3.1. Strategic Behavior Matilde Machado Slides available from: http://www.eco.uc3m.es/mmachado/Teaching/OI-I-MEI/index.html 1 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; Industrial Economics - Matilde Machado 3.1. Strategic Behavior 2 1
3.1. Strategic Behavior Example: Player 2 C D A 3,0 1,-1 Player 1 B 4,2 -2,1 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 Industrial Economics - Matilde Machado 3.1. Strategic Behavior 3 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. Industrial Economics - Matilde Machado 3.1. Strategic Behavior 4 2
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… Industrial Economics - Matilde Machado 3.1. Strategic Behavior 5 3.1. Strategic Behavior player 2 C D player 1 A 3,0 1,-1 B 4,2 -2,1 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. Industrial Economics - Matilde Machado 3.1. Strategic Behavior 6 3
3.1. Strategic Behavior Formally, {B,C} is a Nash equilibrium because: ≥ = argmax U b c ( , ) U a c ( , ) or alternativelly b U i c ( , ) 1 1 1 i ≥ = argmax U ( , ) b c U ( , ) or alternativelly c b d U ( , ) b j 2 2 2 j Industrial Economics - Matilde Machado 3.1. Strategic Behavior 7 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 � Industrial Economics - Matilde Machado 3.1. Strategic Behavior 8 4
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 of the game are the following: Industrial Economics - Matilde Machado 3.1. Strategic Behavior 9 3.1. Strategic Behavior A NA B A A 190,-10 190,0 NA 100,110 100,100 ¿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. R A (A)=A; R A (NA)=A (Advertising is dominant for A) and for B: R B (A)=NA ; R B (NA)=A Industrial Economics - Matilde Machado 3.1. Strategic Behavior 10 5
3.1. Strategic Behavior Generalizing in IO The profit function is continuous in the strategies � of the player and its rival’s. Strategies are going to be actions i.e. 1×1 � vectors Π ≡ profit of firm i when it performs action a i i ( , a a ) i j and its rival action a j where a i ∈ A i and a j ∈ A j . The pair is a Nash equilibrium iff: * * � ( a a , ) i j Industrial Economics - Matilde Machado 3.1. Strategic Behavior 11 3.1. Strategic Behavior The pair is a Nash equilibrium iff: * * ( a a , ) i j ⎧Π ≥ Π ∀ ≠ ∈ i * * i * * ⎪ ( a a , ) ( , a a ) a a A i j i j i i i ⎨Π ≥ Π ∀ ≠ ∈ j * * j * * ⎪ ( a a , ) ( a a , ) a a A ⎩ i j i j j j j Or written a bit differently: ⎧ = Π * i * a arg max ( , a a ) ⎪ i i j a = ⎨ i * * ( a a , ) are best responses respectively = Π i j * j * ⎪ a arg max ( a a , ) j i j ⎩ a j Industrial Economics - Matilde Machado 3.1. Strategic Behavior 12 6
3.1. Strategic Behavior If there are interior solutions, then: ⎧ ∂Π ∂ Π i * * 2 i * * ( a a , ) ( a a , ) = ≤ ⎪ * i j i j a satisfies 0; 0 ∂ ∂ i 2 ⎪ a a i i ⎨ ∂Π ∂ Π j * * 2 j * * ⎪ ( a a , ) ( a a , ) = ≤ i j i j * a satisfies 0; 0 ⎪ ∂ ∂ j 2 a a ⎩ j j Lets assume that A i =A j = ℜ and Π i and Π j are concave (i.e. Π iii <0 and Π jjj <0) for all values of a i and a j ⇒ the FOC’s are enough to derive the Nash equilibrium ( a system of 2 equations and 2 unknowns). Industrial Economics - Matilde Machado 3.1. Strategic Behavior 13 3.1. Strategic Behavior Reaction Function: R i (a j ) is the best response of firm i to the action of firm j, a j . R i (a j ) is defined as: = Π i R a ( ) arg max ( , a a ) i j i j a i or: ∂Π i ( ) = R a ( ), a 0 ∂ i j j a i Then the Nash equilibrium may be defined as : ⎧ = * * ⎪ a R a ( ) Each firm reacts optimally to the action of its rival i i j ⎨ = * * ⎪ a R a ( ) ⎩ j j i Industrial Economics - Matilde Machado 3.1. Strategic Behavior 14 7
3.1. Strategic Behavior Slope of the Reaction function: Let’s write the FOC as : ∂Π i ( ) ( ) = Π = i R a ( ), a R a ( ), a 0 ∂ i j j 1 i j j a i Totally differentiating it (note that it is a function of a j ): ( ) ( ) Π × ′ + Π = i i R a ( ), a R a ( ) R a ( ), a 0 11 i j j i j 12 i j j ( ) Π i R a ( ), a ′ The slope of the ⇔ = − 12 i j j R a ( ) ( ) Reaction Π i j i R a ( ), a function 11 i j j Industrial Economics - Matilde Machado 3.1. Strategic Behavior 15 3.1. Strategic Behavior Since ( ) ′ ⎡ ⎤ Π < ⇒ ⎡ ⎤ = Π i i 0 sign ⎣ R a ( ) ⎦ sign R a ( ), a ⎣ ⎦ 11 i j 12 i j j Strategic Complements: Π > ↑ ⇒↑ Π If i Then i that is the marginal benefit 0 a 12 j 1 from action a i ⇒ ↑ a i a 2 R 2 (a 1 ) The reaction function has positive slope a 1 Industrial Economics - Matilde Machado 3.1. Strategic Behavior 16 8
3.1. Strategic Behavior To find the Nash equilibrium: R 1 (a 2 ) R 2 (a 1 ) a 2 a* 2 a* 1 a 1 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) Industrial Economics - Matilde Machado 3.1. Strategic Behavior 17 3.1. Strategic Behavior Strategic Substitutes: If Π < then ↑ ⇒↓ Π that is the marginal benefit i i a 0 j 1 12 of a i decreases ⇒ ↓ a i i.e. the reaction function has negative slope a 2 R 1 (a 2 ) E=(a* 1 ,a* 2 ) is the Nash equilibrium a* 2 E R 2 (a 1 ) a* 1 a 1 Industrial Economics - Matilde Machado 3.1. Strategic Behavior 18 9
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. Industrial Economics - Matilde Machado 3.1. Strategic Behavior 19 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. Industrial Economics - Matilde Machado 3.1. Strategic Behavior 20 10
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