Introduction to Game Theory and Networks

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Introduction to Game Theoryand Networks

• Networked Life
• CSE 112
• Spring 2007
• Prof. Michael Kearns

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Game Theory

• A mathematical theory designed to model
• how rational individuals should behave
• when individual outcomes are determined by
  collective behavior
• strategic behavior
• Rational 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

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Prisoners Dilemma
cooperate defect
cooperate -1, -1 -10, -0.25
defect -0.25, -10 -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 choose
  between -10 and -8
• so you will also defect
• same logic applies to me
• 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

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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 want to play tails too
• which in turn makes you want to play heads
• etc. etc. etc.
• 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

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The World According to Nash

• A mixed strategy is a distribution on the
  available actions
• e.g. 1/3 rock, 1/3 paper, 1/3 scissors
• Joint mixed strategy for N players a vector P
  (P1, P2, PN)
• Pi is a distribution over the actions for
  player i
• assume everyone knows all the distributions Pj
• but the coin flips used to select from Pi
  known only to i
• private randomness
• two digressions
• mixed strategy simulation in Kings and Pawns?
• can people randomize?
• P is an equilibrium if
• for every player i, Pi is a best response to
  all the other Pj
• 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 Nashs Nobel Prize in
  economics

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Facts about Nash Equilibria

• While there is always at least one, there might
  be many
• zero-sum games all equilibria give the same
  payoffs to each player
• non zero-sum different equilibria may give
  different payoffs!
• Equilibrium is a static notion
• does not suggest how players might learn to play
  equilibrium
• does not suggest how we might choose among
  multiple equilibria
• Nash equilibrium is a strictly competitive notion
• players cannot have pre-play communication
• bargains, side payments, threats, collusions,
  etc. not allowed
• Computing Nash equilibria for large games is
  difficult

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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 to the next move for the player
• number of rows or columns is huge --- but finite!
• Thus, a Nash equilibrium for chess exists!
• its just completely infeasible to compute it
• note can often push randomization inside the
  strategy

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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

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Case Study Interdependent Security Games on
Networks
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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 ModelKunreuther 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_iC)..
• 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 cant 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!

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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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The Price of Anarchy is High
least busy
most busy
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Tipping and Cascading
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Necessary Conditions for Tipping