22 June 2012, Computational Social Choice Seminar, Monica Patriche
We provide a unified presentation of the main models used in the general theory of equilibrium. We also present new equilibrium existence results for abstract economies with a variety of types of preference and constraint correspondences. The abstract economy models of G. Debreu (1952), W. Shafer-H. Sonnenschein (1975) and A. Borglin-H. Keiding (1976) have a finite number of players, like a strategic game. G. Debreu's model uses utility functions but the choice of strategies depends on the constraint correspondences. In contrast, W. Shafer and H. Sonnenschein use the preference correspondences in place of utility functions. The next step was made by N. C. Prabhakar and N. D. Yannelis in 1983 who proposed a model with a countable set of agents and the set of strategies being part of the infinite dimensional topological vector spaces. To ensure the existence of the equilibrium they developed new proof techniques. We also discuss subsequent modifications proposed by G. X. Yuan, who introduced in 1987 a new abstract model of economy with more general constraint correspondences, by W. K. Kim who introduced in 2003 the notion of a generalized quasi-game, and by K. Kim and K. K. Tan who introduced in 2001 the concept of a generalized abstract economy that uses fuzzy constraint correspondences, which take into account the instability of the preferences of an agent due to the uncertainty about the consumer behaviour or market situation.