aa r X i v : . [ ec on . T H ] J un Analytical solution of k th price auction Martin Mihelich ∗ a and Yan Shu † ba Open Pricer b Walnut AlgorithmsJune 22, 2020
Abstract
We provide an exact analytical solution of the Nash equilibrium for the k th price auction by using inverse ofdistribution functions. As applications, we identify the unique symmetric equilibrium where the valuations havepolynomial distribution, fat tail distribution and exponential distributions. key words : Vickrey auctions, k th price auctions, inverse distribution functions In a k th price auction with k or more bidders, the highest bidder wins the bid and pays the k th highest bid as price.The k th price auction has been studied by many researchers in recent years. Readers can refer to [2, 3, 4, 9] forrelated literature. The Revenue Equivalence Theorem (RET) (see [6], [7]) can be used to characterize equilibriumstrategies of k th price auction. Monderer and Tennenholtz [5] proved the uniqueness of the equilibrium strategiesin k th price auctions for k = 3. Under some regularity assumptions, they also provided a characterization equationof the equilibrium bid function (see theorem 2.1 below). Wolfstetter [10] solved the equilibrium k th price auctionsfor a uniform distribution. Recently, Nawar and Sen [11] represented the solution of Monderer and Tennenholtz’scharacterization equation as a finite series involving Catalan numbers. With their representation, they provideda closed form solution of the unique symmetric, increasing equilibrium of a k th price auction for a second degreepolynomial distribution.In this paper, we analysis Monderer and Tennenholtz’s characterization equation by using a method involvinginverse of distribution functions. We provide a new representation of the equilibrium bid function of k th priceauction with this representation. For applications, we extend Nawar and Sen’s results and provide a closed formsolution of a k th price auction for polynomial distribution, fat tail distribution and exponential distributions.After recalling the framework of the problem in Section 2, we prove our main result in Section 3. Then, inSection 4, we compare our result with those of Nawar and Sen. Finally in Section 5 we provide a closed-formsolution of the equilibrium bid function for polynomial distribution, exponential distribution and a class of fat taildistributions. In this section we present our assumptions and recall the result on the uniqueness of the equilibrium strategiesprovided by Monderer and Tennenholtz. Consider a k th price auction with n bidders, where the highest bidderwins, and pays only the k th highest bid. Let k > n > k . We make the following assumptions as in [7]:1. The valuations X i , i = 1 , · · · , n of the bidders are independent and identically distributed with distributionfunction F . ∗ [email protected] † [email protected]
1. The distribution function F is with values in I where I = [0 ,
1] or I = R + .We also assume that:(A) F is k − f := F ′ satisfying ∀ x ∈ I , f ( x ) > f , the quantile function Q := F − exists and iswell-defined on (0 , A ) holds for the case of general k th price auctions. We denote by β i the strategy of the bidder i which determines its bid for any value. A strategyprofile for n bidders is given by ( β , ..., β n ). A strategy profile is symmetric if β i are all equal to a common strategy β . A symmetric strategy is increasing if β is an increasing function. The equilibrium of a k th price auction with asymmetric strategy profile is characterized by the following theorem: Theorem 2.1. [Monderer and Tennenholtz[5]] Let β : [0 , R + . A symmetric strategy profile with commonstrategy β is an equilibrium of the k th price auction if and only if the following two conditions hold.(E1) β is an increasing function.(E2) For all x ∈ [0 , : Z x [ x − β ( y )] F ( y ) n − k [ F ( x ) − F ( y )] k − f ( y ) dy = 0 . (1) Moreover (1) has at most one solution and for such a solution β is differentiable for all x ∈ Supp ( f ) , here Supp ( f ) denotes the support of the distribution with density f . According to Theorem 2.1, for a given F , if we can compute β and show that β is differentiable and increasing,then β is the unique equilibrium bid function. If the equilibrium bid function β exists, β ( X ) is the random variablerepresenting the equilibrium strategy of bidders with valuation X . Furthermore, if β is strictly increasing, togetherwith differentiability of β and the assumption that F has a density, we can deduce that β ( X ) also has a continuousdensity function. We denote ˆ F the distribution function of β ( X ) (that is, ˆ F ( β ( x )) = F ( x )) and ˆ Q := ˆ F − thequantile function. Here we present our main result. We give a closed form solution of equation (1) for k >
3. With this solution,we are able to find a closed form expression of the bid function for some non linear distributions. The key idea is,instead of working directly with the distribution function F as in the literature, we use the quantile function Q ,which can largely simplify the calculus and give a better insight. Theorem 3.1.
Assume that β : [0 , R + is an increasing function and β ( X ) has an increasing distributionfunction with a continuous density. Denote ˜ x = F ( x ) and γ (˜ x ) := Q (˜ x )˜ x n − . Then (1) has a unique solution givenby β ( x ) = γ ( k − (˜ x )( k − (cid:0) n − k − (cid:1) F ( x ) n − k , (2) where the superscript ( t ) is the t th order derivative with respect to ˜ x .Proof. Denote s ( x ) := x Z x [ F ( x ) − F ( y )] k − F ( y ) n − k f ( y ) dy, (3) w ( x ) := Z x β ( y )[ F ( x ) − F ( y )] k − F ( y ) n − k f ( y ) dy. (4)Note that (1) can be written as s ( x ) = w ( x ).Since F ′ ( y ) = f ( y ), making the transformation z = F ( y ) /F ( x ) in (3) we have s ( x ) = xF ( x ) n − Z (1 − z ) k − z n − k dz = xF ( x ) n − B ( n − k + 1 , k − , (5)where B ( c, d ) is the beta function with B ( c, d ) = ( c − d − / ( c + d − c , d . It follows that s ( x ) = xF ( x ) n − ( k − (cid:0) n − k − (cid:1) . (6)2ake the transformation z = F ( y ) in (4). As β is increasing, we have F ( y ) = ˆ F ( β ( y )), so that β ( y ) = ˆ F − ( F ( y )) = ˆ Q ( F ( y )) = ˆ Q ( z ) . Thus, w ( x ) = Z F ( x )0 ˆ Q ( z )[ F ( x ) − z ] k − z n − k dz. (7)As ˜ x = F ( x ), we have x = Q (˜ x ). Denote S (˜ x ) := s ( Q (˜ x )) and W (˜ x ) := w ( Q (˜ x )) . Also recall that γ (˜ x ) := Q (˜ x )˜ x n − . Then from (6): S (˜ x ) = s ( Q (˜ x )) = s ( x ) = γ (˜ x )( k − (cid:0) n − k − (cid:1) . (8)By (7): W (˜ x ) := w ( Q (˜ x )) = w ( x ) = Z ˜ x ˆ Q ( z )(˜ x − z ) k − z n − k dz. (9)According to (1), S (˜ x ) = W (˜ x ). Noticing that assumption (A) ensures that S (˜ x ) is k − S ( k − (˜ x ) = W ( k − (˜ x ) . (10) S ( k − (˜ x ) is known by (8). To find W ( k − (˜ x ), apply Lemma 7.1 of the Appendix. Taking m = n − k and r = k − W ( k − (˜ x ) = ( k − Q (˜ x )˜ x n − k . Together with (8) and (10), we have: γ ( k − (˜ x )( k − (cid:0) n − k − (cid:1) = ( k − Q (˜ x )˜ x n − k . (11)As ˆ F ( β ( x )) = F ( x ) = ˜ x , we have ˆ Q (˜ x ) = ˆ F − (˜ x ) = β ( x ). Using this, (2) follows from (11). Remark 3.1.
By the Leibniz rule for derivation, γ ( k − (˜ x ) = k − X i =0 (cid:18) k − i (cid:19) (˜ x n − ) ( k − − i ) Q ( i ) (˜ x )= k − X i =0 (cid:18) k − i (cid:19) ( n − n − k + i )! ˜ x n − k + i Q ( i ) (˜ x ) Recall that ˜ x = F ( x ) , from equation (2) , we have: β ( x ) = k − X i =0 (cid:18) k − i (cid:19) ( n − k )!( n − k + i )! ˜ x i Q ( i ) (˜ x )= Q (˜ x ) + k − n − k + 1 Q ′ (˜ x ) + k − X i =2 (cid:18) k − i (cid:19) ( n − k )!( n − k + i )! ˜ x i Q ( i ) (˜ x ) (12) Since Q (˜ x ) = F − (˜ x ) = x and Q ′ (˜ x ) = 1 /F ′ ( x ) = 1 /f ( x ) , for k > we have β ( x ) = x + k − n − k + 1 F ( x ) f ( x ) + k − X i =2 (cid:0) k − i (cid:1) F ( x ) i Q ( i ) (˜ x )( n − k + 1) ... ( n − k + i ) = x + k − n − k + 1 F ( x ) f ( x ) + O (cid:18) n (cid:19) . This result coincides with the result of Wolfsteller [10] in O ( n ) . Moreover it agrees with the expression in proposition3 of Gadi Fibich and Arieh Gavious’s work in [1]. Comparison with Nawar and Sen’s result
Applying the revenue equivalence principle for expected payment of a bidder in a k th price auction with n bidders,Nawar and Sen (2018)[11] have obtained the following expression of β ( x ): β ( x ) = ψ k − ( x )( k − (cid:0) n − k − (cid:1) F ( x ) n − k , (13)where ψ ( x ) = Z x yF ( y ) n − f ( y ) dy and ψ t +1 ( x ) = ψ ′ t ( x ) f ( x ) for t = 0 , , ... (14)By making the transformation z = F ( y ) we have ψ ( x ) = R ˜ x Q ( z ) z n − dz .Thus dψ ( x ) d ˜ x = Q (˜ x )˜ x n − = γ (˜ x ) . Also note that as ˜ x = F ( x ), we have d ˜ xdx = F ′ ( x ) = f ( x ). By (14), we have ψ ( x ) = dψ ( x ) dx f ( x ) = dψ ( x ) d ˜ x d ˜ xdx f ( x ) = γ (˜ x ) . Applying the iterative definition of (14) again, we have: ψ ( x ) = dψ ( x ) dx f ( x ) = dψ ( x ) d ˜ x d ˜ xdx f ( x ) = dγ (˜ x ) d ˜ x = γ (1) (˜ x ) . By induction, if ψ t ( x ) = γ ( t − (˜ x ), then ψ t +1 ( x ) = dψ t ( x ) dx f ( x ) = dψ t ( x ) d ˜ x d ˜ xdx f ( x ) = dγ ( t ) (˜ x ) d ˜ x . This shows that ψ t ( x ) = γ ( t − (˜ x ) for t = 1 , , .... So ψ k − ( x ) = γ ( k − (˜ x ) and the expression (13) coincideswith our expression (2). Therefore, our expression (2) is an equivalence representation of Nawar and Sen’s result.Instead of expanding ψ with series about Catalan numbers, we compute it with the quantile function. From thisexpression (2), it is easy to establish the equilibrium for some non linear distributions, and we will detail them inthe next section. In this section we study the equilibrium bid function for some non-linear distributions. First of all, as a corollaryof theorem 3.1, we provide sufficient conditions for the existence and uniqueness of the equilibrium bid function.Then, we will provide a closed-form solution of the equilibrium bid function for polynomial distribution, exponentialdistribution, a class of fat tail distributions.According to theorem 2.1, for some distribution F , if one can show that β found by (2) is an strictly increasingfunction, it will follow that the symmetric strategy profile with common strategy β is an equilibrium of the k thprice auction. Corollary 5.1.
Consider a k th price auction where each X i is i.i.d. on the interval [0 , with quantile function Q .Assume that it holds ∀ i ∈ [ | , k − | ] , ∀ ˜ x ∈ [0 , , Q ( i ) (˜ x ) > . (15) Then the existence and uniqueness of equilibrium bid function is given by (2) and can be rewritten as β ( x ) = x + k − X i =1 (cid:18) k − i (cid:19) ( n − k )!( n − k + i )! ˜ x i Q ( i ) (˜ x ) , (16) with ˜ x = F ( x ) . roof. Equation (16) is direct from equation (12). By equation (15), we deduce that for i = 0 , · · · , k − Q ( i ) (˜ x )is strictly increasing with respect to ˜ x . Together with the fact that ˜ x = F ( x ) is increasing strictly with respectedto x , according to equation (12), we deduce that β ( x ) is strictly increasing. The conclusion follows.For some distributions, the condition Q ( i ) (˜ x ) > x ∈ [0 ,
1) is easy to check but for some it is not. Formany non-linear distributions, the quantile function Q is analytic on [0 , P + ), which is easy to check. Condition ( P + ) : The function Q is analytic on [0 ,
1) with positive coefficient in the representation of powerseries. More precisely, there exist positive α i , such that for x ∈ [0 , Q (˜ x ) = ∞ X i =0 α i ˜ x i . The equality is defined in the sense of power series, reader can refer to any power series literature for a completejustification of technical convergence details. From condition ( P + ), deriving the equation successively, it is easy tosee that equation (15) holds. According to corollary 5.1, β is an equilibrium bid function. To check the condition( P + ), we only need to calculate the Taylor expansion of the quantile function Q on 0, then check the sign of theTaylor coefficient. Here are some examples of applications of corollary 5.1. Example 5.1 (Exponential distribution.) . Let F ( x ) := 1 − e − λx for λ > and x ∈ R + . The equilibrium bidfunction is given by (2) . In fact Q ( x ) := − λ ln(1 − x ) . Moreover Q ( i ) ( x ) = i ! λ (1 − x ) i for i ∈ [ | , k − | ] , which isstrictly positive on [0 , . According to Corollary 5.1, the equilibrium bid function β ( x ) has the expression: β ( x ) = x + 1 λ k − X i =1 (cid:18) k − i (cid:19) ( n − k )!( n − k + i )! ˜ x i i ! ( − ˜ x ) i , where ˜ x = F ( x ) . Example 5.2 (Fat tail distribution.) . Let F ( x ) := 1 − x c for some c > and for x ∈ R + . The equilibrium bidfunction is given by (2) . In fact Q ( x ) := − x ) c . Moreover Q ( i ) ( x ) = c ( c +1) ... ( c + i − x i (1 − ˜ x ) c + i for i ∈ [ | , k − | ] , which isstrictly positive on [0 , . According to Corollary 5.1, the equilibrium bid function β ( x ) has the expression: β ( x ) = x + k − X i =1 (cid:18) k − i (cid:19) ( n − k )!( n − k + i )! c ( c + 1) ... ( c + i − x i (1 − ˜ x ) c + i , where ˜ x = F ( x ) . Theorem 5.2.
Consider a kth price auction where each X i is iid on the interval [0 , , with distribution function F ( x ) = x α where α > . Then there is a unique symmetric equilibrium. The equilibrium common strategy β :[0 , R + is β ( x ) = Γ( n − k + 1)Γ( n − /α )Γ( n − n − k + 1 + 1 /α ) x, (17) where Γ is the Gamma function. In particular, if α = m where m is a positive integer, β ( x ) = ( n − k + m + 1) ... ( n − m )( n − k − +1) ... ( n − x. Proof.
To prove this, we find β using theorem 3.1 and show that it is a strictly increasing function of x . As˜ x = F ( x ) = x α , it follows Q (˜ x ) = F − (˜ x ) = ˜ x /α = x . Then γ (˜ x ) = Q (˜ x )˜ x n − = ˜ x n − /α . Therefore, γ ( k − (˜ x ) = ( n − /α )( n − /α ) ... ( n − k + 1 + 1 /α )˜ x n − k +1 /α = Γ( n − /α )Γ( n − k + 1 + 1 /α ) F ( x ) n − k x. Together with (2), (17) follows. 5
Acknowledgments the reviewers had a key role in the conception of this article and we are grateful for the reviewers for the veryhelpful comments.
Lemma 7.1.
Consider a real valued bounded function ˆ Q : R [0 , . For positive integer r and positive realnumber m , let A r ( u, z ) := ˆ Q ( z )( u − z ) r z m and H r ( u ) := R u A r ( u, z ) dz . Then the ( r + 1) th derivative of H r ( u ) is H ( r +1) r ( u ) = r ! ˆ Q ( u ) u m . (18) Proof.
By the Leibniz rule [12] of differentiating an integral, if H ( u ) := R l ( u ) l ( u ) A ( u, z ) dz , under assumption ofintegrability of ∂ u A ( u, z ), it holds: H ′ ( u ) = [ A ( u, l ( u )) l ′ ( u ) − A ( u, l ( u )) l ′ ( u )] + Z l ( u ) l ( u ) ∂ u A ( u, z ) dz. It is easy to check the integrability of ∂ u ˆ Q ( z )( u − z ) r z m , thus taking l ( u ) = 0 , l ( u ) = u in the previous equation: H ′ r ( u ) = A r ( u, u ) + Z u ∂ u A r ( u, z ) dz. (19)For r = 0, A ( u, u ) = ˆ Q ( u ) u m and ∂ u A ( u, z ) = 0. For r > A r ( u, u ) = 0 and ∂ u A r ( u, z ) = ˆ Q ( z ) r ( u − z ) r − z m = rA r − ( u, z ). Therefore (19) implies H ′ r ( u ) = rH r − ( u ) for r > H ′ ( u ) = ˆ Q ( u ) u m . Thus for t r : H ( t ) r ( u ) = r ( r − ... ( r − t + 1) H r − t ( u ) , so H ( r ) r ( u ) = r ! H ( u ) and therefore H ( r +1) r ( u ) = r ! H ′ ( u ) = r ! ˆ Q ( u ) u m . References [1] G. Fibich and A. Gavious. Large auctions with risk-averse bidders.
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