3.1. Strategic Behavior Matilde Machado Slides available from: - - PDF document

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3.1. Strategic Behavior

Matilde Machado Slides available from:

http://www.eco.uc3m.es/mmachado/Teaching/OI-I-MEI/index.html

Industrial Economics - Matilde Machado 3.1. Strategic Behavior 2

3.1. Strategic Behavior

• The analysis of strategic behavior starts by

formulating a game.

• A game is made of:
• players
• Possible strategies for each player
• Utility functions for each player
• Set of rules such as: simultaneous,

sequential, who plays first;

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3.1. Strategic Behavior

Example: Strategies: player 1 = {A,B} player 2 = {C,D} Utilities: player 1: if he plays A =3 or 1; if B = 4,-2 player 2: if he plays C =0 or 2; if D =-1, 1

• 2,1

4,2 B 1,-1 3,0 A D C

Player 2 Player 1

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3.1. Strategic Behavior

Example (cont.): Rules of the game: each player chooses his strategy independently of the other. But of course, the outcome is a function of both players’ strategies. Therefore, there is an interdependence between their strategies, which is typical of game theory. Solution: The equilibrium concept that is mostly used is Nash equilibrium.

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3.1. Strategic Behavior

Nash equilibrium– a vector of strategies (one strategy for each player) is a Nash equilibrium if none of the players can increase his utility through a unilateral move (that is given the strategy of the other player). …. Back to the game…

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3.1. Strategic Behavior

player 2: If player 1 plays A → best response is C; u=0 If player 1 plays B → best response is C; u=2 ⇒ C is a dominant strategy to player 2 because it is always preferable to D player 1: If player 2 plays C → best response is B; u=4 If player 2 plays D (which he won’t) → best response is A; u=1 {B,C} is the only Nash equilibrium of the game.

• 2,1

4,2 B 1,-1 3,0 A D C

player 2 player 1

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3.1. Strategic Behavior

Formally, {B,C} is a Nash equilibrium because:

1 1 1 2 2 2

( , ) ( , ) or alternativelly ( , ) ( , ) ( , ) or alternativelly c ( , )

argmax argmax

i j

U b c U a c b U i c U b c U b d U b j ≥ = ≥ =

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3.1. Strategic Behavior

In Industrial Organization:

• Players are firms
• Strategies are going to be prices, quantities,

advertising, product quality, R&D, capacity, etc.

• Utilities are going to be profits

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3.1. Strategic Behavior

An example: Assume firms A and B are deciding whether or not to launch an advertising campaign. The campaign will cost 10 000 Euros. In case both firms launch the campaign, firm A gets all the benefits of the campaign and firm B will not benefit from the campaign although it will incur in costs. In case firm B launches the campaign alone, then its profits will increase. The payoffs

• f the game are the following:

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3.1. Strategic Behavior

100,100 100,110

NA

190,0 190,-10

A NA A

B A ¿What will firm A do? That depends on what firm B will do and vice- versa (note that both firms know the payoff matrix). Each firm wishes to maximize profit given what the other firm does and for that needs to take into account what the other firm does. In order to solve for the Nash equilibrium we need to construct the best response functions. RA(A)=A; RA(NA)=A (Advertising is dominant for A) and for B: RB(A)=NA; RB(NA)=A

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3.1. Strategic Behavior

Generalizing in IO

• The profit function is continuous in the strategies
• f the player and its rival’s.
• Strategies are going to be actions i.e. 1×1

vectors profit of firm i when it performs action ai and its rival action aj where ai∈Ai and aj∈Aj.

• The pair

is a Nash equilibrium iff:

( , )

i i j

a a Π ≡

* *

( , )

i j

a a

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3.1. Strategic Behavior

The pair is a Nash equilibrium iff: Or written a bit differently:

* * * * * *

arg max ( , ) ( , ) are best responses respectively arg max ( , )

i j

i i i j a i j j j i j a

a a a a a a a a ⎧ = Π ⎪ = ⎨ = Π ⎪ ⎩

* * * * * * * *

( , ) ( , ) ( , ) ( , )

i i i j i j i i i j j i j i j j j j

a a a a a a A a a a a a a A ⎧Π ≥ Π ∀ ≠ ∈ ⎪ ⎨Π ≥ Π ∀ ≠ ∈ ⎪ ⎩

* *

( , )

i j

a a

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3.1. Strategic Behavior

If there are interior solutions, then: Lets assume that Ai=Aj=ℜ and Πi and Πj are concave (i.e. Πiii<0 and Πjjj<0) for all values of ai and aj ⇒ the FOC’s are enough to derive the Nash equilibrium ( a system of 2 equations and 2 unknowns).

* * 2 * * * 2 * * 2 * * * 2

( , ) ( , ) satisfies 0; ( , ) ( , ) satisfies 0;

i i i j i j i i i j j i j i j j j j

a a a a a a a a a a a a a a ⎧ ∂Π ∂ Π = ≤ ⎪ ∂ ∂ ⎪ ⎨ ∂Π ∂ Π ⎪ = ≤ ⎪ ∂ ∂ ⎩

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3.1. Strategic Behavior

Reaction Function: Ri(aj) is the best response of firm i to

the action of firm j, aj. Ri(aj) is defined as: Then the Nash equilibrium may be defined as :

( )

( ) arg max ( , )

• r:

( ),

i

i i j i j a i i j j i

R a a a R a a a = Π ∂Π = ∂

* * * *

( ) ( )

i i j j j i

a R a a R a ⎧ = ⎪ ⎨ = ⎪ ⎩

Each firm reacts optimally to the action of its rival

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3.1. Strategic Behavior

Slope of the Reaction function: Let’s write the FOC as:

Totally differentiating it (note that it is a function of aj):

( ) ( )

1

( ), ( ),

i i i j j i j j i

R a a R a a a ∂Π = Π = ∂

( ) ( ) ( ) ( )

11 12 12 11

( ), ( ) ( ), ( ), ( ) ( ),

i i i j j i j i j j i i j j i j i i j j

R a a R a R a a R a a R a R a a ′ Π × + Π = Π ′ ⇔ = − Π

The slope of the Reaction function

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3.1. Strategic Behavior

Since Strategic Complements:

If Then that is the marginal benefit from action ai ⇒↑ai

( )

11 12

sign ( ) sign ( ),

i i i j i j j

R a R a a ⎡ ⎤ ′ ⎡ ⎤ Π < ⇒ = Π ⎣ ⎦ ⎣ ⎦

12 i

Π >

1 i j

a ↑ ⇒↑ Π

a2 a1 R2(a1) The reaction function has positive slope

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3.1. Strategic Behavior

To find the Nash equilibrium:

a2 a1 R2(a1) R1(a2) a*1 a*2 Note: prices are usually strategic complements (i.e. if my rival increases its price, I would also like to increase my price) while quantities are usually strategic substitutes (when the rival increases its quantity the best reply is to reduce our production)

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3.1. Strategic Behavior

Strategic Substitutes:

If then that is the marginal benefit

• f ai decreases ⇒↓ai i.e. the reaction function has

negative slope

12 i

Π <

1 i j

a ↑ ⇒↓ Π

a2 a1 R2(a1) R1(a2) a*1 a*2 E E=(a*1,a*2) is the Nash equilibrium

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3.1. Strategic Behavior

Note: In games where actions are simultaneous there is no literal scope for a “reaction” to the rivals’ actions since all the actions happen simultaneously, the equilibrium is attained immediately and therefore no reaction is

• possible. The reaction curve should then be interpreted

as the best response to the rival’s action. Situations different from E (the equilibrium) are never observed in

• reality. When the games are sequential then reaction in

the strict sense to the rival’s action is possible. Oligopoly: Here there are going to exist several firms, less than infinite, competing in a market. These firms are typically large enough to have some market power. The profits of firms will depend not only on their actions but also on their rivals’ actions.

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3.1. Strategic Behavior

Models of Strategic Behavior :

• Cournot model – quantity is the strategic

variable

• Bertrand model – price is the strategic variable
• Stackelberg model – it is the same as Cournot

but sequential

Note: In the standard Monopoly model we obtain the same result regardless of the choice variable of the Monopolist (price or quantity) this no longer holds for the Oligopoly models. The equilibrium depends crucially on the strategic variable.