A Study On First-price Sealed-bid Auctions When Bidders Exhibit Different Attitudes Toward Risk

Auction theory has important practical and research significance.However,the assumptions of symmetric bidders and risk neutral preference in auction theory are somewhat harsh,and most of them are extensive research on asymmetric auctions with risk neutral preference,or symmetric auctions with risk aversion preference.There are few studies on asymmetric auction with risk aversion preference.However,both from the actual situation and the existing empirical studies show that risk aversion and its diversity cannot be ignored in the auction model.Based on the relevant literature and research theory of primary price sealed auction,this paper studies the primary price sealed auction in which bidders have different risk aversion levels.Based on the relevant literature and research theory of first-price sealed-bid auction,this paper studies the first-price auction when bidders exhibit different attitudes towards risk.Firstly,we study the case of two risk-averse bidders who had the same parametric family of(CRRA)functions,but their measures of risk aversion differ.Within a relatively simple informational framework,we have first explicitly derived the asymmetric equilibrium bidding strategies and the optimal strategic markups for both bidders.Next,we analyze the impact of asymmetric risk aversion levels on bidders’ markups and on the expected revenue and allocative efficiency of the auction in two cases.In case 1:we assume that one bidder becomes more risk averse and the other bidder’s risk averse remain unchanged.In case 2:we assume that bidders become more asymmetric in terms of risk aversion.Specifically,on the one hand,when one bidder becomes more risk averse and the other bidder’s risk averse remain unchanged,both bidders reduce their optimal strategic markups,and the rate at which his bid increases is higher than the rate at which his opponent’s bid increases;the allocative efficiency may be precluded;the seller’s expected revenue increases.On the other hand,by allowing risk aversion levels of both bidders to vary simultaneously,we show that as the less bidder becomes less risk averse and the more bidder becomes more risk averse,the less risk-averse bidder reduces his markup when asymmetry in terms of risk aversion between both bidders is sufficiency large;the allocative efficiency may decreases as bidders become more asymmetric in terms of risk aversion;when there is a large gap between both risk aversion levels,the seller’s expected revenue increases as bidders become more asymmetric in terms of risk aversion,but when the gap is within a certain range between both risk aversion levels,the seller’s expected revenue decreases as bidders become more asymmetric in terms of risk aversion.Secondly,we generalize the above work to a more general case where one bidder is characterized by a CRRA utility function while the other has a general concave utility function.We first establish the existence and uniqueness of the optimal strategic markups of two bidders;then we study effects of one bidder ’s risk aversion on bidding behavior.We find that when one bidder becomes more risk averse and the other bidder’s risk averse remain unchanged,both bidders reduce their optimal strategic markups,and an illustrative example shows that the rate at which his markup decreases is higher than the rate at which his opponent’s markup decreases;the allocative efficiency becomes complicated as one bidder becomes more(less)risk averse;we study effects of one bidder’s risk aversion on expected revenue.We find that the seller’s expected revenue increases when one bidder becomes more risk averse,given that the other bidder’s risk averse remain unchanged.Then on the basis of the third chapter,we model the problem that in First-price sealed-bid auctions bidders exhibit different attitudes toward risk and assign different weights of importance to the two criteria of "probability of winning" and "utility upon winning".We determine the asymmetric equilibrium strategies and analyze the impact of bidder’s preferences on markups,and on the expected revenue and on the allocative efficiency of the auction.We find that when the less risk averse bidder prefers "utility",or the more risk averse bidder prefers "winning probability",the indirect effect is no longer positive.While when the more risk averse prefers "utility",the direct effect is uncertain whether it is dominant the indirect effect or not.When the less risk averse bidder prefers "winning probability",the conclusion is similar to that of accommodate the same preference over "utility" and "winning probability".It is worth noting that the difference between the risk aversion levels of the two bidders is smaller than that given the same weight.By assuming one bidder becomes more or less risk averse and the other bidder’s risk averse remain unchanged,we find that when the less risk averse bidder prefers"utility",the direct effect is uncertain whether it is dominant the indirect effect or not.It is noteworthy that when the bidders prefer "utility",the less risk averse bidder has a higher probability of winning are given.Then we analyze the impact of bidder’s preferences on the allocative efficiency of the auction.We find that when bidders preferences "winning probability",the asymmetric risk averse bidders may preclude allocative efficiency.In addition,we also compare the expected revenue of seller with different weights with that of with the same weight.When bidders prefer "winning probability",the difference of risk aversion level between bidders is greater than that when the same weight is given to the bidding criteria.In addition,compared with giving the same weight to the bidding criteria,when the bidder pays more attention to the "utility",the seller’s expected income is greater than that when giving the same weight to the bidding criteria.Finally,we study the case of two risk-averse bidders who have the same parametric family of(CARA)functions,but their measures of risk aversion differ.By using the implicit function theorem,it is proved that the equilibrium markup strategy is an implicit function about the absolute risk aversion coefficient of two bidders.Then,we analyze the impact of risk aversion levels on the expected revenue and on the allocative efficiency of the auction.The changes of risk aversion levels may preclude allocative efficiency and the seller’s expected revenue increase as bidder becomes more risk averse.