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Economics 414 Game Theory Outline aIntroduction aSyllabus aWeb demonstration aExamples Professor Peter Cramton Spring 2005 2 1 About Me: Peter Cramton Course Objectives a B.S. Engineering, Cornell University a Ph.D. Business & Economics, Stanford University a Associate Professor, Yale University, 1984-93 a National Fellow, Hoover Institution, Stanford University, 1992-93 a Professor of Economics, University of Maryland, since 1993 a Chairman, Market Design Inc., since 1995 a Chairman, Spectrum Exchange, since 1999 a President, Criterion Auctions, since 2000 aTo understand the importance of competitive and cooperative factors in a variety of decision problems aTo learn how to structure and analyze these problems from a quantitative perspective 3 Course Outline 4 Logistics aMeet Tuesday and Thursday, 9:30 – 10:45 aProblem Sets (about 6) and Web Exercises [20% of grade] aStrategic-Form Games aExtensive-Form Games aRepeated Games aBayesian Games and Bayesian Equilibrium aDynamic Games of Incomplete Information aBargaining Theory aAuction Theory (and Practice) `Must be own work; don’t look at past solutions `Small discussion groups fine aMidterm Exam [30% of grade] aFinal Exam [50% of grad] `Tuesday, May 17, 8-10 am aOffice Hours Tues 7:30 to 9:30 am `Tydings 4101a `301.405.6987 or [email protected] 5 6 Did you get my email? Web exercises aEmail sent to [email protected] aIf you did not get it, then either: aRegister at http://gametheory.tau.ac.il/ (student’s registration in upper right) aYour login name will be: `University has wrong email address for you `You are not registered for this class (e.g. you are on the waitlist) `Your mail quota is exceeded `CR479U<student e-mail> where <student e-mail> is your full email. aThe class password is: `e139288Zt aI will send an email with assignments 7 Readings 8 Introduction and Examples aMartin J. Osborne, An Introduction to Game Theory, Oxford University Press (2004) [Required] aWeb site: www.cramton.umd.edu 9 Definition 10 Game 1 Game theory is the study of mathematical aEach of three players simultaneously picks a number from [0,1] aA dollar goes to the player whose number is closest to the average of the three numbers aIn case of ties, the dollar is split equally models of conflict and cooperation between intelligent and rational decision makers. aRational: each individual maximizes her expected utility aIntelligent: individual understands situation, including fact that others are intelligent rational decision makers 11 12 Game 1 in Normal Form (Strategic Form) Game 2: Both Pay Auction aplayer i ∈ N = {1,...,n} astrategy si ∈ Si astrategy vector (profile) s = (s1,...,sn) ∈ S = S1×...×Sn apayoff function ui(s):S→ℜ, which maps strategies into real numbers agame in normal form Γ = {S1,...,Sn;u1,...,un} a$10 is auctioned to highest of two bidders aPlayers alternate bidding aAt each stage, bidding player must decide either to raise bid by $1 or to quit aGame ends when one of the two bidders quits in which case the other bidder gets the $10, and both bidders pay the auctioneer their bids 13 14 Game 2 in Extensive Form (Game Tree) Game in Extensive Form aWho plays when? aWhat can they do? aWhat do they know? aWhat are the payoffs? $1 R $2 R 2 Q 1 Q Q Q Q 1 2 $3 R 1 $4 R $5 R 1 ... -3, 6 7, -2 -1, 8 9, 0 0, 10 15 16 Game 4 Game 3 1 l 1: 0 2: 2 1 L R 2 2 r l 3 1 -1 -1 L R 2 r l 2 0 1: 0 2: 2 17 r l 3 1 -1 -1 r 2 0 18 Game 3: How many info sets? Definitions 1 aStrategy: a complete plan of action (what to do in every contingency) aInformation Set: for player i is a collection of decision nodes satisfying two conditions: player i has the move at every node in the collection, and i doesn't know which node in the collection has been reached l 1: 0 2: 2 L R 2 2 r l 3 1 -1 -1 r 2 0 19 Game 4: How many info sets? More Definitions 1 L a Perfect Information: each information set is a single node (Chess, checkers, go, ...) aFinite games of perfect information can be "solved" by backward induction in the extensive form or elimination of weakly dominated strategies in the normal form a Imperfect Information: at some point in the tree some player is not sure of the complete history of the game so far R 2 l 1: 0 2: 2 r l 3 1 -1 -1 r 2 0 21 Game 3: Backward Induction Looking ahead and reasoning back 1 l 1: 0 2: 2 L R 2 2 r l 3 1 -1 -1 20 r 2 0 23 22