Competition and Strategy

Topic 4: Simultaneous Games

(Mixed, continuous strategies)1

David Byrne

Department of Economics

University of Melbourne

1

Recommended reading in DS: In 2nd edition, chapter 5 (123-137) for the first lecture, and chapters 7 and 8 (185-194, 206-216, 233-241, 243-250) for the second lecture. In 3rd edition, chapter 5 (133-141, 143-149) for the first lecture, and chapters 7 and 8 (213-222, 230-239, 262-268) for the second lecture 1 / 45

Continuous Strategies

Few games have truly continuous strategies, but many games allow players to choose from a wide range of strategies

Firms’ price choices

How much money to offer by the first player of the Ultimatum

Game

We cannot use game tables, at least not with a reasonable amount of columns and rows. We thus treat these games as games with continuous strategies as well.

To solve simultaneous games with continuous strategies, we first derive each player’s best response function.

A Nash Equilibrium occurs when each player is playing her best response, i.e., where the best response functions intersect. 2 / 45

Continuous Strategies

Example: Guessing half of the average

Consider a game called “Guessing half of the average”

Two players can choose any number between 0 and 100 including non-integers.

The two players’ guesses are X and Y ; the average is

X +Y

2

Goal: guess half the average of the two players’ guesses

Therefore, player 1’s optimal guess X should satisfy:

X =

1 X +Y

2

2

⇔

3X

Y

Y

=

⇔X =

4

4

3

Similarly, player 2’s best response function is Y =

X

3

3 / 45

Continuous Strategies

Example: Guessing half of the average

To find the Nash Equilibrium graphically, draw the two players’ best response curves:

Y

X best response function

Y best response function

0

0

X

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Continuous Strategies

Example: Guessing half of the average

The Nash Equilibrium can also be derived algebraically:

Insert player 2’s best response into player 1’s best response:

X =

X

X

Y

= 3 =

3

3

9

Hence, player 2’s equilibrium strategy must be X = 0

Insert player 1’s best response into player 2’s best response:

Y =

Y

X

Y

= 3 =

3

3

9

Hence, player 1’s equilibrium strategy must be Y = 0

QUESTION: What is the Nash Equilibrium when there are 5 players? 5 / 45

Continuous Strategies

Example: Price competition

Two restaurants on Lygon Street, Papa Gino’s and Corretto, need to set the prices for pizza.

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Continuous Strategies

Example: Price competition

Two restaurants on Lygon Street, Papa Gino’s (P) and

Corretto (C ), need to set the prices for pizza.

They are each getting menus printed simultaneously so are unaware of their competitor’s choice.

Cost of a pizza for each restaurant is c = $4.

Market research has shown that when Papa Gino’s charges price PP and Corretto’s charges price PC , the number of customers QP and QC are given by

QP = 25 − 2PP + PC

QC = 25 − 2PC + PP

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Continuous Strategies

Example: Price competition

What are Papa Gino’s and Corretto’s profit functions? πP = (PP − c) × QP = (PP − 4) × (25 − 2PP + PC )

πC = (PC − c) × QC = (PC − 4) × (25 − 2PC + PP )

What is Papa Gino’s best response function? That is, what price PP maximises Papa Gino’s profit for any given price PC ?

Need to maximize the continuous function πP wrt PP

Take first derivative of πP wrt PP , set equal to 0 and solve

∂πP

= 25 − 2PP + PC − 2(PP − 4) = 0

∂PP

Solving for Papa Gino’s best response function yields:

PP =

33 + PC

4

8 / 45

Continuous Strategies

Example: Price competition

By symmetry, Papa Gino’s and Corretto’s best response functions are:

PP =

33 + PC

;

4

PC =

33 + PP

4

Recall that the Nash Equilibrium occurs when all players play their best response, and that there are two ways to find it:

1. Graphically

2. Algebraically

9 / 45

Continuous Strategies

Example: Price competition, NE Graphically

To find the Nash Equilibrium graphically, draw the two best response curves.

Pc

P best response function

C best response function

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