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 Games
and Networks
Networked Life
CSE 112
Spring 2004
Prof. 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:
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–
–
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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 Simulation
investment 
least busy carrier
sim time 
• Consistent convergence to a mixed equilibrium
• Larger airlines do not invest at equilibrium!
• Dynamics of influence in the network
most busy carrier
The Tipping Point
least 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
most busy carrier
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.