k- price auctions and Combination-auctions

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1 k- rice auctions and Combination-auctions Martin Mihelich Yan Shu Walnut Algorithms March 6, 219 arxiv: v3 [q-fin.mf] 5 Mar 219 Abstract We rovide for the first time an exact analytical solution of the Nash equilibrium for k- rice auctions. We also introduce a generalization of the second rice auction that maintains the incentive to bid truthfully, therefore aving the way for relacing second rice auctions. key words : 1 Introduction Vickrey auctions, k- rice auctions, Combination-auctions, game theory Second rice auctions, also known as Vickrey auctions [8], are well known and largely used as examles in online or Government auctions because it gives bidders an incentive to bid their true value. Nevertheless, second rice auctions have reached some limits because, inter alia, bidders bid insincerely and the variance of the winning bid often has a large value. A natural generalization of second rice auctions is the k-rice auction, which has been studied by many researchers in the recent years. The reader can refer to [1, 2, 9] for related literature and [3]. In articular, Roger B. Myerson [5] established the Revenue Equivalence Theorem (known as RET theorem in 1981, which characterized the equilibrium strategy. Later in 1998, Monderer and Tenenholtz [4] roved the uniqueness of the equilibrium strategies in k-rice auctions for k 3. Under some regularity assumtions, they also rovided sufficient conditions for the existence of the equilibrium. In 2, Wolfsteller [1] solved the equilibrium k-rice auctions for a uniform distribution and in 214, Nawar and Sen [6] generalized the result of Wolfsteller, and rovided a derivation exression of the k-rice auctions bidding equilibrium for a quadratic distribution. In this aer, we rove two major results. First, in Theorem 3.1, we generalized Wolfsteller and Nawar results by giving a closed form solution of the equilibrium of the k-rice auctions in a general case. The solution takes an easy form which is easily calculable. As alications, this simlifies the discussions of the existence and uniqueness, and we can calculate the equilibrium for classical distributions and recover certain known results. Secondly, we extend the notion of k- rice auctions by introducing a new tye of auction that we will call a "Combination-auctions": the winner ays a centroid of the bids. In fact a natural question is: Are there other combinations that second rice auction which lead to a truthful equilibrium? We demonstrate in Theorem 4.1 that there exists combinations other than the martin.mihelich@walnut.ai yan.shu@walnut.ai 1

2 second rice auction leading to a truthful equilibrium.these new strategies could relace second rice auction... Then we show in Theorem 4.2 that if there exists truthful strategies other than the second rice auctions, then the distribution of the valuations of the bidders is a uniform distribution. We also rovide an alternative roof of the RET theorem with robability tools, which does not involve advanced auctions theory and game theory knowledge. This aer is organized as follows: in Section 2 we state the assumtions and re-establish the RET theorem with a robability aroach. Next, we solve the equilibrium and discuss the uniqueness and the existence, together with alications for classical distributions. Finally, in Section 4 we introduce the "Combination-auctions" and discuss strategies other than second rice auctions which are truthful. 2 Problem formulation and assumtions In this section we resent our assumtions and re-establish the RET theorem. The RET theorem is established in [5] by Roger B. Myerson. The original roof involves heavy advanced auctions theory and game theory insight. Here we rovide an alternative aroach, with robability tools. With our aroach, one can extend the RET theorem for "Combination-autions", which we will introduce in Section 4. The k- rice auction roblem can be formulated as follows: consider a k-rice auctions with n bidders, where the highest bidder wins, and ays only the k-th highest bid. Let s assume that k 2 and n k. We make the assumtions following [7]: 1. The valuations V i, i 1,, n of the bidders are indeendent and identically distributed with distribution function F. 2. The distribution function F is with values in I where I [, 1] or I R +. We also assume that: (A F is k 2 times continuously differentiable and x I F (x >. We remark that for analysis of the 3-rice auction in the literature, the existence and the continuity of the density function are often assumed. It is thus natural to assume (A holds for the case of general k-rice auctions. We also assume that each bidder bids X i g(v i where g is a strictly increasing function. It follows that the bids X i are indeendent variables and we denote their distribution function ˆF and their density ˆf. As g is strictly increasing, ˆF is also strictly increasing on g(i. For n 1 bidders which bid X 1,..., X n 1, we denote Y n 1 max(x 1,..., X n 1 and Y n 2 the second maximum. In general n 1, Y n is the -th maximum. Now we recall the RET theorem for the k-rice auctions (see also [4]: Theorem 2.1 (RET theorem. Let k 3. A risk-neutral strategy g is an equilibrium strategy in the k-rice auction if and only if the following two conditions hold: 1. g is strictly increasing in the interval [, 1]. 2. It holds for all x [, 1]: t (x g(tf(t n k (F(x F(t k 3 F (tdt. 2

3 2.1 An alternative aroach of RET theorem This section rovides a robability aroach of RET theorem. Before roving the theorem, we first show several lemmas with algebra comutation. Lemma 2.2. For all t x and (t, x I 2, denote Then it holds and H (t H(t : P([Y n +1 t] [Y n 1 x]. H(t Proof. It holds for all t x (t, x I 2 : ˆF(t n 1 ( ˆF(x ˆF(t (1 (n 1! (n k!(k 2! ˆF n k (t( ˆF(x ˆF(t k 2 (2 P (Y n +1 t Y n 1 x P (X 1,..., X n 1 t ( n 1 + P (X 1,..., X n 2 t X n 1 ]t, x] 1 ( n P (X 1,..., X n +1 t X n +2,..., X n 1 ]t, x] 2 And equation (1 follows according to the fact that X i are indeendently identically distributed. Now we rove equation (2. Derive equation (1, it follows: ( H (t ˆf(t n 1 (n 1 ˆF(t n 2 ( ˆF(x ˆF(t ( (3 n 1 1 ˆF(t n 1 ( ˆF(x ˆF(t 1 Then ( H (t ˆf(t n 1 (n 1 n 1 ˆF(t n 2 ( ˆF(x ˆF(t ( (4 k 3 n 1 ( ˆF(t n 2 ( ˆF(x ˆF(t A telescoic sum aears and after simlification we get: H (t which is exactly equation (2. Lemma 2.3. For all, m N, it holds: 1 (1 u u m du For n > k 2, it holds: (n 1! (n k!(k 2! ˆF n k (t( ˆF(x ˆF(t k 2 ˆf(t, ( 1 k 2 n 1 ( k 2 3!m! ( + m + 1!. (5 (n k!(k 2!. (6 (n 1!

4 Proof. The equation (5 is direct from integration by arts. We now rove equation (6. for n > k 2, denote ( 1 k 2 ( k 2 A n 1 Consider Now let Observe that ( k 2 r(x ( x n 2 ( 1 n k x n k (1 x k 2. and according to equation (5, it holds R(x r(udu. R(1 ( 1 n k+1 A R(1 ( 1 n k+1(n k!(k 2! (n 1! hence A (n k!(k 2! (n 1! 1 (n 1 (. n 2 k 2 Now we are ready to rove the RET Theorem. Proof. We consider here n 1 bidders which lay with the same rule X i g(v i. The ayoff exression of the n-th bidder for a valuation v and a bid x is: U(x, v (v th (tdt (n 1! (v t (n k!(k 2! ˆF n k (t( ˆF(x ˆF(t k 2 ˆf(tdt (7 The goal is at v fixed to find x g(v which maximizes the ayoff. By develoing (7 we find: ( k 2 (n 1! k 2 U(x, v ( 1 k 2 (n k!(k 2! ˆF (x (v t ˆF n 2 (t ˆf(tdt And then is equivalent to ( 1 k 2 n 1 U (x, v x ( ( k 2 (n 1(v x ˆF n 2 (x + ˆF 1 (x ˆF n 1 (tdt. (8 Now let g 1 be the quantile function of the distribution X, i.e. g 1 ˆF 1. As ˆF is strictly increasing, g 1 exists and differentiable almost everywhere. Let denote ˆF(x a and q the 4

5 quantile function of F. Thus, X g(v imlies that a ˆF(x F(v, therefore we have v q(a. Equality (8 becomes Observe that where ( 1 k 2 n 1 ( ( k 2 a (n 1(q(a g 1 (aa n 2 + a 1 u n 1 g 1 (udu. (9 ( 1 k 2 ( k 2 a a 1 u n 1 g 1 (u du n 1 ( 1 k 2 ( k 2 a 1 u n 1 g 1 (u du n 1 P (a, ug 1 (u du, P (a, u ( 1 k 2 n 1 ( k 2 a 1 u n 1. Since, n k, we deduce that n 1 > for [, k 2]. Thus for all a R, P (a,. Denote a the derivation oerator with resect to a and u the derivation oerator with resect to u. It holds: ( k 2 u P (a, u ( 1 k 2 a 1 u n 2 ( k 2 ( 1 k 2 a 1 u k 2 u n k ( a k 2 ( 1 k 2 a u k 2 u n k a ( (a u k 2 u n k (k 2(a u k 3 u n k. Therefore, alying integration by arts and observing that P (,, it follows that P (a, ug 1 (udu [P (a, ug 1(u] a u P (a, ug 1 (udu According to lemma 2.3 P (a, a P (a, ag 1 (a ( 2(a u k 3 u n k g 1 (udu. (k 2(a u k 3 u n k du 1 (k 2a n 2 (1 u k 3 u n k du a n 2(n k!(k 2!. (n 2! The latter equality together with equation (5 leads to 5

6 (q(a g 1 (aa n 2(n k!(k 2! +a n 2(n k!(k 2! g 1 (a (n 2! (n 2! which is equivalent to (k 2(a u k 3 u n k g 1 (udu (1 q(aa n 2(n k!(k 2! a (k 2(a u k 3 u n k g 1 (udu. (11 (n 2! Notice that the latter equation can be also written as: q(ap (a, a rearranging the terms, it holds: (k 2(a u k 3 u n k g 1 (udu, (k 2(a u k 3 u n k (g 1 (u q(adu. Let x q(a and change the variable u F(t inside the integration, together with the fact that g 1 (u ˆF 1 F(t g(t, we obtain: The roof is comleted. (k 2(F(x F(t k 3 F(t n k (g(t xf (tdt. 3 Analysis of equilibrium In this section, we first give a closed form solution of the equilibrium for k- rice auctions, k 3. Then with the solution, we analyze the existence and uniqueness of the equilibrium. At the end of this section we rovide some examles. 3.1 Solution of the equilibrium Theorem 3.1. We assume that (A holds, then the equilibrium satisfies equation (11. Moreover, equation (11 has unique solution: Proof. We denote for and It holds : and for 1: g 1 (a ak n (n k! (n 2! ( q(aa n 2 (k 2 (q(aa n 2 (k 2 A (a, u (a u u n k g 1 (u G (a, u G (a, u (a u u n k g 1 (u. A (a, a a n k g 1 (a, u n k g 1 (u A (a, a, 6 (a n 2 (k 2. (12 A (a, udu,

7 For > 1 and for all a [, 1]: ( G 1 a+h a (a lim A (a + h, udu A (a, udu h h ( 1 a+h lim A (a + h, udu A (a + h, udu + h h ( 1 a A (a, a + lim A (a + h, u A (a, udu h h A (a, a + ( 1G 1 (a a A (a, udu A (a + h, u A (a, udu We remark that for all b I, b g 1(u du bg 1 (b holds, thus we can switch the integral and the limit in the revious formula. Therefore calculation is valid. Therefore, the ( + 1 th derivative of G on a is G (+1 (a ( 1!G (a ( 1!A (a, a ( 1!a n k g 1 (a. (13 Hence, the (k 2 th derivative of equation(11 gives ( (k 2 ( q(aa n 2(n k!(k 2! a (k 2 (k 2 (a u k 3 u n k g 1 (udu (n 2! Rearranging the terms, we have g 1 (a ak n (n k! (n 2! (k 2G (k 2 k 3 (a (k 2!a n k g 1 (a. ( q(aa n 2 (k 2 (q(aa n 2 (k 2 (a n 2 (k 2. Since equation (11 has a unique solution, according to RET theorem, the equilibrium exists if and only if (q(aa n 2 (k 2 /(a n 2 (k 2 is strictly increasing. 3.2 Examles As alications, we introduce several examles. With the formula from revious section, one can easily recover some classical results. Examle 3.1 (Uniform distribution. q(a a, g(v g 1 (a. g(v g 1 (a ak n (n k! ( q(aa n 2 (k 2 (n 2! ak n (n k! ( a n 1 (k 2 (n 2! n 1 n k + 1 a n 1 n k + 1 v 7

8 Examle 3.2 (3-rice auctions. For k 3, notice that q (a F 1 (a 1/F (v, it follows that: and we found the well known result. g(v g 1 (a a3 n ( q(aa n 2 (n 2 q(a + 1 n 2 aq (a v + 1 F(v n 2 F (v Examle 3.3 ( 4-rice auctions.. For k 4, since q (a 1 F (v 1 F (q(a, it follows that Then it holds : q (a F (q(a F 2 (q(a q (a F (v F 3 (v. a 4 n ( g 1 (a q(aa n 2 (n 2(n 3 a 4 n ( (n 2(n 3a n 4 q(a + 2(n 2a n 3 q (a + a n 2 q (a (n 2(n 3 q(a + 2 a n 3 F (q(a 1 a 2 F (q(a (n 2(n 3 F (q(a 3 Finally, with with v q(a and a F(v: We recover the result of [6], Theorem 2. g(v v + 2 F(v n 3 F (v 1 (n 2(n 3 F 2 (vf (v F (v 3 Examle 3.4 (olynomial distribution.. For F(x : x α with α >, then q(a a 1/α and g(v g 1 (a ak n (n k! ( q(aa n 2 (k 2 (n 2! ak n (n k! ( a n 2+1/α (k 2 (n 2! Γ(n k + 1Γ(n 1 + 1/α Γ(n k /αΓ(n 1 a1/α Γ(n k + 1Γ(n 1 + 1/α Γ(n k /αΓ(n 1 v where Γ is the Gamma function. In articular, if α 1 m where m is a ositive integer, g(v (n 2 + m...(n k + m + 1 v (n 2...(n k + 1 8

9 4 Combination-rice auctions In this section we start by defining a new tye of auction that we call a Combination-rice auction. We demonstrate that a Combination-rice auction could be truthful for linear combinations other than second rice auctions (α 2 1, giving some comanies an incentive to use it instead of second rice auctions. Moreover, we will characterize the distributions for which there exists a linear combination different from α 2 1 which is truthful. 4.1 Nash equilibrium of Combination-rice auctions We call Combination-rice auctions an auction in which the winner will ay a linear combination of the rices bid by the bidders: α 1 Y n α s Y n s+1 where all the α k are ositive satisfying sk1 α k 1. Like in the first art, we consider here n 1 bidders which lay with the same rule X i g(v i. Reasoning as in Section 2, the ayoff exression for the n-th bidder for a valuation v and a bid x can be exressed as a multile integral: U(x, v (x α k t k P (Y n 1 [t 1, t 1 +dt 1 ]... Y n s+1 [t s, t s +dt s ] Y n 1 xdt 1...dt s. k1 Together with s k1 α k 1 we can write: U(x, v α k (x t k P (Y n 1 [t 1, t 1 +dt 1 ]... Y n s+1 [t s, t s +dt s ] Y n 1 xdt 1...dt s k1 then: where U k is the ayoff of the k-rice auctions. U(x, v α k U k (x, v k1 Now combining equation (11 with a simle calculation of U 1 and U 2, we deduce that g 1 is a Nash equilibrium for the combination-rice auctions if and only if g 1 is increasing and the following equation holds: q(aa n 2 ( k2 (n k!(k 2! α α 1 a n 1 (n 2! α 1 (n 1 q(a g 1(a a n 2 + α g 1(a 2 a n 2 g 1 (a Study of truthful equilibrium ( a s g 1 (u α k (k 2(a u k 3 u n k du. k3 (14 The following theorem characterizes the truthful equilibrium for uniform distribution. Theorem 4.1. If the distribution is uniform, there are an infinite number of linear combinations which lead to a truthful strategy. More recisely, adating the notations before, a Combination-rice auction is truthful if and only if the coefficients (α i satisfy the following conditions: 1. It holds α 1 k3 (k 3!(n k!(2k n 3 α k, (n 2! 9

10 2. for all k, α k, and α k 1. k1 Proof. The equilibrium is truthful if and only if a g 1 (a q(a and together with equation (14, it holds: ( s q(aa n 2 (n k!(k 2! α k α 1 a n 1 k3 (n 2! ( a s q(u α k (k 2(a u k 3 u n k du. k3 (15 If the distribution is uniform in [, 1], then q(a a and by relacing the latter formula becomes (k 3!(n k!(2k n 3 α 1 α k. (n 2! k3 Moreover we still have s k1 α k 1 and k α k. We can easily show that this intersection of two hyer lans with the ortion of the sace of α k has an infinite number of solutions. Next theorem characterizes the truthful equilibrium for general distribution. Theorem 4.2. If F is a continuous and strictly increasing function on I [, 1] or I [, + and if it exists a truthful equilibrium different of second rice auctions, then F corresonds to the uniform distribution in [, 1], i.e F(x x for x [, 1]. Proof. As F is continuous on I and strictly increasing, then q exists and is continuous on J [, 1] or J [, 1. Moreover, we have q(. Then, with equation (15 we can easily deduce that q is infinitely differentiable on J \ {}. Moreover by differentiating (15 we can show that q is extendable by continuity in and then by differentiating an other time that q is also extendable by continuity in. With the variable change u av and after simlifying by a n 2, equation (15 becomes: (n!( 2! q(a α α 1 a 3 (n 2! 1 q(av α ( 2(1 v 3 v n dv 3 (16 Then, by deriving the equation (16 two times (the derivation under the integral sum is valid because q is two times continuously differentiable we find a I: q (n!( 2! 1 (a α v 2 q (av α ( 2(1 v 3 v n dv. (17 (n 2! 3 Moreover, according to lemma 5, we have: 3 1 α ( 2(1 v 3 v n dv (n!( 2!. (n 2!

11 Suosing by the absurd that it exists x I such that q (x, and without loss of generality q (x >. By noting a such that q(a max I q(x (or max [,b] q(x b (, 1 if I [, 1 we have that the right hand side of (17 satisfies: 1 v 2 q (a v α ( 2(1 v 3 v n dv < q (a 3 3 which is absurd because there is equality in (16. (n!( 2! (n 2! α, Then a I, q (a and we deduce that we necessarily have I [, 1] and such that q( and q(1 1 we have q(a a which corresonds to an uniform distribution of the valuation of the bidders. 5 Conclusion In this aer we give an exact analytical solution of the k-rice auctions. Moreover we introduce a new tye of auction which could relace the second rice auctions. This article has led us to interesting oen questions that we list here: 1. Is it ossible to find the exact analytical solution of the combination-auctions, i.e solve the equation (11? 2. Here we took constant coefficients α i. What haens if the α i becomes random variables deendent, for examle, on F? 3. For a given F is it ossible to find a combination such that α 2 (or α 2 r, r [, 1 and such that the Nash equilibrium of these combination-auctions noted g minimizes any standard norm between g and the truthful one x x? References [1] Paul Klemerer. Auction Theory: A Guide to the Literature. Microeconomics 9932, University Library of Munich, Germany, [2] Vijay Krishna. Auction Theory (Second Edition. Academic Press, San Diego, second edition edition, 21. [3] Eric Maskin. The unity of auction theory. Journal of Economic Literature, 42(4: , 24. [4] Dov Monderer and Moshe Tennenholtz. K-rice auctions. Games and Economic Behavior, 31(2:22 244, 2. [5] Roger B. Myerson. Otimal auction design. Mathematics of Oerations Research, 6(1:58 73, [6] Abdel-Hameed Nawar and Debariya Sen. k-rice auction revisited [7] John G. Riley and William F. Samuelson. Otimal auctions. The American Economic Review, 71(3: , [8] William Vickrey. Counterseculation, auctions, and cometitive sealed tenders. The Journal of finance, 16(1:8 37,

12 [9] Robert Wilson. Chater 8 strategic analysis of auctions. volume 1 of Handbook of Game Theory with Economic Alications, ages Elsevier, [1] E. Wolfstetter. Third- and higher-rice auctions. In M. Braulke and S. Berninghaus, editors, Beiträge zur Mikro- und zur Makroökonomik: Festschrift für Hans-Jürgen Ramser, ages Sringer Verlag,