Mixed Strategies
Overview
 Principles of mixed strategy equilibria
 Wars of attrition
 All-pay auctions
Tennis Anyone
R
S
Serving
R
S
Serving
R
S
The Game of Tennis
 Server chooses to serve either left or right
 Receiver defends either left or right
 Better chance to get a good return if you
defend in the area the server is serving to
Game Table
Receiver
Server
Left
Right
Left
¼
¾
Right
¾
¼
Game Table
Receiver
Left
Right
Left
¼
¾
Right
¾
¼
Server
For server:
For receiver:
Best response to defend left is to serve right
Best response to defend right is to serve left
Just the opposite
Nash Equilibrium
 Notice that there are no mutual best
responses in this game.
 This means there are no Nash equilibria in
pure strategies
 But games like this always have at least one
Nash equilibrium
 What are we missing?
Extended Game
 Suppose we allow each player to choose
randomizing strategies
 For example, the server might serve left half
the time and right half the time.
 In general, suppose the server serves left a
fraction p of the time
 What is the receiver’s best response?
Calculating Best Responses
 Clearly if p = 1, then the receiver should
defend to the left
 If p = 0, the receiver should defend to the
right.
 The expected payoff to the receiver is:


p x ¾ + (1 – p) x ¼ if defending left
p x ¼ + (1 – p) x ¾ if defending right
 Therefore, she should defend left if

p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾
When to Defend Left
 We said to defend left whenever:

p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾
 Rewriting

p>1–p
 Or

p>½
Receiver’s Best Response
Left
Right
½
p
Server’s Best Response
 Suppose that the receiver goes left with
probability q.
 Clearly, if q = 1, the server should serve right
 If q = 0, the server should serve left.
 More generally, serve left if

¼ x q + ¾ x (1 – q) > ¾ x q + ¼ x (1 – q)
 Simplifying, he should serve left if

q<½
Server’s Best Response
q
½
Right
Left
Putting Things Together
R’s best
response
q
S’s best
response
½
1/2
p
Equilibrium
R’s best
response
q
Mutual best responses
S’s best
response
½
1/2
p
Mixed Strategy Equilibrium
 A mixed strategy equilibrium is a pair of
mixed strategies that are mutual best
responses
 In the tennis example, this occurred when
each player chose a 50-50 mixture of left and
right.
General Properties of Mixed Strategy
Equilibria
 A player chooses his strategy so as to make
his rival indifferent
 A player earns the same expected payoff for
each pure strategy chosen with positive
probability
 Funny property: When a player’s own payoff
from a pure strategy goes up (or down), his
mixture does not change
Generalized Tennis
Receiver
Left
Right
Left
a, 1-a
b, 1-b
Right
c, 1-c
d, 1-d
Server
Suppose c > a, b > d
Suppose 1 – a > 1 – b, 1 - d > 1 – c
(equivalently: b > a, c > d)
Receiver’s Best Response
 Suppose the sender plays left with probability
p, then receiver should play left provided:

(1-a)p + (1-c)(1-p) > (1-b)p + (1-d)(1-p)
 Or:

p >= (c – d)/(c – d + b – a)
Sender’s Best Response
 Same exercise only where the receiver plays
left with probability q.
 The sender should serve left if

aq + b(1 – q) > cq + d(1 – q)
 Or:

q <= (b – d)/(b – d + a – b)
Equilibrium
 In equilibrium, both sides are indifferent
therefore:


p = (c – d)/(c – d + b – a)
q = (b – d)/(b – d + a – b)
Minmax Equilibrium
 Tennis is a constant sum game
 In such games, the mixed strategy
equilibrium is also a minmax strategy


That is, each player plays assuming his
opponent is out to mimimize his payoff (which
he is)
and therefore, the best response is to
maximize this minimum.
Does Game Theory Work?
 Walker and Wooders (2002)



Ten grand slam tennis finals
Coded serves as left or right
Determined who won each point
 Tests:

Equal probability of winning


Pass
Serial independence of choices

Fail
Battle of the Sexes
Chris
Pat
Opera
Fights
Opera
3,1
0,0
Fights
0,0
1,3
Hawk-Dove
Krushchev
Kennedy
Hawk
Dove
Hawk
0, 0
4, 1
Dove
1, 4
2, 2
Wars of Attrition
 Two sides are engaged in a costly conflict
 As long as neither side concedes, it costs
each side 1 per period
 Once one side concedes, the other wins a
prize worth V.

V is a common value and is commonly known
by both parties
 What advice can you give for this game?
Pure Strategy Equilibria
 Suppose that player 1 will concede after t1
periods and player 2 after t2 periods
 Where 0 < t1 < t2
 Is this an equilibrium?

No: 1 should concede immediately in that case
 This is true of any equilibrium of this type
More Pure Strategy Equilibria
 Suppose 1 concedes immediately
 Suppose 2 never concedes
 This is an equilibrium though 2’s strategy is
not credible
Symmetric Pure Strategy Equilibria
 Suppose 1 and 2 will concede at time t.
 Is this an equilibrium?


No – either can make more by waiting a split
second longer to concede
Or, if t is a really long time, better to concede
immediately
Symmetric Equilibrium
 There is a symmetric equilibrium in this
game, but it is in mixed strategies
 Suppose each party concedes with
probability p in each period
 For this to be an equilibrium, it must leave the
other side indifferent between conceding and
not
When to concede
 Suppose up to time t, no one has conceded:


If I concede now, I earn –t
If I wait a split second to concede, I earn:



V – t – e if my rival concedes
– t – e if not
Notice the –t term is irrelevant
 Indifference:


(V – e) x (f/(1 – F)) = - e x (1 – f/(1-F))
f/(1 – F) = 1/V
Hazard Rates
 The term f/(1 – F) is called the hazard rate of
a distribution
 In words, this is the probability that an event
will happen in the next moment given that it
has not happened up until that point
 Used a lot operations research to optimize
fail/repair rates on processes
Mixed Strategy Equilibrium
 The mixed strategy equilibrium says that the
distribution of the probability of concession for
each player has a constant hazard rate, 1/V
 There is only one distribution with this
“memoryless” property of hazard rates
 That is the exponential distribution.
 Therefore, we conclude that concessions will
come exponentially with parameter V.
Observations
 Exponential distributions have no upper
bound---in principle the war of attrition could
go on forever
 Conditional on the war lasting until time t, the
future expected duration of the war is exactly
as long as it was when the war started
 The larger are the stakes (V), the longer the
expected duration of the war
Economic Costs of Wars of Attrition
 The expectation of an exponential distribution
with parameter V is V.


Since both firms pay their bids, it would seem
that the economic costs of the war would be
2V
Twice the value of the item????
 But this neglects the fact that the winner only
has to pay until the loser concedes.
 One can show that the expected total cost if
equal to V.
Big Lesson
 There are no economic profits to be had in a
war of attrition with a symmetric rival.
 Look for the warning signs of wars of attrition
Wars of Attrition in Practice
 Patent races
 R&D races

Browser wars
 Costly negotiations
 Brinkmanship
All-Pay Auctions
 Next consider a situation where expenditures
must be decided up front
 No one gets back expenditures
 Biggest spender wins a price worth V.
 How much to spend?
Pure Strategies
 Suppose you project that your rival will spend
exactly b < V.

Then you should bid just a bit higher
 Suppose you expect your rival will bid b >=V

Then you should stay out of the auction
 But then it was not in the rival’s interest to bid
b >= v in the first place
 Therefore, there is no equilibrium in pure
strategies
Mixed Strategies
 Suppose that I expect my rival will bid
according to the distribution F.
 Then my expected payoffs when I bid B are

V x Pr(Win) – B
 I win when B > rival’s bid

That is, Pr(Win) = F(B)
Best Responding
 My expected payoff is then:

VF(B) – B
 Since I’m supposed to be indifferent over all
B, then


VF(B) – B = k
For some constant k>=0.
 This means

F(B) = (B + k)/V
Equilibrium Mixed Strategy
 Recall

F(B) = (B + k)/V
 For this to be a real randomization, we need it
to be zero at the bottom and 1 at the top.
 Zero at the bottom:

F(0) = k/V, which means k = 0
 One at the top:


F(B1) = B1/V = 1
So B1 = V
Putting Things Together
 F(B) = B/V on [0 , V].
 In words, this means that each side chooses
its bid with equal probability from 0 to V.
Properties of the All-Pay Auction
 The more valuable the prize, the higher the average
bid
 The more valuable the prize, the more diffuse the
bids
 More rivals leads to less aggressive bidding
 There is no economic surplus to firms competing in
this auction


Easy to see: Average bid = V/2
Two firms each pay their bid
 Therefore, expected payment = V, the total value of
the prize.
Big Lesson
 Wars of attrition and all-pay auctions are a
kind of disguised form of Bertrand
competition
 With equally matched opponents, they
compete away all the economic surplus from
the contest
 On the flipside, if selling an item or setting up
competition among suppliers, wars of attrition
and all-pay auctions are extremely attractive.