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Core-Stability in Assignment Markets with Financially Constrained Buyers

We study markets where a set of indivisible items is sold to bidders with unit-demand valuations, subject to a hard budget limit. Without financial constraints and pure quasilinear bidders, this assignment model allows for a simple ascending auction format that maximizes welfare and is incentive-compatible and core-stable. Introducing budget constraints, the ascending auction requires strong additional conditions on the unit-demand preferences to maintain its properties. We show that, without these conditions, we cannot hope for an incentive-compatible and core-stable mechanism. We design an iterative algorithm that depends solely on a trivially verifiable ex-post condition and demand queries, and with appropriate decisions made by an auctioneer, always yields a welfare-maximizing and core-stable outcome. If these conditions do not hold, we cannot hope for incentive-compatibility and computing welfare-maximizing assignments and core-stable prices is hard: Even in the presence of value queries, where bidders reveal their valuations and budgets truthfully, we prove that the problem becomes NP-complete for the assignment market model. The analysis complements complexity results for markets with more complex valuations and shows that even with simple unit-demand bidders the problem becomes intractable. This raises doubts on the efficiency of simple auction designs as they are used in high-stakes markets, where budget constraints typically play a role.

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08/04/2022

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1 Introduction

The idea of a market exchange automatically channeling self-interest toward welfare maximizing outcomes is a central theme in neoclassical economics. The initial conjecture of the “invisible hand” goes back to Adam Smith. Formally, the Arrow–Debreu model showed that under convex preferences and perfect competition there must be a set of Walrasian equilibrium prices (Arrow and Debreu, 1954). In these models, market participants are price-takers, and they sell or buy divisible goods in order to maximize their total value subject to their budget or initial wealth. The more recent stream of research on competitive equilibrium theory assumes indivisible goods and quasilinear utility functions, i.e. buyers maximize value minus the price they pay (their payoff) and there are no budget constraints (Kelso and Crawford, 1982; Gul and Stacchetti, 1999; Bikhchandani and Mamer, 1997; Leme and Wong, 2017; Baldwin and Klemperer, 2019). The underlying question is under which conditions on the preferences markets with indivisible goods can be assumed to be core-stable111The core is the set of feasible outcomes that cannot be improved upon by a subset of the economy’s participants. and welfare-maximizing. This literature focuses on larger markets where bidders are assumed to be price-takers and it emphasizes core-stability over incentive-compatibility.

For two-sided matching markets where quasilinear buyers have unit-demand, referred to as assignment markets, welfare-maximization, core-stability and even incentive-compatibility can be achieved with a polynomial-time auction algorithm (Shapley and Shubik, 1971)

. These auctions can be interpreted as primal-dual algorithms, where the auctioneer specifies a price vector (a demand query) in each round and the bidders respond with their demand set, i.e. the set of goods that maximize their payoff for given prices.

Unfortunately, it is well-known that we cannot hope for such positive results with more general quasilinear preferences. Incentive-compatibility and core-stability are conflicting in general markets with quasilinear utilities (Ausubel and Milgrom, 2006). Even if we give up on incentive-compatibility, only very restricted types of valuations (e.g., substitutes valuations) allow for Walrasian equilibria.222A Walrasian equilibrium describes a competitive equilibrium where supply equals demand and prices are linear (i.e., there is a price for each good) and anonymous (i.e., the price is the same for all participants and there is no price differentiation) (Bikhchandani and Mamer, 1997; Baldwin and Klemperer, 2019; Leme, 2017). As discused in Bikhchandani and Ostroy (2002), under general valuations (allowing for substitutes and complements), competitive equilibrium prices need to be non-linear and personalized and the core can be empty.

Compared to early utility models in general equilibrium theory such as the Fisher markets for divisible items (Eisenberg and Gale, 1959; Orlin, 2010), quasilinear utility functions imply that bidders do not have budget constraints. In many markets, this is too strong an assumption (Dobzinski et al., 2008; Che and Gale, 2000; Dütting et al., 2016). Bidders might well maximize payoff, but they need to respect budget constraints. Spectrum auctions are just one example, where bidders have general valuations with complements and substitutes and they are typically financially constrained (Bichler and Goeree, 2017). Incentive-compatible mechanisms are known to be impossible in multi-object markets (Dobzinski et al., 2008). It is interesting to understand how core-stable and welfare-maximizing prices can be computed in the presence of budget constraints if we assume bidders to be price-takers. This question was recently analyzed for markets that allow for general valuations where the auctioneer has complete information about values and budgets (Bichler and Waldherr, 2021). The result is a spoiler: Computing the welfare-maximizing core-stable outcome is a -hard optimization problem. Such problems are considered intractable, even for very small problem instances.

The intuition behind this result is that the allocation and pricing problems cannot be treated independently anymore. With quasilinear utility functions, the auctioneer first determines the welfare-maximizing outcome and then a corresponding price vector. If budget constraints are binding, then these constraints on the prices need to be considered when computing the welfare-maximizing outcome, which transforms the allocation and pricing problem into a bilevel integer program for general valuations. Considering that budget constraints are a reality in many markets, this casts doubts whether simple market institutions based on polynomial-time algorithms (e.g., simple ascending auctions as used in selling spectrum nowadays) can find a welfare-maximizing outcome even if bidders were price-takers. General preferences as allowed in combinatorial exchanges, might be too much to ask for. A natural question is whether we can at least hope for core-stable and welfare-maximizing outcomes in markets where bidders have unit-demand valuations, and seek to maximize payoff subject budget constraints.

1.1 Contributions

We study the properties that can be achieved in assignment markets with unit-demand bidders who aim to maximize their payoff but have hard budget constraints as illustrated in the previous example. The aim of this work is to compute welfare-maximizing and core-stable outcomes in the presence of such financial constraints. If we cannot achieve incentive-compatibility and core-stability with such simple valuations, also markets with more complex preferences will not satisfy these properties.

We first introduce and analyze an iterative process that always finds a core-stable outcome using only demand queries based on prices and no direct access to valuations. In contrast to Aggarwal et al. (2009), where bidders’ valuations are directly queried, a distinguishing part of our algorithm is that it relies exclusively on demand queries and provides a natural generalization of the auction by Demange et al. (1986). Moreover, we place emphasis on the question of when we can expect the outcome of the auction to not only lie in the core, but also maximize welfare among all core allocations, as welfare maximization is typically a central goal in market design. During the auction process, the auctioneer may sometimes have to make decisions on which buyer to exclude from certain items in subsequent rounds. One of our main results is that, for any market instance, if the auctioneer would be able to guess the right decisions throughout the auction, we terminate in a welfare-maximizing core-allocation. In particular, if the auctioneer does not have to make any such decisions - which is trivial to check ex-post - our result implies that the auction always finds a welfare-maximizing core-outcome. Unfortunately, we do not know ex-ante whether the condition holds, and if it does not, welfare can be arbitrarily low.

Now, it is important to understand whether we can hope for an incentive-compatible and welfare-maximizing core-selecting mechanism without additional conditions beyond the already strong restriction to unit-demand valuations. Unfortunately, the answer to this question is negative. In a novel result, we show that no auction mechanism for the assignment market can be incentive-compatible and core-stable when buyers face budget constraints. If we give up on incentive-compatibility and assume full access to the true valuations (i.e., via value queries) and buyer budgets, we can compute a core-stable and welfare-maximizing outcome.

One might expect that the problem admits a polynomial time solution, since, without the presence of budget constraints, the problem lies in complexity class P. Unfortunately, a main finding of this paper shows that determining core-stable, welfare-maximizing outcomes with financially constrained buyers is an NP-complete optimization problem, even for the assignment market with full access to valuations and budgets. This means, the existence of budget constraints renders the problem of determining welfare-maximizing, core-stable outcomes NP-hard. The hardness proof requires an involved reduction from the maximum independent set problem. One aspect that is making the reduction difficult is that prices need to be considered as continuous variables. These results show that, even for the simplest type of multi-object markets, those with only unit-demand bidders, we cannot expect core-stable and welfare-maximizing outcomes unless additional strong conditions are satisfied that are typically unknown ex-ante.

2 Related Literature

Two-sided matching markets describe markets where buyers want to win at most one item (also known as the unit-demand model) and sellers sell only one item. Buyers and sellers are disjoint sets of agents and each buyer forms exclusive relationships with a seller. Such markets are central to the economic sciences. The well-known marriage model of Gale and Shapley (1962) assumes ordinal preferences and non-transferable utility. Shapley and Shubik (1971) analyzed such markets with quasilinear utility functions and showed that the core of this game is nonempty and encompasses all competitive equilibria. Under the quasilinear utility model, buyers maximize value minus price, while sellers maximize price minus cost. While their setting assumes access to all valuations, Demange et al. (1986) showed that an ascending auction with only demand queries results in a competitive equilibrium at the lowest possible price, i.e. at the competitive equilibrium price vector that is optimal for buyers. In such an auction, the auctioneer specifies a price vector (the demand query) in each round, and buyers respond with their demand set, i.e. the set of goods that maximize payoff at the prices.

The housing market of Shapley and Scarf (1974) is an example of a market without transferable utility or monetary funds. In this market, each agent is endowed with a good or house, and each agent is interested in one house only. The goal of this market is to redistribute ownership of the houses in accordance with the ordinal preferences of the agents. In such housing markets, the core set is nonempty. If no agent is indifferent between any two houses, then the economy has a unique competitive allocation, which is also the unique strict core allocation. An allocation belongs to the strong core, if no coalition of buyers and sellers can make all members as well off and at least one member better off by trading items among themselves. We assume an allocation belongs to the weak core if no coalition can lead to all members’ utilities improved when redistributing items amongst themselves.

Quinzii (1984) generalizes the model of Shapley and Scarf (1974) to one with multiple agents with unit-demand and transferable but non-quasilinear utility. Buyers derive utility from at most one good and a transfer of money. Sellers aim at obtaining the highest possible price above a reservation level. She proved the general existence of the core in her model, and its equivalence to competitive equilibria. In a closely related model, Gale (1984) shows that a competitive equilibrium always exists. These models allow for budgets, but differ from hard budget constraints as examined in our work, where bidders are not permitted to spend more than a certain amount of money. Alaei et al. (2016) provide a structural characterization of utilities in competitive equilibria and a mechanism that is group-strategyproof. These non-quasilinear models assume utility functions that are not necessarily quasilinear, but where small changes of prices do not lead to a discontinuous change of the bidders’ utilities as is the case with hard budget constraints.

Closer to this paper is another line of research that focuses on assignment markets where buyers maximize payoff subject to a hard budget constraint. Aggarwal et al. (2009) show that an extension of the Hungarian algorithm is incentive-compatible and bidder-optimal if the auction is in general position, a rather specific condition that is usually unknown ex-ante and hard to check. Typically, ascending auctions use only demand queries, namely the auctioneer specifies a price and the bidders respond with their demand set. The auction additionally requires value and budget queries, thereby asking for the value of a specific good to a bidder and their budget during the auction. This is quite different from the ascending auctions based on price-based demand queries only, as described in Demange et al. (1986) and the subsequent literature or compared to ascending auctions used in the field. Fujishige and Tamura (2007) consider two-sided markets with budget-constrained bidders whose valuation functions are more general than unit-demand. Their results imply that in the unit-demand setting, there always exists a core allocation. These prior results aim exclusively for core-stability, but do not attempt to maximize welfare as done in this work.

In contrast to competitive equilibrium theory, Henzinger and Loitzenbauer (2015) and its predecessor Dütting et al. (2013) do not aim for core-stability. Note that with hard budget constraints, core-stability does not imply envy-freeness. Consider for example a market with two buyers and one seller selling a single good. If both buyers have the same budget and the same valuation for the good, which exceeds their budget, the only possibility such that no bidder envies the other one is that the good remains unsold. Such an outcome is clearly not in the core, because there is a coalition of buyer and seller who want to deviate. Note that such types of envy cannot arise without binding budget constraints, because if the price is at the value of two bidders, they are indifferent between getting the object or the empty set. It depends on the considered market, whether envy-freeness and bidder-optimality or core-stability should be preferred. While their model appears to be reasonable in cases where all items are sold by one large seller, like ad auctions, it may not seem reasonable for individual sellers to participate in an auction where items remain unsold for the sake of envy-freeness. Finally, van der Laan and Yang (2016) propose an ascending auction for the assignment market that results in an equilibrium under allotment, which is in general not a core-stable outcome.

Core-stability and incentive-compatibility are arguably the most important axioms in market design. Whether one can design assignment markets that satisfy these axioms in the presence of hard budget constraints has not yet been answered. We show that, without strong additional assumptions, this is not possible. Importantly, even with access to all valuations and budgets, the problem is computationally intractable for large market instances.

3 Preliminaries

A two-sided matching market consists of two disjoint sets of agents and , representing bidders and goods . We identify good with the seller owning it, i.e. each seller owns one good. The -item is a dummy item and does not have value to any bidder, meaning that receiving good corresponds no real good. Additionally, the market is defined by each bidder ’s valuation with and budget , as well as each seller ’s reserve values/ ask price .

A price vector is a vector with , assigning price to every good . Bidders have quasi-linear utilities, so if bidder receives item under prices , their utility is , if , and , otherwise. An assignment is represented as a map from bidders to the items they receive, where for all , so only the dummy good may be assigned to more than one bidder. An outcome is a pair , where is an assignment and is a price vector, such that no budget constraint is violated, i.e. for all and only sold items may have a positive price: implies that . For our iterative auction in Section 4,for the sake of simplicity, we assume all reserve prices to be equal to . The results can be easily generalized by starting the auction at the reserve prices, and not at .

In neoclassic economics, a (Benthamite) social welfare function is defined as the sum of cardinally measurable values of all market participants. An optimal allocation of resources is one which maximizes the social welfare in this sense:

This can be written in LP-form as

(1)
s.t.

where the variables in parentheses denote the corresponding duals. This assignment problem is well-known to have an integral optimal solution and can be solved in (Kuhn, 1955). An integral solution corresponds to an assignment via . This notion of utilitarian welfare maximization, i.e. maximizing the sum of participants’ utilities, is widely used in auction theory and competitive equilibrium theory.

New welfare economics, in the tradition of Pareto defies the idea of interpersonal utility comparisons and stipulates ordinal preferences. Pareto efficiency or Pareto optimality is the key design desideratum in this literature. A market outcome is Pareto efficient, if no market participant can be better off without making at least one other participant worse off. With cardinal utilities and interpersonal comparisons a welfare-maximizing outcome is also Pareto efficient. This is because any Pareto-improvement would increase welfare, which is not possible by definition of a welfare-maximizing allocation. It has also been shown that the converse is true (Negishi, 1960). Another design desideratum is that of core-stability.

Definition 1 (Core outcome).

Let be an outcome. A bidder-seller pair is called a blocking pair, if and . is a core outcome, if there are no blocking pairs. We also say that is core-stable in this case.

The idea of a blocking pair is that both bidder and seller would strictly increase their utility, if received item instead of : if pays for item , then still , and at the same time, the profit of seller is increased by .

In the literature, a core outcome is often alternatively defined in the following way: an outcome is in the core if there are no subsets and and an outcome on such that for all and for all (see for example (Zhou, 2017)). These definitions can easily be shown to be equivalent: first suppose that such subsets and do exist. Then it is easy to see that both sets are nonempty. In particular, let and . Then , so . Furthermore, we have , so is a blocking pair. On the other hand, if is a blocking pair, then as in the above paragraph, we can set and get and . Thus we can choose , , and in the alternative definition.

We focus on the problem of finding welfare-maximizing core outcomes:

(2)

If budgets are not binding, i.e., for all bidders and all goods , core-stability coincides with the definition of a competitive equilibrium. For this, let us first define the demand set of bidder , which consist of the most preferred, affordable among all items at prices :

Definition 2 (Competitive equilibrium).

An outcome is a competitive equilibrium, if for all bidders .

The next proposition summarizes well-known equivalences of the different notions for markets where budgets are not binding.

Proposition 1 (Bikhchandani and Mamer (1997)).

Suppose that for all and , and let be an outcome. Then the following statements are equivalent.

  1. is a core outcome.

  2. is a competitive equilibrium.

  3. The variables defined by

    solve the linear program (

    1) and is a corresponding dual solution.

  4. is a welfare-maximizing core outcome.

This equivalence no longer remains true if bidders have binding budgets: in general, a core outcome needs not be a competitive equilibrium, and different core outcomes might generate very different welfare.

Example 1.

For a very simple example, consider two bidders and one item . Suppose that and . Both bidders have the same budget . It is easy to see that there are two core outcomes: either bidder or receives for a price of , while the other bidder does not receive an item. Both core outcomes are no competitive equilibria, since the bidder not receiving does not receive an item in . This bidder thus envies the other. Moreover, one core outcome generates a welfare of , while the other generates a welfare of . Ignoring budgets, the above LP-formulation would assign item to bidder at a price - no such price is feasible when considering the budget constraints.

Besides, as we show, finding a welfare-maximizing core outcome is in general NP-complete, so we cannot expect a simple LP-formulation as above to exist. Note that efficient algorithms for determining core outcomes under budget constraints have been discussed in the literature. However, desirable properties like bidder-optimality and incentive-compatibility are only guaranteed if additional assumptions on the bidders’ preferences are made. Aggarwal et al. (2009) introduced the notion of general position, a sufficient condition for ascending auctions to indeed find the welfare-maximizing core-stable outcome. As this condition has received considerable attention in the literature, we provide a brief discussion:

Definition 3 (Aggarwal et al. (2009)).

Consider a directed bipartite graph with edges between bidders and goods (including dummy good ): For and , there is a

  • forward-edge from to with weight

  • backward-edge from to with weight

  • maximum-price edge from to with weight

  • terminal edge from to the dummy good with weight .

The auction is in general position, if for every bidder , there are no two alternating walks, following alternating forward and backward edges and ending with a distinct maximum-price or terminal edge, having the same total weight.

Example 2.

Consider an auction with two bidders and with . The number of goods and bidders’ valuations may be chosen arbitrarily. Assume is any good. Consider the following path starting from bidder : , where the last edge is a maximum-price edge, with total weight . Now consider the path , where the only edge is a maximum-price edge, with weight . Since , the total weight of both paths is equal, so the auction is not in general position.

As the example shows, the general position condition implies that in an ascending auction, no two bidders may reach their budget limits at the same time. Henzinger and Loitzenbauer (2015) claim the general position condition is rather restrictive, as it excludes, for instance, symmetric bidders. They additionally show that no polynomial-time algorithm can determine whether a set of valuations is in general position. The general position condition is sufficient but not necessary for the existence of a unique bidder-optimal stable matching (Aggarwal et al., 2009), which is thus also welfare-maximizing by our results below. As we will see, there are valuations not in general position, but where a welfare-maximizing core allocation can still be computed efficiently with our auction.

Let us now introduce an iterative auction that always finds a core-stable outcome in markets with budget constrained buyers and, if a simple ex-post condition is satisfied, maximizes welfare among all core outcomes.

4 An Iterative Auction

Our auction is based on the well-known auction by Demange et al. (1986) (denoted as DGS auction from now on), which implements the Hungarian algorithm. Contrary to Aggarwal et al. (2009), where the underlying assumption is that bidders report their valuations and budgets to the auctioneer, in our auction, they only have to report their demand sets at certain prices, similar to other ascending auctions (Mishra and Parkes, 2007). We will provide conditions when reporting demand sets truthfully is incentive-compatible. Thus, we provide a natural generalization of the DGS auction to markets where bidders have binding budget constraints. The simple ascending nature of our auction also naturally motivates an ex-post optimality condition for the returned allocation. Without loss of generality, we will assume in this section.

In the auction process, we may need to “forbid” some bidder to demand a certain item. We model this by introducing subsets of goods for every bidder and define the restricted demand set to be

Note that our definition of the restricted demand set coincides with the definition of demand sets by van der Laan and Yang (2016). The set consists of all affordable items that generate the highest utility among all items in . We introduce the well-known notions of over- and underdemanded sets (Demange et al., 1986; Mishra and Talman, 2006), adjusted to our notion of restricted demand sets.

Definition 4.

Let a price vector and sets with be given. A set is

  • overdemanded, if and , and

  • underdemanded, if for all and .

is minimally over-/underdemanded, if it does not contain a proper over-/underdemanded subset.

Finally, we define the strict budget set of bidder by . It consists of all items with prices strictly less than the bidder’s budget.

4.1 The Auction Algorithm

Algorithm 1 describes our auction. It is based on the following observation.

Lemma 1.

An outcome is in the core if and only if there are sets such that and for all .

Proof.

Suppose first that is a core outcome. Set . Then , since otherwise there would exist an item with generating a higher utility than - this would constitute a blocking pair.

Now let’s assume that there are sets as described with for all . Suppose there is a blocking pair . Then costs strictly less than , so , and generates a higher utility than . This would contradict . Thus, is a core outcome. ∎

Computing a core outcome can thus also be interpreted as computing a “competitive equilibrium” with respect to the restricted demand sets . This is quite similar to the definition of an equilibrium under allotment of van der Laan and Yang (2016). However, they have other requirements on the sets , which, in general, cause their equilibria not to lie in the core.

In view of Lemma 1, the goal of our auction procedure is to determine prices together with sets , such that there are neither over- nor underdemanded sets of items. As observed in Mishra and Talman (2006), this implies existence of an assignment , such that every bidder receives an item in their demand set, and every item with positive price gets assigned to some bidder. The following result is due to the aforementioned work.

Proposition 2.

Suppose that with respect to the , there is no over- or underdemanded set of items. Then there is an assignment such that for all , and for all with , there is some with .

Note that they considers markets without budgets and demand sets without restrictions. However, their proof only uses combinatorial properties of the demand sets, so it can be directly adapted to our setting. Thus, we omit a proof here.

Set and for all bidders . Set , and . Request from all bidders. If and the set
is nonempty, go to Step 3. Otherwise, if there is an overdemanded set, go to Step 4. Else, go to Step 5. Choose a bidder and define . Set , and . For all other bidders , the sets are unchanged. Set and go to Step 2. Choose a minimally overdemanded set . For all , set . The prices for all other goods, as well as the sets remain unchanged. Set and go to Step 2. Compute an assignment , such that for all bidders and . Set and return .
ALGORITHM 1 Iterative Auction

Step 3 of the auction ensures that we do not end up with underdemanded sets of items. Moreover, sets always contain at least all items that cost strictly less than the bidder’s budget . Our proof of correctness is similar to the one by van der Laan and Yang (2016): due to the budget constraints, underdemanded sets of items may appear. We show that Step 3 of the auction takes care of these sets.

Lemma 2.

Let be minimally overdemanded and with . Let prices and sets be given. Then

In particular, is not underdemanded.

The proof of this lemma can be found in the Appendix.

Lemma 3.

For all bidders and all iterations of the algorithm, we have that . In particular, since , .

Proof.

Assume to the contrary that there is a minimal iteration , such that a bidder and a good exist with , but . Then in iteration , Step 3 was executed, since otherwise and , so would not be minimal. Hence, in iteration , we have and in particular . Because Step 3 is executed, we have , so in iteration Step 4 was executed and . Thus, since from iteration to , all prices for all preferred goods of bidder were raised and can still afford at prices , , so . This is a contradiction. ∎

Proposition 3.

For every iteration in the auction, it holds:

  1. if there is a minimally underdemanded set of items , then and Step 3 is executed

  2. if Step 3 is executed, there is no underdemanded set of items with respect to the .

Proof.

We prove this by induction on . For , there clearly is no underdemanded set of items, and Step 3 is not executed.

Suppose now that and that the statement is true for all .

First suppose that there exists an underdemanded set of items . Therefore, by induction, in iteration , Step 4 must have been executed - otherwise, there would not exist an underdemanded set. But then, using the same inductive reasoning, there was no underdemanded set in iteration . It is thus easy to see that, since in iteration only prices for items in were raised, only the demand for those items could decrease, so must be a subset of . By Lemma 2, we have

Thus, since , there must be a bidder with and , but . This implies that , so Step 3 is executed in iteration .

Now suppose that Step 3 is executed in iteration . Then again, in iteration , Step 4 was executed, since otherwise we would have , which implies . By induction, there was no underdemanded set of items in iteration . Note that , so only the demand of a single bidder chosen in Step 3 does change. Since , so , only the demand for items in can decrease. However, for we have again by Lemma 2 that

and, since we only changed , the demand for items in can at most decrease by . Thus, is not underdemanded in iteration . ∎

Employing the previous lemmata, we can proceed to prove correctness of our proposed auction.

Proposition 4.

The auction terminates after a finite number of iterations, and an assignment as is described in Step 5 exists whenever this Step is reached. The returned tuple constitutes a core outcome.

Proof.

Whenever Step 3 is executed, at least one item is removed from the set of one bidder. Hence, Step 3 can only be called a finite number of times. Also, prices can only be increased a finite number of times in Step 4 - if prices of goods go to infinity, they are clearly not overdemanded at some point anymore. Thus, in some iteration , Step 5 is executed. By Lemma 3, there is no underdemanded set in iteration , because otherwise Step 3 would have been executed. Similarly, there is no overdemanded set. Finally, because of Lemma 3, no set is empty, so by Proposition 2, an assignment as required exists. By Lemma 1, is a core outcome. ∎

Example 3.

Consider the example following auction with three bidders and two items and .

Bidder
Bidder
Bidder

The auction proceeds as follows.

In iterations , there is a unique minimally overdemanded set , and is empty. Thus, Step 4 of the auction is executed and the prices for and are raised. In iteration , the set is nonempty which indicates that bidder 1’s budget was tight for at prices . Thus, we forbid to receive item and reset the prices to . Now, in iteration , there is no overdemanded set and is empty. Thus, there exists an assignment with for all , namely , and . It is easily checked that is indeed a core outcome.

Example 4.

Let us now consider an example of an auction where a non-trivial decision has to be made in Step 3.

Bidder
Bidder
Bidder

It is easy to see that after iterations through Step 4 of our auction, we reach prices , where bidder demands , bidder demands and bidder demands . Since is minimally overdemanded, we execute Step once again to reach , where due to the budget constraints, we have and , while bidder still demands . Thus, Step 3 of the auction is executed with , so both bidders or would be valid bidders to choose in Step 3. For the choice , we have , while for the choice , we have . We could thus either remove from , or from . Depending on our choice, we get two different core outcomes, both supported by the prices : one where bidder receives nothing, bidder receives and bidder receives , and one where bidder receives , bidder receives nothing and bidder receives . The total welfare of the former allocation is , while the one of the latter is .

4.2 Economic Properties

The output produced by our iterative auction is not uniquely defined - it may depend on which bidder is chosen whenever Step 3 is executed. Indeed, we prove the following result.

Proposition 5.

Let be an arbitrary core outcome. Then bidders in Step 3 can be chosen in such a way, that for the resulting outcome we have that coefficient-wise, and for all bidders .

The proof can be found in the Appendix.

We say that a core outcome is Pareto optimal for the bidders, if for every core outcome with for some bidder , there is a bidder with . Proposition 5 directly implies that for every core outcome which is Pareto optimal for the bidders, there is an outcome reachable by the auction with for all . Aggarwal et al. (2009) prove that their algorithm for computing a core-stable outcome always finds the bidder-optimal core outcome , whenever the auction is in general position. Here, bidder-optimal means that for every other core outcome we have that for all . Bidder-optimality thus implies Pareto optimality. We show a similar result for our auction: if the bidder to choose in Step 3 of our auction is always unique, our auction also finds a bidder-optimal core outcome.

Corollary 1.

Suppose that whenever Step 3 is executed, , i.e., there is a unique bidder to choose, and let be the uniquely determined outcome of the auction. Then for any core outcome we have that and for all bidders , i.e., is bidder-optimal.

Proof.

Since in every iteration through Step 3, the outcome of the auction is unique. Proposition 5 now directly implies that for every core outcome , we have and . ∎

In particular, if the general position condition is satisfied, it can be shown that never contains more than one bidder. Thus, our auction always finds a bidder-optimal outcome like the auction by Aggarwal et al. (2009) in this case.

Proposition 6.

Suppose the auction is in general position. Then in every iteration through Step 3 of our iterative auction, we have that , and for the unique , we have .

Note that in general that our ex-post condition whenever Step 3 is reached is less demanding than the general position condition, since we do not require , and it is easy to construct examples where , but the ex-post condition is fulfilled. While our condition is only ex-post, it is straight-forward to check for the auctioneer when the auction is actually performed.

Let us now consider welfare-maximization properties of our auction. We first observe that a welfare maximizing core outcome can always be found among the ones which are Pareto optimal for the bidders.

Proposition 7.

Let and be core outcomes. If for all bidders , then

The proofs of Propositions 6 and 7 can be found in the Appendix.

As we described above, by Proposition 5, we can reach any core outcome which is Pareto optimal for the bidders with our auction, and Proposition 7 says that one of them must be welfare-maximizing. Now if we always have , the outcome of our auction is unique which proves our first main result.

Theorem 1.

Bidders in in Step 3 of the auction can be chosen such that the outcome of the auction is a welfare-maximizing core outcome.

In particular, if whenever Step 3 is reached, the unique outcome of the auction is a welfare-maximizing core outcome.

Note that knowledge of the bidders’ demand sets does not suffice in order to always choose the “correct” bidders in Step 3 to reach a welfare-maximizing outcome. Our hardness result in Section 5 implies that even with perfect knowledge of the bidders’ preferences, choosing the correct bidders in Step 3 is NP-hard. However, our simple ex-post condition at least gives the auctioneer a simple certificate of optimality.

If the auction is in general position, then the auction is ex post incentive-compatible, which follows from the original work by Demange et al. (1986) and the paper by Aggarwal et al. (2009). The question is if this algorithm or any other algorithm where the bidders’ preferences are no further restricted can be incentive-compatible. Unfortunately, the answer is no because incentive-compatibility goes against envy-freeness and therefore the core definition as we show next.

Theorem 2.

In assignment markets with payoff-maximizing but budget constrained bidders there is no incentive-compatible mechanism terminating in a core-stable solution for every input.

Proof.

By the direct revelation principle, we may assume that bidders report their exact valuations, as well as their budgets to the auctioneer. Consider a market with three bidders and two items . Let denote a mechanism, mapping the bidders’ reported valuations and budgets to a core-stable outcome with respect to their reports.

We consider instances of the above described market, where all bidders have the same values for both items: for . Let us consider two instances, where the bidders vary their reported budget.

  1. If all bidders report for , then obviously, since there are only two items, one bidder does not receive one: for , there is an with . Without loss of generality, we assume that . It is easy to see that for core-stability to hold, bidders and both receive an item, and that . Bidders and have utility , while bidder has utility .

  2. If bidder reports , and the other bidders report , then clearly bidder receives an item in any core-stable outcome, and without loss of generality . Also the other item must necessarily be assigned to some bidder. Again, without loss of generality, we assume that and . It is easy to see that must be equal to in a core outcome. Additionally, we must have , since otherwise bidder would strictly prefer item to item , which would not be envy-free. Thus, , and bidder has a utility of .

This already shows that is not incentive-compatible: If all bidders’ true budgets are equal to and they report truthfully, bidder has a utility of . However, if bidder misreports , they would receive an item and have a utility of . Note, that in this case, so bidder can still afford the received item. ∎

Note that Theorem 2 does not preclude an incentive-compatible and welfare-maximizing auction (that is not core-stable). Overall, these iterative auctions require bidders to reveal that they are indifferent to not winning the good once the price equals the valuation of a bidder. Only this allows auctioneers to differentiate between a bidder dropping out due to reaching his valuation or his budget. In practice, bidders might not always bid the null set when price reaches value even in an incentive-compatible auction, which can lead to inefficiencies in such iterative auctions.

5 A Sealed-Bid Auction

If in some iteration of the above algorithm Step 3 is reached with , the ascending auction with only demand queries does not necessarily find the welfare-maximizing core-stable outcome. In such a case, a combination of value and demand queries is required in order to obtain the desired assignment. To this end, for the remaining part of this paper, we assume the auctioneer has unlimited access to all valuations for all objects and the budgets of all bidders. Unfortunately, the outcome and pricing problem does not only need a different oracle but it also becomes NP-complete as we will show next. Besides, as we mentioned above, the property of strategyproofness does not hold.

5.1 A MILP Formulation

First, we show that the problem belongs to complexity class NP by modeling it as a mixed integer linear program (MILP). Once a problem is modeled as such, there is a polynomial-time non-deterministic algorithm, where we guess the values of integer variables and solve the resulting linear program (LP) in polynomial time. Bi-linear terms present in the quadratic formulation (q-BC), namely products of continuous prices

and binary variables, can easily be linearized to obtain the resulting MILP.

In this section, an assignment of buyer to seller is denoted as a binary variable , since solving the MILP requires variable definitions. If the resulting assignment assigns the aforementioned pair in this fashion, then , and for all other buyers except , . The equivalence to the previous definitions is . In order to check for the existence of deviating coalitions, two additional binary variables are introduced. Setting represents the case where bidder has a benefit from deviating by trading with seller , and means that the bidder is best satisfied under the current assignment. The second helper binary variable is set to if possesses a sufficient amount of money to purchase , and set to , if the budget of bidder is insufficient to acquire item of seller , namely the set price of item exceeds ’s budget constraint. Variable reflects the case where bidder prefers to trade with seller and has sufficient budget, and the variable is set to 1 if one of the two necessary conditions does not hold.

Utilities of buyers and sellers are defined as previously argued in Section 3. With , we describe the reserve value or ask price of seller . Constraints (1) and (2) represent the utilities of buyers and sellers, respectively. Constraints (5) and (6) guarantee core-stability. We examine all possible deviating combinations of buyer-seller pairs for a given outcome. Constraint (5) examines whether the corresponding payoff a buyer receives in the selected assignment is higher or equal to the alternative assignment in question. In particular, this constraint checks whether an assignment yields a higher payoff for buyer , in which case . Constraint (6) tests whether a seller ’s payoff on the optimal matching is higher or equal to the minimum value between any buyer ’s budget constraint and ’s valuation for the item , which represents the maximum possible payment seller could receive from any buyer. One or both of these conditions need to be true. Put differently, if both buyer and seller had a higher payoff under an alternative assignment , outcome is not core-stable. In essence, core-stability can be expressed as logical or constraint. Constraints (7) and (8) examine whether bidder has a sufficient budget to obtain item under price . Constraint (9) is responsible for handling the value of in an appropriate manner, to reflect whether a deviating coalition of is indeed profitable and budget-feasible, for any positive value . The value of depends on binary values , and we verify our claim by examining the inequalities formed by the different value combinations (the tuple on the left side represents values ):

In cases and , agent has a sufficient budget and can profit from deviating. However, inequality is infeasible, therefore the value of cannot be set to 1 for this combination of and is forced to 0. For the remaining cases, either does not have sufficient budget (), or has no profit from trading with (), or both conditions hold. In all the aforementioned cases, , and thus reflects the case where no deviation is preferable from the buyer’s side.

Constraint (10) then makes sure that if an item is assigned to buyer , then the price is less than the minimum of the budget of this buyer or his value, and it is higher than the reserve price of the seller. We can conclude that the above formulation always results in a the welfare-maximizing core-stable outcome for assignment markets with budget constraints.

5.2 Complexity Analysis

We now proceed to the proof of NP-completeness. The existence of endogenous pricing variables renders the proof non-trivial: there is no standard method of encoding both continuous and discrete variables, for prices and assignment, in a problem instance. Therefore, the proof requires a more complicated structure and special assumptions regarding the handling of item prices. First, we formally define the decision version of the problem.

Maximum Welfare Budget Constrained Stable Bipartite Matching (MBSBM)

Input: Two disjoint sets (sellers) and (buyers) of agents each, a budget for each agent and a reserve value for each seller , a value for each pair of agents and , and a non-negative integer .

Output: Boolean value

Question: Does there exist a stable outcome such that the total value ?

We have already discussed that the problem is in NP, because MBSBM can be modeled as a MILP where all the reserve values of sellers are set to 0, and such problems are contained in NP (Del Pia et al., 2017). We know that for the case when all budgets are too high and therefore non-binding, for all and , the problem is equivalent to the maximum-weight bipartite matching, which admits a polynomial time solution via the Hungarian algorithm (Kuhn, 1955). Core-stable prices can be derived from the duals of the corresponding linear program (Shapley and Shubik, 1971). Therefore, the case of interest that can result in increased computational cost is when budgets are binding for participating buyers. For our problem, we reduce from the Maximum Independent Set (MIS) problem, which is known to be APX-hard, thus implying NP-hardness. Chen et al. (2021) follows a similar approach for fractional matching without transferable utility. However, an environment with partially transferable utility as in our case, requires special attention and leads to significant differences.

Maximum Independent Set (MIS)

Input: A graph , with vertices and edges , and a non-negative integer .

Output: Boolean value

Question: Does there exist an Independent Set (IS) of size at least , where as IS we define a set of vertices no two of which are adjacent?

The proof uses a specific construction in which we introduce an individual vertex and an edge gadget for each vertex and edge of the original MIS problem. In complexity theory, when performing a reduction from computational problem A to problem B, the term gadget refers to a subset of a problem instance of problem B, that simulates the behavior of certain units of problem A. Drawing from graph theory, the vertex and edge gadgets are bipartite graphs where each edge gadget is connected to two vertex gadgets, corresponding to the two endpoints of the original edge. Each vertex has a degree of three and thus each vertex gadget is connected to three edge gadgets. The edge gadget allows for two matchings between buyer and seller nodes, which all lead to the same welfare. The vertex gadgets also allow for two feasible stable matchings, where the welfare differs by one. The edges between an edge and a vertex gadget are such that it is not possible to select the welfare-maximizing matching in two consecutive vertex gadgets of two neighboring vertices, because it would generate pairs of blocking agents in the edge gadget of the connecting edge, i.e. a matching in the edge gadget would not be core-stable. Similar to the original MIS problem, where there cannot be two adjacent vertices in an IS, in our construction, there cannot be two adjacent vertex gadgets with a high welfare matching. While in the MIS problem, we need to find the IS that is maximal, in the MBSBM problem we need to determine the stable matching of the overall bipartite graph that maximizes welfare. Note that both the vertex and edge gadgets contain a pair of buyers that admit the same budget constraint, thus leading to a violation of the general position condition. Let us now discuss the construction and proof in detail.

5.2.1 Construction

Assume an instance of MIS defined on a cubic graph . A cubic graph is a graph in which all vertices have degree three. We define the transformed instance as a bipartite graph , with , and functions for each buyer , and for each seller that represents agents’ payoffs.

  • represents the total set of agents

  • and denote the sets of buyers and sellers respectively

  • represents the potential transactions between buyers and sellers

  • specifies the difference between the true valuation of buyer