A speedy auction using approximated bidders’ preferences
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Abstract
This paper presents a combinatorial auction, which is of particular interest when short completion times are of importance. It is based on a method for approximating the bidders’ preferences over two types of item when complementarity between the two may exist. The resulting approximated preference relation is shown to be complete and transitive at any given price vector. It is shown that an approximated Walrasian equilibrium always exists if all bidders either view the items as substitutes or complements. If the approximated preferences of the bidders comply with the gross substitutes condition, then the set of approximated Walrasian equilibrium prices forms a complete lattice. A process is proposed that is shown to always reach the smallest approximated Walrasian price vector. Simulation results suggest that the approximation procedure works well as the difference between the approximated and true minimal Walrasian prices is small.
Keywords
Approximate auction Approximated preferences Nonquasilinear preferences Combinatorial auction1 Introduction
Auctions are extensively used as a way to determine who gets to buy which good and at what price. It is not uncommon for a seller to simultaneously auction multiple items. Spectrum licenses are often divided into smaller geographical areas rather than one countrywide license, and a company can be sold as several divisions rather than one entity. In recent years, the literature on multiitem auctions, and, in particular, combinatorial auctions, has grown substantially. In a unitdemand setting, Demange et al. (1986) propose a multiitem auction, which is Pareto efficient and strategyproof. Key to their result is to find the unique minimal Walrasian equilibrium price vector, its existence being guaranteed by the lattice structure of equilibrium prices (Demange and Gale 1985; Shapley and Shubik 1972), and to allocate the items in accordance with this price. When allowing bidders to demand multiple units of items, the problem becomes more complex. For homogeneous items, Ausubel (2004) presents an ascendingbid auction, which is efficient and where the outcome of the auction coincides with the outcome of the Vickrey auction. Extending to heterogeneous items, Gul and Stacchetti (2000) designed a generalized version of Demange et al. (1986)’s auction, which also terminates at the unique minimal Walrasian equilibrium price vector.^{1} In their setting, the existence of a Walrasian equilibrium is guaranteed when bidders have gross substitute preferences. The gross substitutes condition was introduced by Kelso and Crawford (1982) and is utilized by Ausubel (2006), who suggests a multiitem auction that reaches the VickreyClarkesGroves outcome and therefore is incentive compatible. Sun and Yang (2006, 2009) introduce the gross substitutes and complements condition, which allows for some complementarity in the bidders’ preferences. The authors show that this condition is sufficient for the existence of competitive equilibrium and propose two auction processes that always find an equilibrium price vector. Sun and Yang (2014) extend their work to the more general case of super additive preferences and show that an equilibrium exists when prices of the packages are allowed to be nonlinear. Ausubel and Milgrom (2002) suggest an ascendingbid proxy auction: each bidder reports a valuation for each package and then commits to bid straightforwardly according to these reports. When bidders have quasilinear preferences in money, and goods are substitutes, the outcome of the proxy auction coincides with the Vickrey auction and sincere bidding is a Nash equilibrium. By allowing prices to differ across packages and bidders, authors such as de Vries et al. (2007) and Mishra and Parkes (2007) propose auction processes that reach the VCG outcome for general valuations.
A possible problem with many auction formats is that they may take a long time to carry out. The auction for British telecom licenses, conducted in 2000, is one example of this as it took two months to complete (Binmore and Klemperer 2002). One reason for long completion times is that many auctions are dynamic processes in which the prices of the items are either only increased or only decreased.^{2} This may result in a timeconsuming process as the starting prices have to be set far below or far above the expected final prices to make sure that the process converges to a desired equilibrium. In some cases, however, short completion times of auctions are very important. One such example is the productmix auction, which was designed to help the Bank of England during the bank run in the autumn of 2007. Due to the outbreak of the financial crisis, the Bank of England wished to allocate loans to commercial banks in a very rapid fashion. Klemperer (2010) proposed a quick auction procedure for allocating two different types of loan to the banks. The idea was that bidders submitted a number of bids consisting of two prices (interest rates), one for each type of loan, and a quantity (same for both loans), which served as an approximation of the bidders’ demand. Based on the supplied quantities of the two loans, prices were determined and the bidders were awarded the loans that gave them the highest, nonnegative profit. In this way, the central bank allocated the loans in a quick fashion.
Quick auctions are not uncommon in the auction literature. Sealedbid auctions, such as the famous Vickrey auction, are well studied examples. However, such auction formats, and many more, are usually analyzed under the assumption that bidders have quasilinear preferences in money. This may be restrictive as it implies that bidders neither exhibit riskaversion, experience wealth effects, nor face financing or budget constraints. If bidders’ preferences are in fact nonlinear in money, this should be taken into account. Optimal auctions, in which bidders exhibit riskaversion, have been studied by Maskin and Riley (1984) and Matthews (1987). Morimoto and Serizawa (2015) analyzed allocation rules for multiple indivisible items, allowing bidders to have nonlinear preferences in money and unit demand. Ausubel and Milgrom (2002) also propose a generalized proxy auction, in which the seller and the bidders have nonlinear but strict preferences over all offers made in the bidding process. This auction is embedded in the matching with contracts model by Hatfield and Milgrom (2005).
Thus far, two problems have been identified: auctions may take a long time to conduct and bidders may not have quasilinear preferences in money. This paper proposes a combinatorial auction which is quick and allows for bidders to have nonlinear preferences in money. In order for the auction to be quick, the bidders report all required information prior to the execution of the auction. Consequently, and similar to sealedbid auctions, the bidders do not participate in a dynamic auction process. Due to the possible high complexity of the bidders’ nonlinear preferences in money, requiring a bidder to report her preferences over money does not seem feasible. Therefore, the bidder will report a fraction of her preferences, which will be used to approximate her preferences. More specifically, a bidder reports two sets of prices that makes her indifferent between the packages that are available in the auction. Using these prices, linear approximations of the bidder’s indifference curves between any two distinct packages will be made. In this context, an indifference curve contains all combinations of prices for the two packages, which makes the bidder indifferent between the packages. By combining the linearly approximated indifference curves, a bidder’s approximated preferences can be constructed.
As suggested in the literature review, linear approximations of bidders’ preferences are not uncommon. Importantly, the quasilinear preferences are contained in the class of preferences corresponding to the approximation procedure of this paper. In particular, if a bidder has quasilinear preferences in money and reports truthfully, the approximated preferences will coincide with the bidder’s true preferences.
It is shown that the approximated preference relation of each bidder is complete and transitive at any price vector. Given the approximated preference relations of the bidders, it is of interest to know whether it is always possible to find an equilibrium assignment. In addition to theoretical interest, equilibrium assignments are particularly important in, for example, spectrum auctions as governments typically want all regions of the country to have coverage. As a bidder’s approximated preferences do not necessarily coincide with her true preferences, the equilibrium concept analyzed in this paper is denoted by an approximated Walrasian equilibrium. It is shown that, if each bidder views the items as substitutes, or complements, then the set of approximated Walrasian equilibrium prices is nonempty. The substitutability (complementarity) only requires that the larger report of prices for the package of two items is strictly smaller (greater) than the sum of the larger reports for the two items separately. It is further shown that imposing the gross substitutes condition on the bidders’ approximated preference relations is sufficient for the set of approximated Walrasian equilibrium prices to form a complete lattice and, hence, to contain unique minimal element. A process is described that can be used to find the unique minimal approximated Walrasian equilibrium price vector. However, the bidders do not actively participate in any intermediate step of this process. Using the bidders’ approximated preferences as input, the process is a structured method for finding the unique minimal approximated Walrasian equilibrium price vector. This price vector may be of particular importance when the auctioneer is concerned with consumer welfare. A government selling spectrum licenses may be interested in assuring low consumer prices. Selling the licenses for the smallest equilibrium prices may aid in achieving this as the resulting producer costs are relatively low. Finally, simulations are conducted that suggest that the approximation procedure works fairly well. In fact, the absolute relative error between a true and approximated minimal Walrasian price is only \(4.8\%\) on average. This is compared to the case when bidders are assumed to have quasilinear preferences, in which case the absolute relative error is \(71.5\%\) on average.
 1.
Each bidder reports prices that makes her indifferent between the available packages.
 2.
These prices are used to construct linear approximations of the bidder’s indifference curves.
 3.
Combining a bidder’s linearly approximated indifference curves, her approximated preferences are constructed.
 4.
Using the approximated preferences as input, a process is used to find the unique minimal approximated Walrasian equilibrium price vector.
 5.
The items are allocated to the bidders in accordance with this price vector.
2 The model
A finite number of bidders, collected in the set \(N=\{1,2,\ldots , n\}\), participate in the auction. A seller wishes to auction two types of indivisible items, called a and b,^{3} of which there may exist multiple copies. Let \(q_a\ge 1\) and \(q_b\ge 1\) denote the finite integer number of copies of each type of item. Copies of the same type are to be sold for some uniform price, \(p_a\) or \(p_b\) depending on the type. In order to sell the items, the seller requires at least some prices \(r_a\ge 0\) and \(r_b\ge 0\) for each type of item. Such prices are referred to as the seller’s reservation prices and imply that \(p_a\ge r_a\) and \(p_b\ge r_b\). Each bidder has the outside option of not acquiring anything in the auction. The outside option is represented by a nullitem, which is denoted 0 and is equal to the empty set. The price of the nullitem is normalized to 0 so \(p_0=r_0=0\). Each bidder is interested in acquiring, at most, one copy of item a and b. Let \(ab=\{a,b\}\) be the combination of one item of each type and let \(p_{ab}=p_a+p_b\) denote its price. The sets of items that the bidders are interested in purchasing are collected in \({\mathcal {I}}=\{0,a,b,ab\}\) and any element \(x\in {\mathcal {I}}\) is referred to as a package. A bidder’s preferences over the packages are determined by the utility generated from consuming the packages and their prices. A consumption bundle is therefore defined to be a pair consisting of a package and a price. For any given prices of the packages, the bidders are hence interested in consuming at least one of the consumption bundles (0, 0), \((a,p_a)\), \((b,p_b)\), or \((ab,p_{ab})\). Each bidder \(i\in N\) has a preference relation, denoted \(R_i\), over all possible consumption bundles. \(R_i\) is complete, transitive, continuous, and finite. Let \(P_i\) be the strict relation and \(I_i\) the indifference relation associated with \(R_i\). The preferences of the bidders satisfy price monotonicity; that is, for any package \(x\in {\mathcal {I}}\) and any two prices \(p'_x,p''_x\in {\mathbb {R}}_{+}\), if \(p'_x>p''_x\), then \((x,p''_x)P_i(x,p'_x)\). Finally, any bidder is indifferent between any two identical consumption bundles. An objective of the auction is to find an assignment of the items to the bidders such that any bidder is assigned either 0, a, b, or ab. While any number of bidders can be assigned the nullitem, an assignment needs to be such that the number of assigned items of any type, a or b, does not exceed the available number of copies of the type. Formally, let \(\mu : N\rightarrow {\mathcal {I}}\) be an assignment such that \(\#N_a\le q_a\) and \(\#N_b\le q_b\), where \(N_a=\{i\in N \mid \mu (i)\in \{a, ab\}\}\) and \(N_b=\{i\in N \mid \mu (i)\in \{b,ab\}\}\), and where \(\mu (i)\) denotes the assignment of bidder \(i\in N\).
3 Approximation of the bidders’ preferences
In order to approximate the true preference relation, \(R_i\), of any bidder \(i\in N\), the bidder makes two reports. The first report, denoted v, consists of one price \(v_j\in {\mathbb {R}}\) for each package \(j\in \{a,b,ab\}\). Recalling that the price of the nullitem is normalized to 0, these reported prices are interpreted as the bidder being indifferent between the consumption bundles (0, 0), \((a,v_a)\), \((b,v_b)\), and, \((ab,v_{ab})\). The second report, z, consists of some other prices \(z_j<v_j\) for each \(j\in \{a,b,ab\}\). The prices in z are interpreted as making the bidder indifferent between the consumption bundles \((a,z_a)\), \((b,z_b)\), and \((ab,z_{ab})\). Note that any price reported for ab need not necessarily equal the sum of the prices reported for the individual items. Moreover, the assumptions on \(R_i\) guarantee the existence of prices that fulfill the requirements of the reports.
Assuming that the bidders report truthfully, the two reports will be used to make linear approximations of the bidders’ indifference curves between any two distinct packages. The approximations will be referred to as the bidders’ approximated indifference curves. The approximated indifference curves will be constructed under the restriction that \(p_{ab}=p_a+p_b\). In line with this, four constants, which are based on the two reports, are defined: \(\alpha _v=v_{ab}v_b\), \(\alpha _z=z_{ab}z_b\), \(\beta _v=v_{ab}v_a\), and \(\beta _z=z_{ab}z_a\). A constant \(\alpha _j\), where \(j\in \{v,z\}\), is interpreted as a price for item a, which would make the bidder indifferent between the consumption bundles \((ab,\alpha _j+j_b)\) and \((b,j_b)\), where \(j_b\) is either the report \(v_b\), or \(z_b\), defined earlier. \(\beta _j\) has the corresponding interpretation for a price of item b. In this way, six pairs of prices, \((p_a,p_b)\), are extracted, with the help of which the approximated indifference curves between any two packages, except 0, are constructed.
In the following, a number of formal concepts will be introduced. In order to ease the understanding of the approximation procedure, an example will accompany these concepts. The example is depicted in Figs. 1 and 2 and is based on a bidder i making the reports of v and z presented in Table 1.
Reports of v and z by bidder i
a  b  ab  \(\alpha _j\)  \(\beta _j\)  

v  10  8  14  6  4 
z  6  5  10  5  4 
In order for the approximated preference relation of a bidder to be meaningful, it is important that, at any given prices of the items, a consistent ranking of the consumption bundles can be constructed. Proposition 1 ensures that this is the case.
Proposition 1
For any given prices of the items, the approximated preference relation of each bidder \(i\in N\) is complete and transitive.
Diagram (b) of Fig. 2 shows the combination of prices for which a certain consumption bundle is uniquely most preferred for the bidder in our example.
For a bidder whose preferences are quasilinear in money, her indifference curves are linear. If prices are reported truthfully, the resulting approximated indifference curves will coincide with the true indifference curves of the bidder. The bidder’s approximated and true preferences will therefore coincide and the quasilinear preferences are thus contained in the class of preferences corresponding to the approximation procedure described in this section. It is difficult to assess how well the approximated preferences approximate the true preferences since this depends on the degree of nonlinearity of the preferences in money and what prices of z are reported. As long as a bidder’s true indifference curves are not linear, there will exist some price p and packages \(x,y\in {\mathcal {I}}\) such that \((x,p)R_i(y,p)\) under the true preferences and \((x,p)\nsucc _i(y,p)\) under the approximated preferences. However, the further away a price vector p is from a true indifference curve between packages x and y, the more likely it is that if \((x,p)R_i(y,p)\), then \((x,p)\succ _i(y,p)\). Moreover, the results from the simulations reported in Sect. 6, perhaps, suggest that this is not a big issue.
4 Existence
Given the approximated preference relations of the bidders, it is interesting to know whether it is always possible to find an equilibrium assignment. A commonly analyzed equilibrium concept is the Walrasian equilibrium. However, as the approximated preferences do not necessarily coincide with the true preferences of the bidders, the equilibrium concept of this paper is denoted by an approximated Walrasian equilibrium. In order to define this formally, let a price vector be denoted by \(p=(0,p_a,p_b)\in {\mathbb {R}}^3\), which contains a price for the nullitem and one price for each type of item. Furthermore, the approximated demand correspondence of a bidder \(i\in N\) is defined as \(D_i(p)=\{x\in {\mathcal {I}} \mid (x,p_x)\succsim _i (y,p_y)\text {for all} y\in {\mathcal {I}}\}\) at any p. If \(x\in D_i(p)\), then package x is said to be demanded by bidder \(i\in N\).
Definition 1
The pair \(\langle p,\mu \rangle \) constitutes an approximated Walrasian equilibrium if: (i) \(\mu (i)\in D_i(p)\) for all \(i\in N\) and (ii) if \(\#N_x<q_x\) for some \(x\in ab\), then \(p_x=r_x\).
Thus, a price vector p and an assignment \(\mu \) constitute an approximated Walrasian equilibrium if each bidder is assigned a package that she demands, and if a copy of an item remains unassigned, then the price of this type of item has to equal the seller’s reservation price for the item.
An approximated Walrasian equilibrium does not always exist. For an excellent example, see Milgrom (2000) and recall that the quasilinear preferences are a special case of the approximated preferences of this paper. However, requiring substitutability, or complementarity, in the bidders’ preferences has been shown to guarantee the existence of equilibrium in the standard model. Kelso and Crawford (1982) required firms’ preferences over workers to comply with the gross substitutes condition to show the existence of a core allocation. This, in turn, implies that a Walrasian equilibrium exists in Gul and Stacchetti (1999, 2000). Analyzing the simultaneous ascending auction, Milgrom (2000) showed that, if objects are mutual substitutes for the bidders, then the objects can be allocated in accordance with a competitive equilibrium. Similarly, in the matching with contracts model, a stable allocation exists if hospitals view contracts as substitutes (Hatfield and Milgrom 2005). Sun and Yang (2006, 2014) showed that an equilibrium also exists when bidders have complementary preferences. The existence of an equilibrium in the first study is guaranteed when bidders’ preferences comply with the gross substitutes and complements condition and prices are linear. In the second study, the more general condition of superadditivity in bidders’ preferences is shown to guarantee the existence of competitive equilibrium when nonlinear pricing is used.
To ensure the existence of an approximated Walrasian equilibrium, we consider both substitutability and complementarity separately. First, we let the bidders treat the packages a and b as substitutes by making the assumption on the reports v that \(v_{ab}<v_a+v_b\) for each bidder \(i\in N\). Then we look at the case of complementarity by requiring that \(v_{ab}>v_a+v_b\) for each bidder \(i\in N\). However, we do not have any requirements regarding the reports of z in either case. Let \({\mathcal {P}}=\{p\in {\mathbb {R}}_+^3 \mid \exists \mu \) s.t. \( \langle p,\mu \rangle \text { is an approximated Walrasian equilibrium}\}\) be the set of approximated equilibrium prices. Proposition 2 asserts that, if \(v_{ab}<v_a+v_b\) for all \(i\in N\), then there exists an approximated Walrasian equilibrium.
Proposition 2
If \(v_{ab}<v_a+v_b\) for each bidder \(i\in N\), then the set of approximated equilibrium prices, \({\mathcal {P}}\), is nonempty.
Similarly, Proposition 3 states that, if \(v_{ab}>v_a+v_b\) for all \(i\in N\), then there exists an approximated Walrasian equilibrium.
Proposition 3
If \(v_{ab}>v_a+v_b\) for each bidder \(i\in N\), then the set of approximated equilibrium prices, \({\mathcal {P}}\), is nonempty.
It turns out that the existence of a unique minimal approximated Walrasian price vector is not guaranteed when either \(v_{ab}<v_a+v_b\), or \(v_{ab}>v_a+v_b\), for all \(i\in N\). The reason, in the first case, is that the approximated indifference curves \(f_2\) and \(f_3\) may be downwardsloping for some bidder. In the second case, the indifference curve between ab and 0 is downwardsloping by construction. Therefore, there may exist an infinite number of minimal approximated Walrasian price vectors along any such, downwardslooping, indifference curve. However, the gross substitutes condition ensures that neither \(f_2\) or \(f_3\) are downwardsloping for any bidder. Following Kelso and Crawford (1982), the gross substitutes condition is defined as:
Definition 2
The approximated preference relation, \(\succsim _i\), of any bidder \(i\in N\), fulfills the gross substitutes condition if, for any two price vectors \(p'\ge p\) and any \(x\in D_i(p)\), there exists \(y\in D_i(p')\) such that \(\{w\in x\mid p_w=p'_w\}\subseteq y\).
The gross substitutes condition implies that a bidder’s demand for an item does not decrease as the prices of any other items are raised and it guarantees that \({\mathcal {P}}\) forms a complete lattice. For any two price vectors \(p',p''\in {\mathbb {R}}^3\), let the meet \(p'\wedge p''\) be defined as a vector \(s\in {\mathbb {R}}^3\) with elements \(s_j=\min \{p'_j,p''_j\}\). Similarly, let the join \(p'\vee p''\) be a vector \(h\in {\mathbb {R}}^3\) with elements \(h_j=\max \{p'_j,p''_j\}\). Any \(S\subseteq {\mathbb {R}}^3\) forms a complete lattice if, for each \(p',p''\in S\), \(s,h\in S\).
Proposition 4
If the gross substitutes condition is fulfilled for the approximated preference relation of each bidder \(i\in N\), then \({\mathcal {P}}\) forms a complete lattice.
Proposition 4 implies that \({\mathcal {P}}\) contains a unique minimal element. Let this unique minimal approximated Walrasian equilibrium price vector be denoted \(p^{min}\).
5 Process
The proposed process can be used to find \(p^{min}\). It is designed as an English auction; starting at some low prices, prices are increased until \(p^{min}\) is reached. As mentioned in Sect. 1, the bidders do not actively participate in any intermediate step of the process. The process uses the approximated preference relations of each bidder as input in order to find \(p^{min}\). As the approximated preferences are constructed prior to running the process, the process can be executed quickly.
Following Gul and Stacchetti (2000), the process will use the bidders’ requirement of the different packages in order to, at least partly, determine how prices should be increased.
Definition 3
Let \(K_N(x,p)=\sum _{i\in N} K_i(x,p)\) be the bidders’ aggregate requirement of any \(x\in {\mathcal {I}}\) at some p. Proposition 5, below, justifies the interest in the requirement function. Most importantly, it asserts that, when, at some p, the bidders’ aggregate requirement for each package is weakly less than the number of existing copies of the items contained in the package, it is possible to assign each bidder a package that she demands. Hence, the first condition for an approximated Walrasian equilibrium is fulfilled at p. As any bidder’s requirement of the nullobject always equals zero, let \(q_0=0\) and naturally \(q_{ab}=q_a+q_b\).
Proposition 5
For a given price vector p, there exists an assignment \(\mu \) such that \(\mu (i)\in D_i(p)\) for all bidders \(i\in N\) if, and only if, \(K_N(x,p)\le q_x\) for all \(x\in {\mathcal {I}}\).
Hence, if \(K_N(x,p)>q_x\) for some package \(x\in {\mathcal {I}}\), then there is more demand for the items contained in x, at p, than the number of available copies of x. To determine the net demand, in terms of aggregate requirement, for any package at some price vector p, the function \(g: {\mathcal {I}}\times {\mathbb {R}}^3\rightarrow {\mathbb {Z}}: g(x,p)=K_N(x,p)q_x\) is defined. Packages with the greatest net demand at p are collected in \(O(p)=\{x\in {\mathcal {I}}\mid g(x,p)\ge g(y,p) \text { for all }y\in {\mathcal {I}}\}\).
Lemma 1
O(p) has a unique minimal element with respect to cardinality denoted \(O_*(p)\).
Lemma 1 is important for describing the process as whenever \(O_*(p)\) contains any of a, b, or ab, in any step of the process, the prices of the items contained in \(O_*(p)\) will be the main focus of the price increase.
A price increase consists of one part determining how much the prices are increased relative to each other and a second part deciding the magnitude. For the first part, \(\delta (p)\in {\mathbb {R}}^3_+\) is introduced, which has elements \(\delta _x(p)\) for each \(x\in \{0,a,b\}\) and p. Let \(p^t\in {\mathbb {R}}^3_+\) denote the price vector at step t of the process. The magnitude of a price increase at any step t is then given by \(\varepsilon (t)= \sup \{e\mid O_*(p^t+e\delta (p^t))=O_*(p^t)\}\). Step 1 of Process 1 checks if it is possible to assign all copies of the items. If this is not possible, it proceeds to Step 2 in which the prices of the items contained in \(O_*(p)\) are raised by equal amounts. However, as the approximated preferences of the bidders are not necessarily quasilinear, such a price increase may not always be possible. To solve this problem, let \(x\ne y\) for \(x,y\in ab\), and \(l_x(t)=\inf \{\delta _x(p^t)\in {\mathbb {R}}_+\mid \delta _0(p^t)=0\), \(\delta _y(p^t)=1\), and \(\varepsilon (t)> 0\}\) is defined. \(l_x(t)\) and \(\delta (p)\) are used to determine the relative price increase of the items.^{4}
Process 1
Set \(t=0\) and let \(p^0=r\)
Step 1: If \(O_*(p^t)=0\) set \(p^t=p^T\) and stop. Otherwise, go to Step 2.
Step 2: Let \(\delta _x(p^t)=1\) if \(x\in O_*(p^t)\) and 0 otherwise.
\(\text {If} = {\left\{ \begin{array}{ll} \varepsilon (t)\ne 0, \text {let } p^{t+1}=p^t+\varepsilon (t)\delta (p^t)\text { and set } t:=t+1 \text { and go to Step 1.} \varepsilon (t)=0,\\ \text {go to Step} 3. \end{array}\right. }\)
Step 3: Let \(\delta _0=0\) and
\(\text {if} = {\left\{ \begin{array}{ll} a,ab\in O_*(p^t), \text { then } \delta _a(p^t)=1 \text { and } \delta _b(p^t)=l_b(t). \\ b\in O_*(p^t), \text { then } \delta _a(p^t)=l_a(t) \text { and } \delta _b(p^t)=1. \end{array}\right. }\)
Let \(p^{t+1}=p^t+\varepsilon (t)\delta (p^t)\text { and set } t:=t+1 \text { and go to Step}\) 1.
Assuming that the bidders’ approximated preferences fulfill the gross substitutes condition, Lemma 2 asserts that Process 1 does not get stuck at any step \(t<T\).
Lemma 2
If the gross substitutes condition is fulfilled for the approximated preference relation of each bidder \(i\in N\) and \(\varepsilon (t)=0\) in Step 2 of Process 1, then \(\varepsilon (t)>0\) in Step 3 of Process 1.
As \(O_*(p^T)=0\), Proposition 5 ensures that the first condition for \(p^T\) to yield an approximated Walrasian equilibrium is fulfilled. Assuming that each bidder’s approximated preference relation complies with the gross substitutes condition, Theorem 1 states that Process 1 always converges to the unique minimal approximated Walrasian equilibrium price vector.
Theorem 1
If the gross substitutes condition is fulfilled for the approximated preference relation of each bidder \(i\in N\), then Process 1 always terminates at \(p^T=p^{min}\).
Bidders’ demand and \(O_*(p^t)\) in example of Process 1
\(p^t\)  \(D_i(p^t)\)  \(D_j(p^t)\)  \(O_*(p^t)\) 

\(p^0\)  ab  ab  ab 
\(p^1\)  b, ab  ab  b 
\(p^2\)  a, b, ab  ab  ab 
\(p^3\)  a, b  b, ab  0 

\(t=0\): As \(O_*(p^0)=\{ab\}\), Process 1 moves to Step 2 where \(\delta _a(p^0)=\delta _b(p^0)=1\) and \(\delta _0(p^0)=0\). Given this \(\delta (p^0)\), it is possible to increase prices and maintain \(O_*(p)=\{ab\}\). Consequently, \(\varepsilon (0)\ne 0\) and prices are raised from \(p^0\) to \(p^1\) in Fig. 3. At \(p^1\), \(O_*(p^1)=\{b\}\) due to the change in bidder i’s demand. Therefore, \(p^1\) is the upper bound for the price increase at this step. Consequently, \(p^1=p^0+\varepsilon (0)\delta (p^0)\) and \(t=1\).

\(t=1\): Since \(O_*(p^1)=\{b\}\), we set \(\delta _b(p^1)=1\) and \(\delta _0(p^1)=\delta _a(p^1)=0\) in Step 2. For this \(\delta (p^1)\), \(\varepsilon (1)=0\) since an increase in \(p_b\) would change \(O_*(p)\) to contain ab as i would change to only demand ab. Therefore, Process 1 proceeds to Step 3. In this step, we find the smallest relative price increase of \(p_a\) to \(p_b\), which makes \(\varepsilon (1)\ne 0\). In Fig. 3, this is given by the slope of the indifference curve of bidder i. \(\delta _a(p^1)\) is therefore adjusted such that \(\delta _a(p^1)=l_a(1)\), which makes \(\varepsilon (1)\ne 0\). The magnitude of the price increase is bounded by the intersection of bidder i’s indifference curves. This is where the demand of bidder i changes. Finally, \(p^2=p^1+\varepsilon (1)\delta (p^1)\) and \(t=2\).

\(t=2\): Now \(O_*(p^2)=\{ab\}\) and the only price increase that is possible, while maintaining \(O_*(p)=\{ab\}\), is to follow bidder i’s indifference curve. \(\delta (p^2)\) is adjusted accordingly and \(p_a\) and \(p_b\) are increased until the packages demanded by bidder j change. Let \(p^3=p^2+\varepsilon (2)\delta (p^2)\) and \(t=3\).

\(t=3\): \(O_*(p^3)=\{0\}\) and item a is sold to i for a price of 6 and b is sold to j for a price of 5.
6 Simulations
Now that we have shown that the approximation procedure of this paper is possible to use from a practical perspective, it is natural to ask how far the unique minimal approximated Walrasian equilibrium price vector is from the true unique minimal Walrasian equilibrium price vector. Measuring this is, probably, the most relevant way to assess how well the approximation procedure of Sect. 3 approximates the bidders’ true preferences since the outcome of an auction is what truly matters to bidders, sellers and auction houses. Simulations are conducted in order to measure this.
Three sets of simulations are carried out. The first set consists in calculating the true unique minimal Walrasian equilibrium price vector. Secondly, the preferences are approximated, by the procedure described in Sect. 3, to calculate the unique minimal approximated Walrasian equilibrium price vector. Thirdly, bidders are assumed to have quasilinear preferences and are only asked to report their valuations for the packages and then the resulting unique minimal Walrasian price vector is calculated. As discussed in Sect. 4, a unique minimal approximated Walrasian price vector does not always exist since some indifference curves may be downwardsloping. This may be true for the true Walrasian price vector as well. In these cases, the minimal Walrasian equilibrium price vector that minimizes \(p_a\) will always be picked for comparison. Calculating a true minimal Walrasian price vector is not trivial due to the nonlinearity of the bidders’ preferences. However, we note that such a price vector, which minimizes at least one of \(p_a\) and \(p_b\), must lie at the intersection of at least two indifference curves. This follows since, if this is not the case, then it would be possible to decrease the “minimal price” sufficiently little, possibly along one indifference curve, without changing any bidder’s demand and, thus, still have a Walrasian equilibrium. Therefore, we calculate all prices that generate an intersection between at least two indifference curves, as well as the reservation prices, and check for the existence of an approximated Walrasian equilibrium to obtain the true (unique) minimal Walrasian price vector.
The simulations are conducted in the following setting: A seller auctions two copies of a and b each to four bidders. The reservation prices are set at \(r_a=r_b=0\). The bidders have private valuations for the packages 0, a, b and ab, denoted by \(pv^i_0\), \(pv^i_a\), \(pv^i_b\) and \(pv^i_{ab}\), for any bidder \(i\in N\). We let \(pv^i_0=0\) for all bidders. The bidders have nonlinear preferences in money and the utility for a package \(x\in \{a,b,ab\}\) is given by \(U_i(x)=pv^i_xp_x^{\alpha }\), for any \(i\in N\). Consequently, \(\alpha \) is the parameter determining the degree of nonlinearity of the bidders’ preferences. We will conduct simulations for \(\alpha =0.6,0.7,\dots ,1.4\). When \(\alpha >1\), bidders exhibit a special case of risk aversion known as aversion to price risk (Mezzetti 2011). Bidders are riskneutral, and have quasilinear preferences, when \(\alpha =1\) and they are seeking price risk when \(\alpha <1\).^{5}\(pv^i_a\) and \(pv^i_b\) are randomly and independently drawn from a uniform distribution on (10, 20). We limit the simulations to the case when bidders view a and b as substitutes. Therefore, let \(pv^i_{max}=max\{pv^i_a,pv^i_b\}\) for each \(i\in N\). In order to ensure that \(v_{ab}<v_a+v_b\), and that ab is desired at some prices, \(pv^i_{ab}\) is randomly and independently drawn from a uniform distribution on \((pv^i_{max},(pv^i_a+pv^i_b)\), if \(\alpha \ge 1\), and randomly and independently drawn from a uniform distribution on \((pv^i_{max},(pv^i_a+pv^i_b)^\frac{1}{\alpha })\) otherwise. The bidders have the same private valuations for the packages in all three sets of simulations. The bidders’ reports that are used for approximating their preferences are generated in the following way: Since \(p_x=(pv^i_x)^{\frac{1}{\alpha }}\) gives \(U_i(x)=0=U_i(0)\), the bidders first report \(v_x=(pv^i_x)^{\frac{1}{\alpha }}\) for each \(x\in \{a,b,ab\}\). To make sure that the reports z are smaller than v, we let \(pv^i_{min}=min\{pv^i_a,pv^i_b\}\) and randomly and independently draw \(c^i\) from a uniform distribution on \((0,pv^i_{min})\) for each \(i\in N\). We then let \(z_x=(pv^i_xc^i)^\frac{1}{\alpha }\) for each \(x\in \{a,b,ab\}\) and bidder \(i\in N\). The simulations were conducted using the stata 15.1 software and 100 simulations were carried out for every \(\alpha \) and each of the three sets of simulations.
True, approximated and Quasilinear average equilibrium prices by each value of \(\alpha \)
\(\alpha \)  Average equilibrium prices  

True  Approximated  Quasilinear  
0.6  48.66  48.59  5.27 
(21.70)  (19.40)  (1.55)  
0.7  26.75  27.00  6.39 
(10.59)  (9.89)  (2.08)  
0.8  18.56  18.74  7.86 
(5.43)  (5.29)  (2.11)  
0.9  13.28  13.38  8.94 
(3.51)  (3.46)  (2.21)  
1.0  10.98  10.98  10.98 
(2.58)  (2.58)  (2.58)  
1.1  8.10  7.97  10.58 
(1.82)  (1.82)  (2.55)  
1.2  6.08  6.03  10.54 
(1.23)  (1.28)  (2.64)  
1.3  4.91  4.84  10.52 
(1.00)  (1.11)  (2.65)  
1.4  3.99  3.99  10.76 
(0.72)  (0.81)  (2.58) 
7 Concluding remarks
This paper has provided a procedure for approximating a bidder’s preferences over two types of items when complementarity between the two may exist. A quick auction procedure is proposed that is shown to always converge to the unique minimal approximated Walrasian equilibrium price vector. The auction procedure is efficient with respect to the approximated preferences of the bidders. Simulation results suggests that the approximation procedure works fairly well as the absolute relative error between the true and approximated minimal Walrasian equilibrium prices is only \(4.8\%\) on average. For future research, it would be desirable to find a, perhaps, similar approximation procedure that can be applied to a more general setting, in which bidders are interested in more than two items. Moreover, the auction process is designed to find the unique minimal approximated Walrasian price vector. Extending the process to the cases when such a price vector is not unique, for example, when bidders view the packages as complements, as in Sun and Yang (2009, 2014), would be another direction for future research. Furthermore, the approximation procedure described in this paper assumes that bidders report truthfully and the auction procedure is not strategyproof. Finding a strategyproof way of conducting a quick auction, when bidders preferences are not necessarily quasilinear, would be of great interest and importance.
Footnotes
 1.
 2.
 3.
To simplify the notation, we let a and b denote both the item and a set containing the item, i.e. \(a\equiv \{a\}\) and \(b\equiv \{b\}\).
 4.
\(\varepsilon (t), l_x(t)\) and \(\delta (p)\) can be identified in a finite number of steps. If Process 1 proceeds to Step 3 at some \(p'\), then \(f_i(p'_x)=p'_y\) for some indifference curve \(i=1,2,3\), \(x,y\in \{ab\}\) and \(x\ne y\). A way to identify \(l_x(t)\) is, thus, to order all indifference curves, for which the above holds true, from smallest to largest by their slopes, m. Starting with the smallest m, set \(\delta _0(p')=0\), \(\delta _a(p'_a)=1\) and \(\delta _b(p'_b)=m\) and check if \(\varepsilon (t)>0\). If not, continue with the second smallest m, and so on until \(\varepsilon (t)>0\), which will happen by Lemma 2. This gives \(l_x(t)\) and \(\delta (p')\). A straight line can be constructed from \(\delta (p')\), with slope \(\frac{p'_b}{p'_a}\). \(\varepsilon (t)\) can be identified by checking the intersection between this straight line and any indifference curve. Since there are only a finite number of indifference curves, this process terminates in a finite number of steps.
 5.
The true and approximated minimal Walrasian equilibrium prices are unique when \(\alpha \le 1\), since no indifference curves are downwardsloping. There exists no unique true or approximated minimal Walrasian equilibrium price when \(\alpha >1\).
Notes
Acknowledgements
Open access funding provided by Uppsala University. I want to thank Federico Echenique, Jörgen Kratz, Jens Gudmundsson, and especially Tommy Andersson for their helpful comments and suggestions.
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