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Figure it Out! An Introduction to Game Theory

Figure it Out! An Introduction to Game Theory. Professor Yan Chen School of Information University of Michigan. Outline. What is a game? History of game theory Best response and Equilibrium Auction Experiments Online Auction Design School Choice. It’s Your Move. It’s Your Move.

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Figure it Out! An Introduction to Game Theory

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  1. Figure it Out!An Introduction to Game Theory Professor Yan Chen School of Information University of Michigan

  2. Outline • What is a game? • History of game theory • Best response and Equilibrium • Auction Experiments • Online Auction Design • School Choice AADL Lecture 2007/07/15

  3. It’s Your Move AADL Lecture 2007/07/15

  4. It’s Your Move AADL Lecture 2007/07/15

  5. It’s Your Move AADL Lecture 2007/07/15

  6. It’s Your Move AADL Lecture 2007/07/15

  7. Problems of strategic choice • The consequences of choices often depend on choices by others • Even when people like each other or are partners (social or business) they may have different interests • Good problem solving requires strategic thought: “If I do X, what will my competitor / spouse / boss do?” AADL Lecture 2007/07/15

  8. What is a game? • A game is being played whenever solving a problem requires people to interact with strategic awareness • Bidding in an auction • Adoption of a new technology standard • Cuban missile crisis • What is not a game? • When strategic awareness of others not important • N = 1: What novel should I read next? • N = infinity: (large) markets AADL Lecture 2007/07/15

  9. Strategic choice tool: Game theory • Short history of game theory: • Cournot (1838) and Edgeworth (1881) • Zermelo (1913): chess-like games can be solved in a (large!) finite number of moves • von Neumann and Morgenstern (1944) • Expected utility theory, zero-sum games, cooperative games, backwards induction • Nash, Harsanyi, Selten: 1994 Nobel Prize for solution concepts in non-cooperative game theory • Aumann and Schelling : 2005 Nobel Prize for game theoretic analysis of conflict and cooperation AADL Lecture 2007/07/15

  10. Strategic choice tool: Game theory • Game theory has been applied to sociology, economics, political science, decision theory, law, evolutionary biology, experimental psychology, military strategy, anthropology … • Where is game theory going to? Behavioral; evolutionary; … • In the abstract, games can describe most multi-agent decision problems AADL Lecture 2007/07/15

  11. Problem representation: strategic form • One way to summarize the problem • Example: Prisoners’ Dilemma • Set of players: N = {Conductor, Tchaikovsky} • Information: common knowledge • Timing: simultaneous move • Set of strategies: Si = {Confess, Not Confess} • Set of payoffs: • If one confesses, the other does not: 0, 15 years in jail • If both confess: each gets 5 years in jail • If neither confess: each gets 1 year in jail AADL Lecture 2007/07/15

  12. Strategic form: Prisoners’ Dilemma Tchaikovsky Conductor AADL Lecture 2007/07/15

  13. Solving games? • For strategic problems, rational choice depends on choices made by others • The main tool is to find an equilibrium: a set of choices by all agents that are mutually rational • There are many different definitions of “rational”, depending on the particulars of the strategic problem • We call the process of finding a reasonable equilibrium “solving the game” AADL Lecture 2007/07/15

  14. Solving a strategic form game: Best response? • A strategy is a best responseto a particular strategy of another player, if it gives the highest payoff against that particular strategy • Is knowing best reply sufficient to find strategic equilibrium? AADL Lecture 2007/07/15

  15. Best reply: Prisoners’ Dilemma Tchaikovsky Conductor AADL Lecture 2007/07/15

  16. Dominant strategy equilibrium • A mild rationality concept: • Dominant strategy axiom: If a player has a dominant strategy, she will use it • Mild: Dominant strategy gets player best payoff possible no matter what others do • If every player has a dominant strategy, the game has a dominant strategy equilibrium (solution) • Problem with dominant strategy equilibrium: in many games there does not exist one AADL Lecture 2007/07/15

  17. Dominance: Prisoners’ Dilemma Tchaikovsky Conductor AADL Lecture 2007/07/15

  18. Efficiency and equilibrium • Game equilibrium is a characterization of the outcome of individually rational behavior • Because of strategic interactions, rational behavior does not always lead to outcomes that are mutually the best • Dominant strategy equilibrium in Prisoner’s Dilemma: (Confess, Confess) • But this is not socially efficient: both players are better off with (Not Confess, Not Confess) • Many applications • Arms race • Tragedy of commons AADL Lecture 2007/07/15

  19. What to do when equilibrium is inefficient? • Can’t always be improved (arms race not an easy problem!) • Opportunities: • Collude / cooperate (sometimes illegal!) • OPEC • marriage • Might involve side payments if not win-win • Design systems to increase trust • Repeated interactions • Build trust • Or create opportunities for punishment! AADL Lecture 2007/07/15

  20. Game Theory and Auctions • We will run four auction experiments • Assistant: • Marco Lorenzon • 4-th grader in September: King Elementary School AADL Lecture 2007/07/15

  21. ‘Sniping’ and the Rule for Ending Second-Price Internet Auctions Source: Axel Ockenfels and Alvin Roth

  22. – Facts and rules • Auctions on the Web for individuals. • 14 million auctions at each time in 4.300 categories. • 58 million registered users. • 40 bids and $2,000 sales per second. AADL Lecture 2007/07/15

  23. Second price rule • Bidders may submit ‘maximum bids’ during one week. • Auctions end with a ‘hard close’. • The highest maximum bid wins. • The price is an increment over the second highest bid. AADL Lecture 2007/07/15

  24. AADL Lecture 2007/07/15

  25. AADL Lecture 2007/07/15

  26. How much and when to bid on eBay? “eBay always recommends bidding the absolute maximum that one is willing to pay for an item early in the auction.” (eBay.com, 2002) AADL Lecture 2007/07/15

  27. The Puzzle

  28. last bid came in at the last second AADL Lecture 2007/07/15

  29. Cumulative distributions of auctions’ last bids on eBay AADL Lecture 2007/07/15

  30. The timing of bids: A puzzle • “Sniping can not be consistent with the presence of private values.” (Bajari and Hortacsu, 2000) • “ ... maybe eBay just makes me giddy …” (Landsburg, 1999) • “ …a particularly intriguing puzzle …” (Varian, 2000) … maybe bidders are just indifferent? … AADL Lecture 2007/07/15

  31. The dangers of sniping eBay’s view (2002) “… if you had bid your maximum amount up front … the outcome would not be based on time.” A seller’s view (Axis Mundi, 1999) “Almost without fail after an auction has closed we receive emails from bidders who claim they were attempting to place a bid and were unable to get into eBay. … All we can do in this regard is to urge you to place your bids early.” AADL Lecture 2007/07/15

  32. Why Sniping?

  33. Theorem 1: Sniping in private value auctions In a private value auction model, sniping can occur in perfect Bayesian equilibria as implicit collusion. “… the essential intuition of the Ockenfels-Roth analysis: bidding high at the last minute and letting chance determine the outcome is better for both players than bidding high early and precipitating a bidding war.” Hal Varian, AADL Lecture 2007/07/15

  34. Varian’s example • Two bidders bid on a pez dispenser. • Same value v > 0. • The seller‘s reservation price is zero. • If both bid v at t < 1, payoffs are zero. • If both bid at t = 1, payoffs are p(1-p)v > 0 • Any early bid starts a bidding war yielding zero payoffs.(Credible, because an early bidding war is an equilibrium.) AADL Lecture 2007/07/15

  35. Theorem 2:Sniping in common value auctions In a common value auction model, sniping can occur in perfect Bayesian equilibria to protect information. • ‘Uninformed’ bidders can incorporate into their bids the information they have gathered from the earlier bids of others. • ‘Informed’ bidders can avoid giving information to ‘uninformed’ bidders through their own early bids. AADL Lecture 2007/07/15

  36. Theorem 3: Sniping as a best response to incremental bidding Out of equilibrium, sniping can occur as a best response to (‘naive’) incremental bidding. • Bidding late would not give the incremental bidder sufficient time to respond. • A sniper competing with an incremental bidder might win the auction at the incremental bidder’s initial (low) bid. AADL Lecture 2007/07/15

  37. Market Design

  38. Is Amazon’s soft close a solution? “We know that bidding may get hot and heavy near the end of many auctions. Our Going, Going, Gone feature ensures that you always have an opportunity to challenge last-second bids. Here's how it works: whenever a bid is cast in the last 10 minutes of an auction, the auction is automatically extended for an additional 10 minutes from the time of the latest bid. This ensures that an auction can't close until 10 ‘bidless’ minutes have passed.“ [Amazon.com, 2002] AADL Lecture 2007/07/15

  39. Theorem 4: Amazon’s soft close In our models, the advantages of sniping are eliminated but the risk remains. As a consequence, sniping (bidding at t = 1) does not occur in perfect Bayesian equilibria. AADL Lecture 2007/07/15

  40. A Natural Experiment

  41. Cumulative distribution of auctions’ last bids over time More experienced bidders on eBay bid later, while experience in Amazon has the opposite effect. AADL Lecture 2007/07/15

  42. Summary: Internet Auction Design Strategic behavior • (Subtle) Strategic incentives substantially affect behavior and outcomes. Robust incentives • Sniping is a rational strategy against sophisticated bidders and against naive inexperienced bidders. Market design • Ceteris paribus a hard close appears to reduce revenues and efficiency. AADL Lecture 2007/07/15

  43. Game Theory and the School Choice Problem • In a school choice problem, there are a number of students each of whom should be assigned a seat at one of a number of schools • Each student has strict preferences over all schools • Each school has a maximum capacity and a strict priority ordering of all students AADL Lecture 2007/07/15

  44. Boston Mechanism • Each student submits a ranking of schools • Each school generates a priority ordering of students based on state and local laws (e.g., walk zone, sibling, etc.) • Student assignment based on submitted preference ranking and priorities in several rounds AADL Lecture 2007/07/15

  45. Boston Mechanism: Assignment Phase • Round 1: only the 1st choices of the studentsare considered. For each school, consider the students who have listed it astheir 1st choice and assign seats of the school to these students one at atime following their priority order until either there are no seats left orthere is no student left who has listed it as her first choice. • Round k: kth choices of the students, following priority order AADL Lecture 2007/07/15

  46. Boston Mechanism: Properties • Truth telling is not a dominant strategy: students might benefit from misrepresenting their preferences by improving ranking of schools for which they have high priority: The Minneapolis algorithm places a very high weight on the firstchoice, with second and third choices being strictly backup options.This is reflected in the advice CPAC gives out to parents, which is tomake the first choice a true favorite and the other two “realistic”,that is, strategic choices. - Glazerman and Meyer (1994) AADL Lecture 2007/07/15

  47. Boston Mechanism: Properties • More manipulation advice: St. Petersburg Times (September 14, 2003) Make a realistic, informed selection on the school you list as your first choice. It's the cleanest shot you will get at a school, but if you aim too high you might miss. Here's why: If the random computer selection rejects your first choice, your chances of getting your second choice school are greatly diminished. That's because you then fall in line behind everyone who wanted your second choice school as their first choice. You can fall even farther back in line as you get bumped down to your third, fourth and fifth choices. AADL Lecture 2007/07/15

  48. Boston Mechanism: Properties • Not Pareto efficient • Not stable: • Justified envy: if there is a student-school pair (i, s) such that • Student i prefers school s to her assignment • Student i has higher priority at school s than some other student who is assigned a seat at school s • Problem: legal actions • Boston’s Children First, et al. v. Boston School Committee (January 25, 2002) AADL Lecture 2007/07/15

  49. Gale-Shapley Mechanism • Each student submits a ranking of schools • Each school generates a priority ordering of students based on state and local laws (e.g., walk zone, sibling, etc.) • Student assignment based on submitted preference ranking and priorities in several rounds AADL Lecture 2007/07/15

  50. Gale-Shapley: Assignment • Round 1: Each student proposes to her 1st choice. Each school rejects the lowest priority students in excess of its capacity and keeps the remaining students on hold • Round k: Each student who was rejected in the previous rounds proposes to her next choice. Each school considers the studentsit has been holding together with its new proposers;it rejects the lowest priority students in excess of its capacityand keeps the remaining students on hold • Algorithm terminateswhen no student isrejected and each student is assigned a seat at her final tentative assignment. AADL Lecture 2007/07/15

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