Interdependent Security Gamesand Networks

Networked LifeCSE 112

Spring 2006Prof. Michael Kearns

Game Theory: Whirlwind Review

• Matrix (normal form) games, mixed strategies, Nash equil.– the basic objects of vanilla game theory– the power of private randomization

• Repeated matrix games– the power of shared history– new equilibria may result

• Correlated equilibria– the power of shared randomization– new equilibria may result– the result of adaptation and learning by players

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 global payoff functions• e.g. financial markets

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• Define the cost of this decision x_i as:

- (x_i C + (1 – x_i)p_i L + S_i L)

• 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 and their probability

or rate of transfer to airline i

• Analysis of terms:– x_i C = C at x_i = 1 (full investment); = 0 at x_i = 0 (no

investment)– (1-x_i)p_i L = 0 at full investment; = p_i L at no investment– S_i L: has no dependence on x_i

• 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[Kearns and Ortiz]

• 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

least busy

most busy

level of investment

simulationtime

The Price of Anarchy is High

Tipping and Cascading

Necessary Conditions for Tipping

Some Obvious Questions• Does the carrier transfer network obey the “universals” of social network

theory?– small diameter, local clustering, heavy tails, etc.

• I don’t know, but probably.• What generally happens with IDS games on such networks?

– Do “connectors” invest or not invest at equilibrium?– Do such networks lead to investing or non-investing equilibria?– Does subsidization of a couple of connectors make everyone invest?

• I don’t know… but it’s just a matter of time.• For standard economic market models, we’ll give answers.