Two key results from auction theory the

Two key results from auction theory, the Revenue Equivalence Theorem and the buy Biotin-HPDP Principle, connect several different types of auction mechanisms through their expected revenues. Amongst the conditions that support the validity of these results, one is particularly important to this study: the auction mechanisms are standard in the sense that the winner is the person bidding the highest amount. There are, however, countless interesting situations where agents allocate their resources and the results are not deterministically reached, i.e., a participant offers the highest bid and yet he is not the winner of the auction. This is the case, e.g., of elections and patent races. In the first example, the candidate that spends more money is not necessarily the one that is elected. In the case of patent races the prototype with the highest investment is not necessarily chosen. These situations are usually treated as a contest where the probability of winning is proportional to the participant\'s bid. Suppose that all resources allocated by a candidate in a public contest (time, discipline, educational background, etc.) can be evaluated in monetary terms. This value may be interpreted as a bid that is paid at any contingency. The bids do not guarantee success but only a probability of success. Therefore, a public contest may be specifically viewed as a first-price all-pay auction whose object being auctioned is not a good but a lottery where the bidder may either win or lose the good. It is reasonable to assume that this lottery establishes a given probability of winning that depends on the participant\'s bid (effort). More specifically, one may assume that the probability of winning follows a stochastic order where higher bids are associated with higher probabilities of winning. This type of auction is also used in charitable events in order to raise funds and is popularly known as the Chinese auction or, more formally, a chance auction. Besides this introduction this paper is organized as follows. In Section 2 a brief literature review is performed; in Section 3 the model is formally presented; in Section 4 equilibrium is derived and its properties are analyzed; in Section 5 the model with linear probability is proposed and, finally, in Section 6 the main conclusions are presented and some extensions are suggested.
Related literature Franke et al. (2009) proposed the design of a contest and its success function through multilevel programming. The idea is to find the contest\'s unique Nash equilibrium and then, based on a preference parameter, obtain a success function where aggregate effort is maximized. In this study the success function depends solely on the participant\'s bid and is optimally determined in each agent\'s utility maximization problem. Kaplan et al. (2002) proposed an all-pay auction where the prize (or ex-post utility) does not depend only on the agent\'s valuation but also on its bid. Although their objectives were different the model proposed in this study is isomorphic to the multiplicatively separable environment proposed by Kaplan et al. (2002). Based on Taylor (1995), Fullerton et al. (2002) proposed a innovation competition model in two stages: first, a contest search type is conducted by a sponsor that afterwards determines the prize through a first-price auction. Hence, in Fullerton et al. (2002), the competition model occurs separately. It is worth mentioning that in the contest stage the success function is endogenously determined. Matros (2006) extends Tullock\'s basic model allowing the contest\'s prize valuation to be asymmetric among the participants. The author concludes that the increase in equilibrium total revenues with the addition of new participants persists in the asymmetric case. It is worth mentioning that this result is maintained even when the addition of new participants may eventually reduce the number of effective competitors. In summary, as the author states,