ECO 340: Micro Theory Intro to Game Theory

with applications to I/O

Sami DakhliaU. of Southern Mississippi

microprof@gmail.com

Game Theory

• 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

Confess Don’t confess

Confess (-10,-10) (0,-11)

Don’t Confess

(-11,0) (-1,-1)Player 1

Player 2

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

Heads Tails

Heads (-1,+1) (+1,-1)

Tails (+1,-1) (-1,+1)Player 1

Player 2

There is no NE in pure strategies. The unique NE is a so-called mixed strategy NE, where players randomize. Given these particular payoffseach player’s optimal strategy is to just flip the coin for a 50%probability of playing Heads.

Subgame-perfect NE

The centipede game

This is a sequential (or dynamic) game. The normal form no longerfully 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 knowledgeof rationality, it is appropriate to use backwards induction to solvethe game. The (sad) prediction is that Player 1 will Accept the first offer and walk away with $1.

A AAAAA

R R R R R R

A A

R1 1 1 12 2 2 2

(100,0) (102,0) (104,0) (106,0)(0,101) (0,103) (0,105) (0,107)

(5x106,5x106)R

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

Enter (q)

Don’t Enter (1-q)

Lowe’s

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-DeterrenceLowe’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).

OligopoliesPlease 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=a-bQ.

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)/2bPrice, finally, is p = a-bQ = a-b(a-c)/2b = (a+c)/2Please draw the graph in (Q,p) space.

OligopoliesIn 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

OligopoliesRepeat 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.

OligopoliesYou 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(a-c))/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).

OligopoliesOur 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

OligopoliesGiven 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

OligopoliesAs 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 microprof@gmail.com with any questions.