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

Figure it Out!An Introduction to Game Theory

Professor Yan Chen School of InformationUniversity of Michigan

© Chen 2007 AADL Lecture 2007/07/15

Outline

What is a game?History of game theoryBest response and EquilibriumAuction ExperimentsOnline Auction DesignSchool Choice


It’s Your Move


It’s Your Move


It’s Your Move


It’s Your Move


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?”


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


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


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


Problem representation: strategic form

One way to summarize the problemExample: 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


Strategic form: Prisoners’ Dilemma

Confess Not Confess

Confess -5, -5 0, -15

Not Confess -15, 0 -1, -1

Tchaikovsky

Conductor


Solving games?

For strategic problems, rational choice depends on choices made by othersThe 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”


Solving a strategic form game: Best response?

A strategy is a best response to 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?


Best reply: Prisoners’ Dilemma

Confess Not Confess

Confess -5, -5 0, -15


Tchaikovsky

Conductor


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


Dominance: Prisoners’ Dilemma

Confess Not Confess

Confess -5, -5 0, -15


Tchaikovsky

Conductor


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


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!


Game Theory and Auctions

We will run four auction experimentsAssistant: Marco Lorenzon 4-th grader in September: King

Elementary School

‘Sniping’ and the Rule for Ending Second-Price Internet Auctions

Source: Axel Ockenfels and Alvin Roth


– 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.


Second price ruleBidders 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.

maximum bid price

bidder 1 bidder 2 high bidder (increment = $1)

$50 $100 2 $51

$70 no change 2 $71

$200 no change 1 $101




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)

The Puzzle


last bid came in at the last second


Cumulative distributions of auctions’ last bids on eBay

30%

40%

50%

60%

70%

80%

90%

100%

60 55 50 45 40 35 30 25 20 15 10 5 0

minutes before auction ends

% o

f sub

mitt

ed la

st b

ids

eBay-Computers eBay-Antiques

share of bids in eBay

last hour 68 %

10 minutes 55 %

5 minutes 50%

1 minute 37 %

10 seconds 12 %


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? …


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.”

Why Sniping?


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,


Varian’s exampleTwo 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.)


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.


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.

Market Design


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]


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.

}{),1(}2{)2,1(}1{)1,0[ nnn

A Natural Experiment


Cumulative distribution of auctions’ last bids over time

30%

40%

50%

60%

70%

80%

90%

100%

60 55 50 45 40 35 30 25 20 15 10 5 0

minutes before auction ends

% o

f sub

mitt

ed la

st b

ids

eBay-Computers eBay-Antiques Amazon-Computers Amazon-Antiques

share of bids in ...

eBay Amazon

last hour 68 % 23 %

10 minutes 55 % 11 %

5 minutes 50% 3 %

1 minute 37 % 0.4 %

10 seconds 12 % 0 %

More experienced bidders on eBay bid later, while experience in Amazon has the opposite effect.


Summary: Internet Auction DesignStrategic behavior

(Subtle) Strategic incentives substantially affect behavior and outcomes.

Robust incentivesSniping is a rational strategy against sophisticated bidders and against naive inexperienced bidders.

Market designCeteris paribus a hard close appears to reduce revenues and efficiency.


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 schoolsEach student has strict preferences over all schoolsEach school has a maximum capacity and a strict priority ordering of all students


Boston Mechanism

Each student submits a ranking of schoolsEach 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


Boston Mechanism: Assignment Phase

Round 1: only the 1st choices of the students are considered. For each school, consider the students who have listed it as their 1st choice and assign seats of the school to these students one at a time following their priority order until either there are no seats left or there is no student left who has listed it as her first choice.

Round k: kth choices of the students, following priority order


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 first choice, with second and third choices being strictly backup options.This is reflected in the advice CPAC gives out to parents, which is to make the first choice a true favorite and the other two “realistic”, that is, strategic choices. - Glazerman and Meyer (1994)



More manipulation advice: St. Petersburg Times (September 14, 2003)

Make a realistic, informed selection on the school you list asyour first choice. It's the cleanest shot you will get at aschool, but if you aim too high you might miss.

Here's why: If the random computer selection rejects your firstchoice, your chances of getting your second choice school aregreatly diminished. That's because you then fall in line behindeveryone who wanted your second choice school as their firstchoice. You can fall even farther back in line as you get

bumpeddown to your third, fourth and fifth choices.



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)


Gale-Shapley Mechanism

Each student submits a ranking of schoolsEach 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


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 students it has been holding together with its new proposers; it rejects the lowest priority students in excess of its capacity and keeps the remaining students on hold

Algorithm terminates when no student is rejected and each student is assigned a seat at her final tentative assignment.


Gale-Shapley: Properties

Truth telling is a dominant strategy: students have no incentive to misrepresent their preferencesStable: eliminates justified envyConstrained efficient: Pareto efficiency incompatible with

stability Gale-Shapley Pareto dominates any other

mechanism which eliminates justified envy


Game Theory and Public Policy

New York City High Schools: switched to the Gale-Shapley mechanism in fall 2004Boston School Committee voted to replace the Boston mechanism with Gale-Shapley in July 2005


Game Theory Readings

Dixit and Nalebuff:Thinking Strategically Fun, intuitive, journalistic International best seller No practice problems

Dutta: Strategies and Games: Theory and Practice Precise lot of practice problems

http://www.amazon.com/exec/obidos/tg/detail/-/0262041693/qid=1076968166/sr=1-1/ref=sr_1_1/103-9367263-5214241?v=glance&s=books

http://www.amazon.com/exec/obidos/tg/detail/-/0262041693/qid=1076968166/sr=1-1/ref=sr_1_1/103-9367263-5214241?v=glance&s=books

Documents

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