Toward a study of bidding processes, part II: Games with capacity limitations

Abstract

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...

Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case

“…As a prelude to Section 4, we show that the limit of this solution converges to Griesmer et al (1967). Taking lim m!0 v 1 (b) and applying L'Hopital's rule yields…”

“…9 Furthermore, Griesmer et al (1967) show that any non-trivial equilibrium must consist of monotonic and di¤erential bid functions, at least without a minimum bid, and Lebrun (2006) shows uniqueness with a minimum bid, making our solution unique overall.…”

“…We note that using similar steps we can arrive at a symmetric equation to (7) Although the di¤erential equation is derived from the …rst-order conditions, any solution to it also satis…es the second-order conditions and hence is an equilibrium bid function. For the uniform distribution without the minimum bid, the second-order was proved by Griesmer et al (1967) and for completeness we provide a simple, more general proof, adapted from Wolfstetter in Appendix A.2. In the following section, we …nd the equilibrium bid functions by solving the di¤erential equations.…”

“…A solution does exist to a special case, dating back to Griesmer et al (1967), with distributions V 1 U [0; 1]; V 2 U [0; ] that has the following equilibrium inverse-bid functions: …”

Asymmetric First-Price Auctions With Uniform Distributions: Analytic Solutions to the General Case

“…It is interesting to compare this with the same conditional value distributions above 2 (without the atoms at m = 2), namely, V 1 U [2; 4] and V 2 U [2; 3]: This is given in Figure 2 and can be obtained from equations (1) by shifting the lower bound from 0 to 2 (hence, it is originally from Griesmer et al,1967). As we see, the presence of a minimum bid, even though it is at the center of both distributions, changes the equilibrium qualitatively by introducing the crossing of the bid functions.…”