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J a m e s H. G r i e s m e r and M a r t i n S h u b i k I n t e r n a t i o n a l B u s i n e s s M a c h i n e s C o r p o r a t i o n T h o m a s J. W a t s o n R e s e a r c h C e n t e r Y o r k t o w n H e i g h t s . N e w Y o r kABSTRACT T h i s p a p e r is the second of a s e r i e s on bidding. A previous p a p e r (published in Mar., 1963 i s s u e of NRLQ) c o n s i d e r e d bidding situations f o r m u l a t e d as t w o -p e r s o n c o n s t a n t -s u m g a m e s ; t h i s a r t i c l e is devoted t o the i m p o r t a n t c a s e of n o n -c o n s t a n t -s u m bidding m o d e l s involving two p a r t i c i p a n t s .Following a n introduction on solution c o n c e p t s , the s i n g l e -s h o t bid is analyzed, both f o r the c a s e of no penalty in t h e c a s e of a t i e , and f o r the c a s e of a penalty. T h e m a j o r i t y of the p a p e r is then devoted t o a study of bidding on a s e q u e n c e of j o b s , using t h r e e different a s s u m p t i o n s about t h e amount of i n f o r m a t i o n r e v e a l e d a f t e r the a w a r d of e a c h job. They a r e , r e s p e c t i v e l y , the s u p e r -s i l e n t bid, w h e r e no information is given; the s i l e n t bid, w h e r e only the n a m e of the winner is announced; a n d the noisy bid, w h e r e c o m p l e t e i n f o r m a t i o n is given. T h e a n a l y s i s is p r i m a r i l y a study of e q u i l i b r i u m solutions u n d e r e a c h a s s u m p t i o n . What is shown is that, in p a s s i n g f r o m one type of bid t o a second i n which m o r e i n f o r m a t i o n is given, new f a m i l i e s of e q u i l i b r i u m solutions a r e added, and existing f a m i l i e s b e c o m e m o r e extensive. CONCEPTS OF SOLUTIONThe minimax solution which appears to be reasonable for two-person constant-sum games is, in general, judged to be too pessimistic a solution to be expected in a non-constantsum game. It can, of course, be calculated for such a game and may serve to provide a lower bound to a player's expectations.Two competing theories of solution have been suggested: an outcome in the form of a set of non-cooperative equilibrium points, or an outcome which jointly maximizes the returns to the bidders as a whole. These are by no means the only solution concepts which have been suggested. In successive sections, we examine several others which appear to be of relevance to situations involving ignorance of the competitors' costs and preferences; to markets where bidding may be repeated over many periods; to bidding processes where there a r e varying degrees of communication between the participants and several other complications. In this paper, we limit ourselves to the two solution concepts just noted, together with a third which provides a lower bound below which we might regard behavior as pathological. The third solution concept is based on the assumption that the main object of each competitor is to do better than the other. This converts a two-person non-constant-sum game into a zero-su...

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