Resource allocation with costly participation

Resource allocation with costly participation ∗ Ali Kakhbod † Massachusetts Institute of T echnology Abstract W e propose a new all- pa y auction for mat in which risk-lovi ng bidders p a y a con- stant fee each time they bid for an object whos e monetary val ue is common knowl- edge a mong the bidders, and bidding fees are the only s ource of benefit for the se ll er . W e sho w that for t he propose d model ther e exists a unique Symmetric Subgame Per - fect Equilib rium (SSPE). The characterized SSPE is stationary when re-entry in the auction is allo w ed, and it is Markov perfect when re-entry is forbidden. Furthe rmore, w e fully characterize the expected rev enue of the seller . Generally , with o r without re-entr y , it is more beneficial for the seller to choo se v (va lue of the object), s (sale price), and c (bidding fee) su ch that v − s c becomes sufficiently large. In particular , when re-entr y is per mitted: the e xpected rev enue of the s e ller is independent of the number o f bidders, d ecreasing in t he sale price, increasing in t h e v alue of the object, and decreasing in the bidding fee; Moreov er , th e seller ’s rev enue is equal to the va lue of the object when pla y e r s are risk ne u tral, and it is st ric tly g reater than the v alue of the object when bidders are risk-loving. W e further show that allowing r e -entry can be important in practice. Be cause , if the seller w ere to run such an auction withou t allo wing re-entry , the auction w o uld last a long time, and for almost all of its dura- tion ha v e only tw o remaining pla y e r s . Th u s, t he s e ll er ’s rev enue r e lies on tho s e t w o pla y e r s being willing to participate, without any breaks , in an auction that might last for t h o usands of round s. Keyw o rds: Pa y-to-bid auctions, risk-lov ing bidders , risk neut r al bidde rs, rev enue management. JEL Classi fi cat ion: C61, C73, D44 ∗ The first draft October 2010. This version December 2015 . † Department of Economics, Ma ssachusetts Institute of T ec hnology (MIT) , C ambridge, MA 021 39, USA. Email: akakhbod@mit .edu . 1 Introduc t ion Ov er the past few y ears, certain internet auctions, commonly referred to a s pa y-to-bid auctions, ha v e seen a rapid rise in popularity . The basic structure of the auction is as follo ws: (i) The price starts at zero. (ii) Each bid costs 1 cent for any bidd e r . (iii) Ea ch bid adds 10 seconds to the game clock, so the auction nev er ends while there are still willing bidders. (iv) If the clock reaches zero, the final bidder wins the object and p ay s the sell p rice. Due to the str ucture of these auctions, when the price is lo w , each pla y er prefers to bid if afterw ards no opponent will enter before the clock runs out. Ho w ev er , if pla yers are too likely to enter in the future, no play er will want to bid. The mechanic of p ay -to-bid auctions (also kno wn as penny auctions) has parallels with the all-pa y auctions and the war -of-attritio n game format, [ 6 ]. B ecause with the classic all-pa y auction, [ 14 ], all bidders pay a p ositiv e sum if the y choose to participate in the auction, regardless of whether they win or lose. The costly action, required to become the winning bidde r , is also reminiscent of the war -of-attrition game for mat. All- pa y auction has been applied to describe rent-seeking, political contests, R &D races, and job-pr omotions. Under full information in one-shot (first price) all-pa y auctions, the full characterization of equilibria is giv en by [ 15 ]. A second-price all-pa y auction, also called w ar of a ttrition, w as propos ed b y [ 16 ] in the context of theoretical biology and, under full infor mation, the full characterization of equilibria of it is giv en b y [ 5 ]. The w ar of attrition game has bee n applied to many problems in economics, most notably industrial competition (e. g. [ 1 ]), public goods pro vision (e.g. [ 7 ]) and Bar gaining (e.g. [ 2 ] and [ 3 ]). Recently , [ 4 ] pro vides an equilibrium pa y off characterizatio n for general class of all-pa y contests. In pa y-to-bid auctions, since the main cost incurred b y bidders come in the form of bidding fees (which are individually small), bidders are not required to place a bid ev ery round in order to sta y in the auction (that is, e ntry and re-entry at any round of the auction is allo w ed). In contrast, re-entry in all-pa y auctions is forbidden. Therefore, a l l- pa y auctions nev er allo w the actual winner to pa y less than the losers, but in pa y-to-bid auctions it happens, in practice, relativ ely often 1 . 1 In more detail, in both all-pay auctions (in p a rticular w ar of attrition) a nd pa y-to-bid auctions, play ers 2 What really sets this particular auction for mat apart fr om other nontraditional auc- tions is the success of its real w orld imple mentatio n which appears to be highly prof- itable for the w ebsite operator/auctioneer (who is the sole seller of goods). I n December 2008, 14 new w ebsites conducted such auctions; b y No v ember 2009, the for mat had in- creased to 35 w ebsites. Ov e r the same time span, traf fic among these sites has increased from 1.2 million to 3.0 million uniq ue visitors per month. For comparison, traffic at eba y .com fluctuated around 75 million unique visitors per month throughout that pe- riod. Pa y-to-bid auctions ha v e gar nered 4% of the traf fic he ld b y the undisputed leader in online auctions, [ 13 ]. 1.1 Risk-lo ving behav ior , H a ving common valuation, and Sw oopo’ s bankr uptcy These internet auctions a re a form of gambling. The bidder dep os its a small fee to pla y , aspiring to a big pay of f of obtaining the item w ell below its v alue, with a major difference that the probabilities of winning are endogenous. Due to the gambling feature of these auctions and the fact that these internet auctions prominently adv ertise themselv es as “entertainment shopping”, the risk-lo ving beha vior (i.e. ha ving a preference for risk) by participants of these auctions is natural 2 . Generally , in these inter ne t auctions the valuatio n of the item is kno wn and the same among all potential bidders. Since all items are new , unopened, and readily a vail- able from inter net retailers, the market prices of these items a re w ell established. I n fact, one could imagine that the value of the object is the lo w est price for which the item ma y be obtained elsewhere. This feature, unlike in the first and second-price auctions, is quite common in all-pay or war -of-attrition auctions, which are the closest relativ es of must pay a cost for the game to continue and a play er wins when other play ers decide not to pay this cost. Ho w ever , they are different in one notable wa y that in a pa y-to-bid auction each play er can bid in any period independent of the history of the game, unlike in a war of attrition, in which all play ers who continue to play must pa y a cost in eac h period. Therefore, unlike in pa y-to-bid a uctions , in w ar of attrition actual winner nev er pay less than the losers. 2 For example, for video game systems (such as the Pla ystation, W ii, or Xbox) the bidde r s ma y not be able to justify to their spouse, their pa rent, or themselv es spending $450 on a Xbox at a reta il store; ho we ver , the potential to win the Xbox early in the auction f or only a fra c tion of that makes it worth the 1 cent gamble, ev en a t unfair odds. 3 the pa y-to-bid auction. Sw oopo.com is one of such w ebsites that w as initially v ery successf ul. According to an A ugust 2009 article from The Economist , Sw oopo h a s 2.5 million registered users and ear ned 32 million dollars in rev enue, in 2008. Data collected by a blogger sho ws that ov e r April and Ma y of 2009 Sw oopo sold items for an a v erage of 1 8 8% of their listed v alue. On 26 March 2011, Sw oopo’s parent company filed for bankruptcy and shut do wn the auction w ebsite. On 8 February 2012, DealDash the longest running pa y-to- bid auction w ebsite in the U.S. acquired the domain Sw oopo.com and the URL currently redirects to Dea lDash’s o wn webs ite. The exit of Sw oopo might suggest: (i) demand for pa y-to-bid auctions fell drastically . (ii) Sw oopo was mismanaged and made a set of poor strategic choices. These e ffects are empirically disentangled in [ 12 ]. In e arly 2008, when penny auctions are launched, there are v ery few online pay -to-bid auctions, and essentially Sw oopo acts as a monopoly . Ov er time, b y increasing the number of visitors, more entrants come into the market, reducing the lev e l of concentration. When Sw oopo exits, the number of market monthly visitors drops temporarily , but soon rises back to around 0.0075 of Inter net traffic , [ 12 ]. This result, therefore, suggests that after Sw oopo’s exit de mand for pa y-to-bid auctions has continued to remain high. 1.2 Contr ibution of the paper This w ork, in continuatio n of [ 11 ], focuses on risk-lo ving pla y ers with C onst a nt A bsolut e Risk Lo ving (CARL) utilities. W e present a sty liz e d model of a pa y-to-bid auction with the follo wing properties: (i) Bid in g fee is small with respect to the value of the object. (ii) Re-entry in the auction ma y be allo w ed (i.e., all the original play ers can participate in any round of the game) or forbidden (ie, in each r ound of the game only pla yers who ha v e participated in the previous round can participat e). W e first e stablis h that, with or without re-entr y permission, there exists a unique symmetric subgame perfect equilibrium, which is stationary when re-entry is perm itted, and it is Markov perfec t 3 when re-entry is forbidden. 3 A Mar ko v perfect equilibrium is a profile of Markov strategies that yields a Nash equilibrium in ev ery proper subgame. A Markov strategy is one that does not depend at all on variables that a re functions of 4 As sho wn in [ 11 ], when re-entry is per mitted, a closed form expr ession of the seller ’s rev enue is computable that becomes independent of the number of pla yers. The closed form expression rev e als that the seller ’s rev enue is increasing in the v alue of the object, decreasing in the biding fee, decreasing the sale price, and decreasing in the risk- lo ving coef ficient. These results are compatible with the relativ ely high v alue d objects sold in the online pa y-to-bid auctions with a lo w bid fe e and a low sale price. Moreo v er , the seller ’s rev enue is exactly equal to the v alue of the object, when pla y e rs are risk neutral, and is strictly greater than the v alue of the object, when p lay ers are risk-lo ving. Furthermore, for suf ficiently small biding fee, it is strictly greater than a seller ’s rev enue from a standar d lottery , for any number of pla y ers. Here, w e sho w , g enerally , with or without re-entry , it is more beneficial for the seller to choose v (v alue of the object), s (sale price), and c (bidding fee) so that v − s c becomes suf ficiently large. Moreov er , whether or not entry or re-entry is pe rmitted in any round of the game, the strategy that each agent follo ws in equilibrium is indepen- dent of her w ealth/budget lev el. Furthermore, in any sub-game starting from ro und t , t ∈ { 1, 2, 3, · · · } , the expected utility of each p lay er i , whose budget lev el in rou nd t is w i , t , is exactly equal to u ( w i , t ) , where u ( · ) denotes the pla y er ’s utility . W e sho w that allo wing re-entry might be important in practice compared to the scenario in which re-entr y is forbidden. Recall that the seller ’s expected rev enue is increasing with respect to the v alue of the object a nd decreasing with respect to the bid fee and the sale price. Giv en this result, w e e stablish that, in the limit, as the v alue of the object gro ws with respect to the bid fee and the sale price, the expected length of the game tends to infinity . Further mor e, with probability approaching to one, in the game in which re-entry is forbidden, the number of pla y ers will be reduced to only tw o remaining pla y ers that pla y the game until it e nds. Moreo v e r , the expected time in which the number of pla y ers reduces to tw o is negligible in comparison to the e ntir e expected length of the game. In sum, when a high v alued object is sold with a lo w bid fee a nd a lo w sale price, an auction in which re-entry is forbidden not only will last on a v e rage for a v e ry long time, the history of the game except those tha t affec t pay offs. 5 but a lso most likely it will only ha v e tw o pla yers who pla y for almost the entire d uration of the game. Therefore, because the auction will end, if e ither of the tw o remaini ng pla y ers chooses to opt out, d ue to an exogenous reason, it is e asy to see why the auction in which re-entry is forbidden is undesirable from the seller ’s point view . 1.3 Organization of the paper The rest of the paper is organized as follo ws. In S ection 2 the model is precisely de- scribed. In S ection 3 w e discuss about related literature. In S ection 4 w e p resent the properties of the proposed model. W e conclude in S ection 5 . 2 The Mode l Consider the follo wing dynamic game with complete information inv olving an object with monetary v alue v and n strategic bidder/agent/ pla y e r/participant. In each round t , t ∈ { 1, 2, 3, · · · } , of the auction/game pla yer i chooses an action from the set of pure strategies/messag es S = { Bid, No B id } , s i , t ∈ S , and observ e s her opponents actions, s j , t , j 6 = i . In each round of the game, play ers submit their messages simultaneously . Pla ying { Bid } is cos tly . Each pla y er immediately pa ys c < v dollars (the bid fee ) to the seller each time she pla ys { Bid } . In any particular round, If a p lay er is the sole pla y er who pla ys { Bid } , she wins the object and pa ys the sale price. The sale price is fixed a nd denoted by s . In any ro und of the game, if more than one pla y er p la y { Bid } , the game continues in the next r ound. It ma y arise a dif ficulty that no participant p lay s { Bid } , in a particular ro und of the game, that w e resolv e it b y the follo wing assumption. Assumption 1 (T ie B reaking ) . We assume if in a round of the game all the pl ayers play { No Bi d } , they resubmit their m essages, ie , they r e-play that peri od of the game . For this model, w e consider the follow ing tw o scenarios: 6 (i) The game with re-entry option. In this scenario, a ll the original pla y e rs are allo w ed to participate/bid at any round of the game, regardless of the history of the game and the w a y that they beha v ed. (ii) The game withou t re-entry option. In this scenario, the set of participating pla y ers in the next round of the game are the ones who ha v e bid. That is, in any round of the game, a pla yer , who does not bid, will be out fore v er . 3 Related work There has been a great deal of recent interest in pa y-to -bid auctions. Most of this w ork of fers the large re v enues earned by w ebsites like Dealdash.com (see [ 8 , 12 , 10 , 13 , 11 ]). The proposed formulation in our paper i s different fro m the formulations in [ 10 , 1 2 , 13 ] in one notable wa y . In [ 10 , 12 , 13 ], the authors a ssume that, in each round t of the game, a leader is selected randomly fro m the pla y ers participating i n round t (ie, the leade r is chosen randomly from the pla yer s who chose to pla y { Bid } in ro und t ). And the leader wins the object, if non of the rema ining pla yers pla y { Bid } in the next round of the game. This for mulation dif fers fro m ours. Because a play er , in our for mulation, wins the object, if no other pla y er bids (ie, no other pla y e r pla y { Bid } in that particular ro und ) 4 . Another notable difference of our model and the models in [ 8 , 1 0 , 12 ] is that, similar to [ 13 ], w e focus on risk-lo ving pla yers while in [ 8 , 10 , 12 ] the authors a ssume pla y ers are risk neutral. In this respect, our paper is closely related to [ 13 ]. But it is, also, dif ferent from [ 13 ], in formulation of the model, such that w e get a more tract able model with a unique symmetric subgame perfect equilibrium, where in [ 13 ], the focus is on stationary equilibria, and there are multiple symmetric equilibria. Another difference of our w ork 4 W e note that the formulation in [ 1 0 , 12 , 13 ] models the online w ebsites, like Dealdash.com, more accurately , but as [ 10 ] e stablishes, the equilibrium a nalysis is c o mpletely intractable . Therefore, in [ 10 , 12 , 1 3 ] the authors focus only on Markov e quilibria, where play ers instead of conditioning their be havior on the whole histories of the game, they only condition their be havior on the current sta te of the game, whereas in our formulation we do not hav e such a ssumptio n. W e also note that, the models in [ 10 , 12 , 13 ] are also different from one another . For exa mple, in [ 10 ] and [ 13 ] (similar to ours), whenev er a play er pla ys { B id } she pa ys the bidding fee, where as [ 12 ] assumes that only the submitting bidder who w a s chosen to be the next leader ha s to incur the bid cost 7 and [ 13 ] (and [ 8 , 10 , 12 ]) is that the main focus of our pape r is to study the effect of ha ving re-entry in any round of the auction, in comparison to the case where re-entry is forbidden. W e further not e that the models in [ 10 , 12 , 13 ] a re also different fr om one another . For example, in [ 10 ] and [ 13 ] (similar to ours) whenev er a pla y er pla ys { B id } she pa ys the bidding fee, ho w ev er , [ 12 ] assumes that only the submitting bidder , who w as chosen to be the next le ader , has to incur the bid cost. In pa y-to-bid auctions each pla yer is a war e of the number of current pla y ers at each round of the game. W e will also assume that the number of pla y ers is common kno wledge. In [ 8 ], the autho rs inv estigate a situation in which pla y ers are not a w are of the exact number of pla y e rs, and they improperly estimate the number of participants. This assumption pro v es to ha v e d ramatic impact on the analysis and, in particular , on the expected rev e nue of the seller . It is often a ssumed that incurred or sunk costs do not directly af fect individual’s decision but ma y ha v e an effect on future decisions 5 . In [ 12 ], the a uthor has built a model of pay -to-bid auction, similar to [ 10 ], in which bid d e rs suff er fr om the sunk cost fallacy. Further , bidders beha v e so naïv ely in the sense that the y do not foresee the ef fect of their loses on their future preferences. Therefore, bidders o v erbid more as they become more monetarily inv ested in the auction. 4 Propertie s of the Game In this section, w e inv estigate the properties of the model pro posed in section 2 . W e first define the utility function of the play ers. 5 There is a large body of literature in psy chology and behavioral economics that address this issue. For example, previous expenditures can affect future decisions by changing one’s rema ining disposable income. The sunk cost fallac y is naturally connected to pay-to-bid auctions. This is due to the fact that in these auctions pla y ers/participa nts spend more and more mone y as time goes on, and p a rticipants who suffer from the sunk cost fallacy will feel a greater a nd grea te r need to justify their loses by winning the prize. 8 4.1 Constant Absolute Risk-Lov ing Utili t y Function A V on Neumann-Morganst e r n utility function u : R → R is said to be Constant Absolute Risk Lov ing (CARL), if the Arro w-Pratt measure of risk R ( x ) = − u ′′ ( x ) u ′ ( x ) is equa l to some constant ρ ( ρ < 0, called risk-lo ving coefficient) for all x . Thus, any CARL utility function has the unique follo wing form (up to an affi ne transfor ma tio n ) u ( x ) = 1 − e − ρ x ρ + K . (1) W e consider pla yers are risk-lo ving and, therefore, their utility functio ns are in the form of ( 1 ). For simplicity , w e focus on the fundamental form in ( 1 ) so that u ( 0 ) = 0, ie , K = 0. Further , w e assume pla y ers do not discount futur e consumption. Remark 1. In the limit whe n ρ tends to 0, the utility function in ( 1 ) converges (point wise) to the risk neutral utility function u ( x ) = x . 4.2 Equilibr ium analysis In the follo wing theorem, w e establish that for the model proposed in S ection 2 there is a unique Symmetric 6 Sub-game Perfect Equilibrium (SSPE), which is stationary 7 when re-entry is allo w ed, and it is Marko v perfect 8 when re-entry is forbidden. 6 In this paper we focus on characteriz ing symmetric equilibria. There might be asymmetric equilibria but is not of our interest. In Corollary 1 we comment on the appropriateness of a nalyzing symmetric equilibria. 7 It is stationary since the strategy each pla yer follows in each roun d t of the ga me depends only on the relev ant state v ariab le s: the number of pla yers n , the sale price s , the bid fee c , and the object’s v a lue v . 8 It is Markov perfec t since the strategy ea c h pla y er follo ws in each round t of the ga me depends only on the relevant state v aria b le s: number of remaining pla yers n t , the sale price s , the bid fee c , and the object’s value v . W e further note that, a Markov perfect equilibrium is a profile of Markov strategies that yields a Nash equilibrium in ev er y proper subgame. A Markov strategy is one that does not depend at all on v aria bles that are functions of the history of the game except those that affect pa yo ffs. 9 Theorem 1. In any symmetric s ub-ga me perfect equilibrium of the game pr oposed in section 2 , with n ≥ 2 pl aye rs, the foll owi ng properties ar e satisfied. I. Both with and without re-entry , in any s ub-game starting from ro und t , t ∈ { 1, 2, 3, · · · } , the expec te d utility of player i, whose wealth/budget l evel is w i , t , is exactly equal to u ( w i , t ) . II. When re-entry is allowed, in each period, each player purely randomizes over { Bid, No Bid } and chooses to play { Bid } wi th the f ollowing stationary pro bability: 1 − n − 1 s u ( c ) u ( v − s ) . Moreover , the s y mmetric sub-game p e rfect equilibri um is unique and stationary . III. When re-entry is not allowed, i n each period, each player purely randomizes over { Bid, No Bid } and chooses to play { Bid } wi th the f ollowing probability: 1 − n t − 1 s u ( c ) u ( v − s ) , where n t is the number of players in round t. Moreover , the s y mmetric sub-game p e rfect equilibri um is unique and Marko v perfect . Proo f. See Appendix. Fro m the abov e theorem, w e directly obtain the follo wing cor ollaries. Corollar y 1. Since the decis ion taken by the players is independe nt of their we al th level, we are able to focus on the symmetric sub-game perfe c t equilibria despite wealth asymm etries. Corollar y 2. Both with and without re-entry , in each peri od of the g ame, the probability of playing { Bid } is stric tly greater than zer o and stri ctly less than one. That is, in each period of the game, each player is indifferent betwee n p laying { Bid } and { No Bid } . Corollar y 3. Theorem 1 is also hold, when players are risk-neutral 9 . In partic ular , when players are risk neutral, in any ro und t , t ∈ { 1, 2, 3, · · · } , of the game: (1) When re-entry i s allowe d , each 9 The risk neutral case can be easily implied b y taking a limit ρ → 0 in ( 1 ), that implies (point-wise) u ( x ) = x for any x . 10 player chooses to play { Bid } with the statio nary probability 1 − n − 1 q c v − s . (2) When re-entry is not allowed, each player chooses to play { Bid } with the pr obability 1 − n t − 1 q c v − s . Corollar y 4. Both with and without r e-entry , the proba bi lity that any bidding p layer 10 wins the object, in any roun d of the game, is equal to u ( c ) u ( v − s ) . 4.3 Rev enue analysis In Theorem 2 , w e focus on computing the seller ’s p r ofit, in equilibrium, only from the bid fees. Theorem 2. Both with and without re-ent ry , in the unique SSPE ch aracterized in Theorem 1 , the expected earning of the seller only from the bid fees is exactly equal to ∞ ∑ t = 1 " n t p t 1 − ( 1 − p t ) n t × c × t − 1 ∏ s = 1  1 − n s p s ( 1 − p s ) n s − 1 1 − ( 1 − p s ) n s  # , (2) where n t denotes the number of remaining pl ayers. Note that when re-ent ry is permitted n t = n, for all t . Proo f. See Appendix. Theorem 3. For a g iven bidding fee c > 0 , both with and without re-entry , in the unique SSPE characterized in Theorem 1 , when v − s c tends to infinity , the e x pected pr ofit of the seller is maximized. Proo f. See Appendix. This result is intuitiv e because risk-lo ving bidders prefer lo w probability a nd po- tentially v ery lucrativ e gambles to gamble that are e q ual in expectations but hav e lo wer v ariance. Moreo v er , the seller is able to ear n large profits ev en when the bid fee and the sale price are small compared to the object’s v alue because the expected number 10 Bidding play er means a play er who plays { Bid } . 11 of bids and the e xpected length of these auctions are large enough to compensate this discrepancy . W e inv estigate more about this property in Theorem 5 . When re-entry is allo w e d, since the SSPE described in Theorem 1 becomes station- ary , it is doable to simplify Eq. ( 2 ) and deriv e its closed for m. A s a result, w e will be able to deriv e a few interesting insights from its expression. As sho wn in [ 11 ], these insights are summarized as follo ws. The expected rev enue of the se lle r is decreasing in the sale price ( s ) , increasing in the v alue of the object ( v ) , and decreasing in the bidding fee ( c ) . Moreo v er , the expected rev enue of the seller is i nd e pendent of the number of pla y ers. In addition, for any number of pla yers, with suf ficiently small biding fee , the seller ’ rev enue is str ictly gr e ater than seller ’s re v enue from any standard lottery 11 . T o be precise, w e ha v e the follo wing theorem. Theorem 4 ([ 11 ]) . Consider the SSPE of the Bi d-No Bid game when re-entry is perm i tted. If players are r isk neutral (ie, ρ = 0 in ( 1 ) ), then I. The expected r e v enue of the seller i s equal to the value of the obje ct ( v ) . If players are r isk-loving (ie, ρ < 0 in ( 1 ) ), then II. The expected r ev e nu e of the sell er is equal to u ( v − s ) c u ( c ) + s. III. The e xpected revenue of the seller is independent of the number of pl aye rs. IV . The e xpected revenue of the seller is strictly greater than the value of the object ( v ) . V . The expected revenue of the seller is strictly incr e asing i n the value of the object ( v ) , de- creasing in the s ale pri c e ( s ) , decreasing in the biding fe e ( c ) , and decreasing in ρ (ie, the seller earns more wh e n players are mor e risk-loving). VI. The maximum r ev enue the sel ler can earn is u ( v ) = 1 − e − ρ v ρ . V .II For any number of players , with sufficiently small biding fe e, the seller ’ s revenue is strictly greater than a standard lottery . 11 The standard lottery is defined as follows. A seller offers lotter y tickets for an object of (monetary) v alue v to n potential b uyers with CARL utility . Cost of buying a ticket for eac h pla yer is e qual to c > 0, and the winner is chosen ra ndomly . 12 4.4 W ith entr y vs. W ithout re-entr y In this section, w e inv estigate the effect of per mitting re-entry in the game proposed in S ection 2 . As sho wn in S e ctio n 4.3 , generally , with or without re-entry , it is more beneficial for the seller to choos e v , s , and c such that v − s c becomes suffic ie ntly large. I n the follo wing theorem, w e characterize the impact of v − s c → ∞ on the auction when re-entry is forbidden. Theorem 5. Without re-entry option, as v − s c tends to infinity the following properties ar e hold. I. The expected number of r ounds of the game tends to infinity . II. When n > 2 , with probability approaching to one, the auction is r e duced to one with only two remaining playe r s before the auction ends. III. The expected time unt il n − 2 players e xit in the auction tends to zer o, with respet to the overall expected length of the auction. Proo f. See Appendix. As the abo v e theor em show s, the expected length of the a uct ion, without re-entry option, tends to infinity when v − s c → ∞ . Fur thermore, in an auction without re-entry , with pr obability approaching to one, the auction is reduced to one with only tw o remain- ing pla y e rs before the auction ends, and the expected time for this to happen becomes arbitrary small compared to the expected length of the auction. Thus, the abo v e the- orem predicts that when v − s c is large not only the game last for a v er y long time but also the game is likely to ha v e only tw o remaining play ers participating in it for almost its entire duration. Therefore, running auctions with re-entry option will be helpful to alleviate this issue because, despite in auctions without re-entr y option, these auctions can be successfully imple mented without any tw o particular p lay ers being willing to participate for the entirety of the auction. 13 5 Conclus i on W e dev eloped a dynamic pa y-to-bid auction for selling an obj ect whose monetary v alue is commo n kno wled ge am ong risk-lo ving bidders, and bidding fees are the only sour ce of benefit for the selle r . W e established that for the propo sed model there exists a unique symmetric subgame perfect equilibrium. The characterized equilibrium is stationary when re-entry in the auction is allo w ed, and it is Marko v perfect when re-entry is for- bidden. Further more, in equilibrium, the strategy chosen b y the pla yers is indepe n d e nt of their w ealth lev el whether or not re-entry is per mitt e d in the auction. When re-entry is per mitted, the e xpect ed rev enue of the seller is independent of the number of buy ers, increasing in the v alue of the object, de creasing in the bid fee, and decreasing in the sale price. Furthermore, the seller ’s expected rev enue , when bidders are risk neutral, is equal to the value of the object, and it is strictly greater than the v alue of the object when bidders are risk lo ving. Moreo v er , the se lle r ’s rev enue is strictly increasing in the degree of risk-lo ving. Thus, when bidders are risk-lo ving, it is more profit a ble to sell expensiv e objects with a low bid fee a nd a low sale price. W e further compared the seller ’s rev enue with a standard lott ery . A s a result, for suf ficiently small biding fee, the selle r ’s rev enue is strictly higher than a standar d lottery , for any number of play ers. W e further sho w ed that allo wing re-entry may be surprisingly important in practice. Because, if the seller w ere to run such an auction without allo wing re-entry , the auction w ould last a long time, and for a lmost all of its duration ha v e only tw o remaining pla yers. Therefore, the selle r ’s rev enue relie s on those tw o pla y e rs being willing to participate, without any breaks, in an auction that might last for thousands of rounds. Referenc es [1] D. Fudenberg, J. T irole, A theory of e x it in duopoly . Econometrica, v ol. 54pp. 934-960, 1986. 14 [2] J. Ordo v er and A. Rubinstein. A sequential concession game with asymmetric information . Quarterly Journa l of Economics. v ol. 1 01, pp. 879-888 , 1986. [3] L. Kornhauser , A. Rubinstein and C. W ilson Reputation and patience in the war of attri- tion . Econometrica. v ol. 56, pp. 15-24, 1989. [4] R. Siegel. All -Pay Cont ests . Econo metrica, v ol. 77, no. 1, pp. 71-92, 2 009. [5] K. Hendricks, A. W eiss, and C. W ilson. The W ar of Attrition in Continuous T ime with Complete Information . Inter natio nal Economic Review . v ol. 29, no. 4, pp. 663-68 0 , 1988. [6] D. Fudenberg, J. T irole, Game Theory , MIT Press, 1991. [7] C. Bliss and B. Rebuf f. Dragon-slaying and ballro om d anci ng: the private supply of a public good . Jour nal of Public Economics. v ol. 25 pp . 1-12, 1984. [8] J. By ers, M. Mitzenmacher , and G. Zerv as. Information Asymm etries in P ay-Pe r-Bid Auctions, How Swoopo Makes Bank . arXiv :1001.0592, 2010. [9] A. Mas-colell, M. Whinston, and J. Green, Microeconomics Theory . Oxford, U.K., Ox- ford Univ ersity Press, 1995. [10] T . Hinnosaar . Penny Auc ti ons are Unpredictable . Unpublished manuscript. A v ailable at http://toomas. hinnosaar .net/, 2010. [11] A. Kakhbod. Pay-to-bid auctions: T o bid or not to bid . Operations Re sear ch Letters, v ol. 41, no 5, pp. 462-467, 2013. [12] N. A ugenblick. Consumer and Producer Behavior in the Market for Penny Auc- tions: A Theoretical an d Em p irical Analysis . Unpublished manuscript. A vailable at http://facult y .haas.berkeley .e du/ned/, 2 0 11. [13] B. C. Platt, J. Price, and H. T appen. The Role of Risk Preferences in Pay-to-Bid Auctions . Management S cience. T o a p pear , 2013. [14] M. Shubik. The Dollar auction game: A paradox in noncooperative behavior and esc alation . Journal of Conflict Resolution. v ol.15, no.1, pp. 109-111, 1971. 15 [15] M. Ba y e , D. Ko v enock, and C. G. de V ries. The all-pay auction with complete informa- tion . Economic Theory , v ol. 8, no. 2, pp. 2 9 1-305, 1996. [16] J. M. Smith. The the ory of g ames and the evolution of animal conflicts . Journal of theo- retical biology , v ol. 47, no. 1 , pp. 209-221, 1974. 16 Appendi x Notation : History of the game up to time t , t ∈ { 1, 2, 3, · · · } , which is common knowledge a mong the pla y ers, is denoted b y h t = ( s 1 , s 2 , · · · , s t − 1 ) , where s k = ( s 1, k , s 2, k , · · · , s n ( t ) , k ) rep- resents the strategy profile reported by the play ers in round k of the game. C onsider a sub-game beginning after history h t . E i , t [ u ( X h t )] is the expected utility of pla y er i in this sub-game and, E i , t [ u ( X h t ) | { Bid } ] ( E i , t [ u ( X h t ) | { No Bid } ] ) is the expected utility of pla y er i in this sub-game giv en that he play s { Bid } ( { N o Bid } ) in r ound t and, X h t is the random v ariable denoting money ea r ned b y the pla y er in equilibrium in the sub-game starting a fter h t . E ach pla y e r at any round t chooses a strategy which is a map fro m any history of the game up to time t to [ 0, 1 ] , a Bernoulli p robability o v e r { Bid, No Bid } . Let p t ( h t ) , 0 ≤ p t ( h t ) ≤ 1, be the probability of choos ing { Bid } after observing history h t of the game. Number of pla y ers present in round t , t ∈ { 1, 2, 3, · · · } of the game is denoted by n t . Note that when re-entry is allo w ed n t = n in any rou nd t , t ∈ { 1, 2 , 3, · · · } . Pr oof of Theor em 1 . W e pr ov e each part of the theorem, separately , a s follo ws. Proof of I. : W e note that in any SSPE, a t any round t , 0 < p t ( h t ) < 1, because, due to the game specification, for example if p t ( h t ) = 1 then for any pla y er it is profi table to unilaterally deviate to { No Bid } (similar arg ume nt holds when p t ( h t ) = 0). Further , it can be sho wn that there exists δ such that at any round t of the game p t ( h t ) > δ . W e pro v e this b y contradictio n. Suppose that there is not such δ , ie, for any δ > 0 there exists history h t , which occurs with positiv e p r obability in equilibrium, such that p t ( h t ) < δ . Thus, by picking δ sufficiently small w e ha v e: E i , t [ u ( X h t ) | { Bid } ] ( a ) > ( 1 − δ ) n − 1 u ( v − s + w i − c ) ≥ u ( v ) n ≥ E i , t [ u ( X h t )] ≥ E i , t [ u ( X h t ) | { No Bid } ] , (3) 17 where ( a ) follo ws since δ is v ery small. Next, w e sho w that in any sub-game perfect equilibrium starting in round t of the game each pla yer earns expected utility u ( w ) where w is the the w ealth of the pla y e r in round t . First, w e sho w this statement is true whe n w = 0, that is, E i , t [ u ( X h t )] = 0 when w = 0 in round t . W e pro v e it b y contradiction. Suppose that there exists history h t , which occurs with positiv e pro bability in equilibrium, such that E i , t [ u ( X h t )] = γ > 0 (4) As w e pro v ed in the abov e , there exists δ such that p t ( h t ) > δ , ∀ t . Further , since, p t ( h t ) ∈ ( 0, 1 ) , ∀ t , ea ch pla y er should be indifferent betw een choosing { Bid } and { No Bid } , in each round t . Also, since E i , t [ u ( X h t )] = γ > 0, there exists h s including h t , ie, h t ⊂ h s , such that E i , t [ u ( X h s )] ≥ γ . Thus, due to the fact pla y er i indifferent be tw e en { Bid } and { No Bid } w e ha v e: E i , s [ u ( X h s )] = E i , s [ u ( X h s ) | { No Bid } ] = Λ i , s E i , s + 1 [ u ( X h s + 1 )] ≥ γ ⇒ E i , s [ u ( X h s + 1 )] ≥ γ Λ i , s , (5) where, Λ i , s is the probabilit y that at least tw o pla y ers (except i ) pla y { Bid } in round s , giv en h s , ie, Λ i , s : = 1 −  n s − 1 1  p s ( h s )( 1 − p s ( h s )) n − 2 , where n s is the number of pla y ers in ro und s , (note that when re-entry is allo w e d n s = n , ∀ s . ) Furthermore, note that 0 < Λ i , s < 1. Using the new lo w e r bound deriv ed in ( 5 ) and, follow ing similar the ar guments w e did to deriv e ( 5 ) imply E i , s + 2 [ u ( X h s + 2 )] ≥ γ Λ i , s Λ i , s + 1 , h s + 2 ⊂ h s + 1 ⊂ h s . (6) 18 Follo wing the abo v e argu ments w e obtain E i , s + r [ u ( X h s + r )] ≥ γ ∏ r − 1 k = 0 Λ i , s + k , (7) where h s + r ⊂ · · · ⊂ h s + 1 ⊂ h s . But notice since for any k ≥ 0, 0 < Λ i , s + k < 1, then b y choosing r suf ficiently large w e can get γ ∏ r − 1 k = 0 Λ i , s + k > u ( v ) , (8) that is a contradiction, since E i , s + r [ u ( X h s + r )] can not exceed u ( v ) . Therefore, E i , t [ u ( X h t )] = 0. (9) No w , suppose that w > 0, then Eq. ( 9 ) along with ( 1 ) imply E i , t [ u ( w + X h t )] = E i , t " 1 − e − ρ ( w + X h t ) ρ # = 1 − E i , t h e − ρ ( w + X h t ) i ρ = 1 − e − ρ w E i , t h e − ρ X h t i ρ = 1 − e − ρ w + e − ρ w  1 − E i , t h e − ρ X h t i  ρ = 1 − e − ρ w ρ + e − ρ w E i , t " 1 − e − ρ X h t ρ # = 1 − e − ρ w ρ + e − ρ w E i , t [ u ( X h t )] = u ( w ) . (10) The abo v e equality completes the proof of the first part of the theorem. Further , note that ( 10 ) follo ws whether re-entry is allo w ed or not. Before pro ving the next part, w e present the follo wing Remark that holds in risk- 19 lo ving utility function. Remark 2 (see [ 9 ]) . Let u ( · ) be a CARL utility function. Su ppose X and Y are two random variables and α is a constant. Then, E [ u ( X + α ) ] ≥ E [ u ( Y + α ) ] ⇐ ⇒ E [ u ( X )] ≥ E [ u ( Y )] Remark 2 states that comparing tw o random v ariables is independent of a const a nt shift when the utility functio ns has a CA RL for m, like what w e defined in ( 1 ). Proof of II. (when r e-entry is allow ed) : The expected utility of play er i if he pla ys { Bid } (with w ealth lev el equal to w i , t at the begging of round t ) in e quilibrium in the sub-game starting a fter history h t is equal to, E i , t [ u ( w i , t + X h t ) | { Bid } ] = u ( w i , t + v − s − c )( 1 − p t ( h t )) n − 1 + n − 1 ∑ k = 1 "  n − 1 k  p t ( h t ) k ( 1 − p t ( h t )) n − 1 − k × E i , t + 1 [ u ( w i , t − c + X h t ∪ s k t + 1 )] # , (11) where s k t + 1 is the strategy profile in round t + 1 that k pla y e rs pla y { B id } and h t ∪ s k t + 1 is the updated history . In ( 11 ), the first term corresponds to the ev ent that pla y er i wins the object, ie, the ev ent in which pla y er i is the only pla yer who play s { Bid } in ro und t and, the second ter m corresponds to the expected utility of play er i from continuing after history h t , equivalently , more than one play er pla y { Bid } . Similarly , if pla yer i pla ys { No Bid } , then E i , t [ u ( w i , t + X h t ) | { No Bid } ] = u ( w i , t )  n − 1 1  ( 1 − p t ( h t )) n − 2 p t ( h t ) + ∑ k 6 = 1 "  n − 1 k  p t ( h t ) k ( 1 − p t ( h t )) n − 1 − k × E i , t + 1 [ u ( w i , t + X h t ∪ s k t + 1 )] # (12) In ( 12 ), the first term cor responds to the ev ent that pla y er j ( j 6 = i ) wins the object , ie, the ev ent in which pla y er j ( j 6 = i ) is the only p lay er who pla ys { Bid } in round t and, the second term corresponds to the expected utility of pla yer i from continuing after history 20 h t , equivalently , more than one play er other than i pla y { Bid } . Since play er i purely randomize s betw een { Bid } and { No Bid } , then pla y er i is indiff erent betw een choosing { Bid } and { No Bid } , that is, ( 11 ) is equal to ( 12 ). No w , using Remark ( 2 ) enable us to simplify ( 11 ) a nd ( 12 ) as follo ws. W ithout loss of generality , w e can set w i , t = c and, consequently , obtain E i , t + 1 [ u ( w i , t − c + X h t ∪ s k t + 1 )]   w i , t = c = E i , t + 1 [ u ( X h t ∪ s k t + 1 )] ( a ) = 0. (13) E i , t + 1 [ u ( w i , t + X h t ∪ s k t + 1 )]   w i = c = E i , t + 1 [ u ( c + X h t ∪ s k t + 1 )] ( b ) = u ( c ) , (14) where ( a ) follo ws from ( 9 ), and ( b ) from ( 10 ). No w , plugging w i = c into ( 11 ) (because of R emark 2 ) and using ( 13 ), Eqs. ( 11 ) can be simplified as follo ws E i , t [ u ( w i , t + X h t ) | { Bid } ]   w i , t = c = u ( v − s ) ( 1 − p t ( h t )) n − 1 (15) and similarly , plugging w i , t = c into ( 12 ) and emplo ying ( 14 ), then ( 12 ) is simplified as 21 follo ws E i , t [ u ( w i , t + X h t ) | { No Bid } ]   w i , t = c = u ( c )  n − 1 1  ( 1 − p t ( h t )) n − 2 p t ( h t ) + n − 1 ∑ k 6 = 1  n − 1 k  p t ( h t ) k ( 1 − p t ( h t )) n − 1 − k u ( c ) = u ( c ) "  n − 1 1  ( 1 − p t ( h t )) n − 2 p t ( h t ) + n − 1 ∑ k 6 = 1  n − 1 k  p t ( h t ) k ( 1 − p t ( h t )) n − 1 − k # = u ( c ) . (16) Finally , since pla y er i is indifferent betw een choosing { Bid } and { No Bid } , then equating ( 15 ) and ( 16 ) giv es that p t ( h t ) = 1 − n − 1 s u ( c ) u ( v − s ) . (17) No w , it is immediate fr om ( 17 ) that the characterized symmetric equilibrium is unique and of course stationary since it is controlled by the ( n , v , s , c ) . Proof of III. (when re-entr y is not allow ed) : Let pla yer i be one of n t (remaining) pla yers who are left to pla y in round t of the game. Then, the expected utility of pla yer i if he pla ys { Bid } (with w ealth lev el equal to w i , t at the begging of ro und t ) in equilibrium in the sub-game starting a fter history h t is equal to, E i , t [ u ( w i , t + X h t ) | { Bid } ] = u ( w i + v − s − c )( 1 − p t ( h t )) n t − 1 + n t − 1 ∑ k = 1 "  n t − 1 k  p t ( h t ) k ( 1 − p t ( h t )) n t − 1 − k × E i , t + 1 [ u ( w i , t − c + X h t ∪ s k t + 1 )] # , (18) where s k t + 1 is the strateg y pro file in round t + 1 that k pla yers of the remaining n t pla y ers of round t , pla y { Bid } and h t ∪ s k t + 1 is the updated history . In ( 18 ), the first term 22 corresponds to the ev ent that pla yer i wins the object, ie, the ev ent in which pla y er i is the only play er (among the remaining pla yers) who pla ys { Bid } in round t and, the second term corresponds to the expected utility of pla yer i from continuing after history h t , equivalently , more than one play er pla y { Bid } . If play er i pla ys { No Bid } , then E i , t [ u ( w i , t + X h t ) | { No Bid } ] = u ( w i , t )   n t − 1 1  ( 1 − p t ( h t )) n t − 2 p t ( h t )  + u ( w i , t ) " ∑ k ≥ 2  n t − k k  ( 1 − p t ( h t )) n t − k p t ( h t ) k # + E i , t + 1 [ u ( w i , t + X h t ∪ s k t + 1 )] ( 1 − p t ( h t )) n t − 1 . (19) In ( 19 ), the first ter m corresponds to the ev ent that pla yer j ( j 6 = i ) wins the object, ie, the ev ent in which pla y er j , one of the remaining pla y ers ( j 6 = i ), is the only pla y er who pla ys { Bid } in round t . The second term is corresponding to the case in which more than tw o pla y e rs among the remaining ones pla y { B id } in round t . Moreo v e r , note that since re-entry is not allo w ed, the first tw o terms of ( 19 ) represent ev ents in which pla y er i will be out from the rest of the game. The third term of ( 19 ) represents the only ev ent in which pla yer i maintains in the game with pla ying { No Bid } , that is the tie breaking case. Since re-entry is not allo w ed, by pla ying { No Bid } pla yer i remains in the game only if all the other remaining pla y ers pla y { No Bid } as w ell. This ev ent is captured b y the last term of ( 19 ). No w , since pla y er i purely randomizes betw een { Bid } and { No Bid } , then pla y er i is indifferent betw een choo sing { Bid } a nd { No Bid } , that is, ( 18 ) is equal to ( 19 ). Again, similar to the case where re-entry is per mitted, b y using Rema rk ( 2 ) and setting w i = c and, w e obtain E i , t + 1 [ u ( w i , t − c + X h t ∪ s k t + 1 )]   w i , t = c = E i , t + 1 [ u ( X h t ∪ s k t + 1 )] ( a ) = 0. (20) 23 E i , t + 1 [ u ( w i , t + X h t ∪ s k t + 1 )]   w i , t = c = E i , t + 1 [ u ( c + X h t ∪ s k t + 1 )] ( b ) = u ( c ) , (21) where ( a ) follo ws from ( 9 ), and ( b ) from ( 10 ). No w , plugging w i , t = c into ( 18 ) and using ( 20 ), Eq s. ( 18 ) is simplified as follo ws E i , t [ u ( w i , t + X h t ) | { Bid } ]   w i , t = c = u ( v − s ) ( 1 − p t ( h t )) n − 1 (22) and similarly , plugging w i , t = c into ( 19 ) and using ( 21 ), then ( 19 ) is simplified as follo ws E i , t [ u ( w i , t + X h t ) | { No Bid } ]   w i = c = u ( c )   n t − 1 1  ( 1 − p t ( h t )) n t − 2 p t ( h t )  + u ( c ) " ∑ k ≥ 2  n t − k k  ( 1 − p t ( h t )) n t − k p t ( h t ) k # + E i , t + 1 [ u ( c + X h t ∪ s k t + 1 )] ( 1 − p t ( h t )) n t − 2 = u ( c ) " ∑ k ≥ 1  n t − k k  ( 1 − p t ( h t )) n t − k p t ( h t ) k # + u ( c )( 1 − p t ( h t )) n t − 1 = u ( c ) . (23) Moreo v er , since play er i is ind ifferent betw e e n choos ing { B id } a nd { No Bid } , then equating ( 22 ) and ( 23 ) giv es that p t ( h t ) = 1 − n t − 1 s u ( c ) u ( v − s ) . (24) Equation ( 24 ) rev eals that the symmetric equilibrium is unique. Also the equilibrium strategy that each pla yer follo ws in each round t of the game depends only on the relev ant state v ariables ( n t , v , s , c ) , therefo re it is Marko v perfect. Pr oof of Theor em 2 . The pro ba bility of the ev ent that the game ends in round t of the 24 game giv en that t is reached is equal to h t = n t p t ( 1 − p t ) n t − 1 1 − ( 1 − p t ) n t (25) h t is called hazard rate at time t . The probabilit y thst a ro und t is reached is the equal to the probability the game doesn’t end in any round s , s < t , which is equal to t − 1 ∏ s = 1 ( 1 − h s ) = t − 1 ∏ s = 1  1 − n p s ( 1 − p s ) n s − 1 1 − ( 1 − p s ) n s  . (26) No w , let Q n t , t denote the expected number of entrants in round t when the number remaining pla yers is n t . Thus, Q n t , t = n t ∑ k = 1  n t k  p k t ( 1 − p t ) n t − k k + ( 1 − p t ) n t Q n t , t = n ∑ k = 0  n t k  p k t ( 1 − p t ) n t − k k + ( 1 − p t ) n t Q n t , t = n p t + ( 1 − p t ) n t Q n t , t (27) Equation ( 27 ) implie s that Q n t , t = n t p t 1 − ( 1 − p t ) n t . (28) The selle r ’s expected ear nings from the bid fees in ro und t is the expected number of entrants times the bid fee times the probability that round t is reached, that is Q n t , t × c × t − 1 ∏ s = 1 ( 1 − h s ) = n t p t 1 − ( 1 − p t ) n t × c × t − 1 ∏ s = 1  1 − n p s ( 1 − p s ) n s − 1 1 − ( 1 − p s ) n s  . Therefore, the seller ’s expected ear ning from the bid fees througho ut the game is exactly equal to ∞ ∑ t = 1 " Q n t , t × c × t − 1 ∏ s = 1 ( 1 − h s ) # = ∞ ∑ t = 1 " n t p t 1 − ( 1 − p t ) n t × c × t − 1 ∏ s = 1  1 − n s p s ( 1 − p s ) n s − 1 1 − ( 1 − p s ) n s  # . 25 Pr oof of Theor em 3 . Keep the bidding fee fixed. As w e pro v ed in Theorem 4 , in equilib- rium, in any round t , the probability that a pla yer exits (pla ys { No Bid } ) whe n there are n t pla y ers remained in the game is uniquely d e termined as a function of the objects’s v alue v , the bid fee c , and the sale price s as p t = 1 − n t − 1 s u ( c ) u ( v − s ) . When v − s c → ∞ , u ( c ) u ( v − s ) → 0 and therefore p t → 1. Notice that, when re-entry is per mitt ed the abo v e probability becomes stationary because n t = n , for all t . Moreo v er , as sho wn in the previous theorem, the seller ’s expected profit (earnings from the bid fees) in ro und t is giv en b y Q n t , t × c × t − 1 ∏ s = 1 ( 1 − h s ) = n t p t 1 − ( 1 − p t ) n t × c × t − 1 ∏ s = 1  1 − n p s ( 1 − p s ) n s − 1 1 − ( 1 − p s ) n s  . Since c is fixed, and p t → 1, it follo ws that the abo v e expression is maximized. Conse- quently , the o v erall seller ’s profit is maximized. Pr oof of Theor em 4 . W e pro v e e ach part of the theorem separately as follo ws. In the follo wing w e first p ro v e the second part of the theorem and then the rest. Proof of II. : As w e pro v ed in Theorem 1 , when re-entry is allo w ed, in equilibrium, in each stage of the game, each pla yer chooses to pla y { Bid } with follo wing s tation ar y probability p = 1 − n − 1 s u ( c ) u ( v − s ) . (29) The stationarity comes fro m the fact that the abov e probabilit y is contr olled b y the con- stant parameters of the model that are n , s , v , and c . Moreo v er , as w e pro v ed in Theorem 2 , the seller ’s expected ear ning only from the bid 26 fees throughout the game is exactly equal to ∞ ∑ t = 1 " n p t 1 − ( 1 − p t ) n × c × t − 1 ∏ s = 1  1 − n p s ( 1 − p s ) n − 1 1 − ( 1 − p s ) n  # . No w , due to the fact that when re-entry is allo w ed the eq uilibrium is stationary (ie, because of ( 29 ), p t = p for any t ∈ { 1, 2, 3, · · · } ), the a bov e quality can be simplified as follo ws. Since the selle r ’s expected rev enue is equal to the sale price, ie, s , plus the seller ’s expected ear ning from the bid fees thr oughout the game, then w e obtain Selle r ’s expected re v enue = s + ∞ ∑ t = 1 "  c n p 1 − ( 1 − p ) n   1 − n p ( 1 − p ) n − 1 1 − ( 1 − p ) n  t − 1 # = s +  c n p 1 − ( 1 − p ) n  ∞ ∑ t = 1  1 − n p ( 1 − p ) n − 1 1 − ( 1 − p ) n  t − 1 = s +  c n p 1 − ( 1 − p ) n  1 − ( 1 − p ) n n p ( 1 − p ) n − 1 = s + c ( 1 − p ) n − 1 ( a ) = s + c u ( v − s ) u ( c ) . (30) Proof of I. : When pla yers are risk neutral (ie, u ( x ) = x ), then ( 30 ) implie s that Selle r ’s expected re v enue (with risk neutral utility) = v . (31) Proof of III. : It is immediate from ( 30 ) that the seller ’s expected rev enue is independent of number of pla y ers. Proof of IV . : 27 No w , w e sho w that when pla y ers are risk-lo ving, ie, ρ 6 = 0, s + c u ( v − s ) u ( c ) > v . (32) T o pr o v e ( 32 ) w e use the follo wing Lemma. Lemma 1. Let u ( · ) be a convex function and x > 0 is a constant. Define f ( α ) : = u (( 1 + α ) x ) − ( 1 + α ) u ( x ) . Then, f ( α ) > 0 . Proo f of Lemma 1 . T o p r ov e Lemma 1 , w e first sho w that u ( x ) x = R x 0 u ′ ( t ) d t x ( a ) < R x 0 u ′ ( x ) d t x = u ′ ( x ) ⇒ u ( x ) < x u ′ ( x ) . (33) where ( a ) follo ws because u ( · ) is increasing. No w , using ( 33 ) w e obtain f ′ ( α ) = x u ′ (( 1 + α ) x ) − u ( x ) > x u ′ ( x ) − u ( x ) > 0. (34) Thus f ( α ) is increasing in α a nd f ( α ) > f ( 0 ) = 0. As a consequence of Lemma 1 w e ha v e x > y ⇒ u ( x ) u ( y ) > x y . (35) because w e can simply set x = ( 1 + α ) y , where α > 0. No w , w e can pro v e ( 32 ) holds as follo ws s + c u ( v − s ) u ( c ) − v = c  u ( v − s ) u ( c ) − v − s c  ( a ) > 0 (36) where ( a ) is correct because v − s > c and ( 35 ). 28 Proof of V . : In the follo wing, w e sho w that the seller ’s rev enue is increasing in v , decreasing in s and decreasing in c . ∂ h s + c u ( v − s ) u ( c ) i ∂ v > 0. ∂ h s + c u ( v − s ) u ( c ) i ∂ s = 1 − c u ′ ( v − s ) u ( c ) < 1 − c u ′ ( c ) u ( c ) ( a ) < 0. ∂ h s + c u ( v − s ) u ( c ) i ∂ c = u ( v − s ) u ( c ) − c u ′ ( c ) u ( v − s ) u ( c ) 2 = u ( v − s ) u ( c ) 2 ( u ( c ) − c u ′ ( c ) ) ( b ) < 0 (37) where ( a ) and ( b ) are both follo w ed b y ( 33 ). Proof of VI. : Because of the results of the previous part, the expected rev enue of the seller is decreas- ing in s and decreasing in c . Therefore, in or de r to find the maximum value of the seller ’s expected rev enue (when v is giv en), in ( 30 ), w e set s = 0 and take a limit when c → 0 as follo ws: max s , c  s + c u ( v − s ) u ( c )  = lim c → 0 c u ( v ) u ( c ) = u ( v ) u ′ ( 0 ) = 1 − e − ρ v ρ = u ( v ) . (38) Proof of VII. : Consider a seller off ering lottery tickets for a p rize /object of (monetary) v alue v to n potential buy ers with C ARL utility . Cost of buying a ticket for each pla y er is equal to c > 0, and the winner is chosen randomly . Thus, the ma ximal price the seller can choose to charge for a (lottery) ticket is the solution of u ( c ) = 1 n u ( v ) . He nce, the optimal (rev enue maximizing) ticket price is c ∗ = u − 1  1 n u ( v )  . Since for all x > 0, u ( x ) > x , thus, for all x > 0, x > u − 1 ( x ) . Therefore, 1 n u ( v ) > u − 1  1 n u ( v )  , and consequently for all n ∈ N , u ( v ) > n × u − 1  1 n u ( v )  = nc ∗ = the maximum selle r ’s rev enue fro m the lottery. As w e ha v e shown, in the previous part, Part (VI), u ( v ) is the maximum seller ’s rev enue in the game pr oposed in Section 2 . Thus, the a bov e inequality implies that, for suf ficiently small biding fee, the seller ’s rev enue is strictly higher than a standard lottery , 29 for any number of pla y ers. Pr oof of Theor em 5 . W e pro v e e ach part of the theorem separately as follo ws. Proof of I. : As we pro v ed in Theorem 4 , in equilibrium, the probability that a play er e xits (pla ys { No Bid } ) when there are n pla yers remainng in the game is uniquely d e termined as a function of the objects’s value v , the bid fee c , and the sale price s as q : = 1 − 1 − n − 1 s u ( c ) u ( v − s ) ! = n − 1 s u ( c ) u ( v − s ) . (39) When v − s c → ∞ , u ( c ) u ( v − s ) → 0 and therefore q → 0. For simplicity of exposition, w e denote u ( c ) u ( v − s ) b y λ and write q n ( λ ) = n − 1 √ λ ⇒ lim λ → 0 q n ( λ ) = 0. Let Z n , λ be a random variable denoting the number of bids (number of pla y ers pla y- ing { Bid } ) in a round without entry if there are n remaining pla y ers and each chooses to pla y { No Bid } with probability q n ( λ ) . Thus, Prob { Z n , λ = m } = ( n m ) ( 1 − q n ( λ )) m q n ( λ ) m 1 − ( q n ( λ )) n , ∀ m ∈ { 1, 2, · · · , n } . (40) Thus, the probability the auction ends in any particular round, giv en that it is reached and has n remaining pla yers , is Prob { Z n , λ = 1 } = n ( 1 − q n ( λ )( q n ( λ )) n − 1 1 − ( q n ( λ )) n . (41) Since as λ → 0, q n ( λ ) → 0, then w e ha v e lim λ → 0 Prob { Z n , λ = 1 } = n ( 1 − q n ( λ )( q n ( λ )) n − 1 1 − ( q n ( λ )) n = 0. (42) Since ( 42 ) holds for any n , it follo ws that the expected length of the auction tends to 30 infinity . Proof of II. : No w w e pro v e the second part of the theorem. It follo ws that lim λ → 0 Prob { Z n , λ = m } Prob { Z n , λ = m + 1 } = lim λ → 0 ( m + 1 ) q n ( λ ) ( n − 1 ) ( 1 − q n ( λ )) = 0. (43 ) Equation ( 35 ) implie s that, while, Prob { Z n , λ = 2 } tends to 0, it is an order of magnitude larger than Prob { Z n , λ = 1 } when λ is small. Now , let the random v ariable T n , m be the number of ro unds without re-entry before an auction with n pla y e rs is reduced to one with no more than m remaining pla y ers. Therefore, T n ,1 means the last round of the game and T n , m < T n ,1 implies that there is some round with few er than m pla y e rs be fore the end of the a uction. W e can write Prob { T n ,2 < T n ,1 } = Pr ob { Z n , λ = 2 } + n ∑ m = 3 Prob { Z n , λ = m } Prob { T m ,2 < T m ,1 } = Prob { Z n , λ = 2 } + n − 1 ∑ m = 3 Prob { Z n , λ = m } Prob { T m ,2 < T m ,1 } + Prob { Z n , λ = n } Prob { T n ,2 < T n ,1 } (44) Equation ( 44 ) implie s that Prob { T n ,2 < T n ,1 } = Prob { Z n , λ = 2 } + ∑ n − 1 m = 3 Prob { Z n , λ = m } Pr ob { T m ,2 < T m ,1 } 1 − Prob { Z n , λ = n } (45) No w , w e sho w that ( 45 ) tends to 1 as λ tends to 0. W e sho w this b y induction o v er n . First, consider the case where n = 3. Then ( 45 ) is simplified as follo ws Prob { T 3,2 < T 3,1 } = Prob { Z 3, λ = 2 } 1 − Prob { Z 3, λ = 3 } = Prob { Z 3, λ = 2 } Prob { Z 3, λ = 2 } + Prob { Z 3, λ = 1 } . (46) Equation ( 43 ) along with ( 46 ) imply that 31 lim λ → 0 Prob { Z 3, λ = 2 } + Prob { Z 3, λ = 1 } Prob { Z 3, λ = 2 } = 1 + lim λ → 0 Prob { Z 3, λ = 1 } Prob { Z 3, λ = 2 } = 1. (47) Therefore, lim λ → 0 Prob { T 3,2 < T 3,1 } = 1. (48) No w , suppose that for m < n , the induction step, lim λ → 0 Prob { T m ,2 < T m ,1 } = 1. The n, equation ( 45 ) yields that lim λ → 0 1 Prob { T n ,2 < T n ,1 } = lim λ → 0 1 − Prob { Z n , λ = n } Prob { Z n , λ = 2 } + ∑ n − 1 m = 3 Prob { Z n , λ = m } Pr ob { T m ,2 < T m ,1 } = lim λ → 0 ∑ n − 1 m = 1 Prob { Z n , λ = m } Prob { Z n , λ = 2 } + ∑ n − 1 m = 3 Prob { Z n , λ = m } Pr ob { T m ,2 < T m ,1 } = lim λ → 0 ∑ n − 1 m = 1 Prob { Z n , λ = m } Prob { Z n , λ = 2 } + ∑ n − 1 m = 3 Prob { Z n , λ = m } [ lim λ → 0 Prob { T m ,2 < T m ,1 } ] = lim λ → 0 ∑ n − 1 m = 1 Prob { Z n , λ = m } Prob { Z n , λ = 2 } + ∑ n − 1 m = 3 Prob { Z n , λ = m } = lim λ → 0 ∑ n − 1 m = 1 Prob { Z n , λ = m } ∑ n − 1 m = 2 Prob { Z n , λ = m } = 1 + lim λ → 0 Prob { Z n , λ = 1 } ∑ n − 1 m = 2 Prob { Z n , λ = m } = 1, (49) where that last equality follo ws because lim λ → 0 Prob { Z n , λ = 1 } ∑ n − 1 m = 2 Prob { Z n , λ = m } = 0 since (b y Sand- wich theorem) 0 ≤ lim λ → 0 Prob { Z n , λ = 1 } ∑ n − 1 m = 2 Prob { Z n , λ = m } ≤ lim λ → 0 Prob { Z n , λ = 1 } Prob { Z n , λ = 2 } = 0, where the last equality follo ws by ( 43 ). Finally , b y ( 49 ), w e conclude that lim λ → 0 Prob { T n ,2 < T n ,1 } = 1. (50) 32 Thus, the pr oof of the second part of the theorem is complete. Proof of III. : No w w e pro v e the last part of the theorem. B y the definition of T n , m , w e ha v e E [ T n , m ] = Prob { Z n , λ ≤ m } + ∑ n − 1 k = m + 1 Prob { Z n , λ = k } E [ T k , m ] 1 − Prob { Z n , λ = n } . (51) Thus, E [ T n ,2 ] E [ T n ,1 ] = Prob { Z n , λ = 1 } + Prob { Z n , λ = 2 } + ∑ n − 1 k = 3 Prob { Z n , λ = k } E [ T k ,2 ] Prob { Z n , λ = 1 } + ∑ n − 1 k = 2 Prob { Z n , λ = k } E [ T k ,1 ] . (52) No w , by induction ov er n w e sho w that lim λ → 0 E [ T n ,2 ] E [ T n ,1 ] = 0. (53) Suppose that n = 3, then lim λ → 0 E [ T 3,2 ] E [ T 3,1 ] = lim λ → 0 Prob { Z 3, λ = 1 } + Prob { Z 3, λ = 2 } Prob { Z 3, λ = 1 } + Prob { Z 3, λ = 2 } E [ T 2,1 ] = 0. (54) The last equality follo ws because lim λ → 0 Prob { Z 3, λ = 1 } Prob { Z 3, λ = 2 } = 0 b y Eq. ( 43 ) and lim λ → 0 E [ T 2,1 ] = ∞ b y the first part of the theorem. No w , suppose that for k < n , the induction step, lim λ → 0 E [ T k ,2 ] E [ T k ,1 ] = 0. (55) 33 Before completing the proof, first w e ha v e the follo wing Lemma that is useful in the sequel. Lemma 2. If 0 ≤ a i and 0 < b i for i = 1 , 2 · · · , n, then ∑ n i = 1 a i ∑ n i = 1 b i ≤ ∑ n i = 1 a i b i . Proo f. The proof is immediate since ∑ n i = 1 a i ∑ n i = 1 b i = n ∑ i = 1 a i ∑ n i = 1 b i ≤ n ∑ i = 1 a i b i . No w , in the follo wing, using the induction step w e sho w the statement is v alid for k = n as w ell. W e do this by emplo ying the Sandwich theorem as follo ws. 0 ≤ lim λ → 0 E [ T n ,2 ] E [ T n ,1 ] ( a ) = lim λ → 0 Prob { Z n , λ = 1 } + Prob { Z n , λ = 2 } + ∑ n − 1 k = 3 Prob { Z n , λ = k } E [ T k ,2 ] Prob { Z n , λ = 1 } + ∑ n − 1 k = 2 Prob { Z n , λ = k } E [ T k ,1 ] ( b ) ≤ lim λ → 0 Prob { Z n , λ = 2 } + ∑ n − 1 k = 3 Prob { Z n , λ = k } E [ T k ,2 ] ∑ n − 1 k = 2 Prob { Z n , λ = k } E [ T k ,1 ] ( c ) ≤ lim λ → 0 " Prob { Z n , λ = 2 } Prob { Z n , λ = 2 } E [ T k ,2 ] + n − 1 ∑ k = 3 Prob { Z n , λ = k } E [ T k ,2 ] Prob { Z n , λ = k } E [ T k ,1 ] # = lim λ → 0 1 E [ T k ,2 ] + n − 1 ∑ k = 3  lim λ → 0 E [ T k ,2 ] E [ T k ,1 ]  ( d ) = 0. (56) where (a) is follo w ed b y ( 52 ), (b) is correct since lim λ → 0 Prob { Z n , λ = 1 } Prob { Z n , λ = 1 } + ∑ n − 1 k = 2 Prob { Z n , λ = k } E [ T k ,1 ] = 0, (c) is corr e ct b y Lemma 2 , and finally (d) is follo w ed b y ( 55 ). By the abo v e equality the proof of the third part of the theorem is complete. 34