Market Games
In monopoly and perfectly competitive markets there is no strategic
interaction
Microeconomics II
In oligopolistic markets there is strategic interaction
Strategic Form Games: Applications
◮
there are finitely many firms each with a significant influence on market
price and/or how much the other firms sell
We have seen one example, Cournot Duopoly, in which firms compete by
choosing how much to produce
Levent Koçkesen
Firms competing in markets have other strategic choices, such as
Koç University
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price
advertising
product design
We will look at some commonly used models of market games, starting
with a recap of the Cournot model
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Cournot Oligopoly Model
In any Nash equilibrium q∗ ∈ S of the Cournot oligopoly game:
qi ≥ 0, i = 1, 2, . . . , n
Given the total output in the market, price adjusts to clear the market
Inverse (market) demand function: p(Q)
◮ Q = ∑n
i=1 qi
◮ p : R+ → R+
◮ p′ < 0
Cost function of firm i: ci (qi )
◮ c i : R+ → R+
◮ c′ > 0, c′′ ≥ 0
i
i
p(Q∗ ) > c′i (q∗i ),
if q∗i > 0
p′ (Q∗ )q∗i + p(Q∗ ) − c′i (q∗i ) ≤ 0,
with equality if q∗i > 0
Q∗ = 0 implies that p(0) − c′i (0) ≤ 0, for all i, contradicting our assumption that
p(0) > c′i (0), for some i. Now suppose that q∗i > 0. First order condition and
the assumption that p′ < 0 immediately implies p(Q∗ ) > c′i (q∗i ).
N = {1, 2, . . . , n}
Si = R+
ui (q) = p(Q)qi − ci (qi ) for each q ∈ S
Applications
Q∗ > 0
2
Let q∗ = (q∗1 , q∗2 , . . . , q∗n ) be a Nash equilibrium. Then the following first order
condition must hold for each i:
p(0) > c′i (0), for some i
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1
Proof.
The strategic form is given by
3
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Proposition
Firms independently and simultaneously choose how much to produce:
2
Applications
Cournot Oligopoly Model
There are n ≥ 2 firms producing a homogenous product
1
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Levent Koçkesen (Koç University)
Applications
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Cournot Oligopoly with Linear Demand and Cost Functions
Claim
Best response correspondence of firm i = 1, 2, . . . , n is given by
Let
p(Q) =
(
a − bQ, Q ≤ a/b
0,
Q > a/b
Bi (qi ) =
and for each i = 1, 2, . . . , n
( a−c−b
∑j6=i qj
,
2b
0,
∑j6=i qj <
∑j6=i qj ≥
a−c
b
a−c
b
There is a unique, symmetric Nash equilibrium in which
ci (qi ) = cqi
q∗i =
where a > c ≥ 0 and b > 0
Therefore, payoff function of firm i = 1, 2 is given by
a−c
,
(n + 1)b
i = 1, 2, . . . , n
Proof.
(
(a − c − bQ)qi , Q ≤ a/b
ui (q) =
−cqi ,
Q > a/b
Exercise
Exercise
Find the set of rationalizable strategies for n = 2 and n = 3.
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Price Competition Models
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Bertrand Duopoly with Homogeneous Products
There are two firms in a market producing a homogenous product
Market demand function is given by Q(p)
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1
Bertrand Oligopoly with Homogeneous Products
2
Bertrand Oligopoly with Differentiated Products
Continuous and strictly decreasing
Firms independently and simultaneously choose prices: pi ≥ 0, i = 1, 2
Known as Bertrand models, after Joseph Bertrand (1883)
Two main models:
The one with the lower price captures the entire market
In case of a tie they share the market equally
qi (pi , pj ) =


if pi < pj ,
Q(pi ),
1
2 Q(pi ), if pi = pj ,


0,
if pi > pj .
Cost function of firm i: ci (qi ) = cqi , with c ≥ 0
◮ There exists a p > c such that Q(p) > 0
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Applications
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Levent Koçkesen (Koç University)
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Bertrand Duopoly with Homogeneous Products
Proposition
There is a unique Nash equilibrium in which p∗1 = p∗2 = c.
Strategic form of the game:
Proof.
p∗1 , p∗2 = c is a Nash equilibrium: easy.
1
N = {1, 2}
2
S1 = S2 = R+
3
Payoff function of player i = 1, 2: For any (p1 , p2 ) ∈ R2+
Let (p∗1 , p∗2 ) be a Nash equilibrium.
p∗1 , p∗2 ≥ c: otherwise at least one is getting negative payoff, which could
be avoided by choosing a price equal to c.


(pi − c)Q(pi ), if pi < pj ,
ui (p1 , p2 ) = 21 (pi − c)Q(pi ), if pi = pj ,


0,
if pi > pj .
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Applications
p∗i > p∗j ≥ c cannot be a Nash equilibrium:
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p∗j = c ⇒ firm j could increase its price a little and increase its payoff (since
Q(c) > 0)
p∗j > c ⇒ then firm i could increase its payoff by choosing a price between
c and p∗j .
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Applications
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Bertrand Duopoly with Differentiated Products
Firms’ products are not perfect substitutes
Demand function for firm i is given by: qi (pi , p−i )
Proof contd.
p∗i = p∗j > c cannot be an equilibrium:
◮
Q(p∗i ) > 0 ⇒ firm i can set its price equal to
Example:
3p∗i
4
+ 4c and increase its payoff:
There are two firms. Firm i 6= j’s demand function is given by:
3p∗ c
3p∗ c
1
( i + − c)Q( i + ) > (p∗i − c)Q(p∗i )
4
4
4
4
2
◮
qi (p1 , p2 ) =
Q(p∗i ) = 0 ⇒ firm i can set its price equal to p and obtain positive profit.
(
a − bpi + pj , a − bpi + pj ≥ 0
0,
otherwise
Assume that b > 1
Therefore, it must be that p∗i = p∗j = c, completing the proof.
Cost function of firm i is cqi
Exercise
Formulate as a strategic form game and find its Nash equilibria.
There are several models of product differentiation and pricing that lead to
particular demand functions
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Firm 1 chooses price p1 and Firm 2 chooses p2
The Linear City Model
Consumer x receives a payoff of
Continuum of consumers (of measure one) are located uniformly along a
linear city of length one
v(0, x) = u − t(0 − x)2 − p1 ,
v(1, x) = u − t(1 − x)2 − p2 ,
if she purchases from Firm 1
if she purchases from Firm 2
zero,
if she does not purchase
Let
be the consumer who is indifferent between purchasing from Firm 1
and Firm 2
Location of a consumer indexed by x ∈ [0, 1].
x∗
Firms are located along [0, 1] interval as well, each producing a
homogenous product at unit cost c > 0.
x∗ =
Each consumer has unit demand. Consumer x who purchases from a firm
located at s receives a benefit equal to
v(s, x) = u − t(s − x)2
where t > 0.
Buy from Firm 1
Suppose there are only two firms and they are already located at each
end of the city: Firm 1 at 0 and Firm 2 at 1.
Firm 1
0
Firm 2
0
Applications

1,
pj −pi +t
2t ,


0,
pi < pj − t
pi ∈ [pj − t, pj + t]
pi > pj + t
pj − pi + t
2t
pi ≥ 0
Applications
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The best response correspondence of Firm i 6= j is given by


pj + t,
pj +t+c
2 ,


pj − t,
pj ≤ c − t
pj ∈ (c − t, c + 3t)
pj ≥ c + 3t
Fix a pj and let p∗i be a best response of firm i to pj . There are three
possibilities:
s.t. pi ∈ [pj − t, pj + t]
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Applications
Claim
Proof
Therefore, firm i’s best response correspondence can be obtained by
solving
pi
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Bi (pj ) =
We can limit best responses for Firm i to interval [pj − t, pj + t]
◮ if pi < pj − t, can increase price and payoff;
◮ if pi > pj + t, can lower price and payoff remains same or increases (the
latter if pj > c − t).
max (pi − c)
1
We assume that u > c + 1.25t so that this does not happen in equilibrium
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For prices at which everybody purchases, demand function for Firm i 6= j
is given by

qi (p1 , p2 ) =
Buy from Firm 2
x∗
It is possible that some consumers choose not to buy at all.
1
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p2 − p1 + t
2t
if x∗ has a non-negative payoff, then
◮ all customers with x < x∗ buy from Firm 1
◮ all customers with x > x∗ buy from Firm 2
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1
p∗i ∈ (pj − t, pj + t)
2
p∗i = pj − t → ui = pj − t − c
3
p∗i = pj + t → ui = 0
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Proof cont.
If pj ≤ c − t, then we know that the solution cannot be in the interior.
Comparing the payoffs at the boundaries, we have that p∗i = pj + t. Similarly, if
pj ≥ c + 3t, the solution must be at one of the boundaries. Comparing the
payoffs, we get p∗i = pj − t.
Proof cont.
If p∗i ∈ (pj − t, pj + t), then the following first order condition must hold:
c + t + pj − 2p∗i = 0
which is solved at
pj + t + c
p∗i =
2
p∗i ∈ (pj − t, pj + t) implies that pj ∈ (c − t, c + 3t). Conversely, let us assume
that pj ∈ (c − t, c + 3t). Then,
ui
pj + t + c
, pj
2
Therefore,
=
pj +t+c
2
Applications
The unique Nash equilibrium is given by p∗1 = p∗2 = c + t
Exercise
Find the Nash equilibria without the assumption u > c + 1.25t.
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Models of Spatial Competition
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Let’s assume the distribution is uniform
Each voter θ has single peaked preferences %θ over the policy space
x %θ y ⇔ |x − θ| ≤ |y − θ|,
tax rates
product characteristics
environmental regulations
∀(x, y) ∈ P2 and θ ∈ Θ
Candidates announce a policy in P
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Firms choose product characteristic and care about how much they sell
Political parties choose policy positions and care about whether they win
the election and which position gets implemented
Applications
10% tax rate vs. 25% tax rate
pro-EU vs anti-EU
Voters vote sincerely, i.e., voter θ votes for the candidate whose policy is
closest to θ
Players have preferences over P and the choices of the individuals
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Applied to electoral competition by Downs (1957)
P = [0, 1]:
Given players’ choices, each individual chooses a player
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There is a continuum of voters distributed over the type set Θ = [0, 1]
Players (firms, political parties, etc.) choose strategies in P
◮
Applications
Originally due to Hotelling (1929)
income
taste
political inclination
Individuals have preferences (which may depend on their types) over
some outcome set P
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Electoral Competition: Downs Model
There is a population of individuals (consumers, voters, etc.)
characterized by some trait (type) in a set Θ, e.g.,
◮
λ1 (pj − t − pi ) = 0
Claim
if, and only if, pj ∈ (c − t, c + 3t).
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c + t + pj − 2pi + λ1 − λ2 = 0
λ2 (pi − pj − t) = 0
(pj + t − c)2
=
8t
>0
> pj − t − c
p∗i
We could alternatively derive the best response correspondence using the
Kuhn-Tucker conditions: λ1 , λ2 ≥ 0 and
if indifferent they vote for each with equal probabilities
Candidate with the highest amount of votes wins
Candidates care only about the outcome of the election
◮ preferences: win ≻ tie ≻ lose
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Nash Equilibrium
Strategic Form of the Game
Suppose p∗1 , p∗2 is a Nash equilibrium. Then
1
N = {1, 2}
2
S1 = S2 = [0, 1]
3
For any (p1 , p2 ) ∈ [0, 1]2
1
Outcome must be a tie
2
p∗1 6= p∗2 ?
3
p∗1 = p∗2 6= 1/2?
◮


1, if i wins at (p1 , p2 )
ui (p1 , p2 ) = 21 , if there is a tie at (p1 , p2 )


0, if i loses at (p1 , p2 )
Whatever your opponent chooses you can always guarantee a tie
0
0
p1
0
Vote for 2
p1 +p2
2
p2
1/2
p2
1
p2
Say the two candidates choose 0 < p1 < p2 < 1
Vote for 1
p1
p1
1
1/2
The only candidate for equilibrium is p∗1 = p∗2 = 1/2, which is indeed an
equilibrium.
1
Proposition
The unique Nash equilibrium of the game is (p∗1 , p∗2 ) = (1/2, 1/2)
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Extensions
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Ideological Candidates: Wittman Model
Exercise (Median Voter Theorem)
Consider the above model but assume that the voters are distributed over [0, 1]
according to a distribution function F that has a continuous density function f
with f (x) > 0 for all x ∈ [0, 1]. Define the median of the distribution F as
m = F −1 (1/2).
1
Show that the unique Nash equilibrium of the game is (m, m).
2
Is (m, m) also a weakly dominant strategy equilibrium?
Downs model assumes candidates care only about winning
But politicians themselves may care about the implemented policy
Candidate i has an ideal policy θ ∈ Θ and payoff function:
ui (x, θi ) = −|x − θi |,
∀x ∈ P
For any p ∈ [0, 1]n , let πi (p) be the probability that candidate i wins.
We assume that candidate i with type θi maximizes her expected payoff:
Other Models
n
Models with participation costs
ui (p, θi ) = ∑ πj (p)(−|pj − θi |)
Models with more than two players
j=1
Models with multidimensional policy space
Models with ideological candidates: Wittman (1973) Model
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Wittman Model: Example
Wittman Model
Two candidates: θ1 = 0, θ2 = 1
The form of the function πi depends on other modeling choices such as the
amount of information candidates are assumed to have:
Voters are distributed over [0, 1] according to F
◮ has a continuous density function f with f (x) > 0 for all x ∈ [0, 1]
◮ Median of F : m ∈ (0, 1)
Strategic Form
1
Certainty: Candidates know F and hence m
2
Uncertainty: Candidates are uncertain about F .
Exercise
1
N = {1, 2}
2
S1 = S2 = [0, 1]
3
Payoff functions:
u1 (p1 , p2 ) = π1 (p)(−|p1 |) + π2 (p))(−|p2 |)
u2 (p1 , p2 ) = π1 (p)(−|p1 − 1|) + π2 (p))(−|p2 − 1|)
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1
Find the set of Nash equilibria of the Wittman model under certainty.
2
Now suppose candidates are uncertain about F . In particular, they believe
that the median is uniformly distributed over the interval [µ − a, µ + a],
where µ − a > 0 and µ + a < 1. Find the set of Nash equilibria of this
game.
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