Transcript
3.1. Strategic Behavior Matilde Machado
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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:
Player 1
Player 2
A
C 3,0
D 1,-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
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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
player 1
A
C 3,0
D 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
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3.1. Strategic Behavior Formally, {B,C} is a Nash equilibrium because: U1 (b, c) ≥ U1 ( a, c) or alternativelly b = arg maxU1 (i, c) i
U 2 (b, c) ≥ U 2 (b, d ) or alternativelly c = arg maxU 2 (b, j ) 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 of the game are the following: Industrial Economics - Matilde Machado
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3.1. Strategic Behavior B
A
NA
A
190,-10
190,0
NA
100,110 100,100
A
¿What will firm A do? That depends on what firm B will do and viceversa (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 Industrial Economics - Matilde Machado
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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 i (ai , a j ) ≡ profit of firm i when it performs action ai and its rival action aj where ai∈Ai and aj∈Aj. The pair (ai* , a*j ) is a Nash equilibrium iff:
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3.1. Strategic Behavior The pair
( ai* , a*j )
is a Nash equilibrium iff:
Π i (ai* , a*j ) ≥ Π i (ai , a*j ) ∀ai ≠ ai* ∈ Ai j * * j * * Π (ai , a j ) ≥ Π (ai , a j ) ∀a j ≠ a j ∈ A j Or written a bit differently: ai* = arg max Π i (ai , a*j ) ai = (ai* , a*j ) are best responses respectively * j * a j = arg max Π (ai , a j ) aj
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3.1. Strategic Behavior If there are interior solutions, then: * ∂Π i (ai* , a*j ) ∂ 2Π i (ai* , a*j ) a satisfies = 0; ≤0 i ∂ai ∂ai2 j * * 2 j * * a* satisfies ∂Π (ai , a j ) = 0; ∂ Π (ai , a j ) ≤ 0 j ∂a j ∂a 2j
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). Industrial Economics - Matilde Machado
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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. Industrial Economics - Matilde Machado
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