Abstract
In Ch.5 we have already considered the contributions to game theory by von Neumann, indubitably the modern founding father of this theory. In economic analysis, game theory has been very useful in two branches: the study of market forms different from perfect competition and monopoly, and the study of the interaction between coalitions in complex organizations, such as multinational firms. Of course, this book is interested only in applications to market forms, and some anticipations are already contained in the short presentation, in § 1.2, of Cournot’s contributions. After the book by von Neumann and Morgenstern (1944), the most important ideas elaborated by game theorists are those introduced by Nash (1950), probably up to now the most fruitful ones in analyzing non cooperative, many-person games, together with a theorem about the non emptiness of the core of a game, obtained by Scarf (1967).
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References
See § 26.4.
See Ch.5.
See Myerson (1986, p.4-5).
Debreu (1982, § 2) calls such games ‘simultaneous optimization approach’. It is also interesting to note that the proof of the existence of a competitive general equilibrium, as was proposed by Arrow and Debreu (1954), consisted in transforming their model into an equivalent many—person game.
0r utility function, payment function, loss function, cost function,….
Whatever this means: a sum of money, a (cardinal) utility level, a certain quantity of one specific commodity, a vector of goods,….
See § 12.9.
on the stability problem of Nash’s solutions, see Karp (1992).
To clear the market it seems necessary, as in Walrasian general equilibrium, for an auctioneer to be at work to implement an equilibrium price. Maybe the auctioneer is a committee formed by the set of all firms; but a caveat’ is put by Arrow’s impossibility theorem (see § 21.5).
0f course, it is important to suppose that total capacity, namely, Ek yZ, satisfies (20.2).
The differentiability conditions of Assumptions 20.1,20.2 are not needed.
0f course, it is equally possible to express every firm's profit as a function of price instead of all outputs; actually, this is just a matter of simplicity.
By the intervention of an auctioneer?
For instance, look at the market for television sets.
The choice of price as the strategic variable of the firm is in accordance with the formulation proposed long ago by Bertrand (1883). Two modern formulations of duopoly in the Bertrand tradition are Kreps and Scheinkman (1983), and Deneckere and Kovenok (1996). See also Madden (1998) on the analysis by Kreps and Scheinkman.
Very likely, nobody buys a television set at the price of one million dollars, when there are other good models of television sets on sale at the price of one thousand dollars!
They are cost functions, capacity limitations, and market demand functions.
Since all fundamentals are stationary, when extended indefinitely into the future the oligopoly model becomes a supergame, assuming that the same firms are always active (no entry and no exit).
Ceteris paribus’, in the infinite horizon case certain results may change with respect to the finite horizon case.
As previously said, oligopoly theory is only a partial equilibrium theory.
More realistically, every firm chooses its period t price knowing the whole history of all past prices chosen by the active firms.
See the Mathematical Appendix to Ch.24, § 24.4.1.
See § 1.2.
See § 12.9.
Because, for instance, the differentiated goods are non durable.
To be considered in Ch.26
The so—called ‘folk theorem’ of the theory of repeated games is in part responsible for the ad hoc’ character of dynamic oligopoly models under stationary fundamentals. Loosely speaking, the folk theorem states that in an infinitely repeated game almost every strategy, capable of giving to every firm a profit greater than the minimum profit it can obtain in the corresponding one stage game, can be a solution, on condition that firms discount factors are sufficiently large.
Böhm shows that different price normalization rules have an influence on noncompetitive general equilibria; on this point see also Dierker and Grodal (1999).
This line of research was pioneered by Triffin (1940).
The Edgeworth’s core and Pareto’s efficient allocations look very similar to this notion.
Those players, also called ‘dummy’ or redundant players, who contribute nothing to any coalition; an agent j is powerless when, for every coalition K,one has v(K U {j}) =v(K).
See Shapley (1953).
These formulae were generalized by Aumann and Shapley (1974) to cooperative games containing infinitely many agents.
See Aubin ( 1993, Proposition 13. 5, p. 215 ).
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Nicola, P. (2000). Game Theory and Oligopoly. In: Mainstream Mathematical Economics in the 20th Century. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04238-0_20
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