Discontinuous Games and Endogenous Sharing Rules

Abstract

UNIVERSITY OF CALIFORNIA, BERKELEY Department o f Economics Berkeley, California Working Paper 8756 DISCONTINUOUS GAMES ENDOGENOUS SHARING RULES AND Leo K. Simon and. W i l l i a m R. October 1, Key words: Zame D i s c o n t i n u o u s games, e x i s t e n c e , Nash e q u i l i b r i u m , H o t e l l i n g , Bertrand, sharing r u l e s , r a t i o n i n g . Abstract We propose a d i f f e r e n t approach t o the k i n d s of economic problems t h a t l e a d t o d i s c o n t i n u o u s games. We t a k e the view t h a t the under- l y i n g p a y o f f s f o r t h e s e problems are o n l y p a r t i a l l y determined, r a t h e r than discontinuous. At p o i n t s where t i e s occur, we propose t h a t t h e s h a r i n g r u l e s h o u l d be determined endogenously, i . e . , as p a r t o f the s o l u t i o n t o the model r a t h e r t h a n as p a r t of the des- c r i p t i o n o f the model. T h i s l e a d s us t o d e f i n e a game with an en- dogenous s h a r i n g r u l e . I t c o n s i s t s of a s t r a t e g y space f o r each of a f i n i t e number o f p l a y e r s , t o g e t h e r with a p a y o f f correspondence, i n t e r p r e t e d as the union of a l l p o s s i b l e s h a r i n g r u l e s . A s o l u t i o n f o r such a game i s a s e l e c t i o n from the p a y o f f correspondence t o - g e t h e r with a s t r a t e g y p r o f i l e s a t i s f y i n g t h e u s u a l (Nash) b e s t response c r i t e r i o n . Our p r i n c i p a l r e s u l t i s t h a t such a s o l u t i o n always e x i s t s . JEL C l a s s i f i c a t i o n :