Static Games of Complete
Information
.
Games and best responses
• Bob is a florist
• One bunch of flowers sells for $10, but to sell 100
bunches market price should be zero
• Thus, each extra bunch reduces market price by a
dime
• Cost is $2 a bunch
Games and best responses
.
$ 10
Price
$2
80
Quantity
100
Games and best responses
• What is Bob’s optimal quantity as a
monopolist?
• It turns out to be 40 units
• Suppose now Ann is contemplating entry
• How will the total quantity be allocated
between the two?
Games and best responses
• It depends on how they expect each other to act
• Suppose Ann is a sophisticated thinker and
responds optimally to Bob
• Ann’s best response to Bob is (80-QB)/2
• What should Bob do?
• Note:
Bob wants the total quantity to be the midpoint between
Ann’s quantity (QA) and the break-even point (80)
Games and best responses
•
Demand curve facing Bob is
$ 10
Price
$2
QA
(80+QA)/2
Bob
80
100
Quantity
• To maximize profits, Bob wants the total quantity to be the midpoint
between Ann’s quantity (QA) and the break-even point (80)
Games and best responses
• Four scenarios for Bob’s reasoning
Scenario
Description
Bob’s
conjecture
QB
QA
Naive
Ignores Ann (sticks to monopoly output)
None
40
20
Primitive
Reasons that Ann reacts to Bob’s previous
output
QA=20
30
25
Sophisticated
Reasons that Ann will deduce that Bob
considers Ann’s reaction to Bob’s
monopoly output
QA=25
Both Bob and Ann believe that the other
understands the logic of best response
QA=(80- QB)/2
Ultra sophisticated
27
1
2
26
2
3
26
1
4
26
2
3
What if Bob’s quantity is a given?
• How many bunches should Bob bring,
knowing how Ann would react to him?
• Now, total quantity is (80-QB)/2+QB=
(80+QB)/2
• Thus, mkt price is $10-$0.10{(80+QB)/2}
• Bob’s optimal quantity is 40; and so Ann’s
quantity is 20
• A significant first-mover advantage!
• Why does Bob produce monopoly quantity?
Strategic-Form Games
• Elements of strategic-form games:
- finite set of players, i  I ={1,2,…,I}
- pure-strategy space Si for each player i
- payoff functions ui(s) for each player i and for each
strategy profile s=(s1, s2,…, sI)
• Each player goal is to maximize his own payoff, and NOT
to ‘beat’ other players
2
ui ( s ) =0
• A Zero-Sum game is where 
i 1
• Most applications in the Social Sciences are non zero-sum
games
Mixed-Strategies
• A mixed-strategy σi is a probability distribution
over pure-strategies
• σi(si) is probability that σi assigns to si
• Space of player i’s mixed-strategies is ∑i
• Space of mixed-strategy profiles is ∑= i i with
elements σ
• Each player’s randomization is independent of
those of other players
I

 j ( s j ) ui ( s)
• Player i’s payoff to profile σ is sS  
j 1


Computing payoff from a mixedstrategy
• Player 1 plays along rows; 2 along columns
L
M
R
U
4, 3
5, 1
6, 2
M
2, 1
8, 4
3, 6
D
3, 0
9, 6
2, 8
• Let σ1={⅓, ⅓, ⅓} and σ2={0, ½, ½ }.
• What is player 1’s payoff?
Dominated Strategies
• Is there an obvious prediction about how the
previous game will be played?
• Iterated Strict Dominance predicts (U, L)
• Let “–i” denote all players other than i
• Then strategies of other players s-i  S-i and a
strategy profile is (si , s-i )
• Defn: Pure strategy si is strictly dominated for
player i if there exists  i/  ∑i such that,
ui ( i/ , si )  ui ( si , si ) for all s-i  S-i
Some implications of dominance
1. A mixed-strategy that assigns positive
probability to a dominated pure strategy,
is dominated
2. A mixed-strategy my be dominated even
though it assigns positive probability only
to undominated pure strategies.
•
Examples: -Prisoner’s Dilemma
-Second-Price Auction
Nash Equilibrium
• Defn: A mixed-strategy profile σ* is a Nash
Equilibrium if, for all players i,
ui ( i* ,  *i )  ui ( si ,  *i ) for all for all s-i  S-i
• Common examples in economics:
- Cournot quantity-setting game
- Bertrand price-setting game
Assumptions
• Rationality
• Players aim to maximize their payoffs
• Players are perfect calculators
• Common Knowledge
• Each player knows the rules of the game
• Each player knows that each player knows the
rules
• Each player knows that each player knows that each player
knows the rules
• Each player knows that each player knows that each player knows that each
player knows the rules
•
Each player knows that each player knows that each player knows that each player knows that each player knows the rules
• Etc. Etc. Etc.
Properties and implications
• Property 1: A player must be indifferent between
all pure strategies in the support of a mixedstrategy
• Property 2: One only needs to check for purestrategy deviations
• Property 3: If a single strategy survives iterated
deletion of strictly dominated strategies it is a
Nash Equilibrium
• Question: Why play mixed strategies when all
pure strategies in the support have same payoff?
Further comments on N.E.
• In many games pure strategy NE do not
exist
-example is game of “Matching Pennies”
• Sometimes there are multiple Nash
Equilibria
- “Battle of the Sexes”; “Chicken”
• Schelling’s theory of “focal points”
• Pre-play communication
Baseball anyone?
• 1986 baseball National League
championship series
• The New York Mets won a crucial game
against he Houston Astros
• Len Dykstra hit Dave Smith’s second pitch
for a two-run home run
• Later the two players talked about this
critical play
Analysis of a home run
• Dykstra said, “He threw me a fastball on the first
pitch and I fouled it off. I had a gut feeling then
that he’d throw me a forkball next, and he did. I
got a pitch I saw real well, and I hit it real well”.
• Smith said, “What it boils down to is that, it was
a bad pitch selection…if I had to do it over again,
it would be [another] fastball”.
• Would Dykstra not have been prepared for a
fastball?
• Again, randomization is the only way to go
But how do you randomize?
• In a game of tennis, suppose receiver’s forehand
is stronger than backhand
• Consider following probabilities of successfully
returning serve
Server’s Aim
Forehand Backhand
Receiver’s Move
Forehand
90%
20%
Backhand
30%
60%
Reducing receiver’s effectiveness
by randomizing
• Suppose server tosses a coin before each serve
• Aims to forehand or backhand according to coin turning
heads/tails
• When receiver moves to forehand, his successful return
rate is 55%
• When receiver moves to backhand, his successful return
rate is 45%
• Given server’s randomization, receiver should move to
forehand
• The server has already an improved outcome compared
to serving the same way all the time!!
What is server’s best mix?
• Consider following graph
Percentage successful returns
90
60
48
30
20
0
40
Percentage of times server aims serve to forehand
100
The mixing probabilities
• The 40:60 mixture of forehands to
backhands is the equilibrium
• This mixture is the only one that cannot be
exploited by the receiver to his own
advantage
• With this mixture the receiver does equally
well with either of his choices
• Both ensure the receiver a success rate of
48%
Nash bargaining
• Two players, N=1, 2, bargain over the partition of a
cake
• The set of Agreements, A, has elements
A={(a1, a2)єR2: ai≥0 for i=1, 2}
• The event of Disagreement is D
• Players have utilities ui, i=1, 2, s.t. ui: A {D}→ R
• Let, S={(u1(a), u2(a)) for a є A}, and d=(u1(D), u2(D))
• A Bargaining Problem B is the set of pairs <S, d>,
and the Bargaining Solution is a function f :B → R2
that assigns to each <S, d> єB an element of S
Nash’s axioms
1. INV (invariance to equiv utility representations):
If <S/, d/> is obtained from <S, d> by si   i si  i ,
then, fi (S/, d/)=  i f i ( S , d )  i
2. SYM (symmetry)
If <S, d> is symmetric, then f1(S, d)= f2(S, d)
3. PAR (Pareto efficiency)
Suppose <S, d> is a bargaining problem, with si ,
ti єS , and ti >si , for i=1,2, then f(S, d)≠s
4. IIA (independence of irrelevant alternatives)
If <S, d> and <T, d> are bargaining problems
with S  T and f(T, d) єS, then f(S, d) = f(T, d).
Nash bargaining solution
Theorem (Nash 1950)
There is a unique bargaining solution fN:B → R2
satisfying the above axioms given by
fN(S, d)= arg max (s1  d1 )( s2  d 2 )
( d1 , d 2 )( s1 , s2 )S
Application: Splitting a dollar
• Set of agreements is
A={(a1, a2)єR2: a1+ a2≤1 and ai≥0 for i=1, 2}
• Players are risk-averse, i.e. ui are concave
• Disagreement point is d=(u1(0), u2(0))=(0, 0)
• So <S, d> is a bargaining problem
Result 1: With equal risk-aversion, players are
symmetric & SYM, PAR give (u(1/2), u(1/2))
Role of risk-aversion
• Let player 2 be more risk-averse than player 1
• Let player 2’s utility be v2 =h u2, where h is concave, and v1
= u1
• Let <S/, d/> be bargaining problem with utilities vi
• The optimizing program for <S, d> gives solution zu where zu
solves:
max u1 ( z )u 2 (1  z )
0 z 1
• The optimizing program for <S/, d/> gives solution zv where
zv solves:
max v1 ( z )v2 (1  z )
0 z 1
Role of risk-aversion
• The first program gives,
u1/ ( z ) u2/ (1  z )

u1 ( z ) u2 (1  z )
• The second program gives,
u1/ ( z ) h / [u2 (1  z )]u2/ (1  z )

u1 ( z )
h[u2 (1  z )]
• Result 2: If player 2 becomes more risk-averse,
then Player 1’s share of the dollar in the Nash
solution increases.
Existence of NE
Theorem (Nash 1950)
Every finite strategic-form game has a mixedstrategy equilibrium.
Sketch of Proof:
1. Use players’ Reaction Correspondences r(σ)
2. Realize that a NE is a Fixed Point of r(σ)
3. Show that conditions for Kakutani’s Fixed
Point Theorem are satisfied in this case
Existence of NE (cont)
Theorem (Shizuo Kakutani 1941)
If :
(i) ∑  R n is compact and convex
(ii) A correspondence r(σ) :∑→∑ is non-empty and
convex
(iii) r(σ) has a closed graph
Then r(σ) has a fixed point, that is, there exists σ*  
such that, r(σ*)= σ*
Nash equilibrium is a fixed point
• For any strategy profile σ, player i’s reaction
correspondence ri maps σ to the mixed strategy that
maximizes his payoff, given his opponents play σ-i . Thus,
ri (σ)= ri (σi , σ-i ) = arg max ui(σ/i , σ-i )
 i/  i
• Each player i=1,2,..n, has a reaction function
• Let us form the n-tuple, r(σ)=(r1 (σ), r2 (σ),…, rn(σ)), where
r(σ)є∑
• Suppose there exists σ* є∑ such that, for all players, σ*i
=ri(σ*)= ri(σ*i , σ*-i), then σ* is a Nash equilibrium
• But σ*i =ri(σ*) for all i, means that σ*=r(σ*), i.e. σ* is a fixed
point of the reaction correspondence r(.)
Applying Kakutani
• ∑ is compact and convex:
1. Mixed strategies convexify the strategy space, and
so each ∑i is a convex space of dimension (#Si-1)
2. Theorem (Heine-Borel):
Closed & bounded subsets in Rn are compact
• r(σ) is non-empty:
1. Theorem (Weierstrass):
A continuous real-valued function defined on a
compact space achieves its maximum values
2. Player’s i’s utility functions are continuous in σi
Applying Kakutani
• r(σ) is convex:
Suppose σ/, σ// є r(σ) . ui is linear, therefore for λє(0, 1)
ui(λσi/+ (1- λ)σi//, σ-i)= λ ui(σi/, σ-i) + (1- λ)ui(σi//, σ-i). Thus, if
σi/, σi// are best responses to σ-i, then so is their weighted
average. Thus, λσi/+ (1- λ)σi// є r(σ)
• r(σ) has a closed graph:
n
n
If ( , ˆ )  ( , ˆ ) with ˆ n  r ( n ) , then ˆ  r ( )