エッジワース復興 (The Edgeworthian Revival)

Separating the Convex Hull from the Origin -- from Debreu-Scarf (1963)

「エッジワース復興 (The Edgeworthian Revival)」というのは、ワルラス的な競争均衡と、それとは別の交換プロセス（得にゲーム理論) から得られる解との関係を考えてみようという努力を指す。1960年代と1970年代にこの試みで使われるようになった、お気に入りの数学ツール――measure 理論とnon-standard 分析――はほとんどの経済学者が慣れ親しんだどんなものよりも、すさまじくむずかしかった。

　努力の中心は「エッジワースの仮説」を証明することだっった。これは競争の水準（ここではエージェントの数として定式化）を無限大に増やしたら、core はワルラス均衡の集合に縮小する、という仮説だ。1963 年に、ジェラール・ドブリューとハーバート・E・スカーフが、 set the ball rolling with their proof of core convergence within the context of a "replicated" economy (i.e. arbitrarily large numbers of agents of each type). In 1964, Robert Aumann proved the equivalence of the Edgeworthian core and the Walrasian equilibria when we have a continuum (uncountably infinite number) of agents. This "new" definition of "perfect competition" required the introduction of measure theory -- notably Lyapunov's Theorem -- into economics. Edgeworth's conjecture in more general forms has been pursued by other economists since (esp. Truman Bewley (1973), Werner Hildenbrand (1974), Donald J. Brown and Abraham Robinson (1972, 1974), Robert M. Anderson (1978)).

The innovations continued. New characterizations of competitive equilibrium were also achieved during this time. We already know, from Aumann (1964) that competitive equilibrium can be characterized as a core with an infinite number of agents. Later economists characterized equilibrium as limiting cases of other game-theoretic solution concepts, e.g. with the set of fair net trades by David Schmeidler and Karl Vind (1972), with the Shapley value by Robert J. Aumann and Lloyd S. Shapley (1974), with the bargaining set by Andreu Mas-Colell (1989), for instance . Unfortunately, these solution concepts are for "cooperative" games and, furthermore, require infinite number of agents. Efforts have been made throughout , while we usually like to think that Walrasian equilibrium is non-cooperative and can be achieved with less than infinite number of agents.

The Founder