Mixed Strategy Nash Equilibrium: Understanding Players' Randomization in Game Theory, Study notes of Game Theory

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Mixed strategy Nash equilibrium

Felix Munoz-Garcia

Strategy and Game Theory - Washington State University

Looking back...

So far we have been able to Önd the NE of a relatively large class of games with complete information: Games with two or several (n > 2) players. Games where players select among discrete or continuous actions. But, can we assure that all complete information games where players select their actions simultaneously have a NE? We couldnít Önd a NE for the matching pennies game!! (Next slide) We will be able to claim existence of a NE if we allow players to randomize their actions.

Another example

Here we have another example of an anti-coordination game with no psNE:

80 , 20 0 , 100

10 , 90 60 , 40

Street Corner Park Street Corner

Park

Police Officer

Surprise! Drug Dealer

We need to allow players randomize their choices (i.e., to play mixed strategies).

Mixed strategy Nash equilibrium

Harrington: Chapter 7, Watson: Chapter 11. First, note that if a player plays more than one strategy with strictly positive probability, then he must be indi§erent between the strategies he plays with strictly positive probability. Notation: "non-degenerate" mixed strategies denotes a set of strategies that a player plays with strictly positive probability. Whereas "degenerate" mixed strategy is just a pure strategy (because of degenerate probability distribution concentrates all its probability weight at a single point).

Degenerate Probability Distributions

Example of a degenerate probability distribution Prob.

Output, q 0 q = 8 units

1

The player (e.g., Örm) puts all probability weight (100%) on only one of its possible actions: q = 8.

DeÖnition of msNE:

Consider a strategy proÖle σ = ( σ 1 , σ 2 , ..., σ n ) where σ i is a mixed strategy for player i. σ is a msNE if and only if

ui ( σ i , σ i )  ui (s^0 i , σ i ) for all s i^0 2 Si and for all i

That is, σ i is a best response of player i to the strategy proÖle σ i of the other N 1 players, σ i = BRi ( σ i ).

Example 1:Matching pennies

Matching pennies

Player 2 q 1 q Heads Tails Player 1 p Heads 1 , 1 1 , 1 1 p Tails 1 , 1 1 , 1

Two alternative interpretations of playersí randomization: If player 1 is using a mixed strategy, it must be that he indi§erent between Heads and Tails Alternatively, if player 1 is indi§erent between Heads and Tails, it must be that player 2 mixes with such probability q such that player 1 is made indi§erent between Heads and Tails: EU 1 (H) = EU 1 (T ) () 1 q + ( 1 q)( 1 ) = ( 1 )q + 1 ( 1 q)

Matching pennies

Matching pennies (example of a normal form game with no psNE):

Player 2 q 1 q Heads Tails Player 1 p Heads 1 , 1 1 , 1 1 p Tails 1 , 1 1 , 1

Solving for the EU comparison, we obtain

EU 1 (H) = EU 1 (T ) () 1 q + ( 1 q)( 1 ) = ( 1 )q + 1 ( 1 q)

q =

! Graphical Interpretation

Matching pennies

(Player 2 ) q

0

1

(Player 1 ) p 1

q = ½

BR 1 ( q)

From 1 st^ and 2 nd^ steps

From 3 rd^ and 4 th^ steps

Heads

Heads

Tails

Matching pennies

Similarly, if player 2 is using a mixed strategy, it must be that he is indi§erent between Heads and Tails:

EU 2 (H) = EU 2 (T )

( 1 )p + 1 ( 1 p) = 1 p + ( 1 )( 1 p) () p = (^12) (See Ögure after next slide)

Matching pennies

(Player 2 ) q

0

1

(Player 1 ) p p = ½^1

Heads

Heads

Tails

q = 1 for all p < ½ ( 3 rd^ and 4 th^ steps)

q = 0 for all p > ½ ( 1 st

d and 2 nd^ Steps)

BR 2 ( p)

Matching pennies

We can represent these BRFs as follows: Player 1

BR 1 (q) =

8 < :

Heads if q > (^12) fHeads, Tailsg if q = (^12) Tails if q < (^12)

Player 1 is indi§erent between Heads and Tails when q is exactly q = (^12) Player 2

BR 2 (p) =

8 < :

Tails if p > (^12) fHeads, Tailsg if p = (^12) Heads if p < (^12)

Player 2 is indi§erent between Heads and Tails when p is exactly p = (^12)

Matching pennies

Therefore, the msNE of this game can be represented as  1 2

H,

T

H,

T

where the Örst parenthesis refers to player 1(row player), and the player 2(column player).

Battle of the sexes

2. Battle of the sexes (example of a normal form game with 2 psNE already!):

3 , 1 0 , 0 0 , 0 1 , 3

Football Opera Football Opera

Husband

Wife

p

q

1 - p

1 - q

If the Husband is using a mixed strategy, it must be that he indi§erent between Football and Opera: EU 1 (F ) = EU 1 (O) 3 q + 0 ( 1 q) = 0 q + 1 ( 1 q) 3 q = 1 q

4 q = 1 =) q =