Game Theory: Whirlwind Review

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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

2
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

3
Interdependent Security Gamesand Networks

• Networked Life
• CSE 112
• Spring 2004
• Prof. Michael Kearns

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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 gtgt 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

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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!

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Abstract Features of the Game

• Direct and indirect sources of risk
• Investment reduces/eliminates direct risk only
• Risk is of a catastrophic event (L gtgt 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.

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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

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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)

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Network Visualization
Airport to airport
Carrier to carrier
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Results of Simulation
least busy carrier
most busy carrier

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

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The Tipping Point
least 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

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Some Obvious Questions

• Does the carrier transfer network obey the
  universals of social network theory?
• small diameter, local clustering, heavy tails,
  etc.
• I dont 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 dont know but its just a matter of time.
• For standard economic market models, well give
  answers.