Introduction to Game Theory and Networks Networked Life CSE 112 Spring 2007 Prof. Michael Kearns

Game Theory

• A mathematical theory designed to model: – how

rational individuals

should behave – when individual outcomes are determined by

collective

– strategic behavior • Rational behavior usually means selfish --- but not always • Rich history, flourished during the Cold War • Traditionally viewed as a subject of economics • Subsequently applied by many fields – evolutionary biology, social psychology • Perhaps the branch of pure math most widely examined outside of the “hard” sciences

Prisoner’s Dilemma

cooperate defect cooperate -1 , -1 defect -0.25

, -10 -10 , -0.25

-8 , -8 • Cooperate = deny the crime; defect = confess guilt of both • Claim that ( defect , defect ) is an

equilibrium:

– if I am definitely going to defect , you – so you will also defect – same logic applies to me choose between -10 and -8 • Note

unilateral

nature of equilibrium: – I fix a behavior or strategy for you, then choose my best response • Claim: no other pair of strategies is an equilibrium • But we would have been so much better off

cooperating…

Penny Matching

heads tails heads 1 0 , , 0 1 tails 0 , 1 1 , 0 • What are the equilibrium strategies now?

• There are none!

– if I play heads then you will of course play tails – but that makes me – which in turn makes you – etc. etc. etc.

want to play tails want to play too heads • But what if we can each (privately)

flip coins?

– the strategy pair ( 1/2 , 1/2 ) is an equilibrium • Such randomized strategies are called

mixed strategies

The World According to Nash

• A mixed strategy is a – e.g. 1/3 rock, 1/3 paper, 1/3 scissors • Joint mixed strategy for N players: a vector P = (P[1], P[2],… P[N]): – P[i] is a distribution over the actions for player i – assume

everyone distribution

knows all the on the available actions

distributions

P[j] – but the “coin flips” used to

select

• “private randomness” from P[i] known – two digressions: • mixed strategy simulation in Kings and Pawns?

• can people randomize?

only

to i • P is an equilibrium if: – for every player i, P[i] is a

best response

to all the other P[j] • Nash 1950: every game has a mixed strategy equilibrium – no matter how many rows and columns there are – in fact, no matter how many players there are • Thus known as a

Nash equilibrium

• A major reason for Nash’s Nobel Prize in economics

Facts about Nash Equilibria

• While there is always at least • Equilibrium is a

static

notion

one

, there might be – does not suggest how players might

learn many

– zero-sum games: all equilibria give the same payoffs to each player – non zero-sum: different equilibria may give different payoffs!

to play equilibrium – does not suggest how we might

choose

• Computing among multiple equilibria • Nash equilibrium is a

strictly competitive

notion – players cannot have “pre-play communication” – bargains, side payments, threats, collusions, etc. not allowed Nash equilibria for large games is difficult

Digression: Board Games and Game Theory

• What does game theory say about richer games?

– tic-tac-toe, checkers, backgammon, go,… – these are all games of

complete information

with

state

– incomplete information: poker • Imagine an absurdly large “game matrix” for chess: – each row/column represents a complete strategy for playing – strategy = a mapping from

every possible board configuration

move for the player – number of rows or columns is huge --- but finite!

• Thus, a Nash equilibrium for chess exists!

– it’s just completely infeasible to

compute

it – note: can often “push” randomization “inside” the strategy to the next

Games on Networks

• Matrix game “networks” • Vertices are the players • Keeping the normal (tabular) form – is expensive (exponential in N) – misses the point • Most strategic/economic settings have much more

structure

– asymmetry in connections – local and global structure – special properties of payoffs • Two broad types of structure: – special structure of the network • e.g. geographically local connections – special payoff functions • e.g. financial markets

Case Study: Interdependent Security Games on Networks

The Airline Security Problem

• Imagine an expensive new bomb-screening technology – large cost C to invest in new technology – cost of a mid-air explosion: L >> C • There are

two sources

of explosion risk to an airline: – risk from

directly

checked baggage: new technology can reduce this – risk from

transferred

baggage: new technology does nothing – transferred baggage

not

re-screened (except for El Al airlines) • This is a “game”… – each player (airline) must choose between I(nvesting) or N(ot) • partial investment ~ mixed strategy – (negative) payoff to player (cost of action) depends on

all others

• …on a network – the network of transfers between air carriers – not the complete graph – best thought of as a

weighted

network

The IDS Model

[Kunreuther and Heal]

• Let x_i be the fraction of the investment C airline i makes • p_i: probability of explosion due to directly checked bag • S_i: probability of “catching” a bomb from someone else – a straightforward function of all the “neighboring” airlines j – incorporates both their investment decision j (x_j) and their probability or rate of transfer to airline i • Payoff structure (qualititative, can be made quantitative): – increasing x_i reduces “effective direct risk” below p_i… – …but at increasing cost (x_i*C)..

– …and does nothing to reduce effective indirect risk S_i, which can only be reduced by the investments of others – network structure influences S_i • Typical strategic incentives: – when your neighbors are under-investing, your incentive to invest is low • basic problem: so much indirect risk already that you can’t help yourself much – when your neighbors are all fully investing, your incentive to invest is high • because your fate is in your own control now --- can reduce your only remaining source of risk • What are the Nash equilibria?

– fully connected network with uniform transfer rates:

full

investment or

no

investment by all parties!

Abstract Features of the Game

• Direct and indirect sources of risk • Investment reduces/eliminates direct risk only • Risk is of a

catastrophic

event (L >> C) – can effectively occur only once • May only have incentive to invest if enough others do!

• Note: much more involved network interaction than info transmittal, message forwarding, search, etc.

Other IDS Settings

• Fire prevention – catastrophic event: destruction of condo – investment decision: fire sprinkler in unit • Corporate malfeasance (Arthur Anderson) – catastrophic event: bankruptcy – “investment” decision: risk management/ethics practice • Computer security – catastrophic event: erasure of shared disk – investment decision: upgrade of anti-virus software • Vaccination – catastrophic event: contraction of disease – investment decision: vaccination – incentives are

reversed

in this setting

An Experimental Study

• Data: – 35K N. American civilian flight itineraries reserved on 8/26/02 – each indicates full itinerary: airports, carriers, flight numbers – assume all direct risk probabilities p_i are small and equal – carrier-to-carrier xfer rates used for risk xfer probabilities • The simulation: – carrier i begins at random investment level x_i in [0,1] – at each time step, for every carrier i: • carrier i computes costs of full and no investment

unilaterally

• adjusts investment level x_i in direction of improvement (gradient)

Network Visualization

Airport to airport Carrier to carrier

The Price of Anarchy is High

least busy level of investment simulation time most busy

Tipping and Cascading

Necessary Conditions for Tipping