Game Theory Nash equilibrium, Correlated equilibrium

1
Game Theory Nash equilibrium, Correlated
equilibrium

• Univ. Prof.dr. M.C.W. Janssen
• University of Vienna
• Winter semester 2008-9
• Week 43 (October 20, 22)

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Nash equilibrium

• A strategy combination (profile) (si ,s-i ) is
  a Nash equilibrium if for all players
• ui (si ,s-i ) ? ui (si ,s-i ) for all si ?Si
• Pure strategies are a special case
• If there is a mixed strategy equilibrium, the
  inequality must be binding (and player should be
  indifferent between any of the pure strategies
  that receive positive weight)
• NE is strict if each player has a strict best
  response to equilibrium strategies of other
  players
• Has to be in pure strategies

3
Some properties

• If game is dominance solvable, then the outcome
  must be a Nash equilibrium
• If the game has a unique Nash equilibrium, it
  does not imply that real people will always
  choose it
• PD game where defection yields only marginally
  higher pay-offs
• Many games have multiple Nash equilibria, and
  then there is no obvious way to play
• Pareto-dominance vs. risk dominance in Stag Hunt
  game
• Focal points (Schelling, 1960 ex. Of battle of
  Sexes game)

4
Games with only mixed strategy equilibria

• Matching pennies
• Inspection game
• Tennis, penalty shooting in football
• Mark Walker and John Wooders (AER 2001)
• Mixed strategies in continuous games
• Price dispersion in markets
• Continuous version of exercise 1.12
• There is a good rationale for mixing
• If your behaviour can be predicted, you loose out.

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Existence of Nash equilibrium

• Every finite game has a mixed strategy
  equilibrium (Nash 1950)
• If strategy spaces Si are nonempty compact convex
  sets and the pay-off function ui(si ,s-i ) is
  continuous in all its arguments and quasi-concave
  in si, then there exists a pure strategy
  equilibrium (Debreu 1952)
• What is quasi-concavity?
• Why does continuous version of exercise 1.12 not
  have a pure strategy equilibrium?

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Nash Equilibrium Existence Proof I

• Basically, application of Kakutanis fixed point
  theorem. A correpondence r S? S has a fixed
  point if
• S is compact, convex, nonempty subset of
  Euclidean space
• r is defined for all s ? S (non-empty)
• r is convex for all s ? S
• r has a closed graph (is upper-hemicontinuous),
  i.e., if (sn, sn) ? (s, s) with sn ? r(sn), then
  s ? r(s) ,

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Nash Equilibrium Existence Proof II

• Check that conditions are satisfied
• Interpret r as the Cartesian product of all best
  response correspondences ri
• Best response correspondence ri has only s-i as
  argument
• Si is a simplex of dimension n-1
• Pay-off ui(si ,s-i) is linear and therefore
  continuous and thus attains maximum on compact
  space
• As ui(asi(1- a)si,s-i)aui(si,s-i)(1-
  a)ui(si,s-i), asi(1- a)sI is a best response
  if both si and sI are
• Suppose condition 4 is violated, then for the
  sequences considered it must be the case that
  there is an e gt 0 and a sI such that ui(si,s-i)
  gt ui(si,s-i)3e. However, because of the way the
  sequence is defined and the continuity of u(.,.),
  for large n ui(si,sn-i) gt ui(si,s-i) - e gt
  ui(si,s-i)2e gt ui(sni,sn-i)e
  but this violates the optimality of sni
  against sn-i

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Correlated equilibrium

• Nash equilibrium assumes that players make
  independent decisions
• Sometimes, however, players may be able to use
  some commonly observable info to base their
  decision on (weather, or so) and be better off
• Examples
• a. convex hull of Nash eq. pay-offs attainable
  (ex. 3,3)
• b. higher pay-offs possible if three outcomes
  A,B,C are equally likely and row player can only
  distinguish A from BC and column player can only
  distinguish C from AB

a. Same signal
b. Different signals
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Definition correlated equilibrium

• Define O,Hi,p, where
• O is the state space of the random device
• Hi is the information partition for player I if
  true state is ??O, then player i is informed
  state is hi(?)
• p is probability measure on state space O
• Bayes rule applies, i.e., p(??hi) p(?)/p(hi)
• Strategy is a rule mapping histories to actions
• si hi ? A
• Strategy combination s is a correlated
  equilibrium relative to O,Hi,p iff ???hi
  p(??hi)ui(si(?),s-i(?)) ? ?? p(??hi)ui(si(?),s-
  i(?)) for all i, hi and si

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Equivalent definition

• Any probability distribution p over the set of
  pure strategies S is a correlated equilibrium iff
  ?s-i?S-i p(s-i?si)ui(si,s-i) ? ? s-i?S-i
  p(s-i?si)ui(si,s-i) for all i, si and si with
  p(si) gt 0
• Check idea of equivalence with the example on
  previous slide

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Properties correlated equilibria

• Nash equilibria can always be attained as
  correlated equilibria
• Set of correlated equilibria is convex
• Check equivalent definition
• If p and p are correlated equilibria, then for
  p ap(1-a)p, p(s-i?si) ?p(s-i?si)
  (1-?)p(s-i?si) for some ? ? (0,1)
• Contains at least convex hull of all NE

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Games differ from single decision-making

• Burning money may benefit players
• In upper game normal outcome is (U,L) with u1
• If row player commits to burning two utils if he
  plays U, then outcome is (D,R) with u3
• Neglecting information may benefit players
• Example correlated eq. next sheet

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Correlated equilibrium example
B
A
C

• Row, Column and matrix player
• Unique Nash equilibrium (D,L,A)
• If there is the following random divide players
  can do better. Throw a coin, outcome is observed
  by row and column player only. If Head they play
  U,L, if Tails they play D,R. B is optimal
  response for matrix player iff he does not know
  the outcome. If B is played both (U,L) and (D,R)
  are NE. If matrix player observes outcome coin,
  this outcome is not feasible.