Introduction to Game Theory

1
Introduction to Game Theory

• Networked Life
• CSE 112
• Spring 2006
• 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

4
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

• If gt 2 actions, mixed strategy is a distribution
  on them
• e.g. 1/3 rock, 1/3 paper, 1/3 scissors
• Might also have gt 2 players
• A general mixed strategy is 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
• P is an equilibrium if
• for every 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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Hawks and Doves
hawk dove
hawk (V-C)/2, (V-C)/2 V, 0
dove 0, V V/2, V/2

• Two parties confront over a resource of value V
• May simply display aggression, or actually have a
  fight
• Cost of losing a fight C gt V
• Assume parties are equally likely to win or lose
• There are three Nash equilibria
• (hawk, dove), (dove, hawk) and (V/C hawk, V/C
  hawk)
• Alternative interpretation for C gtgt V
• the Kansas Cornfield Intersection game (a.k.a.
  Chicken)
• hawk speed through intersection, dove yield

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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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Repeated Games

• Nash equilibrium analyzes one-shot games
• we meet for the first time, play once, and
  separate forever
• Natural extension repeated games
• we play the same game (e.g. Prisoners Dilemma)
  many times in a row
• like a board game, where the state is the
  history of play so far
• strategy a mapping from the history so far to
  your next move
• So repeated games also have a Nash equilibrium
• may be different from the one-shot equilibrium!
• depends on the game and details of the setting
• We are approaching learning in games
• natural to adapt your behavior (strategy) based
  on play so far

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Repeated Prisoners Dilemma

• If we play for R rounds, and both know R
• (always defect, always defect) still the only
  Nash equilibrium
• argue by backwards induction
• If uncertainty about R is introduced (e.g. random
  stopping)
• cooperation and tit-for-tat can become equilibria
  
• If computational restrictions are placed on our
  strategies
• as long as were too feeble to count, cooperative
  equilibria arise
• formally lt log(R) states in a finite automaton
• a form of bounded rationality

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The Folk Theorem

• Take any one-shot, two-player game
• Suppose that (u,v) are the (expected) payoffs
  under some mixed strategy pair (P1,P2) for
  the two players
• (P1, P2) not necessarily a Nash equilibrium
• but (u,v) gives better payoffs than the security
  levels
• security level what a player can get no matter
  what the other does
• example sec. level is (-8, -8) in Prisoners
  Dilemma (-1,-1) is better
• Then there is always a Nash equilibrium for the
  infinite repeated game giving payoffs (u,v)
• makes use of the concept of threats
• Partial resolution of the difficulties of Nash
  equilibria

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Correlated Equilibrium

• In a Nash equilibrium (P1,P2)
• player 2 knows the distribution P1
• but doesnt know the random bits player 1 uses
  to select from P1
• equilibrium relies on private randomization
• Suppose now we also allow public (shared)
  randomization
• so strategy might say things like if private
  bits 100110 and shared bits 110100110, then
  play hawk
• Then two strategies are in correlated equilibrium
  if
• knowing only your strategy and the shared bits,
  my strategy is a best response, and vice-versa
• Nash is the special case of no shared bits

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Hawks and Doves Revisited
hawk dove
hawk (V-C)/2, (V-C)/2 V, 0
dove 0, V V/2, V/2

• There are three Nash equilibria
• (hawk, dove), (dove, hawk) and (V/C hawk, V/C
  hawk)
• Alternative interpretation for C gtgt V
• the Kansas Cornfield Intersection game (a.k.a.
  Chicken)
• hawk speed through intersection, dove yield
• Correlated equilibrium the traffic signal
• if the shared bit is green to me, I am playing
  hawk
• if the shared bit is red to me, I will play dove
• you play the symmetric strategy
• splits waiting time between us --- a different
  outcome than Nash

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Correlated Equilibrium Facts

• Always exists
• all Nash equilibria are correlated equilibria
• all probability distributions over Nash
  equilibria are C.E.
• and some more things are C.E. as well
• a broader concept than Nash
• Technical advantages of correlated equilibria
• often easier to compute than Nash
• Conceptual advantages
• correlated behavior is a fact of the real world
• model a limited form of cooperation
• more general cooperation becomes extremely
  complex and messy
• Breaking news (late 90s now)
• CE is the natural convergence notion for
  rational learning in games!

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Next Up

• Have so far examined simple games between two
  players
• Strategic interaction on the smallest network
• two vertices with a single link between them
• much richer interaction than just info
  transmission, messages, etc.
• Classical game theory generalizes to many players
• e.g. Nash equilibria always exist in multi-player
  matrix games
• but this fails to capture/exploit/examine
  structured interaction
• We need specific models for networked games
• games on networks local interaction
• shared information economies, financial markets
• voting systems
• evolutionary games