University of Verona
Academic Year 2007/2008
Applied Game Theory
Luca Zarri
Lecture 4
Focus on:
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Mixed Strategies
What are they?
What is the difference between pure and
mixed strategies?
Main reference: Gibbons (Ch. 1)
Mixed Strategies
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In the ‘Matching Pennies’ game there are
no Nash Equilibria in pure strategies
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The key feature of this kind of games is
that each player would like to know what
the opponent will do without letting know
to her what he will do (the same holds for
poker, baseball, battle..)
Mixed Strategies
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Remark: in simultaneous-move games
with complete information, pure strategies
are simply the actions that a player can
take
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For example in Matching Pennies the pure
strategies for a player are simply Heads
and Tails
Mixed Strategies
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In Matching Pennies, a mixed strategy for
player i is the probability distribution (q,
1-q), where q is the probability to play
Heads and (1-q) is the probability to play
Tails (with 0≤q≤1)
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Hence, the mixed strategy (0,1) is simply
the pure strategy Tails and
the mixed strategy (1,0) is simply the
pure strategy Heads
Mixed Strategies
Definition
In the normal form game
G = {S1, …., Sn; u1, …, un },
let us suppose that S = {si1, …, siK }.
A mixed strategy for player i is a
probability distribution pi = (pi1, …, piK),
with 0 ≤pik≤ 1 for k = 1,…, K and
pi1+.. + piK = 1
Mixed Strategies and Nash Equilibrium
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In order to extend the definition of Nash Equilibrium
(so that mixed strategies can be taken into account)
we simply need to require that the mixed strategy of
each player is the best reply to the mixed strategies
of the other players
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Since every pure strategy can be represented as a
mixed strategy that assigns probability zero to all the
other pure strategies of the player, the new definition
includes the previous one
Mixed Strategies and Nash Equilibrium
Definition (for 2-player games)
In the normal form game with two players
G = {S1,S2; u1,u2}, the mixed strategies
(p1* ,p2*) are a Nash Equilibrium if the
mixed strategy of each player is a best
reply to the mixed strategy of the other
player
Mixed Strategies: Interpretation
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The Mixed Strategy of player j can be
interpreted as the effect of the
uncertainty of player i over the choice of
a (pure) strategy on the part of player j
Existence of Nash Equilibria
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Does a Nash Equilibrium necessarily exist
in a game?
Yes, for the class of games we have been
considering, the answer is positive under
fairly broad circumstances, provided that
we are willing to accept equilibria in which
players randomize
Existence of Nash Equilibria
Theorem (Nash 1950)
In the normal form game with n players
G = {S1, …., Sn; u1, …, un }, if n is finite
and si is finite for every i, then at least
one Nash Equilibrium exists (possibly in
mixed strategies)
Mixed Strategies
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In general, qualitatively speaking, we can have:
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1. one Nash Equilibrium in pure strategies (e.g.
the Prisoner’s Dilemma)
2. one Nash Equilibrium in mixed strategies (e.g.
the Matching Pennies)
3. two Nash Equilibria in pure strategies and one
equilibrium in mixed strategies (e.g. the Battle of
Sexes, the Game of Assurance or the Chicken
Game – see the movie ‘Rebel Without A Cause’
(1955), with James Dean)
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