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

• Axiomatic approaches– to bargaining– to voting and social choice

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

Interdependent Security Gamesand Networks

Networked LifeCSE 112

Spring 2004Prof. Michael Kearns

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• 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• 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

Results of Simulationleast busy carrier

most busy carrier• Consistent convergence to a mixed equilibrium• Larger airlines do not invest at equilibrium!• Dynamics of influence in the network

sim time

invest

ment

The Tipping Pointleast busy carrier

most busy carrier• Fix (subsidize) 3 largest airlines at full investment• Now consistently converge to global, full investment!• Largest 2 do not tip; cascading effects• Permits consideration of policy issues

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.