ECO 340: Micro Theory
Intro to Game Theory
with applications to I/O
Sami Dakhlia
U. of Southern Mississippi
[email protected]
Game Theory
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What is a game?
Some examples
Mixed Strategy Nash Equilibrium
Subgame-Perfect Nash Equilibrium
Applications to Industrial Organization
– Entry Deterrence
– Oligopolies
What is a game?
In its so-called “normal form”, a game
consists of three elements:
– A set of players
– A set of strategies for each player
– A payoff function for every possible
outcome
Some examples
The prisoner’s dilemma
Player 2
Player 1
Confess
Don’t
confess
Confess
(-10,-10)
(0,-11)
Don’t
Confess
(-11,0)
(-1,-1)
The Nash Eq. (NE) is (confess, confess), regardless of guilt.
This, you will note, is not Pareto efficient.
Question: how does the Mafia deal with this dilemma?
Some examples
Matching Pennies
Player 2
Player 1
Heads
Tails
Heads
(-1,+1)
(+1,-1)
Tails
(+1,-1)
(-1,+1)
There is no NE in pure strategies.
The unique NE is a so-called mixed strategy NE,
where players randomize. Given these particular payoffs
each player’s optimal strategy is to just flip the coin for a 50%
probability of playing Heads.
Subgame-perfect NE
The centipede game
1
A
R
2
A
(100,0) (0,101)
R
1
A
(102,0)
R
2
A
(0,103)
R
1
A
(104,0)
R
2
A
(0,105)
R
1
A
(106,0)
R
2
R
(5x106,5x106)
A
(0,107)
This is a sequential (or dynamic) game. The normal form no longer
fully captures the game, which is why it is now represented in its
full arborescence. This is known as the extensive form of the game.
Given our assumptions of full rationality and common knowledge
of rationality, it is appropriate to use backwards induction to solve
the game. The (sad) prediction is that Player 1 will Accept the first
offer and walk away with $1.
Applications to I/O
• Entry-Deterrence
• Oligopolies
Entry-Deterrence
Suppose that Lowe’s is the sole supermarket in a
particular regional market and that Home Depot has
plans to build a competing store. Of course, Lowe’s
does not like this idea one bit and promises to be a
most aggressive competitor should Home Depot
proceed with their plan, indeed so aggressive that
Home Depot’s investment would likely yield
insufficient returns for a positive profit. The game is
thus as follows: Home Depot can choose to enter the
market, after which Lowe’s can choose whether to
follow up on their threat or whether to simply share
the market.
Entry-Deterrence
To find the NE, it is sufficient to look at the game’s normal form:
Home Depot
Lowe’s
Enter
(q)
Don’t Enter
(1-q)
Fight
(p)
(2,-1)
(10,0)
Don’t Fight
(1-p)
(4,4)
(10,0)
Clearly, there are two NE in pure strategies: (DF,E) and (F,DE).
Since Nash Equilibria generically come in odd numbers, an
equilibrium in mixed strategies is lurking here as well.
Entry-Deterrence
Lowe’s will fight for sure (p=1) if the expected payoff of doing so
exceeds the expected payoff of not fighting: 2q + 10(1-q) > 4 q +
10(1-q), that is, if q<0, which is not possible. On the other hand,
Lowe’s is indifferent between F and DF when q=0.
Home Depot will enter if -p+4(1-p) ≥ 0p+0(1-p), that is, p≤4/5.
Please plot the Best-Response correspondences in (p,q) space.
As you can see, there is actually a whole continuum of mixed
strategy Nash equilibria.
So, of all these equilibria, which one is most likely? We can answer
this by recalling that our game is a sequential one and that we can
thus use the concept of subgame-perfection to narrow down our
selection. Go ahead and draw the extensive form of the game and
use backwards induction. The only SPNE is (DF,E).
Oligopolies
Please recall the canonical model of profit maximizing monopoly:
A firm produces output Q under constant marginal cost (MC) c.
Consumer behavior is described by the inverse demand function p=abQ.
The objective function is ∏ = Revenue - Cost = pQ-cQ and Q is the
firm’s choice variable.
The first-order condition for an optimum is d∏/dQ = 0. Read this as
marginal profit must be equal to 0. By the same token, marginal
revenue must be equal to marginal cost: MR=MC.
Since Revenue = pQ = (a-bQ)Q = aQ-bQ2, MR = a-2bQ.
Since Cost = cQ, MC = c.
MR=MC implies a-2bQ=c, that is, Q = (a-c)/2b
Price, finally, is p = a-bQ = a-b(a-c)/2b = (a+c)/2
Please draw the graph in (Q,p) space.
Oligopolies
In the case of a Cournot Duopoly, the model allows for a second
firm producing the same good. The first firm produces quantity Q1
and the second firm produces quantity Q2. Thus, total output
Q=Q1+Q2. Both firms operate under constant marginal cost (MC)
c1=c2=c.
Consumer behavior continues to be described by the inverse
demand function p=a-bQ.
The objective function of firm 1 is ∏1 = pQ1-cQ1 and Q1 is the
firm’s choice variable.
Since Revenue = pQ1 = (a-b(Q1+Q2))Q1 = aQ1-bQ12-bQ1Q2,
MR1 = a-2bQ1-bQ2.
MR1=MC1 implies a-2bQ1-bQ2 =c, that is,
Q1 = (a-c)/2b - 0.5Q2
Oligopolies
Repeat the same exercise for firm 2 and you should obtain
Q2 = (a-c)/2b - 0.5Q1
Please keep in mind that this is a game, where the set of players
is {Firm1, Firm2}, the set of strategies for Firmi is + (Qi  [0,))
and that the payoff vectors are given by the profit functions.
Please plot the Best-Response correspondences in (Q1,Q2)
space, while keeping in mind that firms cannot produce negative
outputs.
The intersection of the two curves defines the Cournot-Nash
equilibrium.
Oligopolies
You can determine this equilibrium algebraically, by
simultaneously solving both equations Q1 = (a-c)/2b - 0.5Q2 and
Q2 = (a-c)/2b - 0.5Q1
If all goes well, you should get Q1=Q2=(a-c)/3b, thus Q=(2(ac))/3b and p=(1/3)a+(2/3)c.
As an exercise, please also calculate the firms’ profits at the
Cournot NE. Show that this is indeed a NE in the sense that no
firm has an incentive to produce more or less. Start with a
numerical example if it makes your life easier (say a=100, b=1,
c=20).
Oligopolies
Our final application is the Stackelberg Duopoly, an extension of
the Cournot Duopoly to a dynamic two-period framework. I.e.,
the two firms choose their strategies (Qi) sequentially, with Firm1
moving first (the leader) and Firm2 moving last (the follower).
Using backwards induction, we first solve the problem for Firm2.
Using the same approach as in the Cournot game, we find that
Q2 = (a-c)/2b - 0.5Q1
Oligopolies
Given common knowledge of payoffs and common knowledge of
rationality, Firm1 correctly anticipates Firm2’s best response and
can embed it into its own profit maximization:
∏1 = pQ1-cQ1 = (a-c-bQ)Q1 = (a-c-b(Q1+Q2))Q1
= (a-c-b(Q1+ (a-c)/2b - 0.5Q1))Q1
= (a-c- bQ1- (a-c)/2 + 0.5bQ1)Q1
= ((a-c)/2 - 0.5bQ1)Q1
The first-order condition for an optimum is thus ((a-c)/2 - 0.5bQ1)
-0.5bQ1=0, that is:
Q1=(a-c)/2b
We can now plug this result back into Firm2’s best response to
get Q2=(a-c)/4b
Oligopolies
As an exercise, please calculate the firms’ profits.
And be sure to understand the paradox of first-mover
advantage! (The follower has the luxury of waiting and
observing the leader’s strategy and can then optimize;
nevertheless, the leader makes twice as much profit. How
come?)
Can you think of other “games” where the first mover has an
advantage? Can you think of games where it is better to move
last?
Please remember to email me at [email protected] with
any questions.