless of their actual types do not bid 5
6 If P c , then nobody bids If P c +, c , then let s solve P c + s + 3 s + Bidder i 3 does not bid when s i s ; when s i > s, he bids bs i c + If P c +, then bidders i 3 each bid 3 s i + s i s s i + 3 s i + bs i c + 3 s i + 3 s i + In the first case, if P c , then a bidder i 3 with signal s i epects no one else to bid, and believes that the security is worth in epectation c + s i P, and therefore does not bid. Bidder i {, } believes the security to be worth in epectation c + s i + + < P and therefore does not bid either. For the second case, a bid above c is dominated; a bid by bidder i 3 equal to b > s wins whenever s j < for all j {3, 4,..., } {i}, which is with probability 3, and therefore delivers epected payoff of c 3 + si c s i 3 s 3 Differentiating with respect to gives s s i 3 3 s 3si 3 3 which is positive for < s i and negative for > s i, so bs i is a best-response among bids above P. Bidding P or slightly above P would give epected payoff s i s 3 3 s s si s 3 s which is positive for s i > s and negative for s i < s. So not bidding is a best-response for s i s, and bs i a best-response for s i > s. Bidder i {, } who is not the first-round winner would, by 6
7 bidding b, earn epected payoff c + si + + c s i + 3 s s and so prefers not to bid. Bidder i {, } who is the first-round winner is effectively bidding bs by not making a new bid; but since { d d s i + 3 } s 3 s i + 3 s < 0 he prefers bs to a higher bid, and does not bid again. For the third case, a bid of b by bidder i 3 gives epected payoff 3 c + si c s 3 i 3 which has derivative 3 s i 3 3, which once again is positive for < s i and negative for > s i ; setting s i gives strictly positive payoff, so bs i is a best-response. For bidder i {, } whether or not the first-round winner, since the first-round winner will for sure be outbid if he does not bid again, but counting the premium received as a sunk benefit, bidding b gives payoff c + si + + c s i + 3 so not bidding is a best-response Bidder s Epected Payoff From Failing To Punish et, we show that if s s and either bidder or wins the first round at price P, his epected payoff including the first-round premium is 0. This ensures that i once the price passes P and bidder believes s, then assuming s as well, bidder is willing to stay in until P ; and ii bidder cannot gain by failing to punish a deviation by bidder, since he epects bidder to stay in until P, and even if s s, winning the first round would therefore give bidder zero epected payoff. If ρ, then P c , so if P P, nobody will bid in the second round; in 7
8 that case, the first-round winner wins the security for sure regardless of others signals, so if s s, his epected payoff is c + + c ρ ρ which, combined with winning the premium ρ in the first round, gives a net epected payoff of 0. If ρ <, then when P P, s solves c + + ρ c + s + 3 s + and therefore s ρ. So with probability s, nobody bids in the second round, and the security is worth, in epectation assuming s s, c + s +. So the first-round winner i {, } at price P, assuming s s, gets epected payoff gross of winning the premium s c + s + c s + s s s ρ ρ and so including winning the premium, the first-round winner would get e-ante epected payoff 0. First Round Strategies Are Equilibrium Strategies Finally, we need to show the strategies given earlier for the first round are indeed best-responses. First, consider bidder at P 0. If he drops out immediately leading to the first round being determined by a random tiebreaker, he wins the premium with probability instead of, and does not affect second-round play, so this is not a profitable deviation. Following a deviation by bidder, bidder believes s, and epects bidder to stay active until the price reaches P. Dropping out at price P would lead to CG and nonnegative payoffs. Dropping out before or after P would lead to CG3 and zero payoffs; staying in until P would lead to zero payoffs or negative payoffs if s <. So sticking to equilibrium strategies is a best-response. As for bidder representing all bidders i >, dropping out immediately leads to nonnegative strictly positive if s > 0 epected payoffs in the second round. ot dropping out immediately, bidder would epect bidder to remain active until P and then drop out; dropping out before 8
9 then would lead to CG and zero payoffs; winning the first round at price P would likewise lead to CG and zero payoffs or negative if s <. So having already failed to drop out at P 0, it remains a best-response to stay active until P P. If bidder fails to drop out at P P, bidder then believes s ; provided s as well, bidder epects zero payoffs if he stays in and wins at price P, strictly positive payoffs if he stays in and wins at a lower price than that; dropping out prior to P would lead to CG3, and zero payoffs for bidder. So having stayed in past P 0, bidder has no reason to drop out prior to P. Once the price passes P, the winner of the first round is assured negative payoffs, so both players best-respond by dropping out immediately. As mentioned above, any history not eplicitly addressed in the strategies above could only be reached via simultaneous deviations by multiple bidders, and therefore cannot affect play on the equilibrium path; we therefore do not specify what happens at these histories. What if 3? If 3, define { P ma P, c } 6ρ If P P, modify both bidder s strategy following a deviation by bidder to be, stay in till P, drop out at P, and drop out immediately at any P > P, and bidder s strategy following a deviation by himself to be, stay in till P, do not drop out at P, drop out immediately at any P > P. Play at all histories other than CG3 are otherwise unchanged. In CG3 when 3, bidders effectively play the drainage tract auction equilibrium considered in Engelbrecht-Wiggans, Milgrom and Weber 983, Milgrom and Weber 98b, and Hendricks, Porter and Wilson 994, where the one bidder with private information plays a pure strategy and the uninformed bidders play mied strategies and earn zero epected profit. We will construct such an equilibrium where the round-one winner plays a mied strategy while the third bidder whiever of bidders and lost the first round does not bid. Let i denote the identity of the first-round winner, and j whichever of and is not i. Recall that this continuation game includes the common beliefs that s s. Let s 3 solve P c+ 3 +s 3, or s 3 0 if P < c + 3. Define a function Γ : s 3, R + by Γt c t + s 3 t 9
10 and note that Γ t 3 Γs 3 P. s 3 t > 0 for t s 3,, so Γ is strictly increasing. Also note that Define a probability distribution G with support P, Γ by if Γ G ep Γ dt if P, Γ c+ 3 +Γ t t 0 if < P ote that this distribution has a point mass at P. Bidding strategies are as follows: Bidder 3 bids Γs 3 if s 3 > s 3, and does not bid if s 3 s 3 If s i, bidder i mies such that the CDF of his bids is G with the point mass at P indicating no new bid; if s i <, bidder i does not bid Bidder j does not bid To see this is an equilibrium, we first show that if bidder 3 plays this strategy, bidder i is indifferent between not bidding and bidding any number in the support of b 3. Treat the first-round bonus won by bidder i as a sunk benefit, and note that since bidder 3 never bids eactly P not bidding is identical to bidding P Γs 3 for bidder i. By bidding Γ for s 3,, bidder i wins when s 3 < ; if bidder i has signal s i and believes s j, this gives epected payoff c si + Γ c si + c s 3 3 s i s 3 3 s i 6 s 3 If s i, then this is the same for all, so bidder i is indifferent among all bids in the range of Γ and not bidding effectively bidding P, and is therefore best-responding by playing a mied strategy. As for bidder j, by those same calculations, if s i, bidding Γ would give an epected payoff of 3 s j 6 s 3 G, which is nonpositive for any and any s j, so j best-responds by not bidding. Finally, knowing the value of the security is c s 3, bidder 3 solves the maimization problem ma b P,Γ {0} ln Gb c s 3 b 0
11 Taking the derivative with respect to b gives gb Gb c s 3 b c Γ b b c s 3 b This is positive when Γ b < s 3, or b < Γs 3 and negative when b > Γs 3 ; so bidder 3 s log-profit problem is strictly quasiconcave and solved at b Γs 3. For s 3 > s 3, c s 3 Γs 3 c s 3 c 3 + s 3 + s 3 s 3 3 s 3 s 3 s 3 > 0 so bidder 3 prefers bidding to not bidding; and for any s 3 s 3, c s 3 P, so bidder 3 prefers not bidding to bidding. What we need from CG3 is for bidder j the one who is not the first-round winner to earn 0 payoffs which is true since he doesn t bid, and if P P, for player i to earn non-postiive e-ante payoffs. To see the latter, note that means s 3 c s 3 P P c + 3 6ρ ; bidder i s second-round profit is therefore + 6ρ 3 s i 6 s 3 ρ and therefore even counting the first-round bonus, bidder i s profits are not positive. What if? For, the game is modified in the following way: First-round strategies are the same, but with P and P modified to be P c+ and P c + + ρ + ρ If bidder deviates in the first round but drops out at a price P P, leaving bidder as the first-round winner, CG is replaced by a drainage tract auction where s and s U0,. Bidder s strategy is to bid Γ s i c + + s if this is above P, and not bid otherwise. Bidder s strategy is to mi over not bidding and bidding in the range of Γ, such that the
12 distribution of his bids is if Γ G ep Γ dt if P, Γ c+ +Γ t t 0 if < P with the point mass at P corresponding to not bidding. If bidder deviates in the first round and wins when bidder drops out at price P, CG is replaced by a different drainage tract auction where s and s U0,. Bidder s strategy is to bid Γ 3 s c + + s + ρ s if s > ρ, and to not bid if s ρ ; bidder mies such that the distribution of his bids is if Γ 3 G 3 ep Γ dt if P, Γ c+ +Γ 3 3 t t 0 if < P If both bidders deviate in the first round, leading to either bidder winning at a price P < P or either bidder winning a price P > P, beliefs are s s ; if P < c +, both bidders bid c + in the second round, and if P c +, neither bidder bids in the second round. In the case where deviated but still dropped out at P P, losing the premium, bidder s strategy implies that a bid of Γ by bidder gives epected payoff c + s + Γ s so if s, bidder earns zero payoffs from any bid and is thus content to mi. Given bidder s mied strategy, bidder having already won the premium and believing that s, by bidding b, earns additional payoff G b c + + s b. Taking the log and then the first- order condition gives, which is positive when b < Γ s c+ +Γ b b c+ and negative +s b when b > Γ s, making Γ s a best-response when it is greater than P and making not bidding, equivalent to bidding P, a best-response when Γ s < P.
13 In the case where bidder won the first round at price P, the logic is similar, but now bidding Γ 3 earns bidder a second-round payoff of c + s + Γ 3 c + s + c + ρ s ρ so if s, any second-round bid gives the same payoff of ρ, eactly countering the win of the premium in the first round. Since bidder won the premium, not bidding is the same as bidding P, and gives the same payoff of ρ. Again given bidder s miing strategy, Γ 3 is a best-response for bidder. If bidder deviates, bidder epects to drop out of the first round at price P and earn positive payoff in the second round; by dropping out earlier, he would induce bidder to bid c + in the second round, so he would earn 0. So it is credible for bidder to stay in until P. If bidder deviates, then, he epects to either drop out before P, in which case he earns nothing; or to win the first round at P, in which case he wins the premium ρ but then earns payoff ρ in the second round. So deviating in the first round is not profitable for bidder. Second-round play on the equilibrium path is the same symmetric equilibrium as in the > cases. References. Engelbrecht-Wiggans, Richard, Paul Milgrom, and Robert Weber, Competitive Bidding and Proprietary Information, Journal of Mathematical Economics 983, Hendricks, Kenneth, Robert Porter, and Charles Wilson, Auctions for Oil and Gas Leases with an Informed Bidder and a Random Reservation Price, Econometrica 6 994, Milgrom, Paul, and Robert Weber, The Value of Information in a Sealed-Bid Auction, Journal of Mathematical Economics 0 98b,
Chapter 7. Sealed-bid Auctions
Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 100-3 90-99 21 80-89 14 70-79 4 0-69 11
The distribution of grades was as follows. ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY Range Numbers 100-3 90-99 21 80-89 14 70-79 4 0-69 11 Question 1: 30 points Games
More informationCournot s model of oligopoly
Cournot s model of oligopoly Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If firms total output is Q then market price is P(Q),
More informationNext Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept 25 2007 Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the price-discriminating
More informationA Theory of Auction and Competitive Bidding
A Theory of Auction and Competitive Bidding Paul Milgrom and Rober Weber Econometrica 1982 Presented by Yifang Guo Duke University November 3, 2011 Yifang Guo (Duke University) Envelope Theorems Nov. 3,
More informationOptimal Auctions Continued
Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for
More information8 Modeling network traffic using game theory
8 Modeling network traffic using game theory Network represented as a weighted graph; each edge has a designated travel time that may depend on the amount of traffic it contains (some edges sensitive to
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationRent-Seeking and Corruption
Rent-Seeking and Corruption Holger Sieg University of Pennsylvania October 1, 2015 Corruption Inefficiencies in the provision of public goods and services arise due to corruption and rent-seeking of public
More informationChapter 9. Auctions. 9.1 Types of Auctions
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/
More informationAsymmetric Information
Chapter 12 Asymmetric Information CHAPTER SUMMARY In situations of asymmetric information, the allocation of resources will not be economically efficient. The asymmetry can be resolved directly through
More informationHybrid Auctions Revisited
Hybrid Auctions Revisited Dan Levin and Lixin Ye, Abstract We examine hybrid auctions with affiliated private values and risk-averse bidders, and show that the optimal hybrid auction trades off the benefit
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationLecture V: Mixed Strategies
Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.3-1.3.A 1 The Advantage of Mixed Strategies Consider the following Rock-Paper-Scissors game: Note that
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationColored Hats and Logic Puzzles
Colored Hats and Logic Puzzles Alex Zorn January 21, 2013 1 Introduction In this talk we ll discuss a collection of logic puzzles/games in which a number of people are given colored hats, and they try
More informationInternet Advertising and the Generalized Second Price Auction:
Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords Ben Edelman, Harvard Michael Ostrovsky, Stanford GSB Michael Schwarz, Yahoo! Research A Few
More informationGames of Incomplete Information
Games of Incomplete Information Yan Chen November 16, 2005 Games of Incomplete Information Auction Experiments Auction Theory Administrative stuff Games of Incomplete Information Several basic concepts:
More information0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)
September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 28-29: 1, 4, 5, 6 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium When players move sequentially and have private information, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational. The problem is
More informationOverview: Auctions and Bidding. Examples of Auctions
Overview: Auctions and Bidding Introduction to Auctions Open-outcry: ascending, descending Sealed-bid: first-price, second-price Private Value Auctions Common Value Auctions Winner s curse Auction design
More informationA Study of Second Price Internet Ad Auctions. Andrew Nahlik
A Study of Second Price Internet Ad Auctions Andrew Nahlik 1 Introduction Internet search engine advertising is becoming increasingly relevant as consumers shift to a digital world. In 2010 the Internet
More informationGame Theory and Poker
Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are
More informationManagerial Economics
Managerial Economics Unit 8: Auctions Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 8 - Auctions 1 / 40 Objectives Explain how managers can apply game
More informationLATE AND MULTIPLE BIDDING IN SECOND PRICE INTERNET AUCTIONS: THEORY AND EVIDENCE CONCERNING DIFFERENT RULES FOR ENDING AN AUCTION
LATE AND MULTIPLE BIDDING IN SECOND PRICE INTERNET AUCTIONS: THEORY AND EVIDENCE CONCERNING DIFFERENT RULES FOR ENDING AN AUCTION AXEL OCKENFELS ALVIN E. ROTH CESIFO WORKING PAPER NO. 992 CATEGORY 9: INDUSTRIAL
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More informationMicroeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3
Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting
More informationEquilibrium Bids in Sponsored Search. Auctions: Theory and Evidence
Equilibrium Bids in Sponsored Search Auctions: Theory and Evidence Tilman Börgers Ingemar Cox Martin Pesendorfer Vaclav Petricek September 2008 We are grateful to Sébastien Lahaie, David Pennock, Isabelle
More informationI d Rather Stay Stupid: The Advantage of Having Low Utility
I d Rather Stay Stupid: The Advantage of Having Low Utility Lior Seeman Department of Computer Science Cornell University lseeman@cs.cornell.edu Abstract Motivated by cost of computation in game theory,
More informationMarkus K. Brunnermeier
Institutional tut Finance Financial Crises, Risk Management and Liquidity Markus K. Brunnermeier Preceptor: Dong BeomChoi Princeton University 1 Market Making Limit Orders Limit order price contingent
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationECON 40050 Game Theory Exam 1 - Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.
1 ECON 40050 Game Theory Exam 1 - Answer Key Instructions: 1) You may use a pen or pencil, a hand-held nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,
More informationCHAPTER NINE WORLD S SIMPLEST 5 PIP SCALPING METHOD
CHAPTER NINE WORLD S SIMPLEST 5 PIP SCALPING METHOD For a book that was intended to help you build your confidence, it s spent a long time instead discussing various trading methods and systems. But there
More informationMinimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example
Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games
More informationThe GMAT Guru. Prime Factorization: Theory and Practice
. Prime Factorization: Theory and Practice The following is an ecerpt from The GMAT Guru Guide, available eclusively to clients of The GMAT Guru. If you would like more information about GMAT Guru services,
More information6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning
6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning Daron Acemoglu and Asu Ozdaglar MIT November 23, 25 and 30, 2009 1 Introduction
More informationSpecial Notice. Rules. Weiss Schwarz Comprehensive Rules ver. 1.64 Last updated: October 15 th 2014. 1. Outline of the Game
Weiss Schwarz Comprehensive Rules ver. 1.64 Last updated: October 15 th 2014 Contents Page 1. Outline of the Game. 1 2. Characteristics of a Card. 2 3. Zones of the Game... 4 4. Basic Concept... 6 5. Setting
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09-074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationThe Basics of Game Theory
Sloan School of Management 15.010/15.011 Massachusetts Institute of Technology RECITATION NOTES #7 The Basics of Game Theory Friday - November 5, 2004 OUTLINE OF TODAY S RECITATION 1. Game theory definitions:
More informationPoker with a Three Card Deck 1
Poker with a Three Card Deck We start with a three card deck containing one Ace, one King and one Queen. Alice and Bob are each dealt one card at random. There is a pot of $ P (and we assume P 0). Alice
More informationAnglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam
Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam Christiaan van Bochove Lars Boerner Daniel Quint School of Business & Economics Discussion Paper Economics 2012/3 ANGLO-DUTCH PREMIUM AUCTIONS
More informationFinancial Markets. Itay Goldstein. Wharton School, University of Pennsylvania
Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting
More informationEasy Casino Profits. Congratulations!!
Easy Casino Profits The Easy Way To Beat The Online Casinos Everytime! www.easycasinoprofits.com Disclaimer The authors of this ebook do not promote illegal, underage gambling or gambling to those living
More informationAN ANALYSIS OF A WAR-LIKE CARD GAME. Introduction
AN ANALYSIS OF A WAR-LIKE CARD GAME BORIS ALEXEEV AND JACOB TSIMERMAN Abstract. In his book Mathematical Mind-Benders, Peter Winkler poses the following open problem, originally due to the first author:
More informationAUCTIONS IN THE REAL ESTATE MARKET A REVIEW
AUCTIONS IN THE REAL ESTATE MARKET A REVIEW Samuel Azasu Building & Real Estate Economics KTH Stockholm February 2006 Working Paper No. 55 Section for Building and Real Estate Economics Department of Real
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More informationOptimal Auctions. Jonathan Levin 1. Winter 2009. Economics 285 Market Design. 1 These slides are based on Paul Milgrom s.
Optimal Auctions Jonathan Levin 1 Economics 285 Market Design Winter 29 1 These slides are based on Paul Milgrom s. onathan Levin Optimal Auctions Winter 29 1 / 25 Optimal Auctions What auction rules lead
More informationFINAL EXAM, Econ 171, March, 2015, with answers
FINAL EXAM, Econ 171, March, 2015, with answers There are 9 questions. Answer any 8 of them. Good luck! Problem 1. (True or False) If a player has a dominant strategy in a simultaneous-move game, then
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationEconomics 335, Spring 1999 Problem Set #7
Economics 335, Spring 1999 Problem Set #7 Name: 1. A monopolist has two sets of customers, group 1 and group 2. The inverse demand for group 1 may be described by P 1 = 200? Q 1, where P 1 is the price
More informationFUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 6 Introduction to Corporate Bonds
FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 6 Introduction to Corporate Bonds Today: I. Equity is a call on firm value II. Senior Debt III. Junior Debt IV. Convertible Debt V. Variance
More informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More informationOnline Appendix to Stochastic Imitative Game Dynamics with Committed Agents
Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents William H. Sandholm January 6, 22 O.. Imitative protocols, mean dynamics, and equilibrium selection In this section, we consider
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More informationAuctioning Keywords in Online Search
Auctioning Keywords in Online Search Jianqing Chen The Uniersity of Calgary iachen@ucalgary.ca De Liu Uniersity of Kentucky de.liu@uky.edu Andrew B. Whinston Uniersity of Texas at Austin abw@uts.cc.utexas.edu
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -
More informationCapital Structure. Itay Goldstein. Wharton School, University of Pennsylvania
Capital Structure Itay Goldstein Wharton School, University of Pennsylvania 1 Debt and Equity There are two main types of financing: debt and equity. Consider a two-period world with dates 0 and 1. At
More informationAn Analysis of the War of Attrition and the All-Pay Auction*
journal of economic theory 72, 343362 (1997) article no. ET962208 An Analysis of the War of Attrition and the All-Pay Auction* Vijay Krishna Department of Economics, Pennsylvania State University, University
More informationThe Ascending Auction Paradox
The Ascending Auction Paradox Lawrence M. Ausubel and Jesse A. Schwartz* University of Maryland July 05, 1999 Abstract We examine uniform-price, multiple-unit auctions that are dynamic rather than sealedbid,
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More information25 Integers: Addition and Subtraction
25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers
More informationWeek 7 - Game Theory and Industrial Organisation
Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More informationNPV with an Option to Delay and Uncertainty
NPV with an Option to Delay and Uncertainty In all the examples and the theory so far, we have assumed perfect foresight when making investment decisions and analyzing future variables. We have also ignored
More informationGame Theory: Supermodular Games 1
Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution
More informationBad Reputation. March 26, 2002
Bad Reputation Jeffrey C. Ely Juuso Välimäki March 26, 2002 Abstract We study the classical reputation model in which a long-run player faces a sequence of short-run players and would like to establish
More informationJust Visiting. Visiting. Just. Sell. Sell $20 $20. Buy. Buy $20 $20. Sell. Sell. Just Visiting. Visiting. Just
Down South East North Up West Julian D. A. Wiseman 1986 to 1991, and 2015. Available at http://www.jdawiseman.com/hexagonal_thing.html North West Up Down East South ONE NORTH 1N 1N ONE NORTH ONE UP 1U
More informationOn Best-Response Bidding in GSP Auctions
08-056 On Best-Response Bidding in GSP Auctions Matthew Cary Aparna Das Benjamin Edelman Ioannis Giotis Kurtis Heimerl Anna R. Karlin Claire Mathieu Michael Schwarz Copyright 2008 by Matthew Cary, Aparna
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationNonlinear Systems of Ordinary Differential Equations
Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally
More information0 0 such that f x L whenever x a
Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:
More informationVickrey-Dutch Procurement Auction for Multiple Items
Vickrey-Dutch Procurement Auction for Multiple Items Debasis Mishra Dharmaraj Veeramani First version: August 2004, This version: March 2006 Abstract We consider a setting where there is a manufacturer
More informationDecision & Risk Analysis Lecture 6. Risk and Utility
Risk and Utility Risk - Introduction Payoff Game 1 $14.50 0.5 0.5 $30 - $1 EMV 30*0.5+(-1)*0.5= 14.5 Game 2 Which game will you play? Which game is risky? $50.00 Figure 13.1 0.5 0.5 $2,000 - $1,900 EMV
More informationLecture 28 Economics 181 International Trade
Lecture 28 Economics 181 International Trade I. Introduction to Strategic Trade Policy If much of world trade is in differentiated products (ie manufactures) characterized by increasing returns to scale,
More informationWhy Plaintiffs Attorneys Use Contingent and Defense Attorneys Fixed Fee Contracts
Why Plaintiffs Attorneys Use Contingent and Defense Attorneys Fixed Fee Contracts Winand Emons 1 Claude Fluet 2 1 Department of Economics University of Bern, CEPR, CIRPEE 2 Université du Québec à Montréal,
More informationA short note on American option prices
A short note on American option Filip Lindskog April 27, 2012 1 The set-up An American call option with strike price K written on some stock gives the holder the right to buy a share of the stock (exercise
More informationThe winner of the various auctions will get 2 cents per candy.
Candy Auction Jar contains an integer number of candies The winner of the various auctions will get 2 cents per candy. You are bidding for the money in these envelopes, not the candy! Playing for real
More informationAbout the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
More informationJournal Of Financial And Strategic Decisions Volume 8 Number 1 Spring 1995 INTEREST RATE SWAPS: A MANAGERIAL COMPENSATION APPROACH
Journal Of Financial And Strategic Decisions Volume 8 Number 1 Spring 1995 INTEREST RATE SWAPS: A MANAGERIAL COMPENSATION APPROACH John L. Scott * and Maneesh Sharma * Abstract The market for interest
More informationDissolving a partnership (un)fairly
Dissolving a partnership (un)fairly John Morgan Haas School of Business and Department of Economics, University of California, Berkeley Berkeley, CA 947-19, USA (e-mail: morgan@haas.berkeley.edu) Received:
More informationBackward Induction and Subgame Perfection
Backward Induction and Subgame Perfection In extensive-form games, we can have a Nash equilibrium profile of strategies where player 2 s strategy is a best response to player 1 s strategy, but where she
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationFUNDING INVESTMENTS FINANCE 738, Spring 2008, Prof. Musto Class 4 Market Making
FUNDING INVESTMENTS FINANCE 738, Spring 2008, Prof. Musto Class 4 Market Making Only Game In Town, by Walter Bagehot (actually, Jack Treynor) Seems like any trading idea would be worth trying Tiny amount
More informationWhen is Reputation Bad? 1
When is Reputation Bad? 1 Jeffrey Ely Drew Fudenberg David K Levine 2 First Version: April 22, 2002 This Version: November 20, 2005 Abstract: In traditional reputation theory, the ability to build a reputation
More informationFrom the probabilities that the company uses to move drivers from state to state the next year, we get the following transition matrix:
MAT 121 Solutions to Take-Home Exam 2 Problem 1 Car Insurance a) The 3 states in this Markov Chain correspond to the 3 groups used by the insurance company to classify their drivers: G 0, G 1, and G 2
More informationChristmas Gift Exchange Games
Christmas Gift Exchange Games Arpita Ghosh 1 and Mohammad Mahdian 1 Yahoo! Research Santa Clara, CA, USA. Email: arpita@yahoo-inc.com, mahdian@alum.mit.edu Abstract. The Christmas gift exchange is a popular
More informationFinance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model
Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model These notes consider the single-period model in Kyle (1985) Continuous Auctions and Insider Trading, Econometrica 15,
More informationCustomer Service Quality and Incomplete Information in Mobile Telecommunications: A Game Theoretical Approach to Consumer Protection.
Customer Service Quality and Incomplete Information in Mobile Telecommunications: A Game Theoretical Approach to Consumer Protection.* Rafael López Zorzano, Universidad Complutense, Spain. Teodosio Pérez-Amaral,
More informationUCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory
UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More information1 Nonzero sum games and Nash equilibria
princeton univ. F 14 cos 521: Advanced Algorithm Design Lecture 19: Equilibria and algorithms Lecturer: Sanjeev Arora Scribe: Economic and game-theoretic reasoning specifically, how agents respond to economic
More informationAn Introduction to Sponsored Search Advertising
An Introduction to Sponsored Search Advertising Susan Athey Market Design Prepared in collaboration with Jonathan Levin (Stanford) Sponsored Search Auctions Google revenue in 2008: $21,795,550,000. Hal
More informationCommon Knowledge: Formalizing the Social Applications
Common Knowledge: Formalizing the Social Applications 1 Today and Thursday we ll take a step in the direction of formalizing the social puzzles, such as omission commission. 2 First, a reminder of the
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
More informationPART A: For each worker, determine that worker's marginal product of labor.
ECON 3310 Homework #4 - Solutions 1: Suppose the following indicates how many units of output y you can produce per hour with different levels of labor input (given your current factory capacity): PART
More informationJournal Of Business And Economics Research Volume 1, Number 2
Bidder Experience In Online Auctions: Effects On Bidding Choices And Revenue Sidne G. Ward, (E-mail: wards@umkc.edu), University of Missouri Kansas City John M. Clark, (E-mail: clarkjm@umkc.edu), University
More informationLECTURES ON REAL OPTIONS: PART I BASIC CONCEPTS
LECTURES ON REAL OPTIONS: PART I BASIC CONCEPTS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART I August, 2008 1 / 44 Introduction
More informationLESSON 7. Leads and Signals. General Concepts. General Introduction. Group Activities. Sample Deals
LESSON 7 Leads and Signals General Concepts General Introduction Group Activities Sample Deals 330 More Commonly Used Conventions in the 21st Century General Concepts This lesson covers the defenders conventional
More information