Mixed Strategies, Expected Utility and Existence of Nash equilibrium

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Mixed Strategies, Expected Utility andExistence
of Nash equilibrium

• Guilherme Carmona
• Winter 2007

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Mixed Strategies

• Pure strategy element of strategy set
• Mixed strategy Probability distribution over
  strategy set
• Can define Nash equilibrium in mixed strategies
  or Mixed strategy Nash equilibrium
• new game with new strategies and payoffs

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Mixed strategies What for?

• Mathematical point of view
• Needed if pure strategy equilibria do not exist
• May coexist with pure strategy NE
• Practical point of view
• Be unpredictable
• Tennis service
• Penalty kicks in football
• War

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Finite Games

• Let Si sij, j 1,,Mi
• Mixed strategy pij, j 1,,Mi such that
• 0 ? pij ? 1 for all j 1,,Mi
• ?j1Mi pij 1
• pij is the probability that player i plays sij
• Support of a mixed strategy supp(pi)sij ? Si
  pij gt 0
• Payoffs expected utility

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Expected Utility Theorem

• Players care about s(s1,...,sn)
• But choose p(p1,...,pn)
• So, they must have preferences over ps
• Is there a way to have preferences over ps that
  reflect preferences over ss?
• Yes von Neumann-Morgensterns expected utility
  theorem.

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Expected Utility Theorem

• Let U be an utility function over mixed
  strategies
• Assume U is continuous and
• Independence axiom for all p,p,p and all a in
  (0,1), U(p) gt or U(p) if and only if U(a
  p(1-a) p) gt or U(a p(1-a)p).
• If U satisfies these assumptions, then there
  exist v(s)s in S such that U(p) ?sp(s)v(s).

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Observations

• Mixed strategies can dominate pure strategies
• if these are good only against specific
  strategies by the opponent
• If a player plays a mixed strategy in equilibrium
  then he is indifferent between all strategies in
  its support
• No strategy not in support can give a higher
  payoff

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Finding a mixed strategy equilibrium

• Examples
• Best responses
• Solving equations
• central condition indifference between strategies

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Philosophical Problems

• Why should players choose a mixed equilibrium
  strategy if they dont care about what to play?
• Why not?
• Alternative interpretation (Harsanyi) limit of
  pure strategy equilibria in perturbed games
• Experiments results indicate
• Players often do not mix, but play pure
  strategies
• Distribution of strategies over population of
  players often corresponds closely to predicted
  mixed equilibrium

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Infinite Games

• Strategy sets Si Li,Hi
• Mixed strategy is distribution Fi on Si
• How to find?
• Tricks and experience.
• Example One-unit Bertrand model

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Existence of Nash equilibrium in finite games

• Nash Theorem In every finite normal form game
  there exists at least one mixed strategy
  equilibrium
• which may be a pure strategy equilibrium
• Statement is wrong with pure instead of mixed
  strategy equilibrium
• Proof special case of DGF-theorem, but original
  proof was independent

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Number of Equilibria

• In almost all normal form games, there is an
  finite and odd number of Nash equilibria

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Existence of Nash equilibrium in infinite games

• Debreu-Glicksberg-Fan TheoremLet G be a normal
  form game with
• compact and convex strategy sets,
• payoffs continuous in all players strategies,
• payoffs quasi-concave in own strategy.
• Then a pure strategy Nash equilibrium exists
• Implies Nash Theorem

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Existence of Nash equilibrium in infinite games
(2)

• Glicksberg TheoremLet G be a normal form game
  with
• compact strategy sets,
• payoffs continuous in all players strategies.
• Then a mixed strategy Nash equilibrium exists
• which may be a pure strategy equilibrium

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Exercises

• PS 1 4, 5, 9, 10, 12, 13