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Game theory Week #1: Introduction

Prof. Pingzhong Tang iiis.tsinghua.edu.cn/~kenshin/gt/ Weibo: 唐平中 THU Pingzhong Tang 1

Personnel

• • Instructor • Pingzhong Tang (FIT 1-608-5) • [email protected]

• Office hour: Monday (12:15 –13:00) Teaching assistants • Weiran Shen, PhD student (1 st • [email protected]

• Office hour: TBD • year) Yuan Deng, Yao class student (4 th • [email protected]

• Office hour: TBD year) 4/28/2020 Pingzhong Tang 2

Contents

• Based on slides and notes by the instructor • Reference #1: An introduction to game theory. • Reference #2: Multi-agent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. • Reference #3: Markus Mobuis notes@Harvard 4/28/2020 Pingzhong Tang 3

Grading

• • • • • 10% Participation 20% Competition • Repeated prisoners’ dilemma • Auction revenue 30% Assignments, essay • Last year: the art of being stupid 40% Exam Will be adjusted over time 4/28/2020 Pingzhong Tang 4

Prerequisite

• Knowledge on • Calculus • Probability • Algorithm (e.g., Linear programming, complexity) • Programming 4/28/2020 Pingzhong Tang 5

Discussion: what is game theory?

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Game theory

• • Decision making by multiple, selfish agents • Single agent decision = AI • Multi-agent decision = game theory Themes • How to formulate multi-agent interactions • Model • What (rational) agents should (would) • Analysis (prediction) behave • How to incentivize rational agents to well behave • Design 4/28/2020 Pingzhong Tang 7

Two parts of the course

• • First part: models and analysis • Normal form games • Extensive form games • Bayesian games Second part: design • Mechanism design • Voting • Auction 4/28/2020 Pingzhong Tang 8

Why do computer scientists care about game theory?

• • • • John von Neumann = Father of computer science + Father of game theorist • So CS and GT are bothers… CS can be applied to compute GT solutions GT can be applied to design CS protocols/algorithms Rigorous economic thinking • Lifelong benefit • A smarter person 4/28/2020 Pingzhong Tang 9

Examples

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#1. Equilibrium and commitment

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Prisoner’s Dilemma

• Two criminals has been caught • District attorney has evidence to convict them of a minor crime (1 year in jail); knows that they committed a major crime together (3 years in jail) but cannot prove it • Offers each of them a deal: – If both defect ( 背叛 ), they each get a 1 year reduction – If only one defects, that one gets 3 years reduction defect defect cooperate

-2, -2 0, -3

cooperate 4/28/2020

-3, 0

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-1, -1

12

What if one player can make a commitment?

• I promise that I will defect/cooperate • What will I promise?

defect cooperate defect cooperate

-2, -2 0, -3 -3, 0 -1, -1

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What if the game is played repeatedly?

• How would you play?

defect cooperate defect cooperate

-2, -2 0, -3 -3, 0 -1, -1

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A modified game

• Equilibrium play?

T B L

1, 0 2, 1

R

3, 2 4, 0

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A modified game

• I promise that I will play T/B • What will I promise?

T B L

1, 0 2, 1

R

3, 2 4, 0

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Matching pennies

• I promise that I will play H/T • What will I promise?

H T H

1, -1 -1, 1

T

-1, 1 1, -1

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Real world application: security

• • • An airport 10 entrances, 5 policemen How to assign policemen to entrances? 4/28/2020 Pingzhong Tang 18

#2. Selfish routine

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System optimal flow

Adding A



B

System optimal flow

Better network, worse outcome!

Conclusion

• • • Equilibrium outcome is inefficient Further thought • Objective: social cost = sum of everyone’s cost • What is the system optimal/ equilibrium flow now?

Further thought • Can we bound equilibrium/system-optimal for any networks?

Tim Roughgarden. Selfish Routing.

Gödel Prize

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#3. Matching

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Stable marriage model

• •

Players

Set of men M, with typical man m 

M

Set of women W, with typical woman w 

W

|M|=|W| One-to-one matching: each man matched to one woman, and vice-versa

Preferences

Each man has strict preferences over women, and vice versa

Matching

• • • A

matching

is a set of pairs (m,w) such that each individual has one partner.

A matching is • •

stable

if Every individual is matched with a partner.

There is no man-woman pair, each of whom would prefer to match with each other rather than their assigned partner.

If such a pair exists, the match is

unstable

.

Examples

• • • • Two men m,m’ and two women w,w’

Example 1

• m prefers w to w’ and m’ prefers w’ to w • w prefers m to m’ and w’ prefers m’ to m • Unique stable match: (m,w) and (m’,w’)

Example 2

• • m prefers w to w’ and m’ prefers w’ to w w prefers m’ to m and w’ prefers m to m’ • Two stable matches {(m,w),(m’,w’)} and {(m,w’),(m’,w)} • First match is better for the men, second for the women.

Is there always a stable match?

Gale-Shapley Algorithm

• • • Men and women rank all potential partners

Algorithm

• Each man proposes to highest woman on his list • Women make a “tentative match” based on their preferred offer, and reject other offers, or all if none are acceptable.

• Each rejected man removes woman from his list, and makes a new offer.

• Continue until no more rejections or offers, at which point implement tentative matches.

This is the man-proposing algorithm; there is also a “woman proposing” version .

GS in pictures

Properties of men-propose matching

• • • • Stable O(n 2 ) Worst for all women, among all stable matchings Truthful for men, not truthful for women • Find an example where woman can benefit from lying • Further interesting things about stable matching: • All stable matchings form a lattice • Men and women have strictly opposite preferences among the stable matchings 4/28/2020 Pingzhong Tang 35

Importance of stability

• • • Roth (2000) surveys 17 major matching markets, 10 of which are stable All stable markets survive over time For the remaining 7 unstable markets, only 2 survive 4/28/2020 Pingzhong Tang 36

Application to Kidney Exchanges- Nobel Prize 2012

• • • • More than 75,000 people in the United States are waiting to receive a kidney transplant.

There is a shortage of donors • Deceased donors • Living donors • In 2005, 4200 patients died on the wait list, 10,000 in 2010 Problem is not just straight supply and demand • Donor kidney needs to be compatible with the patient.

• So sometimes patient has a living donor, but can’t use the kidney because of incompatibility.

• Maybe two patients could trade donor kidneys, or several patients could engage in a kidney exchange.

Matching theory can also help us understand this problem, and make optimal use of a limited pool of donors.

#4. Facility location game – Where to build a library in THU?

Single-peaked preference

Minimizing social cost: Median mechanism

1 2 3 5

Another objective: minimizing the maximum cost

• • Algorithmic solution: • Given x=(x 1 ,…x n ) • Output: OPT(x)=(leftmost(x)+rightmost(x))/2 • Not truthful: • x=(0,1), player 2 will misreport: 1  2 An algorithmic and game-theoretical solution: • Given x=(x 1 ,…x n ) • Output: leftmost(x) • Truthful • 2-approximation: no more than twice of algo. sol.

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Surprisingly

• • Theorem: No deterministic truthful mechanism can do better than 2-approximation.

Proof. • Consider n=2, x=(0, 1). WLOG, consider a truthful mechanism f(x)=1/2+c, with c≥0; • Now consider x’=(0, 1/2+c). • In this case, OPT(x’)=1/4+c/2, OPT cost= ¼+c/2.

• To achieve better than 2-approximation: f(x’) ~ (0, ½+c).

• Contradict to truthfulness: • In x’, player 2 misreports 1/2+c  1, • Location moves to 1/2+c by f(x). Better for player 2. □ 4/28/2020 Pingzhong Tang 41

Randomization

• Theorem : The following mechanism is truthful and a 3/2 approximation: given x, return leftmost(x) with ¼ prob, rightmost(x) with ¼, and OPT(x) with ½.

Theorem : No truthful mechanism (including randomization) can do better than 3/2.

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Thinking

• • What about there are two facilities?

What about everyone has a garden (interval) within which the library can not be built?

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#5. Auctions

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Dutch flower auction (Aalsmeer)

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Japan Bluefin Tuna Auction

• 34usd (Spain) vs 46 usd (Japan) • Record price: 1.7 M USD, 220 KG • Fish market = 43 football courts • Largest seafood market in the word 4/28/2020 Pingzhong Tang 46

# 5. 1

st

price and 2

nd

price auctions

• • 1 st • price auction : Everyone secretly bids a price, the highest bidder wins, paying his bid 2 nd • price auction : Everyone secretly bids a price, the highest bidder wins, paying the 2 nd highest bid

Which one is better?

• Definitions of good : • Simple to play • Higher revenue for the seller • Higher utility for the buyer 4/28/2020 Pingzhong Tang 48

334

Proof of truthfulness in

11 Protocols for Multiagent Resource Allocation: Auctions

2 nd -price auction

(a) Bidding honestly, i has the highest bid.

(b) i bids higher and still wins.

(c) i bids lower and still wins.

(d) i bids even lower and loses.

(e) Bidding honestly, i does not have the high est bid.

(f) i bids lower and still loses.

(g) i bids higher and still loses.

(h) i bids even higher and wins.

Figure 11.1: A case analysis to show that honest bidding is a dominant strategy in a second-price auction with independent private values.

would sometimes lose by bidding dishonestly in this case. Now consider the other case, where i ’s valuation is less than at least one other bidder’s bid. In this case i would lose and pay zero (Figure 11.1e). If he bid less, he would still lose and pay zero (Figure 11.1f). If he bid more, either he would still lose and pay zero (Figure 11.1g) or he would win and pay more than his valuation (Figure 11.1h), achieving negative utility. Thus again, i cannot gain, and would sometimes lose by bidding dishonestly in this case.

Notice that this proof does not depend on the agents’ risk attitudes. Thus, an agent’s dominant strategy in a second-price auction is the same regardless of whether the agent is risk neutral, risk averse or risk seeking.

In the IPV case, we can identify strong relationships between the second-price auction and Japanese and English auctions. Consider first the comparison between second-price and Japanese auctions. In both cases the bidder must select a number (in the sealed-bid case the number is the one written down, and in the Japanese case it is the price at which the agent will drop out); the bidder with highest amount wins, and pays the amount selected by the second-highest bidder. The difference between the auctions is that information about other agents’ bid amounts is disclosed in the Japanese auction. In the sealed-bid auction an agent’s bid amount must be selected without knowing anything about the amounts selected by others, whereas in the 3. Figure 11.1d is oversimplified: the winner will not always pay i ’s bid in this case. (Do you see why?) Uncorrected manuscript of

Multiagent Systems

, published by Cambridge University Press Revision 1.1 © Shoham & Leyton-Brown, 2009, 2010.

Which auction makes more money?

• • • • 2 nd • • price Truthful revenue=2 nd 1 st • • • highest value price is not truthful Bidder value drawn iid from a uniform [0,1] Bidder with value v will bid (n-1)v/n Revenue= (n-1)/n *highest value Fact: • E(highest among n~uniform [0,1])=n/(n+1) • E(2 nd highest among n~uniform [0,1])=(n-1)/(n+1) Thus: • E(2 nd • highest value)=E((n-1)/n *highest value) Expected revenue equivalence

Failure of 2

nd

price auction

• • Second price can be arbitrarily far from optimal Why?

Which auction is revenue-optimal then?

• Solved by Myerson, 1981 • Nobel Prize 2007 • n~uniform[0,1], second price auction with 0.5 reserve • In general: • Does not necessarily sell to the highest bidder • Does not necessarily sell the item at all

Application: Selling advertisement

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#6. Play a game…

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Travelers’ dilemma

• • • • Two travelers lost luggage Exactly the same Asked to report an integer value [2,100] • Both report m , they receive m • When M reports m , N reportes n, and m