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Unformatted text preview: Economics 405/505 Introduction to Game Theory Prof. Rui Zhao University at Albany Some Applications of Nash Equilibrium and Best Response Functions Cournot Oligopoly The outcome of the competition among oligopoly firms depends on the market demand for firms’ output, the firms’ costs of production, the number of firms, and the way in which firms respond to rivals’ actions. We now dis cuss quantity competition where firms compete by setting quantities, which is referred to as Cournot oligopoly games. Duopoly Consider a simple example of Cournot oligopoly. Two firms. Each can choose a nonnegative quantity, q i . The market price p depends on the total output Q = q 1 + q 2 , say p = p ( Q ) . Each firm incurs a unit cost of production c i . The game: • players: two firms. • actions: firm i chooses quantity q i ≥ . • payoffs: firm i ’s payoff is its profit, pq i c i q i . Assume p ( Q ) = A Q, where A is a positive number. To find the Nash Cournot equilibrium, we first find the best response functions of the firms. Firm 1’s payoff: u 1 ( q 1 ,q 2 ) = ( A q 1 q 2 ) q 1 c 1 q 1 . The best choice q 1 ( q 2 ) given firm 2’s quantity q 2 is given as solution to ∂u 1 ( q 1 ,q 2 ) ∂q 1 = A q 1 q 2 q 1 c 1 = 0 which gives (1) q 1 ( q 2 ) = A c 1 q 2 2 . Similarly, we can find out firm 2’s best response function: (2) q 2 ( q 1 ) = A c 2 q 1 2 . The Nash equilibrium is given by solving Eq. (1) and Eq. (2), which turns out to be: 1 q * 1 = A + c 2 2 c 1 3 and q * 2 = A + c 1 2 c 2 3 Provision of Public Good Players: two individuals. Each has wealth w > 0. Strategies: each individual i can choose to contribute g i dollars to the provision of a public good, where 0 ≤ g i ≤ w ....
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This note was uploaded on 09/14/2011 for the course ECON 505 taught by Professor Zhao during the Spring '11 term at SUNY Albany.
 Spring '11
 Zhao
 Game Theory, Oligopoly

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