All-pay auctions with risk-averse players
|
|
- Mabel Barrett
- 5 years ago
- Views:
Transcription
1 Int J Game Theory 2006) 34: DOI /s ORIGINAL ARTICLE All-pay auctions with risk-aerse players Gadi Fibich Arieh Gaious Aner Sela Accepted: 28 August 2006 / Published online: 23 September 2006 Springer-Verlag 2006 Abstract We study independent priate-alue all-pay auctions with risk-aerse players. We show that: 1) Players with low alues bid lower and players with high alues bid higher than they would bid in the risk neutral case. 2) Players with low alues bid lower and players with high alues bid higher than they would bid in a first-price auction. 3) Players expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-aerse players and the seller s expected reenue. In particular, we show that in all-pay auctions the seller s expected payoff in the risk-aerse case may be either higher or lower than in the risk neutral case. Keywords Priate-alue auctions Risk aersion Perturbation analysis JEL Classification: D44 D72 D82 G. Fibich B) School of Mathematical Sciences, Tel Ai Uniersity, Tel Ai 69978, Israel fibich@math.tau.ac.il A. Gaious Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gurion Uniersity, P.O. Box 653, Beer-Shea 84105, Israel ariehg@bgumail.bgu.ac.il A. Sela Department of Economics, Ben-Gurion Uniersity, P.O. Box 653, Beer-Shea 84105, Israel anersela@bgu.ac.il
2 584 G. Fibich et al. 1 Introduction Consider n players who compete for a single item. Eery player submits a bid and the player with the highest bid receies the item. All players bear a cost of bidding which is an increasing function of their bids. This setup, which is called an all-pay auction, is commonly used to model applications such as job-promotion competitions, R&D competitions, political campaigns, political lobbying, sport competitions, etc. The literature on contests and particularly on all-pay auctions has dealt mostly with risk-neutral players. 1 In contrast to all-pay auctions, seeral studies on the classical auction mechanisms first-price and second-price auctions) with risk-aerse players appear in the literature. In independent priate-alue second-price auctions, risk aersion has no effect on a player s optimal strategy which remains to bid her own aluation for the object. In independent priate-alue first-price auctions, on the other hand, risk aersion makes players bid more aggressiely Maskin and Riley 1984). Since the risk-neutral) seller is indifferent to the first-price and second-price auctions when players are risk neutral, 2 she prefers the first-price auction to the second-price auction when players are risk aerse. Howeer, the seller s preference relations for auction mechanisms with risk-aerse players do not imply anything about the players preference relations for these auctions, since under risk aersion the combined reenue of the seller and the players is not a constant. Indeed, Matthews 1987) showed that risk aerse players with constant absolute risk aersion are indifferent to first and second-price auctions, and that players prefer the first-price auction if they hae increasing absolute risk aersion and the second price auction if they hae decreasing absolute risk aersion. 3 In this paper we analyze the role of risk aersion in all-pay auctions by comparing the situation where all players are risk neutral henceforth referred to as the status quo), with the case where players are risk-aerse. In Sect. 2 we show that a risk-aerse player with a low aluation bids less aggressiely than in the status quo situation. On the other hand, a risk-aerse player with a high aluation bids more aggressiely than in the status quo. This behaior can be explained as follows. When a player s alue is small, she is most likely to lose. Therefore, as she becomes more risk aerse, she is willing to pay less, that is, she bids less aggressiely. On the other hand, when a player s alue is ery high, she is afraid of losing the object, therefore, she bids more aggressiely. These results are consistent with the experimental studies of Barut et al. 2002) and Noussair and Siler 2005), who obsered that players in single-unit and multiple-unit 1 All-pay auctions with linear cost functions and incomplete information about the players alues include, among others: Weber 1985), Hilman and Riley 1989), Krishna and Morgan 1997), Kaplan et al. 2002). All-pay auctions with complete information about the players alues include, among others: Tullock 1980), Dasgupta 1986), Dixit 1987), Baye et al. 1993, 1996). 2 This follows from the Reenue Equialence Theorem Vickrey 1961; Myerson 1981; Riley and Samuelson 1981). 3 This result was generalized by Monderer and Tennenholtz 2000) to all k-price auctions.
3 All-pay auctions with risk-aerse players 585 all pay auctions with low alues tend to bid below the risk-neutral equilibrium, and those with large alues tend to bid aboe the risk-neutral equilibrium. We can learn much about the all-pay auction with risk aerse players by comparing it to the first-price auction. Although the first-price auction is a classical auction whereas the all-pay auction is a contest, these models are similar since in both the highest player wins for sure and pays her bid. Intuitiely, one can expect that as in the risk-neutral case, the equilibrium bids of risk aerse players in all-pay auctions should be lower than in first-price auctions. We show that, indeed, in all-pay auctions, low types bid less aggressiely than they bid in firstprice auctions. Howeer, high types bid more aggressiely in all-pay auctions than they bid in first-price auctions. In light of the aboe comparison of the players bids in first-price auctions and all-pay auctions, it is not clear in which auction the player s expected utility is larger. Neertheless, we show that, independent of the distribution of the players aluations and the number of players, the expected utility of a risk-aerse player in the first-price auction is always larger than in the all-pay auction. Consequently, a risk-aerse player will prefer the first-price auction to the all-pay auction. We note that Eso and White 2004) proed that bidders would prefer the first-price auction to the all-pay auction under symmetric, affiliated alues and decreasing absolute risk aersion DARA). Therefore, the present study shows that this result remains true if one replaces the assumption of affiliation with the stronger assumption of independence, but relaxes the assumption of DARA to any type of risk aersion. Rigorous analysis of all-pay auctions with risk-aerse players is limited since usually explicit expressions for the equilibrium strategies with risk-aerse players cannot be obtained. In order to oercome this difficulty, in Sect. 3 we consider the case of weakly risk-aerse players. The presence of the small risk-aersion parameter allows us to employ perturbation analysis, one of the most powerful tools in applied mathematics, to calculate explicit approximations of the equilibrium strategies of risk-aerse players and the seller s expected reenue. 4 The high accuracy of the explicit approximations of the equilibrium bids is illustrated by an example with two weakly risk-aerse players. We show that een when the risk-aersion parameter is not small, the agreement between the explicit approximations obtained by the perturbation analysis and the exact alues obtained by numerical analysis is quite remarkable. 5 The approximate solutions in Sect. 3 show, for example, that risk aersion can lead to an increase, as well as a decrease, in the seller s expected reenue in all pay auctions. In addition, they show that, roughly speaking, weak risk aersion leads to a larger departure from reenue equialence than weak asymmetry. Altogether, the combination of the quantitatie results of risk-aersion in allpay auctions, together with the qualitatie results gien in Sect. 2 proide a clear picture of the behaior of risk-aerse players in all-pay auctions. 4 Fibich and Gaious 2003) and Fibich et al. 2004) employed perturbation analysis to study asymmetric auctions. 5 This is, more often than not, the case in perturbation analysis see, e.g., Bender and Orszag 1978).
4 586 G. Fibich et al. Remark For clarity, the proofs are delegated to the Appendices and presented in the order in which they are proed. The results in Sect. 2 are presented in a different order in which they are proed in the Appendices, in order to better present the economic results. 2 All-pay auctions with risk-aerse players Consider n players that compete to acquire a single object in an all-pay auction. The aluation of each player for the object is independently distributed according to a distribution function F) on the interal [, ], where 0. Each player submits a bid b and pays her bid regardless of whether she wins or not, but only the highest player wins the object. Each player s utility is gien by a function U b) which is twice continuously differentiable, monotonically increasing, normalized such that U0) = 0, and satisfies U < 0 i.e., riskaerse players). Gien that the equilibrium bid function b) is monotonically increasing, we can define the equilibrium inerse bid function = b). The maximization problem of player i with aluation is gien by max V, b) = b Fn 1 b))u b) + 1 F n 1 b)))u b). Differentiating with respect to b gies the first-order condition 0 = V b = n 1)Fn 2 b))f b)) b)[u b) U b)] F n 1 b))[u b) U b)] U b). Therefore, the inerse bid function satisfies the ordinary differential equation Fb))[U b) U b)] b) = n 1)f b))[u b) U b)] U b) + n 1)F n 2 b))f b))[u b) U b)], 1) subject to the initial condition 0) =. Equation 1) is exact in the risk-neutral case Ux) = c x where c is a constant. In that case, its solution is gien by b all rn ) = Fn 1 ) F n 1 s)ds. 2)
5 All-pay auctions with risk-aerse players 587 For comparison, the equilibrium bid in a first price auction with risk-neutral bidders is gien by Therefore, it immediately follows that b 1st rn ) = 1 F n 1 ) ball rn ). 3) b all rn ) <b1st rn ), < <. 4) Although there are no explicit solutions of Eq. 1) for a general utility function U, we can derie some qualitatie results by comparing the equilibrium bids in the risk-aerse and the risk-neutral cases. These results are in the spirit of the ones obtained by Maskin and Riley 1984), who showed that in a first-price auction the equilibrium bid of a risk-aerse player is higher than the equilibrium bid of a risk-neutral player with the same type, that is, b 1st rn ) <b1st ), <. 5) We also show how relation 4) is affected by risk aersion, by comparing the bids of risk-aerse bidders in all-pay auctions with the ones in first-price auctions, denoted by b 1st ). We first show that risk aersion affects low type players to bid less aggressiely: Proposition 1 In an all-pay auction the equilibrium bid of a risk-aerse player with low type is smaller than the equilibrium bid of a risk-neutral player with the same type, i.e., Proof See Appendix B. b all ) <b all rn ), 0 < 1. 6) The following result shows that risk-aersion affects high type players and low type players quite differently: Proposition 2 In an all-pay auction, the equilibrium bid of a risk-aerse player with high type is higher than the equilibrium bid of a risk-neutral player with the same type, i.e., Proof From Eq. 3) it follows that b all ) >b all rn ), ) b all rn ) = b1st ). 8) rn
6 588 G. Fibich et al. Similarly, from Eq. 5) it follows that b 1st rn ) <b1st ). 9) By Proposition 4, the equilibrium bid of a risk-aerse player with type in an all-pay auction is larger than in a first-price auction, that is, b 1st ) <b all ). 10) The combination of the three inequalities 8)+9)+10) completes the proof. Since in an all-pay auction a player pays her bid regardless of whether she wins, whereas in a first-price auction she pays only if she wins, it seems natural that players will bid more carefully i.e., hae lower bids) in all-pay auctions than in first-price auctions. Indeed, the bid of a risk-neutral player in an all-pay auction is smaller than her bid in a first-price auction, see Eq. 5), and we can expect this relation to be een stronger for risk-aerse players. Howeer, as Propositions 3 and 4 show, the relation of bids in first-price and all-pay auctions with risk-aerse players is more complex: Proposition 3 The equilibrium bid of a risk-aerse player with sufficiently low type in an all-pay auction is smaller than her bid in a first-price auction. Proof From Proposition 1, Eq. 3) and 9) we hae that b all ) <b all rn ) <b1st rn ) <b1st ). 11) Proposition 4 The equilibrium bid of a risk-aerse player with sufficiently high type in an all-pay auction is larger than in a first-price auction. Proof See Appendix C. Example [ ] 1 Consider two players where each player s aluation is distributed on 0, 1 according to the uniform distribution function F) =. Assume that each player s utility function is Ux) = x εx 2. In Fig. 1 we show the equilibrium bids of risk neutral and risk aerse bidders in first-price and all-pay auctions. The results illustrate our finding that 1) Players in the all pay auction with low alues bid lower and players with high alues bid higher than they would bid in the risk neutral case. 2) Players in the all pay auction with low alues bid lower and players with high alues bid higher than they would bid in a first-price auction. Propositions 3 and 4 show that there is no dominance relation among the bids in first-price and all-pay auctions. Neertheless, first-price auctions dominate all-pay auctions from the player s point of iew: Proposition 5 The expected utility of a risk-aerse player with type < in the first-price auction is larger than her expected payoff in the all-pay auction. Proof See Appendix A.
7 All-pay auctions with risk-aerse players 589 ε = 0.25 ε = bids first price all pay bids first price all pay Fig. 1 Bids of risk-aerse players solid lines) and of risk-neutral players dashed lines) in all-pay auctions and in first-price auctions 3 All-pay auctions with weakly risk-aerse players The results of the preious section leae many open questions. For example, because of the complex way that risk aersion affects the equilibrium bids, it is not clear whether, oerall, risk aersion leads to an increase or a decrease in the seller s expected reenue. In addition, the tools that we used in the preious section, which are standard in auction theory, typically proide qualitatie results e.g., which of two possibilities is larger), but do not gie a quantitatie estimate e.g., by how much). In order to address such questions, we consider the case of weak risk aersion, 6 i.e., U x. This is the case, for example, for players with a constant absolute risk aersion CARA) utility function Ux) =[1 exp ɛx)]/ɛ, or for players with constant relatie risk aersion CRRA) utility function Ux) = x 1 ɛ,if0<ɛ 1. Therefore, in general, the utility function of weakly riskaerse players can be written as Ux) = x + εux) + Oε 2 ), ε 1. 12) Thus, ε is the risk aersion parameter and ε 1 implies weak risk aersion. Note that u0) = 0 and u < 0. On the other hand, u can be either positie or negatie gien that u x) > 1 ε ) since in either case U = x+ɛu is monotonically increasing. The existence of a small risk aersion parameter enables us to use perturbation methods to calculate explicit approximations to the bidding strategies: 6 The assumption of weak risk aersion is quite reasonable. Indeed, while most people would prefer to receie $500 dollar with probability 1 rather than $1,000 with probability 1/2, much fewer would prefer receiing $300 dollar with probability 1 rather than $1,000 with probability 1/2.
8 590 G. Fibich et al. Proposition 6 The symmetric equilibrium bid function in an all-pay auction with weakly risk-aerse players is gien by b all ) = b all rn ) + εball 1 ) + Oε2 ), where b all rn ) is the equilibrium bid in the risk-neutral case 2), and ) [ ) ) ] b all 1 ) = u b all rn ) + F n 1 ) u b all rn ) u b all rn ) Proof See Appendix D. F n 1 s)u s b all rn s) ) ds. 13) We thus found an explicit expression for εb all 1 ), i.e., the leading-order effect of risk-aersion on the equilibrium strategy. Roughly speaking, for a 10% leel risk aersion, we calculated the corresponding 10% change in the equilibrium strategy with 1% accuracy. Example 2 The results of our perturbation analysis can be illustrated by the following example. Consider two players where each player s aluation is distributed on [ 0, 1 ] according to the distribution function F) = α. Assume that each player s utility function is Ux) = x εx 2. From Proposition 6 the equilibrium bid function in the all-pay auction is gien by b all ) = α 1 + α 1+α + ε α 2 + α 2+α + α ) 1 + α 2+2α + O ε 2). 14) In Fig. 2 we compare the approximation 14) with the exact bid functions i.e., the numerical solutions of equation 1)), for the case α = 1. Atɛ = 0.25, the approximations are almost indistinguishable from the exact bids. Although when ɛ = 0.5 the risk-aersion parameter is not small, 7 the agreement between the explicit approximations and the exact alues is quite remarkable. 8 In addition for proiding quantitatie predictions for the equilibrium bids, the explicit approximations obtained in Proposition 6 can be used to approximate the seller s expected reenue under risk aersion: 7 In fact, ɛ = 0.5 is the largest possible alue of ɛ for which U = x ɛx 2 is monotonically increasing. 8 Such good agreement was also obsered in numerous other comparisons that we made with different distribution functions and utility functions.
9 All-pay auctions with risk-aerse players 591 ε = 0.25 ε = 0.5 bids bids Fig. 2 Bids of risk-aerse buyers solid lines) and their explicit approximation [Eq. 14), dotted lines] in all-pay auctions Proposition 7 In an all-pay auction with weakly risk-aerse players, the seller s expected reenue is gien by R all = R rn + εn + 1 F n 1 ) [ F n 1 )u b all rn )) ) u b all rn )) ]f ) d F n 1 )1 F))u b all rn where R rn is the expected reenue in the risk-neutral case. Proof See Appendix E. )) d + Oε2 ), 15) As we hae said, unlike first price auctions, the effect of risk aersion on the seller s reenue in all pay auctions is not obious, since it lowers the bids for low alues but increases the bids for large alues. Indeed, the result of Proposition 7 shows that risk-aersion can lead to an increase, as well as to a decrease, of the seller s expected reenue in all-pay auctions: Example 3 Consider n = 2 risk aerse players with distribution functions F) = α in [0, 1], such that Ux) = x ɛx 2. Substituting ux) = x 2 in 15) and integrating gies R all = R rn + ɛ R + Oɛ 2 ), R = 2 α) α α + 3 α 2 ) α + 2).
10 592 G. Fibich et al. We thus see that depending on the alue of α, R can be either positie or negatie. Hence, we conclude that risk-aersion can lead to an increase, as well as to a decrease, of the seller s expected reenue in all-pay auctions. An immediate, yet important consequence from Proposition 7 is as follows: Proposition 8 An Oɛ) risk aersion leads to an Oɛ) difference in the seller s reenue among different auction mechanisms. Proof Since risk-aersion does not affect the reenue in a second price auction, the result follows from Proposition 7. In Fibich et al. 2004) we showed that if ɛ is the leel of asymmetry among the distribution functions of the players aluations, then weak asymmetry only leads to an Oɛ 2 ) difference in the seller s reenue among different auction mechanisms. Hence, Proposition 8 shows that, roughly speaking, weak risk aersion leads to a larger reenue differences among different auction mechanisms than weak asymmetry. We can also use the explicit expression obtained in Proposition 6 to analyze the effect of weak risk aersion on the players expected utility. Proposition 9 The expected utility of a weakly risk aerse player with type in an all-pay auction is gien by V all ) = V rn ) + ε F n 1 s)u s b all rn s) ) ds + Oε 2 ), where V rn ) = Fn 1 ) d is the expected utility in the risk-neutral case. Proof See Appendix F. Note that the difference between the expected payoffs of a weakly riskaerse player and a risk-neutral player does not depend on the alue of u, but depends on the alue of u. That is, if the utility function of a risk-aerse player Ux) always larger or equal than the utility function of a risk-neutral player U rn x) = x, it does not necessarily imply that the expected utility of the risk-aerse player is larger than the expected utility of the risk-neutral player. A natural question that arises is whether in the case of weak-risk aersion one cannot simply approximate the bidding functions using the risk-neutral expressions. In other words, when ɛ is small, is there an adantage for the approximation b all ; ɛ) b all rn ) + ɛball 1 ) oer the continuous approximation b all ; ɛ) b all ; ɛ = 0) = b all rn )? The answer is that the accuracy of the first approximation is Oɛ 2 ), whereas that of the second approximation is only Oɛ). Therefore, the first approximation is significantly more accurate when ɛ is moderately small but not negligible). Indeed, comparison of Figs. 1 and 2 shows that the exact) bids in the risk-aerse case are well-approximated with the explicit approximation that we deried, but are not well-approximated with the bids in the risk-neutral case.
11 All-pay auctions with risk-aerse players 593 Acknowledgments We would like to thank an anonymous referee for many useful comments. A Proof of proposition 5 The proof here is similar to the one in Milgrom and Weber 1982) and Matthews 1987), who used it to obtain similar results. When all players follow their equilibrium bidding strategies, a player s expected utility gien that his type is and that he plays as if his type is t is V all t ) = F n 1 t)u b all t) ) + 1 F n 1 t))u b all t) ), V 1st t ) = F n 1 t)u b 1st t) ), 16) for all-pay and first-price auctions, respectiely. Let V all ) = V all ) and V 1st ) = V 1st ). By a standard argument, in equilibrium V j t ) t = 0, j = all, 1st. 17) t= Therefore, V j )) = F n 1 )U b j )), j = all, 1st. 18) In addition, V all ) = V 1st ), since in both auctions the lowest type expects a zero utility. We proe by negation. Assume that for some type, < <, we hae V all ) V 1st ). Then, by 16) it follows that b 1st ) >b all ). From the concaity of U it follows that U b all )) < U b 1st )). Thus V all ) ) < V 1st ) ). Let y = V all V 1st. Then, y) = 0, and for < <, y) 0 implies that y ) <0. Therefore, it follows that y < 0for < <. To complete the proof, we now proe that y ) = V all ) V 1st ) <0. Since y) <0for < <, we only need to proe that it is not possible to hae y ) = 0. Assume, therefore, by negation that y ) = 0. We will show that this implies that y ) = 0 and y ) >0, which is in contradiction with the fact that y < 0for < <. Indeed, y ) = 0 U b all )) = U ) b 1st ) b 1st ) = b all ) V all) ) = V 1st) ) y ) = 0. In addition, substituting t = = in 17) gies that b all) ) b 1st) n 1)f ) ) ) = U b) U b all ) > 0.
12 594 G. Fibich et al. Therefore, by 18), y ) = V all) ) V 1st) ) [ ) = U b all ) b all) ) b 1st) ] ) > 0. B Proof of Proposition 1 Since b all ) = b all rn ) = 0, we can proe the result by showing that Let us first note that 17) implies that ) b all ) b all ) ) <b all rn ) ), 0 < 1. 19) = n 1)F n 2 U b all ) ) U b all ) ) )f ) F n 1 )U b all ) ) + 1 F n 1 ))U b all ) ). In particular, in the case of risk neutrality ) b all rn ) = n 1)F n 1 )f ). Therefore, b all ) ) ) b all rn ) n 1)F n 1 )f ) U b all ) ) U b all ) ) = F n 1 )U b all ) ) + 1 F n 1 ))U b all ). 20) ) Let us begin with the case when > 0. Since b all ) = 0, then b all ) ) ) b all rn ) [ U) U0) U ] x) n 1)F n 1 = )f ) U = 0) U 0) 1, = where 0 < x <. By the concaity of U, U x) U 0) < 1. Therefore, we proed 19) for > 0. To proe 19) when = 0, we first expand, ) ) ) U b all ) = U b all ) + U b all ) + 2 ) 2 U b all ) + O 3 ), ) ) U b all ) = U b all )) + U b all ) + O 2 ).
13 All-pay auctions with risk-aerse players 595 Therefore, U b all )) U b all )) F n 1 )U b all )) + 1 F n 1 ))U b all )) = U b all )) U b all )) + O 3 ) U b all )) + F n 1 ) [ U b all )) + O 2 ) ] + 2 U b all )) 2 U = b all )) + O3 ) 1 + F n 1 ) U b all )) U b all )) + O2 ) [ ][ ] = 1 + U b all )) 2 U b all )) + O2 ) 1 F n 1 ) U b all )) U b all )) + O2 ) = + 2 U b all [ ] )) 1 U b all )) 2 Fn 1 ) + O 3 ). Therefore, by 20), b all ) ) b all rn ) ) n 1)F n 1 )f ) = 2 U b all ) )) 1 U b all )) 2 Fn 1 ) + O 3 )<0. C Proof of Proposition 4 By Proposition 5, the expected utility of a risk-aerse player with type in the first-price auction is larger than her expected payoff in the all-pay auction V all ) <V 1st )). Since V j ) = U b j )) for j = all, 1st, see Eq. 16), and since U is monotonically increasing, the result follows. D Proof of Proposition 6 We can write the equilibrium bid as b) = rn b) + ε 1 b) + Oε 2 ), where rn b) is the inerse function of the risk-neural equilibrium strategy in all-pay auctions 2 ). For clarity, we drop the superscript all. We first note that when ε 1, Fb)) = F rn ) + ε 1 F rn ) + Oε 2 ), f b)) = f rn ) + ε 1 f rn ) + Oε 2 ), Ub) b) U b) = b) + ε[ub) b) u b)] = rn b) + ε[ 1 b) + u rn b) b) u b)]+oε 2 ), U b) b) U b) = ε[u b) b) u b)] = ε[u rn b) b) u b)]+oε 2 ).
14 596 G. Fibich et al. Substitution in 1) and expanding in a power series in ε, the equation for the O1) term is identical to the one in the risk-neutral case and therefore is automatically satisfied. The equation for the Oε) terms is 1 b) = F rnb))[u rn b) b) u b)] n 1)f rn b)) rn b) u b) + n 1)F n 2 rn b))f rn b)) rn b) n 2) 1 b) n 1)F n 1 rn b)) rn b) 1 f rn b)) n 1)F n 2 rn b))f 2 rn b)) rn b) [ 1b) + u rn b) b) u b)] n 1)F n 2 rn b))f rn b)) 2 rn b), subject to 1 0) = 0. Since, by 1), rn b) = 1 n 1)F n 2 rn b))f rn b)) rn b), 21) the equation for 1 b) can be rewritten as 1 b) + 1b)Bb) = Gb) 22) where Bb) = [ rn b) rn b) + f rn b)) f rn b)) rn b) + n 2) f ] rnb)) F rn b)) rn b), and [ ] Gb) = rn { b) u rn b) b) u b) n 1)F n 2 rn b))f rn b)) rn b) ) } + F n 1 rn b)) u rn b) b) u b) + u b). 23) The solution of 22) is gien by 1 b) = e brn b B C 1 b rn b Gx)e brn x B dx,
15 All-pay auctions with risk-aerse players 597 where b rn = b rn ). It is easy to erify that see 21)) e brn b B = rn b) rn b rn ). brn Thus, as b 0, rn b) and e b B. Therefore it follows that C 1 = brn 0 Gx)e brn x B dx and that b 1 b) = rn b) 0 Gx)/ rn x) dx. In addition, we note that if we differentiate the identity = b; ε); ε) with respect to ε and set ε = 0, we get that 1 b rn )) + rn b rn))b 1 ) = 0or b 1 ) = 1 / rn b). Thus, we get that b 1 ) = b rn ) 0 Gx)/ rn x) dx. Substitution of G from 23) gies b 1 ) = b rn ) 0 {[ ] u rn b) b) u b) F n 1 rn b))) ) F n 1 rn b)) u rn b) b) u b) } u b) db. A few more technical calculations complete the proof. E Proof of Proposition 7 The seller s reenue is gien by R all = n bs)f s) ds. Substituting b = b rn + εb 1 + Oε 2 ), we hae
16 598 G. Fibich et al. R all = n b rn + εb 1 ) f s)ds + Oε 2 ) = n = R rn + εn b 1 f s) ds + Oε 2 ). Substituting b 1 from 13) yields b 1 f s) ds = + b rn f s)ds + εn 1 F n 1 ))u b rn ))f ) d F n 1 )u b rn ))f ) d Integrating by parts the double integral gies = Therefore, the result follows. F n 1 s)u s b rn s)) ds f ) d. F n 1 s)u s b rn s)) ds f ) d F n 1 )1 F))u b rn )) d. b 1 f s)ds + Oε 2 ) F Proof of Proposition 9 The expected utility for a player with type in all-pay auctions in equilibrium is gien by V all ) = F n 1 )U b)) +[1 F n 1 )]U b)). In the case of weak risk aersion 12), V all ) = F n 1 ) b) [ ) + ε F n 1 ) u b)) u b)) ] + u b)) + Oε 2 ).
17 All-pay auctions with risk-aerse players 599 Using the relation b) = b rn ) + εb 1 ) + Oε 2 ), we hae V all ) = Vrn all { ) [ ) ε b 1 ) F n 1 ) u b rn )) u b rn )) ]} + u b rn )) + Oε 2 ). By the reenue equialence theorem, V all rn ) = V rn) = Fn 1 s) ds is independent of the auction mechanism. Substituting 13) in the last equation yields the result. References Barut Y, Koenock D, Noussair C 2002) A comparison of multiple-unit all-pay and winner-pay auctions under incomplete information. Int Econ Re 433): Baye M, Koenock D, de Vries C 1993) Rigging the lobbying process. Am Econ Re 83: Baye M, Koenock D, de Vries C 1996) The all-pay auction with complete information. Econ Theory 8: Bender CM, Orszag S 1978) Adanced mathematical methods for scientists and engineers. McGraw-Hill, New York Dasgupta P 1986) The theory of technological competition. In: Stiglitz J, Mathewson G eds) New deelopments in the analysis of market structure. MIT, Cambridge Press Dixit A 1987) Strategic behaior in contests. Am Econ Re 775): Eso P, White L 2004) Precautionary Bidding in auctions. Econometrica 72:77 92 Fibich G, Gaious A 2003) Asymmetric first-price auctions a perturbation approach. Math Oper Res 28: Fibich G, Gaious A, Sela A 2004) Reenue equialence in asymmetric auctions. J Econ Theory 115: Hilman A, Riley JG 1989) Politically contestable rents and transfers 1. Econ Polit 1:17 39 Kaplan T, Luski I, Sela A, Wettstein D 2002) All-pay auctions with ariable rewards. J Ind Econ L4): Krishna V, Morgan J 1997) An analysis of the war of attrition and the all-pay auction. J Econ Theory 72: Maskin E, Riley JG 1984) Optimal auctions with risk aerse buyers. Econometrica 6: Matthews S 1987) Comparing auctions for risk aerse players: a player s point of iew. Econometrica 55: Milgrom P, Weber R 1982) A theory of auctions and competitie bidding. Econometrica 50: Monderer D, Tennenholtz M 2000) K-price auctions. Games Econ Beha 31: Myerson RB 1981) Optimal auction design. Math Oper Res 6:58 73 Noussair C, Siler J 2005) Behaior in all-pay auctions with incomplete information. Games Econ Beha Forthcoming) Riley JG, Samuelson WF 1981) Optimal auctions. Am Econ Re 71: Tullock G 1980) Efficient rent-seeking. In: Buchanan J. et al ed) Towards a theory of the rentseeking. A&M Uniersity Press, College station Vickrey W 1961) Counterspeculation, auctions, and competitie sealed tenders. J Finan 16:8 37 Weber R 1985) Auctions and competitie bidding. In: Young HP ed) Fair Allocation. American Mathematical Society, Proidence
All-Pay Auctions with Risk-Averse Players
All-Pay Auctions with Risk-Aerse Players Gadi Fibich Arieh Gaious Aner Sela December 17th, 2005 Abstract We study independent priate-alue all-pay auctions with risk-aerse players. We show that: 1) Players
More informationAuction Theory. Philip Selin. U.U.D.M. Project Report 2016:27. Department of Mathematics Uppsala University
U.U.D.M. Project Report 2016:27 Auction Theory Philip Selin Examensarbete i matematik, 15 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Juni 2016 Department of Mathematics Uppsala Uniersity
More informationAuction Theory Lecture Note, David McAdams, Fall Bilateral Trade
Auction Theory Lecture Note, Daid McAdams, Fall 2008 1 Bilateral Trade ** Reised 10-17-08: An error in the discussion after Theorem 4 has been corrected. We shall use the example of bilateral trade to
More informationOptimal auctions with endogenous budgets
Optimal auctions with endogenous budgets Brian Baisa and Stanisla Rabinoich September 14, 2015 Abstract We study the benchmark independent priate alue auction setting when bidders hae endogenously determined
More informationGame Theory Solutions to Problem Set 11
Game Theory Solutions to Problem Set. A seller owns an object that a buyer wants to buy. The alue of the object to the seller is c: The alue of the object to the buyer is priate information. The buyer
More informationKIER DISCUSSION PAPER SERIES
KIER DISCUSSION PAPER SERIES KYOTO INSTITUTE OF ECONOMIC RESEARCH http://www.kier.kyoto-u.ac.jp/index.html Discussion Paper No. 657 The Buy Price in Auctions with Discrete Type Distributions Yusuke Inami
More informationOnline Appendix for The E ect of Diversi cation on Price Informativeness and Governance
Online Appendix for The E ect of Diersi cation on Price Informatieness and Goernance B Goernance: Full Analysis B. Goernance Through Exit: Full Analysis This section analyzes the exit model of Section.
More informationInformative advertising under duopoly
Informatie adertising under duopoly Scott McCracken June 6, 2011 Abstract We consider a two-stage duopoly model of costless adertising: in the first stage each firm simultaneously chooses the accuracy
More informationNBER WORKING PAPER SERIES BIDDING WITH SECURITIES: AUCTIONS AND SECURITY DESIGN. Peter M. DeMarzo Ilan Kremer Andrzej Skrzypacz
NBER WORKING PAPER SERIES BIDDING WITH SECURITIES: AUCTIONS AND SECURITY DESIGN Peter M. DeMarzo Ilan Kremer Andrzej Skrzypacz Working Paper 10891 http://www.nber.org/papers/w10891 NATIONAL BUREAU OF ECONOMIC
More informationRevenue Equivalence and Income Taxation
Journal of Economics and Finance Volume 24 Number 1 Spring 2000 Pages 56-63 Revenue Equivalence and Income Taxation Veronika Grimm and Ulrich Schmidt* Abstract This paper considers the classical independent
More informationNotes on Auctions. Theorem 1 In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy.
Notes on Auctions Second Price Sealed Bid Auctions These are the easiest auctions to analyze. Theorem In a second price sealed bid auction bidding your valuation is always a weakly dominant strategy. Proof
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012 The Revenue Equivalence Theorem Note: This is a only a draft
More informationAll Equilibrium Revenues in Buy Price Auctions
All Equilibrium Revenues in Buy Price Auctions Yusuke Inami Graduate School of Economics, Kyoto University This version: January 009 Abstract This note considers second-price, sealed-bid auctions with
More informationAuctions That Implement Efficient Investments
Auctions That Implement Efficient Investments Kentaro Tomoeda October 31, 215 Abstract This article analyzes the implementability of efficient investments for two commonly used mechanisms in single-item
More informationIndependent Private Value Auctions
John Nachbar April 16, 214 ndependent Private Value Auctions The following notes are based on the treatment in Krishna (29); see also Milgrom (24). focus on only the simplest auction environments. Consider
More informationThe FedEx Problem (Working Paper)
The FedEx Problem (Working Paper) Amos Fiat Kira Goldner Anna R. Karlin Elias Koutsoupias June 6, 216 Remember that Time is Money Abstract Benjamin Franklin in Adice to a Young Tradesman (1748) Consider
More information1 Theory of Auctions. 1.1 Independent Private Value Auctions
1 Theory of Auctions 1.1 Independent Private Value Auctions for the moment consider an environment in which there is a single seller who wants to sell one indivisible unit of output to one of n buyers
More informationMarch 30, Why do economists (and increasingly, engineers and computer scientists) study auctions?
March 3, 215 Steven A. Matthews, A Technical Primer on Auction Theory I: Independent Private Values, Northwestern University CMSEMS Discussion Paper No. 196, May, 1995. This paper is posted on the course
More informationSignaling in an English Auction: Ex ante versus Interim Analysis
Signaling in an English Auction: Ex ante versus Interim Analysis Peyman Khezr School of Economics University of Sydney and Abhijit Sengupta School of Economics University of Sydney Abstract This paper
More informationOptimal Auctions. Game Theory Course: Jackson, Leyton-Brown & Shoham
Game Theory Course: Jackson, Leyton-Brown & Shoham So far we have considered efficient auctions What about maximizing the seller s revenue? she may be willing to risk failing to sell the good she may be
More informationECON Microeconomics II IRYNA DUDNYK. Auctions.
Auctions. What is an auction? When and whhy do we need auctions? Auction is a mechanism of allocating a particular object at a certain price. Allocating part concerns who will get the object and the price
More informationComparing Allocations under Asymmetric Information: Coase Theorem Revisited
Comparing Allocations under Asymmetric Information: Coase Theorem Revisited Shingo Ishiguro Graduate School of Economics, Osaka University 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan August 2002
More informationAuction is a commonly used way of allocating indivisible
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 16. BIDDING STRATEGY AND AUCTION DESIGN Auction is a commonly used way of allocating indivisible goods among interested buyers. Used cameras, Salvator Mundi, and
More informationMeans of Payment and Timing of Mergers and Acquisitions in a Dynamic Economy
Means of Payment and Timing of Mergers and Acquisitions in a Dynamic Economy Alexander S. Gorbenko London Business School Andrey Malenko MIT Sloan School of Management This ersion: January 2014 We are
More informationDiscriminatory Information Disclosure
Discriminatory Information Disclosure Li, Hao Uniersity of British Columbia Xianwen Shi Uniersity of Toronto First Version: June 2, 29 This ersion: May 21, 213 Abstract We consider a price discrimination
More informationRevenue Equivalence and Mechanism Design
Equivalence and Design Daniel R. 1 1 Department of Economics University of Maryland, College Park. September 2017 / Econ415 IPV, Total Surplus Background the mechanism designer The fact that there are
More informationFee versus royalty licensing in a Cournot duopoly model
Economics Letters 60 (998) 55 6 Fee versus royalty licensing in a Cournot duopoly model X. Henry Wang* Department of Economics, University of Missouri, Columbia, MO 65, USA Received 6 February 997; accepted
More informationGames of Incomplete Information ( 資訊不全賽局 ) Games of Incomplete Information
1 Games of Incomplete Information ( 資訊不全賽局 ) Wang 2012/12/13 (Lecture 9, Micro Theory I) Simultaneous Move Games An Example One or more players know preferences only probabilistically (cf. Harsanyi, 1976-77)
More informationSingle-Parameter Mechanisms
Algorithmic Game Theory, Summer 25 Single-Parameter Mechanisms Lecture 9 (6 pages) Instructor: Xiaohui Bei In the previous lecture, we learned basic concepts about mechanism design. The goal in this area
More information1 Auctions. 1.1 Notation (Symmetric IPV) Independent private values setting with symmetric riskneutral buyers, no budget constraints.
1 Auctions 1.1 Notation (Symmetric IPV) Ancient market mechanisms. use. A lot of varieties. Widespread in Independent private values setting with symmetric riskneutral buyers, no budget constraints. Simple
More informationUp till now, we ve mostly been analyzing auctions under the following assumptions:
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 7 Sept 27 2007 Tuesday: Amit Gandhi on empirical auction stuff p till now, we ve mostly been analyzing auctions under the following assumptions:
More informationAuctions: Types and Equilibriums
Auctions: Types and Equilibriums Emrah Cem and Samira Farhin University of Texas at Dallas emrah.cem@utdallas.edu samira.farhin@utdallas.edu April 25, 2013 Emrah Cem and Samira Farhin (UTD) Auctions April
More informationHurdle Rates and Project Development Efforts. Sunil Dutta University of California, Berkeley Qintao Fan University of California, Berkeley
THE ACCOUNTING REVIEW Vol. 84, No. 2 2009 pp. 405 432 DOI: 10.2308/ accr.2009.84.2.405 Hurdle Rates and Project Deelopment Efforts Sunil Dutta Uniersity of California, Bereley Qintao Fan Uniersity of California,
More informationProject Selection: Commitment and Competition
Project Selection: Commitment and Competition Vidya Atal Montclair State Uniersity Talia Bar Uniersity of Connecticut Sidhartha Gordon Sciences Po Working Paper 014-8 July 014 365 Fairfield Way, Unit 1063
More informationRobust Trading Mechanisms with Budget Surplus and Partial Trade
Robust Trading Mechanisms with Budget Surplus and Partial Trade Jesse A. Schwartz Kennesaw State University Quan Wen Vanderbilt University May 2012 Abstract In a bilateral bargaining problem with private
More informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationCommittees and rent-seeking effort under probabilistic voting
Public Choice 112: 345 350, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands. 345 Committees and rent-seeking effort under probabilistic voting J. ATSU AMEGASHIE Department of Economics,
More informationEfficient Dissolution of Partnerships and the Structure of Control
Efficient Dissolution of Partnerships and the Structure of Control Emanuel Ornelas and John L. Turner January 29, 2004 Abstract In this paper, we study efficient dissolution of partnerships in a context
More informationCS364A: Algorithmic Game Theory Lecture #3: Myerson s Lemma
CS364A: Algorithmic Game Theory Lecture #3: Myerson s Lemma Tim Roughgarden September 3, 23 The Story So Far Last time, we introduced the Vickrey auction and proved that it enjoys three desirable and different
More informationECON20710 Lecture Auction as a Bayesian Game
ECON7 Lecture Auction as a Bayesian Game Hanzhe Zhang Tuesday, November 3, Introduction Auction theory has been a particularly successful application of game theory ideas to the real world, with its uses
More informationWe examine the impact of risk aversion on bidding behavior in first-price auctions.
Risk Aversion We examine the impact of risk aversion on bidding behavior in first-price auctions. Assume there is no entry fee or reserve. Note: Risk aversion does not affect bidding in SPA because there,
More informationAuctions in the wild: Bidding with securities. Abhay Aneja & Laura Boudreau PHDBA 279B 1/30/14
Auctions in the wild: Bidding with securities Abhay Aneja & Laura Boudreau PHDBA 279B 1/30/14 Structure of presentation Brief introduction to auction theory First- and second-price auctions Revenue Equivalence
More informationCS 573: Algorithmic Game Theory Lecture date: March 26th, 2008
CS 573: Algorithmic Game Theory Lecture date: March 26th, 28 Instructor: Chandra Chekuri Scribe: Qi Li Contents Overview: Auctions in the Bayesian setting 1 1 Single item auction 1 1.1 Setting............................................
More informationThe Value of the Freezeout Option
1 The Value of the Freezeout Option Zohar Goshen and Zi Wiener The Hebrew Uniersity of Jerusalem Jerusalem, 91905 ISRAEL msgoshen@mscc.huji.ac.il mswiener@mscc.huji.ac.il Introduction According to Delaware
More informationGame Theory Lecture #16
Game Theory Lecture #16 Outline: Auctions Mechanism Design Vickrey-Clarke-Groves Mechanism Optimizing Social Welfare Goal: Entice players to select outcome which optimizes social welfare Examples: Traffic
More informationAUCTIONEER ESTIMATES AND CREDULOUS BUYERS REVISITED. November Preliminary, comments welcome.
AUCTIONEER ESTIMATES AND CREDULOUS BUYERS REVISITED Alex Gershkov and Flavio Toxvaerd November 2004. Preliminary, comments welcome. Abstract. This paper revisits recent empirical research on buyer credulity
More informationQuality Upgrades and (the Loss of) Market Power in a Dynamic Monopoly Model
Quality Upgrades and (the Loss of) Market Power in a Dynamic Monopoly Model James J. Anton Duke Uniersity Gary Biglaiser 1 Uniersity of North Carolina, Chapel Hill February 2007 PRELIMINARY- Comments Welcome
More informationRecap First-Price Revenue Equivalence Optimal Auctions. Auction Theory II. Lecture 19. Auction Theory II Lecture 19, Slide 1
Auction Theory II Lecture 19 Auction Theory II Lecture 19, Slide 1 Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 2 Motivation
More informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationIdeal Bootstrapping and Exact Recombination: Applications to Auction Experiments
Ideal Bootstrapping and Exact Recombination: Applications to Auction Experiments Carl T. Bergstrom University of Washington, Seattle, WA Theodore C. Bergstrom University of California, Santa Barbara Rodney
More informationLoss Aversion and Insider Trading
4 Loss Aersion and Insider Trading SAMUEL OUZAN * [Preliminary ersion. Please do not quote] First ersion, April 6 th 014 ABSTRACT This study analyses equilibrium trading strategies and market quality in
More informationLong run equilibria in an asymmetric oligopoly
Economic Theory 14, 705 715 (1999) Long run equilibria in an asymmetric oligopoly Yasuhito Tanaka Faculty of Law, Chuo University, 742-1, Higashinakano, Hachioji, Tokyo, 192-03, JAPAN (e-mail: yasuhito@tamacc.chuo-u.ac.jp)
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationSandra Ludwig; Philipp C. Wichardt und Hanke Wickhorst: Overconfidence Can Improve an Agent s Relative and Absolute Performance in Contests
Sandra Ludwig; Philipp C. Wichardt und Hanke Wickhorst: Overconfidence Can Improve an Agent s Relative and Absolute Performance in Contests Munich Discussion Paper No. 2010-35 Department of Economics University
More informationHybrid Markets, Tick Size and Investor Welfare 1
Hybrid Markets, Tick Size and Inestor Welfare Egenia Portniaguina Michael F. Price College of Business Uniersity of Oklahoma Dan Bernhardt Department of Economics, Uniersity of Illinois Eric Hughson Leeds
More informationAuction Theory: Some Basics
Auction Theory: Some Basics Arunava Sen Indian Statistical Institute, New Delhi ICRIER Conference on Telecom, March 7, 2014 Outline Outline Single Good Problem Outline Single Good Problem First Price Auction
More informationAuctions. Microeconomics II. Auction Formats. Auction Formats. Many economic transactions are conducted through auctions treasury bills.
Auctions Microeconomics II Auctions Levent Koçkesen Koç University Many economic transactions are conducted through auctions treasury bills art work foreign exchange antiques publicly owned companies cars
More informationA folk theorem for one-shot Bertrand games
Economics Letters 6 (999) 9 6 A folk theorem for one-shot Bertrand games Michael R. Baye *, John Morgan a, b a Indiana University, Kelley School of Business, 309 East Tenth St., Bloomington, IN 4740-70,
More informationOn the existence of coalition-proof Bertrand equilibrium
Econ Theory Bull (2013) 1:21 31 DOI 10.1007/s40505-013-0011-7 RESEARCH ARTICLE On the existence of coalition-proof Bertrand equilibrium R. R. Routledge Received: 13 March 2013 / Accepted: 21 March 2013
More informationAuctions. Agenda. Definition. Syllabus: Mansfield, chapter 15 Jehle, chapter 9
Auctions Syllabus: Mansfield, chapter 15 Jehle, chapter 9 1 Agenda Types of auctions Bidding behavior Buyer s maximization problem Seller s maximization problem Introducing risk aversion Winner s curse
More informationForeign Bidders Going Once, Going Twice... Government Procurement Auctions with Tariffs
Foreign Bidders Going Once, Going Twice... Government Procurement Auctions with Tariffs Matthew T. Cole (Florida International University) Ronald B. Davies (University College Dublin) Working Paper: Comments
More informationQuality, Upgrades and Equilibrium in a Dynamic Monopoly Market
Quality, Upgrades and Equilibrium in a Dynamic Monopoly Market James J. Anton and Gary Biglaiser August, 200 Abstract We examine an in nite horizon model of quality growth for a durable goods monopoly.
More informationQuality, Upgrades and Equilibrium in a Dynamic Monopoly Market
Quality, Upgrades and Equilibrium in a Dynamic Monopoly Market James J. Anton and Gary Biglaiser April 23, 200 Abstract We examine an in nite horizon model of quality growth for a durable goods monopoly.
More information6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More informationw E(Q w) w/100 E(Q w) w/
14.03 Fall 2000 Problem Set 7 Solutions Theory: 1. If used cars sell for $1,000 and non-defective cars have a value of $6,000, then all cars in the used market must be defective. Hence the value of a defective
More informationISSN BWPEF Uninformative Equilibrium in Uniform Price Auctions. Arup Daripa Birkbeck, University of London.
ISSN 1745-8587 Birkbeck Working Papers in Economics & Finance School of Economics, Mathematics and Statistics BWPEF 0701 Uninformative Equilibrium in Uniform Price Auctions Arup Daripa Birkbeck, University
More informationDay 3. Myerson: What s Optimal
Day 3. Myerson: What s Optimal 1 Recap Last time, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] and density f i residual valuation
More informationAll-Pay Contests. (Ron Siegel; Econometrica, 2009) PhDBA 279B 13 Feb Hyo (Hyoseok) Kang First-year BPP
All-Pay Contests (Ron Siegel; Econometrica, 2009) PhDBA 279B 13 Feb 2014 Hyo (Hyoseok) Kang First-year BPP Outline 1 Introduction All-Pay Contests An Example 2 Main Analysis The Model Generic Contests
More informationProblem Set 3: Suggested Solutions
Microeconomics: Pricing 3E00 Fall 06. True or false: Problem Set 3: Suggested Solutions (a) Since a durable goods monopolist prices at the monopoly price in her last period of operation, the prices must
More informationBijections for a class of labeled plane trees
Bijections for a class of labeled plane trees Nancy S. S. Gu,2, Center for Combinatorics Nankai Uniersity Tianjin 0007 PR China Helmut Prodinger 2 Department of Mathematical Sciences Stellenbosch Uniersity
More informationSequential Auctions and Auction Revenue
Sequential Auctions and Auction Revenue David J. Salant Toulouse School of Economics and Auction Technologies Luís Cabral New York University November 2018 Abstract. We consider the problem of a seller
More informationCESifo Working Paper Series
CESifo Working Paper Series DISORGANIZATION AND FINANCIAL COLLAPSE Dalia Marin Monika Schnitzer* Working Paper No. 339 September 000 CESifo Poschingerstr. 5 81679 Munich Germany Phone: +49 (89) 94-1410/145
More informationMicroeconomic Theory III Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIT 14.123 (2009) by
More informationA lower bound on seller revenue in single buyer monopoly auctions
A lower bound on seller revenue in single buyer monopoly auctions Omer Tamuz October 7, 213 Abstract We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with
More informationSequential Investment, Hold-up, and Strategic Delay
Sequential Investment, Hold-up, and Strategic Delay Juyan Zhang and Yi Zhang December 20, 2010 Abstract We investigate hold-up with simultaneous and sequential investment. We show that if the encouragement
More informationAuction theory. Filip An. U.U.D.M. Project Report 2018:35. Department of Mathematics Uppsala University
U.U.D.M. Project Report 28:35 Auction theory Filip An Examensarbete i matematik, 5 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Augusti 28 Department of Mathematics Uppsala University
More informationThe Value of Information in Asymmetric All-Pay Auctions
The Value of Information in Asymmetric All-Pay Auctions Christian Seel Maastricht University, Department of Economics This version: October 14, 2013 Abstract This paper analyzes a two-player all-pay auction
More informationSignaling Games. Farhad Ghassemi
Signaling Games Farhad Ghassemi Abstract - We give an overview of signaling games and their relevant solution concept, perfect Bayesian equilibrium. We introduce an example of signaling games and analyze
More informationAuctions 1: Common auctions & Revenue equivalence & Optimal mechanisms. 1 Notable features of auctions. use. A lot of varieties.
1 Notable features of auctions Ancient market mechanisms. use. A lot of varieties. Widespread in Auctions 1: Common auctions & Revenue equivalence & Optimal mechanisms Simple and transparent games (mechanisms).
More informationChapter 3. Dynamic discrete games and auctions: an introduction
Chapter 3. Dynamic discrete games and auctions: an introduction Joan Llull Structural Micro. IDEA PhD Program I. Dynamic Discrete Games with Imperfect Information A. Motivating example: firm entry and
More informationSocial Network Analysis
Lecture IV Auctions Kyumars Sheykh Esmaili Where Are Auctions Appropriate? Where sellers do not have a good estimate of the buyers true values for an item, and where buyers do not know each other s values
More informationMechanism Design and Auctions
Mechanism Design and Auctions Game Theory Algorithmic Game Theory 1 TOC Mechanism Design Basics Myerson s Lemma Revenue-Maximizing Auctions Near-Optimal Auctions Multi-Parameter Mechanism Design and the
More informationThe Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims
he Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims Naoto Kunitomo and Akihiko akahashi August 1999 (Fourth Reision) Abstract We propose a new methodology for the aluation
More informationParkes Auction Theory 1. Auction Theory. Jacomo Corbo. School of Engineering and Applied Science, Harvard University
Parkes Auction Theory 1 Auction Theory Jacomo Corbo School of Engineering and Applied Science, Harvard University CS 286r Spring 2007 Parkes Auction Theory 2 Auctions: A Special Case of Mech. Design Allocation
More informationUniversity of Hong Kong
University of Hong Kong ECON6036 Game Theory and Applications Problem Set I 1 Nash equilibrium, pure and mixed equilibrium 1. This exercise asks you to work through the characterization of all the Nash
More informationColumbia University. Department of Economics Discussion Paper Series. Bidding With Securities: Comment. Yeon-Koo Che Jinwoo Kim
Columbia University Department of Economics Discussion Paper Series Bidding With Securities: Comment Yeon-Koo Che Jinwoo Kim Discussion Paper No.: 0809-10 Department of Economics Columbia University New
More informationOn Forchheimer s Model of Dominant Firm Price Leadership
On Forchheimer s Model of Dominant Firm Price Leadership Attila Tasnádi Department of Mathematics, Budapest University of Economic Sciences and Public Administration, H-1093 Budapest, Fővám tér 8, Hungary
More informationEmpirical Tests of Information Aggregation
Empirical Tests of Information Aggregation Pai-Ling Yin First Draft: October 2002 This Draft: June 2005 Abstract This paper proposes tests to empirically examine whether auction prices aggregate information
More informationAuditing in the Presence of Outside Sources of Information
Journal of Accounting Research Vol. 39 No. 3 December 2001 Printed in U.S.A. Auditing in the Presence of Outside Sources of Information MARK BAGNOLI, MARK PENNO, AND SUSAN G. WATTS Received 29 December
More informationPricing Services Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions?
Uniersity of Pennsylania ScholarlyCommons Operations, Information and Decisions Papers Wharton Faculty Research 0 Pricing Serices Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions? Gerard.
More informationPartial privatization as a source of trade gains
Partial privatization as a source of trade gains Kenji Fujiwara School of Economics, Kwansei Gakuin University April 12, 2008 Abstract A model of mixed oligopoly is constructed in which a Home public firm
More informationProblem Set 3: Suggested Solutions
Microeconomics: Pricing 3E Fall 5. True or false: Problem Set 3: Suggested Solutions (a) Since a durable goods monopolist prices at the monopoly price in her last period of operation, the prices must be
More informationWorking Paper. R&D and market entry timing with incomplete information
- preliminary and incomplete, please do not cite - Working Paper R&D and market entry timing with incomplete information Andreas Frick Heidrun C. Hoppe-Wewetzer Georgios Katsenos June 28, 2016 Abstract
More informationSubjects: What is an auction? Auction formats. True values & known values. Relationships between auction formats
Auctions Subjects: What is an auction? Auction formats True values & known values Relationships between auction formats Auctions as a game and strategies to win. All-pay auctions What is an auction? An
More informationEconS Games with Incomplete Information II and Auction Theory
EconS 424 - Games with Incomplete Information II and Auction Theory Félix Muñoz-García Washington State University fmunoz@wsu.edu April 28, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 9 April
More informationBACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL. James A. Ligon * University of Alabama. and. Paul D. Thistle University of Nevada Las Vegas
mhbr\brpam.v10d 7-17-07 BACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL James A. Ligon * University of Alabama and Paul D. Thistle University of Nevada Las Vegas Thistle s research was supported by a grant
More informationHW Consider the following game:
HW 1 1. Consider the following game: 2. HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child,
More informationOptimal Fees in Internet Auctions
Optimal Fees in Internet Auctions Alexander Matros a,, Andriy Zapechelnyuk b a Department of Economics, University of Pittsburgh, PA, USA b Kyiv School of Economics, Kyiv, Ukraine January 14, 2008 Abstract
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationExpected utility inequalities: theory and applications
Economic Theory (2008) 36:147 158 DOI 10.1007/s00199-007-0272-1 RESEARCH ARTICLE Expected utility inequalities: theory and applications Eduardo Zambrano Received: 6 July 2006 / Accepted: 13 July 2007 /
More information