3.1. Strategic Behavior

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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
• 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

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

• player 2 If player 1 plays A ? best response is
  C u0
• If player 1 plays B ? best response is C
  u2
• ? 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 u4
• If player 2 plays D (which he wont) ? best
  response is A u1
• B,C is the only Nash equilibrium of the game.

C D
A 3,0 1,-1
B 4,2 -2,1
player 1
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3.1. Strategic Behavior

• Formally, B,C is a Nash equilibrium because
  

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3.1. Strategic Behavior

• In Industrial Organization
• Players are firms
• Strategies are going to be prices, quantities,
  advertising, product quality, RD, 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

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3.1. Strategic Behavior
A NA
A 190,-10 190,0
NA 100,110 100,100
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 of the player and its rivals.
• Strategies are going to be actions i.e. 11
  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

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3.1. Strategic Behavior

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

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3.1. Strategic Behavior

• If there are interior solutions, then
• Lets assume that AiAj? and ?i and ?j are
  concave (i.e. ?iiilt0 and ?jjjlt0) for all values
  of ai and aj ? the FOCs are enough to derive the
  Nash equilibrium ( a system of 2 equations and 2
  unknowns).

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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.