MANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec31

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1 MANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec31 e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 2007 INFORMS Electronic Companion Temporary and Permanent Buyout Prices in Online Auctions by Jérémie Gallien and Shobhit Gupta, Management Science 2007, 53(5) Technical Supplement EC.1. Proof of Theorem 1 Consider a bidder A with type t in an auction where eery other bidder uses strategy, where is an arbitrary threshold function. If A is not the first bidder, the first bidder has either placed a bid or exercised the option immediately (following strategy ), so that the buyout option is no longer aailable to bidder A. In that case bidder A s weakly dominant strategy is to bid his true aluation, as shown in Vickrey (1961). His bid submission time in t T will not affect his utility in any way, so that bidding at any time in t T constitutes then a best response. Suppose now that A is the first bidder, so that the buyout option is aailable to him. We introduce the following notation for the three possible actions he may take at time t: bidt: Bid in the auction at time t (in which case it is a dominant strategy for him to bid his aluation ); buyt: Buyout at time t; waitt : Wait for t time units before deciding to bid (if the auction is still open) or buy out (if the option is still aailable). We define the utility of bidder A with type t and taking action a bidt buyt waitt as U a t. If bidder A chooses bidt, i.e. bids immediately, the buyout option disappears. Following strategy, all subsequent bidders will bid their true aluation. Denoting by NtT the random number of bidders arriing in interal t T and Nt the cumulatie number of arrials up to t, this implies: EU bidt t N t = 0NtT= e T t Fx NtT dx (EC1) and using the model assumption that NtT is Poisson with parameter T t, we obtain the expected utility of the first bidders when bidding his aluation upon his arrial at t: EU bidt t N t = 0 = e +T t e T tf x dx (EC2) B 1 t (EC3) Conditional on A being the first bidder (i.e., N t = 0), the utility from exercising the buyout option immediately is EU buyt t N t = 0 = p (EC4) The key to the proof of Theorem 1 is the following Lemma, which establishes that bidder A s expected utility from acting immediately upon his arrial (i.e., choosing either bidt or buyt) is always as large as that obtained from waiting, i.e., EU waitt t N t = 0: Lemma 1. EU waitt t N t = 0 maxb 1 t p. Proof. Let E = N t = 0 be the eent that no bidder arries in the interal t. In this case bidder A remains the first bidder so that EU bid t N t = 0 E = e t EU bid N = 0 = e t B 1 (EC5) ec1

2 ec2 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions and the buyout option is still aailable thus EU waitt t N t = 0 E = e t maxb 1 p The complementary eent E = N t > 0 corresponds to one or more arrials occurring in the interal t. In that case the buyout option is no longer aailable, so that EU waitt t N t = 0 E = EU bid t N t = 0 E = e t EU bid N t = 0 E (EC6) Note that the eent E includes the eent that one of the bidders who arried during t exercised the buyout option, in which case bidder A s utility is zero. The expected utility of the first bidder A if he waits up to time >tis thus EU waitt t N t = 0 = e t maxb 1 p PE + EU bid N t = 0 E P E (EC7) By the law of conditional expectation, we also hae B 1 t = EU bidt t N t = 0 E PE + EU bidt t N t = 0 E P E (EC8) Define as the eent that the first bidder, say B, arriing in t with type B t B (where t B t ) has aluation t B < B. Note that P E 0; in particular, P E = 0if t B for all t B t. Then (EC8) can be rewritten as B 1 t = EU bidt t N t = 0 E PE + EU bidt t N t = 0 E P E P E (EC9) + EU bidt t N t = 0 E P E P E (EC10) where is the complementary eent. Conditional on the eent E (i.e., Nt= 0), the expected utility of bidding is same whether A bids at time t or, i.e., EU bidt t N t = 0 E = EU bid t N t = 0 E (EC11) Now consider the case when the eent E occurs, i.e., the bidder B with type B t B has aluation t B < B. If bidder A bids in the auction at time t then the buyout option disappears and so B also bids in the auction; howeer if bidder A waits up to, then the buyout option is still present at time t B t and bidder B, following strategy, exercises the buyout option. As a result, we hae EU bidt t N t = 0 E >EU bid t N t = 0 E = 0 (EC12) where EU bidt t N t = 0 E >0 because B. Additionally, conditional on the eent E we hae EU bidt t N t = 0 E = EU bid t N t = 0 E (EC13) This can be explained as follows: The eent implies that either (1) B >; in this case the expected utility from bidding is zero irrespectie of when bidder A bids, or (2) B t B ; in this case bidder B, following strategy, bids in the auction immediately if A waits up to and thus the buyout option disappears. If A bids at time t then also the buyout option disappears and thus, irrespectie of when A bids, the buyout option is not exercised. Hence the expected utility from bidding for A is same from both actions bidt and bid. Using (EC13), (EC11), and (EC12) in (EC10) we get B 1 t EU bid t N t = 0 E PE + EU bid t N t = 0 E P E P E + EU bid t N t = 0 E P E P E = EU bid t N t = 0 E PE + EU bid t N t = 0 E P E (EC14)

3 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec3 Furthermore e t B 1 B 1 t (EC15) because, although both sides of the aboe inequality hae the same time discounting, the right-hand side is conditioned on N t = 0 and the left-hand side is conditioned on N = 0 (implying fewer competing bidders). Additionally, as indicated earlier in (EC5), we hae e t B 1 = EU bid t N t = 0 E. Equation (EC14) and inequality (EC15) thus imply together that EU bid t N t = 0 E B 1 t (EC16) Consider now the following two cases: Case 1: p B 1. Equation (EC7) becomes then EU waitt t N t = 0 = e t B 1 PE + e t EU bid N t = 0 E P E = EU bid t N t = 0 E PE + EU bid t N t = 0 E P E B 1 t maxb 1 t p where the second equality follows from (EC5) and (EC6) and the first inequality follows from (EC14). Case 2: p>b 1. In this case note that e t p>e t B 1 B 1 t EU bid t N t = 0 E = e t EU bid N t = 0 E (EC17) where the second and third inequalities follow from (EC15) and (EC16) respectiely, and the final equality follows from (EC6). Equation (EC7) thus implies EU waitt t N t = 0 = e t p PE + EU bid N t = 0 E P E <e t p PE + e t p P E = e t p < p maxb 1 t p where the first inequality follows from (EC17) and the second inequality from the law of total probability. Because Cases 1 and 2 aboe are exhaustie, the proof is complete. We hae thus established that the best response for bidder A, if she sees the buyout option, is to act immediately upon her arrial. Defining now t EU buyt t N t = 0 EU bidt t N t = 0 as the expected utility difference from exercising the buyout option and placing a bid immediately for the first bidder, Equations (EC1) and (EC4) imply t = p e +T t e T tf x dx (EC18) Note that t is continuous and differentiable on + 0T, and it is increasing in for all t 0T because t = 1 e 1 F T t > 0 (EC19) Assuming without loss of generality that p implies that t 0 for all t 0T, which combined with (EC19) proes the existence of a unique t + such that tt = 0. Defining tmp t mint and denoting R the set of best-response strategies to the symmetric profile, we hae thus proen that tmp R. But because the characterization of tmp proided by tt = 0 does not depend on the choice of as can be seen from Equation (EC18), we also hae tmp R tmp, completing the proof of Theorem 1.

4 ec4 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions EC.2. Proof of Proposition 1 Let t be the solution of t t p = e +T t e T tx /m dx me +T t = e T tt /m 1 (EC20) T t which is the same equation as (2) except that Fx has been replaced by x /m. The solution of (EC20) is gien as m ( ( t = p ) W e e +T t +p T t/m +T t + e +T t) T t If t for some t then (2) and (EC20) are equialent because Fx= x /m, x, and t = t. It thus follows that tmp t = t = mint. If t > for some t then we hae t t t e +T t e T tx /m dx t e +T t e T tf x dx = p because Fx x /m, x +. Furthermore, note that e +T t et tx /m dx is increasing in for all t and this combined with the fact that t is the solution of (EC20) implies that t t >. Thus tmp t = = mint. EC.3. Proof of Theorem 2 Consider a bidder A with type t arriing in the auction when the buyout option is present. If A is desperate then his strictly dominant strategy is to exercise the buyout option immediately and thus any strategy where the bidder waits cannot be an equilibrium of the game G. Suppose now that A is not desperate. We next show that the utility from bidding immediately is strictly greater than the utility from waiting for w units of time (where 0 <w<t t) and then bidding, i.e., in the notation of Theorem 1, EU bidt t >EU bidt+w t for all w 0T t, where denotes the eent that the buyout option is present when A first arries (at time t). The utility from bidding is calculated assuming the subsequent nondesperate bidders follow any arbitrary strategy whereas the desperate bidders follow their dominant strategy, which is to exercise the buyout option immediately, if aailable, and to not participate in the auction otherwise. Note that, as before, bidders are assumed to be rational; therefore, if and when they choose to bid in the auction, they will bid their true aluation (which is their weakly dominant strategy because this is a second-price auction). As in the proof of Theorem 1, the law of conditional expectation implies EU bidt t = EU bidt t P + EU bidt t P (EC21) where denotes the eent that the buyout option is exercised by a desperate bidder if bidder A waits up to time t + w. Note that the eent {first bidder arriing in t t + w is desperate} and thus we hae P Pfirst bidder arriing in t t + w is desperate>0 (EC22) Because desperate bidders do not participate in the auction if the buyout option is not present, we hae EU bidt t >EU bidt+w t = 0 (EC23) Bidder A bids in the auction at t + w, if it is still open, and hence the buyout option disappears at t +w. Thus no desperate bidder arriing in the interal t+wt participates in the auction. In addition if the eent does not occur then no desperate bidder arriing in the interal t t +w participates in the auction. The presence of desperate bidders therefore does not affect the expected utility of bidder A if the eent D occurs and hence the analysis used to obtain (EC14) can be essentially repeated, with minor modifications to incorporate the fact that subsequent bidders follow some arbitrary strategy to get EU bidt t EU bidt+w t (EC24)

5 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec5 Using (EC24), (EC23), and (EC22) in (EC21), we obtain EU bidt t >EU bidt+w t P + EU bidt+w t P (EC25) = EU bidt+w t (EC26) which proes that bidder A is strictly better off bidding immediately. Therefore, any strategy whereby a bidder, who arries when the buyout option is present, waits for w units of time (w >0) is not a Bayesian Nash equilibrium of G. The second result of the theorem inoles characterizing a threshold function such that the strategy whereby nondesperate bidders play is a Bayesian Nash equilibrium of the game G. If the first bidder arriing in the auction is desperate then his dominant strategy is to exercise the buyout option immediately and the auction ends. Otherwise if first bidder A with type t is nondesperate, then the analysis of Theorem 1 can be exactly repeated to show that the best response strategy of A is to bid immediately if his aluation tmpt and to exercise the buyout option immediately otherwise. The threshold aluation tmp = mint where t is the unique solution in + of t t p = e 1 +T t e 1 T tf x dx (EC27) which is the same as (EC2) except that is replaced by 1 because the arrial rate of nondesperate bidders in the game G is 1. Thus the strategy tmp is a Bayesian Nash equilibrium of G and, in addition, as 0, the right-hand side of (EC27) conerges to the right-hand side of (EC2) and it follows that lim 0 tmp = tmp. We hae thus shown that in any equilibrium of G a bidder, who arries when the buyout option is present, acts (bids/buyouts) immediately, and her behaior is characterized by the threshold function tmp. Once a bid has been placed in the auction, the buyout option disappears and, consequently, for all subsequent bidders a weakly dominant strategy is to bid their true aluation. This proes that for any >0, tmp is the unique (see 4.1 for a discussion of what uniqueness means for this analysis) Bayesian Nash equilibrium of the game G, thus implying that tmp is the unique equilibrium of a temporary buyout-price auction that is robust to the payoff perturbation discussed aboe. EC.4. Proof of Theorem 3 Consider a bidder A with type t and information I t in an auction where all other bidders play strategy, where 0T 0 is a continuous function such that t0 is nondecreasing in t and ti is nonincreasing in I for any t. We first derie bidder A s utility if he bids his true aluation at time T. Following strategy all other bidders also bid at T and thus I = 0 0T. Now bidder A wins the auction if no bidder exercises the buyout option and if eery bidder has a aluation less than bidder A s aluation, i.e., the eent A wins = min 0 for eery bidder 0} where the notation indicates the aluation of a bidder arriing at time. Also, because the auction is open at t, it can be inferred that all bidders 0 with 0t hae aluation 0 and thus the arrial rate of bidders at any 0t is F 0. Then probability that A wins the auction is Pr ( Nt A wins t i Nt ˆt i NtT ) Fmint = i 0 Ft i 0 NtT Fminˆt i 0 (EC28) where Ntis a counting process denoting the number of bidder arrials in 0tin a nonhomogeneous Poisson process with arrial rate = F 0 and 0t; t i Nt are the corresponding arrial epochs. NtT is a counting process denoting the number of arrials in t T in a Poisson process with arrial rate and ˆt i NtT are the arrial epochs. The first term of the product in (EC28) is the probability that the eent ti mint i 0 ti t i 0 occurs for all bidders arriing in 0t. The second term is the probability that the aluation ˆt i of eery bidder arriing in t T satisfies ˆt i minˆt i 0. Here, and in the remainder of the paper, we assume that if k = 0 then k = 1.

6 ec6 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions Conditional on bidder A winning, the distribution of the highest bid, 1, among the other Nt+ NtT bidders arriing at epochs t i Nt and ˆt i NtT is F 1 A winst i Nt ˆt i NtT Nt x = NtT Fmint i 0 x Fminˆt j 0 x Fmint j 0 j=1 Fminˆt j 0 Using the aboe distribution function, we hae E [ 1 A wins t i Nt ˆt i NtT ] ( = 1 F 1 A winsti Nt 0 = ˆt i NtT F 1 A winst i Nt ˆt i NtT x ) dx x dx x 0 and thus the expected discounted utility from bidding at T for bidder A is [ ( Nt NtT EU bidt t 0 = E e T t Fmint i 0 x Fminˆt Ft i 0 j 0 x dx j=1 (EC29) where the expectation on the right-hand side is oer t i Nt and ˆt i NtT. The rest of the proof proceeds as follows: We first show using Lemma 2 that if bidder A bids in the auction he must do so at time T, and next proe that the bidder is weakly better off making a decision immediately in Lemma 3. Consequently we derie in Lemma 4 the best-response strategy R of a bidder when all other bidders play and then characterize a threshold function prm such that prm R prm thus establishing that prm is an equilibrium strategy. Lemma 2. EU bid t 0 EU bidt t 0 for all t T. Proof. This result is a direct implication of the assumption that prm t I t is nonincreasing in I t and admits the following justification: Although bidding earlier does not increase the utility of a bidder, it reeals information about his aluation to other bidders, who can use this information to their adantage. More formally, suppose that the bidder A bids immediately, i.e., = t while all other bidders, following strategy, bid at T and hence I =, t T. Then the probability that bidder A wins, from (EC28), is Pr ( Nt A wins t i Nt ˆt i NtT ) Fmin prm t i 0 = F prm t i 0 NtT Fmin prm ˆt i and the corresponding expected utility from bidding for bidder A is [ ( Nt NtT Fmin EU bidt t 0 = E e T t prm t i 0 x Fmin F prm t i 0 prm ˆt j xdx Because threshold ti is nonincreasing in I, we hae j=1 (EC30) Fmin prm ˆt j x Fmin prm ˆt j 0 x j = 1 2NtT (EC31) Comparing (EC30) with (EC29) using (EC31) we obtain EU bidt t 0 EU bidt t 0 If the first bidder bids at some t<<t then I =, T and then the aboe analysis can be repeated to obtain EU bid t 0 EU bidt t 0 More generally, if bidder A places any bid in the auction (not necessarily his true aluation) at time < T then I > 0 and the threshold aluation in T is lower than if A bids at time T. Thus the probability that the buyout option is exercised by a bidder in T is higher, and because bidder A gets zero utility in such an eent he will not bid in the auction at any time <T. Moreoer if a bidder is bidding at time T then his weakly dominant strategy is to bid his true aluation. We next establish that bidder A is weakly better off making a decision immediately upon his arrial, i.e., he instantaneously decides either to exercise the buyout option immediately or place a bid in the auction at time T.

7 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec7 Lemma 3. When facing bidders who follow strategy, bidder A is weakly better off making a decision immediately i.e., EU waitt t 0 maxeu buyt t 0 EU bidt t 0. Proof. Here we gie an intuitie argument: a formal proof can be constructed on the lines of the proof of Lemma 1. We hae already proen in Lemma 2 that if a bidder decides to bid in the auction, she must do so at time T and thus her expected utility from bidding is independent of when she makes the decision to bid. Indeed, if we let EU bidt t 0 denote the utility of a bidder of type t 0 who waits up to time >tand then decides to place a bid in the auction at time T, then it can shown that EU bidt t 0 = EU bidt t 0 where we recall that EU bidt t 0 denotes the utility of a bidder t 0 who decides immediately to place a bid at time T. Additionally, although the buyout price remains constant throughout the auction, waiting decreases the bidder s utility from exercising the buyout option because of his time-sensitiity. Thus if a bidder waits before making a decision, his expected utility from bidding remains constant but the utility from exercising the buying option decreases and so he is weakly better off making a decision immediately. Thus we hae shown, in Lemma 2 and Lemma 3, that bidder A must choose, at time t, one of the two actions bidt buyt. We now show that the best response strategy R to is indeed a threshold strategy. Lemma 4. The best response strategy to is Buyout at p immediately if >min t 0 R t 0 Bid at time T if min t 0 where t 0 is such that EU buyt t 0 t 0 = EU bidt t 0 t 0 (EC32) Note that Lemma 4 only specifies the equilibrium path behaior of bidders. The best response strategy is completely specified by choosing a continuous threshold function t I, which is nonincreasing in I for all t and such that t 0 satisfies (EC32). Proof. The deriatie of the expected utility from bidding EU bidt t 0 with respect to is ( Nt EU bidt t 0 = E [e T t Fmint i 0 Ft i 0 NtT j=1 Fminˆt j 0 dx (EC33) For eery realization of Nt, NtT, t i Nt, and ˆt i NtT, the term inside the expectation on the right-hand side of (EC33) is nonnegatie and less than 1. Thus 0 /EU bidt t 0 < 1 for all + and t 0T. The utility from exercising the buyout option EU buyt t 0 = p and thus 0 EU bidt t 0 < 1 = EU buyt t 0 (EC34) Assuming, without loss of generality, that p we get EU buyt t0 = p 0 = EU bidt t 0. This together with (EC34) implies that there exists a unique aluation t 0 such that EU buyt t 0 t 0 = EU bidt t 0 t 0 (EC35) where the notation t 0 indicates the dependence of this aluation on strategy and the fact that this corresponds to the case when I = 0. Thus a bidder t 0 with t 0 bids in the auction at time T because EU buyt t 0 EU bidt t 0. On the other hand, if > t 0 then EU buyt t 0 > EU bidt t 0 and thus it is profitable for bidder A to exercise the buyout option.

8 ec8 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions Next we characterize a continuous threshold function prm t I such that prm t 0 is nondecreasing in t, prm t I is nonincreasing in I for all t and is such that prm R prm (EC36) Now notice that R prm is also a threshold strategy and indeed (EC36) holds, if prm t I = min prm t I for all t, I. For I = 0 substituting prm t 0 = min prm t 0 in (EC32) implies that prm t 0 = t where t must satisfy [ ( t t p = E t e T t Nt Fmint i x Ft i NtT j=1 Fminˆt j x dx (EC37) for all t 0T. For I>0, choosing any prm t I, which is nonincreasing in I for all t, suffices and so we set prm t I = prm t I. We thus obtain that if a threshold function prm t I is nonincreasing in I for all t and prm t 0 = mint is nondecreasing in t where is the solution of (EC37), then the corresponding strategy prm defines an equilibrium. We next proe the existence of prm. First, consider the following equation obtained by substituting Fminˆt j x with Fx in (EC37) for all bidders arriing in the interal t T : [ ( t t p = E t e T t Nt Fmint i x Ft i NtT j=1 Fxdx (EC38) Note that the right hand side of (EC38) at any time t depends only on 0t whereas the right-hand side of (EC37) is the function of 0T. Howeer if the solution t to (EC38) is nondecreasing in t then minˆt j x = x for all x t, ˆt j t T and thus t also soles (EC37). Lemma 5. For any >0, there exists a solution to (EC38) in the interal 0T. Proof. Define [ ( t Gt = E t e T t Nt Fmint i x F t i NtT j=1 Fxdx + p Using this notation, (EC38) becomes t = Gt (EC39) which we seek to sole on the interal 0T. To proe the existence of a solution of the aboe equation, we use the following theorem (Theorem 4.1 in Smart 1974), which is a slight generalization of Schauder s (1930) fixed point theorem. Theorem EC.1. Let M be a nonempty conex subset of a normed space B. Let T be a continuous mapping of M into a compact set M. Then T has a fixed point. Using the aboe theorem, we show that (EC39) has a solution on the interal 0T for any >0. For M>0 let F be the set of continuous functions with domain 0T and range q q > that satisfy the following Lipschitz condition t t Mt t t t 0T (EC40) Let = G F, i.e., G maps the set F to. We first proe the following lemma. Lemma 6. If q>p e /1 e and M>e 2 + q/1 e then F.

9 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec9 Proof. For any F, t 0T we hae p Gt p + e q (EC41) Clearly, if q>p e /1 e then Gt q for all F, t 0T. Next we show that for any F Gt Gt Mt t tt 0T (EC42) Indeed for any F and t 0T consider [ ( t Gt = E t e T t Nt Fmint i x F t i NtT j=1 Fxdx + p (EC43) where the expectation E t is with respect to the number Nt and epochs t 1 t Nt of arrials in 0t of a nonhomogeneous Poisson process with rate F t, and number NtT of arrials in t T of a Poisson process with rate. Let [ ( t Nt NtT EHt t = Gt p = E t e T t Fmint i x Fxdx F t i j=1 Now to calculate EHt t, we condition on the number of arrials in the interal t t + t where t > 0 and small, and is such that t + t T. For the sake of breity, let ( Nt t i Nt ) = Fmint i x F t i (EC44) First suppose that there was arrial in t t + t; an eent that has probability t because the arrial process is Poisson with rate. Then the conditional expectation is EHt t Ntt+ t = 1 [ ( t = E e T t ( t i Nt ) Fx Fx l dx PNt+ t T = l (EC45) Note that here we first calculate the expected utility gien Nt + t T = l and then sum oer all possible l. The expectation E t on the right-hand side is oer Nt and t i Nt. Now if there was no arrial in t t + t, an eent that has probability 1 t + ot where ot indicates any function f t such that lim t 0 f t/t = 0, we get EHt t Ntt+ t = 0 = [ E e T t ( t ( t i Nt ) Fx l dx PNt+ t T = l (EC46) The probability of more than one arrial in an interal of length t is ot and thus the unconditional expectation of Ht t becomes: EHt t = EHt t Ntt+ t = 1 t + EHt t Ntt+ t = 0 1 t + ot (EC47) Substituting for the terms, we get ( [ ( t EHtt = E e T t ( ) t i Nt ) Fx Fx l dx PNt+tT =l t ( [ + E e T t ( t ( ) t i Nt ) Fx l dx PNt+tT =l 1 t+ot (EC48) Similar to the aboe analysis, we next condition on the number of arrials in t t to ealuate EHt, where for ease of exposition the notation t = t + t is used.

10 ec10 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions Now suppose that there is arrial in the interal t t. For small t, the probability of this eent is F tt, because the arrial process at t is nonhomogeneous Poisson with rate F t. We then hae EHt t Ntt = 1 [ ( t = E e T t ( t i Nt )Fmint x F t Fx l dx PNt T= l (EC49) Now if there was no arrial in t t, an eent with probability 1 F tt + ot, weget [ ( t EHt t Ntt = 0 = E e T t ( t i Nt ) Fx l dx PNt T= l (EC50) The unconditional expectation of Ht t is EHt t = EHt t Ntt = 1F tt + EHt t Ntt = 01 F tt + ot (EC51) Substituting for the terms we get EHt t [ = E e T t [ + E e T t ( t ( t ( t i Nt ) Fmint x Fx l dx PNt T= lf tt F t ( t i Nt ) Fx l dx By the definition of the function G we hae PNt T= l1 F tt + ot Gt Gt = EHt t EHt t Substituting (EC48) and (EC52) in the aboe expression we obtain Gt Gt [ = E e T t ( t [ + E e T t ( [ E ( [ E ( t e T t ( t (EC52) ( t i Nt ) Fmint x Fx l dx PNt T= lf tt F t e T t ( t ( t i Nt ) Fx l dx PNt T= l1 F tt + ot ( ) t i Nt ) Fx Fx l dx PNt T= l t ( ) t i Nt ) Fx l dx PNt T= l 1 t + ot Simplifying the aboe expression we get the following bound Gt Gt [ ( t E e T ( ( t ti Nt ) Fmint xf x l Fx l+1) dx PNt T= lt [ ( t + E e T t ( t i Nt ) Fmin xf x l dx PNt T= lt t ( [ ( t + E e T t ( t i Nt ) Fx l dx PNt T= l) t (EC53)

11 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions [ ( + t1 F t t E e T t ( t i Nt ) Fx l dx PNt T= l 0 + t [ ( t E e T t ( t i Nt ) Fx l dx PNt T= l 0 t ) + (e T t e T t EHt t + ot ec11 Now noting that tt T and Ht t q t 0T, the aboe bound can be further simplified to obtain Gt Gt e 2qt + 2t tt +t t+e t 1q + ot (EC54) Because F we hae t t Mt t, t, t 0T. We use this and rearrange terms to obtain Gt Gt e q2 + + Mt + ot (EC55) Thus if we choose M>e q2 + + M, which can be rearranged to obtain M>e 2 + q/ 1 e, we get that Gt Gt Mt t 0 t t t t 0T (EC56) where = sup > 0 o/ < M e 2 + q/1 e. Note that >0 because lim 0 o/ = 0. Because is independent of t, it follows from (EC56) that Gt Gt Mt t t t t t 0T (EC57) where is as defined aboe. Now for any tt 0T, assuming without loss of generality that t >t, we hae Gt Gt Gt Gt + +Gt n Gt M+ +Mt n t=mt t where n =t t/. The second inequality follows from (EC57). Thus we hae shown that for any F This combined with the fact that for all F Gt Gt Mt t tt 0T (EC58) Gt q t 0T (EC59) implies that the set = G F F. Thus if we choose q>p e /1 e and M>e 2 + q/1 e the operator G maps F to F. We next show that the set is compact in where is the space of continuous functions with domain 0T. Lemma 7. is compact in. Proof. First note that because q>p e /1 e it follows from Lemma 6 that Gt q for all F and t 0T. It follows from (EC58) that for any >0 and all t t < /M Gt Gt Mt t < F tt 0T (EC60) proing that the set = G F is equicontinuous on the interal 0T ( 7.22 in Rudin 1976). The compactness of set then follows from the Arzelà-Ascoli Theorem (Theorem 3(3.I) in Kantoroich and Akilo 1964).

12 ec12 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions We next show that the operator G is continuous on the set F. Lemma 8. G is continuous on the set F. Proof. Let denote the sup norm, i.e., = max t t (EC61) 0 t T Let >0 be gien. For any F, t 0T we hae [ ( t Gt Gt E t e T t [ t + E t Nt ( Nt Fmin t i x F t i Fmint i x F t i NtT j=1 Nt Fmin t i x F t i ) NtT Fxdx j=1 Fxdx] (EC62) where Nt (respectiely Nt) is the number of arrials in a nonhomogeneous Poisson process with rate F (respectiely F ) for 0t, and the corresponding arrial times are gien by t i Nt (respectiely t Nt i ). NtT is the number of arrials in the interal t T of a Poisson process with arrial rate. If we assume that all bidders in t T bid in the auction, the first term on the right-hand side of (EC62) can be interpreted as the difference in the expected utility for a bidder with aluation t if eery bidder of type with 0t has aluation as opposed to haing aluation. This term is positie only if there are one or more arrials in the interal 0t in a nonhomogeneous Poisson process with rate F F. Now, because F is a gien continuous function for any >0, there exists a > 0 such that for all F, < F F < 0T Thus the rate F F can be upper bounded by for all 0T if <. Now the probability that there is one or more arrial in the interal 0t in a Poisson process with rate is t +o. Thus probability of at least one arrial in the interal 0t in a nonhomogeneous Poisson process with rate F F can be upper bounded by t + o. In addition an arrial in the aboe mentioned process can lead to a difference in expected utility, which is bounded aboe by q. Hence the first term in (EC62) can be upper bounded by qt +o. In addition because Fmin t i x Fˆt i and F 1, the second term of (EC62) is bounded aboe by t t. We thus get Gt Gt qt + o +t t qt + o + Define 1 = /3qT and 2 = sup o</3q ( 2 > 0 because lim 0 o/ = 0) and let = min 1 2. Thus for = min /3, we hae for all F such that < G G = sup t 0T Gt Gt qt + o + </3 + /3 + /3 = and hence the operator G is continuous on the set F. Thus G is a continuous mapping of F (which is nonempty and conex) into a compact set F and hence, by Theorem EC.1, G has a fixed point, i.e., there exists a solution to the equation (EC39) on the interal 0T. Because Lemma 5 holds for any >0, it proes that a solution to (EC38) exists on the interal 0T. We next show that t satisfying (EC38) on the interal 0T, is nondecreasing in t. For that we first need the following result. Let EH t be the right-hand side of (EC38), i.e., [ ( EH t = E t e T t Nt Fmint i x Ft i NtT j=1 Fxdx (EC63)

13 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec13 Lemma 9. E t H t > E t H t for t >t; tt 0T. Proof. Let t >t for some tt 0T. Suppose that there are k bidders in 0t and l bidders in t T and they arrie at time 0 <t 1 <t 2 < <t k <t and t<ˆt 1 < ˆt 2 < < ˆt l <T respectiely. Suppose also that j of the l bidders 0 j l arrie in the interal t t. Then from (EC63), we hae ( EH t kljt i k ˆt j l = e T t For this arrial stream of bidders, Ht is ( EH t klt i k ˆt j l = e T t k Fmint j i x k Ft Fminˆt i x j i Fˆt i k Fmint i x k Ft i l ) Fxdx Because Fminˆt i x/f ˆt i Fx for i = 1j and e T t >e T t, we hae EH t kljt i k ˆt j l >EH t klt i k ˆt j l l i=j+1 ) Fxdx (EC64) The inequality (EC64) is true for any realization of the random bidder arrial process and hence the inequality holds if we take the expectation oer the arrial process. Thus, we hae EH t > EH t We next use Lemma 9 to proe t is nondecreasing in t. Lemma 10. The solution t of (EC38) is nondecreasing in t. Proof. By definition of t, we hae t0 p = EHt t (EC65) Assume for contradiction that t < t dt for some t 0T and dt > 0 (and such that t dt 0T). Then there exists d > 0, such that t0 = t dt 0 d Using Lemma 9, we hae EHt t > EHtt dt Substituting (EC66) in (EC65) and using (EC67), we get By the definition of t dt, we hae Using (EC69) in (EC68), we get (EC66) (EC67) t dt d p = EHt dt d t > EHt dt d t dt (EC68) t dt p = EHt dt t dt EHt dt t dt EHt dt d t dt > d (EC69) Howeer, by arguments similar to those in the proof of Lemma 4, it can be shown that / EH t 1 for t 0T, which contradicts the aboe result. Hence t is nondecreasing in t. Thus we hae shown that t, satisfying (EC38) on the interal 0T, is nondecreasing in t. Ifwe ignore the zero probability eent of a bidder arriing first to the auction at time T then, as argued before, t also satisfies (EC37). Next, consider any function prm t I that is nonincreasing in I and satisfies prm t 0 = mint where t is the solution of (EC37) (and (EC38)) on the interal 0T, which exists by Lemma 5. Then Lemma 10 shows that prm t 0 is nondecreasing in t and thus, as argued before, the corresponding strategy prm satisfies prm R prm and hence defines a Bayesian Nash equilibrium for the online auction game with a permanent buyout price p.

14 ec14 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions EC.5. Proof of Theorem 4 For any continuous function 0T 0 such that ti is nonincreasing in I for all t, suppose that normal (noncommon alue) bidders follow the strategy Buyout at time t if >I t I t Bid at time T if p< I for all t T Bid at any time in t T if p The common alue bidders are assumed to play the following strategy: { Exercise buyout option at t T if >c I at any t T c c t I = Bid true aluation at T otherwise where c t 0 = t0 and c t I = I>0t. That is, such a bidder continuously monitors the auction and exercises the buyout option at the first time t T when > c I ; otherwise he bids his true aluation at T. We next show that if normal bidders play then there exists at least one bidder who has an incentie to deiate thus proing that the strategy does not define a Bayesian Nash equilibrium of the game G. Indeed, consider a bidder, e.g. A, of type t 0 with aluation <<p. We proe that EU bidt t EU bid t t T where the equality holds only at = T. Thus bidder A has an incentie to deiate from the strategy and bid at time T. Using the law of conditional expectation, we hae EU bidt t = EU bidt t P + EU bidt t P (EC70) where is the eent that at least one common alue bidder arries in the auction (notice that P>0). Now notice that if bidder A bids at time t then the threshold c for a common alue bidder becomes for the remaining duration of the auction. As a consequence, a common alue bidder will exercise the buyout option with probability 1 and thus EU bidt t = 0 because the bidder A does not get the product in this case. Similarly, we hae EU bidt t = EU bidt t P + EU bidt t P (EC71) If bidder A bids at time T then EU bidt t >0. To see this, consider the eent E that no other normal bidders arrie in the auction and all common alue bidders arriing in the auction hae aluation less than. If eent E occurs the information I t = 0 for all t<t and consequently all common alue bidders use the threshold function t0. Because <p t0, this implies that all bidders will bid in the auction and then, because bidder A has the highest aluation, she wins the auction and gains positie utility. Also note that PE >0 and thus EU bidt t >0. If eent occurs then no common alue bidders arrie in the auction and thus the auction only has normal bidders. Howeer, we hae shown in Lemma 2 that in the presence of normal bidders EU bid t 0 EU bidt t 0 for all t T and thus we hae that EU bidt t EU bidt t This combined with the aboe arguments and using (EC70) and (EC71) yields that EU bidt t > EU bidt t. The same analysis can be repeated to show that for any <T EU bidt t > EU bid t and thus bidder A is strictly better off bidding at time T. Hence for any threshold the game G does not hae an equilibrium of the form. The aboe analysis can be extended to show that indeed any strategy where a bidder places a bid before time T does not form a Bayesian Nash equilibrium of G. On the other hand, the proof of Theorem 3 still holds for this case and thus a strategy where normal bidders play strategy prm still defines a Bayesian Nash equilibrium of G where prm is as defined in Theorem 3.

15 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec15 EC.6. Proof of Proposition 2 Let t be the solution of (EC38), i.e., t p = E t [ e T t ( t Nt Fmint i x Ft i NtT j=1 Fxdx (EC72) for all t 0T for some >0 and small. As discussed in the proof of Theorem 3, t is nondecreasing in t and thus it follows from (EC29) that the expected utility from bidding for a bidder with type t 0, assuming other bidders follow strategy, is [ E t U bidt t 0 = E e T t ( Nt Fmint i x Ft i NtT j=1 Fxdx Therefore, (EC72) can be restated as t p = E t U bidt tt0 (EC73) for all t 0T. For any tt + t 0T, must satisfy t + t t = EU bidt t + t t + t 0 EU bidt tt0 (EC74) t t Consider a bidder A with type tt and information I t = 0. Because the auction is running at time t, all bidders of type i t i 0 arriing in the interal 0t hae aluation i t i. Thus for bidder A the arrial process of other bidders is 1. nonhomogeneous Poisson process in 0t with arrial rate = F prm for 0t, 2. homogeneous Poisson process in t T with arrial rate. Now to calculate EU bidt tt0, we condition on the number of arrials in the interal t t + t (Figure EC.1). To reduce notational complexity, let ( Nt t i Nt ) = Fmint i x Ft i (EC75) Suppose that there was arrial in t t + t; this eent has probability t + ot, where ot indicates any function f t such that lim t 0 f t/t = 0, because the arrial process is Poisson with rate. Then the conditional expected utility is EU bidt tt0 Ntt+ t = 1 = [ ( t E e T t ( t i Nt ) Fx Fx l dx PNt+ t T = l (EC76) Figure EC.1 Threshold Valuation prm Threshold aluation ν(t + t) ν(t) 0 t t + t T Auction time line N(t) bidders 0/1 bidder N(t + t,t) bidders

16 ec16 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions Note that here we first calculate the expected utility gien Nt + t T = l and then sum oer all possible l. The expectation E t on the right-hand side is oer Nt and t i Nt. Now, if there was no arrial in t t + t, an eent which has probability 1 t + ot, weget EU bidt tt0 Ntt+ t = 0 [ ( t = E e T t ( t i Nt ) Fx l dx PNt+ t T = l (EC77) The probability of more than one arrial in an interal of length t is ot and thus the unconditional expected utility from bidding becomes EU bidt tt0 = EU bidt tt0 Ntt+ t = 1 t + EU bidt tt0 Ntt+ t = 0 1 t + ot (EC78) Substituting for the terms, we get EU bidt tt0 [ ( t = E e T t ( t i Nt ) Fx l+1 dx PNt+ t T = l t [ + E e T t ( t ( t i Nt ) Fx l dx PNt+ t T = l 1 t + ot (EC79) Similar to the aboe analysis, we next condition on the number of arrials in t t to calculate EU bidt t t 0, where to reduce notational complexity the notation t = t + t is used. Consider a bidder B with type t t and information I t = 0. Because the auction is running at time t, all bidders of type i t i 0 arriing in the interal 0t hae aluation i t i. Thus for bidder B the arrial process of other bidders is 1. nonhomogeneous Poisson process in 0t with arrial rate = F for 0t, 2. homogeneous Poisson process in t T with arrial rate. Now suppose that there is arrial in the interal t t. For small t, the probability of this eent is F tt + ot, because the arrial rate at t is F t. Therefore, EU bidt t t 0 Ntt = 1 [ ( t = E e T t ( t i Nt )Fmintx Fx l dx PNt T= l Ft If there was no arrial in t t, an eent with probability 1 F t + ot), we get [ ( t EU bidt t t 0Ntt =0= E e T t ( t i Nt ) Fx l dx PNt T =l (EC80) (EC81) Because the probability of more than one arrial in an interal of length t is ot, the unconditional expected utility is EU bidt t t 0 = EU bidt t t 0 Ntt = 1F tt + EU bidt t t 0 Ntt = 01 F tt + ot Substituting we get EU bidt t t 0 [ ( t = E e T t ( t i Nt [ + E e T t ( t )Fmintx Fx l dx PNt T= lf tt Ft PNt T= l1 F tt + ot ( t i Nt ) Fx l dx (EC82)

17 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec17 Consider now the error associated with substituting Fminx with Fx in the first term of the right-hand side of (EC82). The substitution effectiely assumes that one bidder has aluation in the interal t instead of t. When calculating the maximum aluation among the bidders, this assumption can lead to a difference dt t, which is bounded as follows: 0 dt t t t Mt t where we hae used the fact that satisfies the Lipschitz condition for constant M. Thus, if we let T 1 be the first term in Equation (EC82) and T1 a be the corresponding approximate expression, then we hae 0 T a 1 T 1 Ee T t t t PrN t T= l F prm t e T t MFtt 2 and the error due to the approximation is thus ot. Therefore, (EC82) can be rewritten as [ ( t EU bid t t = E e T t Fx l+1 dx PrN t T= lt ( [ + E e T t ( t ) Fx l dx PrN t T =l1 F t +ot (EC83) Subtracting Equation (EC79) from Equation (EC83), diiding by t and taking the limit t 0, we get, after simplification, d t + 1 Ftt P = dt 1 e +1 FtT t Recall that bidder aluations are assumed to be uniformly distributed and thus Fx= x /m for x. Substituting t = 0 in (EC72), we get the initial alue for the aboe differential equation 0 = P m ( ) ) W ( e +T P T + me +T + e +T T m where, as before, W is Lambert s W function. EC.7. Empirical Analysis of Bid Times The outcome predictions in Theorem 1 and Theorem 3, for temporary and permanent buyout-price auctions respectiely, suggest the following two erifiable hypotheses about the timing of bids placed in the corresponding buyout-price auction, which we test using data from ebay and Yahoo! auctions: Hypothesis 1. The first actiity (bid/buyout) in a temporary buyout-price auction occurs earlier than in a standard auction (without a buyout price). Hypothesis 2. The aerage bid time in a permanent buyout price auction is higher than in a standard auction (without a buyout price). In practice, bidding in online auctions is affected by seeral factors including the number of competing auctions, presence of a resere price, seller s feedback ratings, leel of bidders rationality and experience and their different incenties and, in all likelihood, the bidding data we collect from auction websites depends on some or all of these factors. Howeer, for the purpose of this analysis we assume that the difference in bid times (as obsered below) is a consequence primarily of the presence of a buyout price. For testing the hypotheses we collect bidding data for auctions belonging to the Consumer Electronics category, which hae the string ipod EC1 in their title. This criterion gies data on auctions EC1 The search is not case-sensitie.

18 ec18 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions for ipods and its accessories (including headphones, ipod skins, chargers, cases); this market was chosen for seeral reasons. First, the olume transacted is high as compared to other products. Second, most of these items are aailable through fixed-price mechanisms and hence bidder aluations should hae well-defined upper and lower bounds. Furthermore, these items will most likely hae little or no common alue and so the independent priate aluation assumption should hold for bidders participating in such auctions. Further details of the data collection process are proided in Gupta (2006). EC.7.1. ebay Data. For all auctions where at least one bid is placed, the time elapsed in the auction (expressed as a fraction of the auction length) before the first actiity occurs (bid/buyout) is recorded. Note that for auctions where the buyout option is exercised we use the scheduled auction ending time, as opposed to the actual ending time (which is the time of buyout exercise), to calculate the auction duration. We ignore all auctions where the buyout price (bp) is too close to the starting price (sp) (bp/sp < 125), because in such a case the auction effectiely becomes a fixed-price mechanism. We also filter out auctions where the buyout price is much higher than the winning bid wb, in particular wb/bp < 08, because in such cases the buyout price may be set too high for any bidder to eer exercise it. Note that, although the winning bid is a random ariable depending on the bidder arrial process and bidder aluations, we assume that for auctions of the aboe category the ariance of the winning bid is low enough so that it seres as a good indicator of the actual price of the auctioned item. From the data it is obsered that the time of the first bid/buyout decreases as the winning bid increases for auctions with and without a buyout price. Furthermore, because in our data set almost 90% of the buyout-price auctions hae a winning bid of less than $30, for the purpose of this analysis we only consider auctions where the winning bid is less than $30. Table EC.1 summarizes the descriptie statistics of the data. The number in parentheses is the standard deiation associated with the corresponding mean. Obsere that the aerage time of the first actiity in auctions without a buyout option is higher than in auctions with a buyout option. Furthermore, the aerage winning bid in buyout-price auctions is lower and, because the first bid time tends to decrease as the winning bid increases, adjusting the data to ensure that the aerage winning bid is the same for both cases would further lower the aerage first actiity time in buyout-price auctions (or alternately increase the aerage first bid time in auctions without a buyout price). The two sample t-test gies a p-alue of implying that the hypothesis that the aerage first actiity time in buyout price auctions is lower than the first bid time in auctions without a buyout price can be accepted. In a separate analysis we find that in our data set, the buyout option is exercised in about 53% of the auctions in which it is present with a mean exercise time, expressed as a fraction of the auction duration, of (standard deiation = 00064). The presence of a buyout option thus decreases the waiting time of the auction participants to about three-quarters of what they would hae experienced if the buyout option was not present. EC.7.2. Yahoo! Data. Collecting bid timing data for auctions where the buyout price is exercised requires obtaining the scheduled auction ending time, as opposed to the actual ending time (the time the buyout price is exercised), which would necessitate tracking auctions that hae not closed. This would, howeer, take a prohibitiely long time on Yahoo! because the number of open auctions in the market segment of our interest that satisfy certain conditions on the buyout price, starting price, and winning bid are ery small. For buyout-price auctions, we thus restrict our analysis to auctions where the buyout price is not exercised. Note that this biases the data because we are more likely to get auctions where the buyout price is higher than what bidders expect to pay for getting the product. We partially correct this bias by only considering, as in the earlier case, auctions where the ratio of Table EC.1 wb/bp 08 Summary of Auction Data with bp/sp 125, and No buyout price Buyout price Number of auctions 3,414 1,954 Mean winning bid $5.56 $4.93 Mean first actiity time (0.0041) (0.0067)

19 Gallien and Gupta: Temporary and Permanent Buyout Prices in Online Auctions ec19 Table EC.2Summary of Auction Sample with wb > $100, bp/sp 125, and wb/bp 08 No buyout price Buyout price (not exercised) Number of auctions Mean winning bid $ $ Mean bid time ( ) (0.0187) the winning bid to the buyout price is high enough (wb/sp 08); and as before, we ignore auctions where the buyout price is too close to the starting price in particular we require bp/sp 125. The median winning bid in the complete Yahoo! data (including auctions where the buyout option is exercised) is $152.51, as compared to $5.95 for ebay data, which presumably happens because the search for the string ipod on Yahoo! leads to more auctions where ipod s are sold as opposed to some of its cheaper accessories. Indeed it turns out that almost 95% of buyout price auctions satisfying the aboe criterion (buyout option not exercised, wb/sp 08, bp/sp 125) hae a winning bid greater than $100 and hence we only consider auctions where the winning bid is greater than $100. The important statistics of the restricted data set are summarized in Table EC.2. Obsere that the aerage bid time in buyout-price auctions is higher than in auctions without a buyout price. Howeer a two sample t-test returns a p-alue of , which suggests that although Hypothesis 2 in EC.7 can be accepted based on this data set; it is with a much lower confidence leel than the earlier case. A separate analysis reeals that similar to ebay data the aerage bid time usually decreases as the winning bid increases. Also obsere that for this data set, the aerage winning bid is higher in buyout-price auctions, and hence the aerage bid time would further increase for buyout price auctions (or alternately, decrease for auctions without a buyout price) if the data were adjusted to ensure that the aerage winning bid is same for both cases. One of the main reasons the result in this case is not as conclusie as in the temporary case is because some Yahoo! auctions use a slightly different auction mechanism than the one assumed in this paper; in particular, whereas we assume a fixed auction end time, some Yahoo! auctions hae a floating deadline that extends if a bid is placed near the end of the auction. The presence of a floating deadline may alter bidding strategy in particular, the strategy bid just near the end of the auction cannot be played because wheneer a bid is placed in the auction, the deadline is automatically extended; see also Roth and Ockenfels (2002) who empirically test the effect of a floating deadline on bid times. The primary motiation of bidding late in an auction with a buyout price is to preent bidders from utilizing the information proided by one s bid; howeer, in an auction where the deadline extends when a bid is placed, other bidders always get some time to respond to a bid irrespectie of when it is placed, thus decreasing the incentie of bidding late in the auction. Furthermore, as discussed before, our data is biased towards auctions with a buyout price that is higher than what bidders expect to pay for getting the product. As mentioned aboe, an important reason why bidders bid late in a permanent buyout-price auction is that information about a bidder s aluation may cause another bidder to exercise the buyout option; howeer a ery highly priced buyout option has a small probability of exercise and hence decreases the incentie of bidding late. In summary, bidding data from ebay and Yahoo! suggests that the hypotheses adocated by equilibrium strategies tmp and prm for the temporary and permanent case respectiely can be accepted, albeit with a lower degree of confidence for a permanent option than a temporary option. This implies that bidders do seem to follow strategies tmp and prm in temporary and permanent buyout-price auctions respectiely, although, as mentioned before, this simple empirical analysis does ignore other factors that impact bidder behaior. An extensie study of these factors, howeer, is beyond the scope of this paper. EC.8. Deriation of Results Shown in Table 2 We will use the following lemma to derie the results in Table 2. Lemma 11. Consider a continuous function gp 0 0 and let p = arg max p X gp. Suppose gp is such that lim gp = gp (in the sup norm) where gp is also continuous on the set. Then if g has a unique maximizer p = arg max p X gp, lim p = p.