Mixed Strategies

Mixed Strategies

Player 2

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

Mixed Strategies

Player 2

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

Mixed Strategies

Definition: A mixed strategy of a player in a simultaneous move game is a probability distribution over the player’s actions

In matching pennies a mixed strategy will be i = (i(H), i(T)), where 0 ≤ i(.) ≤ 1

Mixed Strategies

Player 2

q 1 - q

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

Where 0 q 1

Mixed Strategies

Player 2

q 1 - q

Head Tail Expected Payoff

Player 1

Head 1, -1 -1, 1 2q - 1

Tail -1, 1 1, -1 1 – 2q

Mixed StrategiesPlayer 2

1 – 2q1, -1-1, 1Tail

2q - 1 -1, 11, -1Head p

Player 1 1-p

Expected Payoff

TailHead

1 - qq

Expected 1 - 2p 2p-1 Payoff

Mixed Strategies

1 – 2q > 2q – 1 if and only if q < ½

player 1’s Best pure-strategy response is:

- Tail if q < ½- Head if q > ½- Indifferent between H and T if q = ½

Mixed Strategies

Player 2

q 1 - q

Head Tail

Player 1

p Head 1, -1 -1, 1

1 – p Tail -1, 1 1, -1

Where 0 p 1

Mixed Strategies

E1(Payoff) = pq*1 + p(1 - q)*(-1) + (1 – p)q*(-1) + (1 – p)(1 – q) * 1

= (1 – 2q) + p(4q – 2)

Maximize E1(Payoff) choosing p.If 4q – 2 < 0 [q < ½] p = 0 (Tail) is best

responseIf 4q – 2 > 0 [q > ½] p = 1 (Head) is best

responseIf 4q – 2 = 0 [q = ½] any p in [0, 1] is a best

response

Mixed Strategies

E2(Payoff) = pq*(-1) + p(1 - q)*1 + (1 – p)q*1 + (1 – p)(1 – q) *(-1)

= (2p - 1) + q(2 – 4p)

Maximize E2(Payoff) choosing q.If 2 - 4p < 0 [p > ½] q = 0 (Tail) is best

responseIf 2 - 4p > 0 [p < ½] q = 1 (Head) is best

responseIf 2 - 4p = 0 [p = ½] any q in [0, 1] is a best

response

Mixed Strategies

p

q

1

1

1/2

1/2

b1(q)

b2(p)

The unique Nash equilibrium is in mixed-strategy:

(1, 2) = ((1/2,1/2), (1/2,1/2))

Mixed Strategies

Definition: The mixed strategy profile * in a simultaneous-move game with VNM preferences is a mixed strategy Nash equilibrium if, for each player i and every mixed strategy i of player i, the expected payoff to player i of * is at least as large as the expected payoff to player i of (i, *-i) according to a payoff function whose expected value represents player i’s preferences over lotteries.

Mixed Strategies

Equivalently, for each player i,

Ui(*) ≥ Ui (i, *-i) for every mixed strategy profile i of player i,

Where Ui() is player i’s expected payoff to the mixed strategy profile

Mixed Strategies

Alternative definition: The mixed strategy profile * is a mixed strategy Nash equilibrium if and only if *i is in Bi(*-i) for every player i.

Mixed Strategies

A player’s expected payoff to the mixed strategy profile is a weighted average of her expected payoffs to all mixed strategy profiles of the type (ai, -i), where the weight attached to (ai, -i) is the probability i(ai) assigned to ai by player i’s mixed strategy i

ii

i

A

iiiii aEaU

,

Where Ai is player i’s set of actions (pure strategies)

Mixed Strategies

MSNE Proposition: A mixed strategy profile * in a strategic game in which each player has finitely many actions is a mixed strategy Nash equilibrium if and only if, for each player i,

• The expected payoff, given *-i, to every action to which *i assigns positive probability is the same,

• The expected payoff, given *-i, to every action to which *i assigns zero probability is at most the expected payoff to any action to which *i assigns positive probability.

(See page 116 in Osborne.)

So actions which the player is mixing between must yield the same expected payoff. Those that are not being mixed, must not yield a higher expected payoff than those that are.

Mixed Strategies

American

q 1 - q

Enter Stay out

United

p Enter -50, -50 100, 0

1 – p Stay out 0, 100 0, 0

Mixed Strategies•Suppose both airlines mix between both strategies.

•United’s expected payoff from entering and staying out must be the same:

-50q +100(1-q) = 0q + 0(1-q) --> q = 2/3

•American’s expected payoff from entering and staying out must be the same:

-50p +100(1-p) = 0p + 0(1-p) --> p = 2/3

•Symmetric expected payoffs are thus:-50(2/3)(2/3) +100(2/3)(1/3) + 0(1/3)(2/3)+0(1/3)(1/3) = 0

•Note that equalizing the conditional expected payoffs gives you the interior solution (if it exists) while maximizing the unconditional expected payoffs will give you ALL NE.

•ALL NE are thus {((1,0),(0,1)); ((0,1),(1,0)); ((2/3,1/3),(2/3,1/3)) }

Mixed Strategies

Proposition: Every simultaneous-move game with vNM preferences and a finite number of players in which each player has finitely many actions has at least one Nash equilibrium, possibly involving mixed strategies.

Mixed StrategiesAsymmetric game

0, 00, 100Stay out1 – p

150, 0-50, -50Enterp

United

Stay outEnter

1 - qq

American

Asymmetric United/American Solution

Consider the unconditional expected payoff of United:E[UU] = -50pq + 150p(1-q) + 0(1-p)q + 0(1-p)(1-q) = -200pq + 150p = p(150-200q)So United’s Best Response correspondence is:•If 150-200q > 0 <=> q < 3/4 ==> p=1.•If 150-200q < 0 <=> q > 3/4 ==> p=0.•If 150-200q = 0 <=> q = 3/4 ==> p [0,1].

Consider the unconditional expected payoff of American:E[UA] = -50pq + 100q(1-p) + 0(1-q)p + 0(1-p)(1-q) = -150pq + 100q = q(100-150p)So American’s Best Response correspondence is:•If 100-150p > 0 <=> p < 2/3 ==> p=1.•If 100-150p < 0 <=> p > 2/3 ==> p=0.•If 100-150p = 0 <=> p = 2/3 ==> p [0,1].

Graph the BR correspondences (in p,q space) to find ALL NE.

Mixed StrategiesAsymmetric game

• Pure-strategy Nash equilibrium:

(Enter, Stay out)

(Stay out, Enter)

• Mixed-strategy Nash equilibrium:

(U, A) = ((2/3,1/3), (3/4,1/4))

Mixed Strategies

Definition: In a strategic game with vNM preferences, player i’s mixed strategy i strictly dominates her action a’i if

Ui(i, a-i) > ui(a’i, a-i) for every a-i

Mixed Strategies

3, .0, .B

0, .4, .M

1, .1, .TRL

Does this game have any dominated pure strategies? No, but if the row player mixes equally between M and B, then if the column player plays L, row gets 4(1/2)+0(1/2) = 2 if she mixes while just 1 if she plays T. If column plays R, row gets 0(1/2)+3(1/2) = 3/2 if she mixes, while again just 1 by playing T.

Thus T is strictly dominated by a mixed strategy.

Mixed Strategies

20, 1515, 103, 25B

15, 10 10, 1010, 15M

25, 320, 105, 5T

RCL

What are the NE (pure and mixed) of this game?

Method of finding all mixed-strategy Nash equilibrium

• For each player i, choose a subset Si of her set Ai of actions.

• Check whether there exists a mixed strategy profile such that (1) the set of actions to which each strategy i assigns positive probability is Si and (2) satisfies the conditions in proposition 116.2 in Osborne.

• Repeat the analysis for every collection of subsets of the players’ sets of actions

Mixed Strategies

1, 32, 40, 0S

0,10, 04, 2B

XSB

Mixed Strategies

• Potential types of equilibria:– 1) Player one plays 1 strategy, Player two plays 1 strategy.

• These are pure strategy NE.

– 2) Player one plays 1 strategy, Player two plays 2 strategies.• One plays a pure strategy, Two mixes on BS, BX, or SX

– 3) Player one plays 1 strategy, Player two plays 3 strategies.• One plays a pure strategy, Two mixes on BSX

– 4) Player one plays 2 strategies, Player two plays 1 strategy.• One mixes on BS, Two plays a pure strategy

– 5) Player one plays 2 strategies, Player two plays 2 strategies.• One mixes on BS, Two plays BS, BX, or SX

– 6) Player one plays 2 strategies, Player two plays 3 strategies.• One mixes on BS, Two mixes on BSX