Divisible Goods Research Articles

OPTIMAL MULTI-ATTRIBUTE AUCTIONS FOR DIVISIBLE GOODS

In this paper, the problem of multi-attribute procurement auction of divisible goods is investigated, and an optimal mechanism of multi-attribute auction, which enables negotiation on several attributes in addition to the price of a divisible good, is designed. First, several important assumptions of multi-attribute procurement auctions of divisible goods are given, and the buyer's utility function and the supplier's utility function are defined. Then, a set of sufficient conditions for feasible multi-attribute auction mechanism are given. Based on these conditions, an optimal model of multi-attribute procurement auction whose goal is to maximize the buyer's expected utility, is established. By solving this model, the optimal allocation strategies are obtained. Third, the properties of the optimal multi-attribute auction are discussed. Finally, a multi-attribute auction example about the steam coal procurement is given to show how to implement the optimal multi-attribute auction mechanism.

Incentive mechanism for allocating total permitted pollution discharge capacity and evaluating the validity of free allocation

In this paper, the validity evaluation method of free allocation is studied under the background of allocating the total permitted pollution discharge capacity (TPPDC). First, the free allocation method’s localization and players’ strategic behaviors are analyzed. Second, an incentive mechanism for allocating TPPDC is proposed based on the uniform price auction of divisible goods, and an evaluation model to evaluate the validity for the free allocation of TPPDC is established, and the feasibility and reasonableness of this evaluation model are proved. Third, the evaluation criterion and the evaluation process are given. Finally, this evaluation model is applied to the environmental planning of Han River basin, and the validity evaluation result of free allocation is obtained by analyzing the statistical data of chemical oxygen demand (COD) in 14 counties and cities of Hubei Province, China. The results of our study will provide the government environmental management departments with theoretical basis and scientific methods to effectively implement the pollutant gross control system and design the related environmental policies.

A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good

We address the problem of allocating a single divisible good to a number of agents. The agents have concave valuation functions parameterized by a scalar type. The agents report only the type. The goal is to find allocatively efficient, strategy proof, nearly budget balanced mechanisms within the Groves class. Near budget balance is attained by returning as much of the received payments as rebates to agents. Two performance criteria are of interest: the maximum ratio of budget surplus to efficient surplus, and the expected budget surplus, within the class of linear rebate functions. The goal is to minimize them. Assuming that the valuation functions are known, we show that both problems reduce to convex optimization problems, where the convex constraint sets are characterized by a continuum of half-plane constraints parameterized by the vector of reported types. We then propose a randomized relaxation of these problems by sampling constraints. The relaxed problem is a linear programming problem (LP). We then identify the number of samples needed for “near-feasibility” of the relaxed constraint set. Under some conditions on the valuation function, we show that value of the approximate LP is close to the optimal value. Simulation results show significant improvements of our proposed method over the Vickrey-Clarke-Groves (VCG) mechanism without rebates. In the special case of indivisible goods, the mechanisms in this paper fall back to those proposed by Moulin, by Guo and Conitzer, and by Gujar and Narahari, without any need for randomization. Extension of the proposed mechanisms to situations when the valuation functions are not known to the central planner are also discussed.

Discrete Bids and Empirical Inference in Divisible Good Auctions

I examine a model of a uniform price auction of a perfectly divisible good with private informa tion in which the bidders submit discrete bidpoints rather than continuous downward sloping demand functions. 1 characterize necessary conditions for equilibrium bidding. The characterization reveals a close relationship between bidding in multiunit auctions and oligopolistic behaviour. I demonstrate that a recently proposed indirect approach to the revenue comparisons of discriminatory and uniform price auc tions is not valid if bid functions have steps. In particular, bidders may bid above their marginal valuation in a uniform price auction. In order to demonstrate that discrete bidding can have important consequences for empirical analysis I use my model to examine a data set consisting of individual bids in uniform price treasury auctions of the Czech government. I propose an alternative method for evaluating the perfor mance of the employed mechanism. My results suggest that the uniform price auction performs well, both in terms of efficiency of the allocation and in terms of revenue maximization. I estimate that the em ployed mechanism failed to extract at most 3 basis points in terms of the annual yield of T-bills worth of expected surplus while implementing an allocation resulting in almost all the efficient surplus. Failing to account for discreteness of bids would in my application result in overestimating the unextracted revenue by more than 50%.

Mechanism Choice and Strategic Bidding in Divisible Good Auctions: An Empirical Analysis of the Turkish Treasury Auction Market

We propose an estimation method to bound bidders’ marginal valuations in discriminatory auctions using individual bid-level data and apply the method to data from the Turkish Treasury auction market. Using estimated bounds on marginal values, we compute an upper bound on the inefficiency of realized allocations as well as bounds on how much additional revenue could have been realized in a counterfactual uniform price or Vickrey auction. We conclude that switching from a discriminatory auction to a uniform price or Vickrey auction would not significantly increase revenue. Moreover, such a switch would increase bidder expected surplus by at most 0.02 percent.

MECHANISM DESIGN FOR OPTIMAL AUCTION OF DIVISIBLE GOODS

In this paper, the auction of divisible goods is investigated and a mechanism design method for optimal auction of divisible goods is presented. First, the definitions of feasible allocations and divisible goods auctions are given based on several important assumptions of divisible goods auction. Second, an optimal auction mechanism of divisible goods is designed, and a method of how to use the uniform price auction to implement the optimal auctions is discussed under the background of allocating the total permitted pollution discharge capacity (TPPDC). Lastly, this method is applied to the environmental planning of Wuhan City Circle in Hubei Province, China.

Natural oligopolies with exogenous sunk costs: A non-Suttonian result

In a spatial competition model with exogenous fixed costs and divisible goods, we obtain non-Suttonian results. When the economy is infinitely replicated, the number of firms does go to infinity but, as consumers’ income goes to infinity, the equilibrium number of firms tends toward a finite value. This occurs because the global demand to each firm becomes in the limit infinitely sensitive to price differentials since they give then rise to infinitely large differences in purchase expenditure.

An efficient Nash-implementation mechanism for network resource allocation

We propose a mechanism for auctioning bundles of multiple divisible goods in a network where buyers want the same amount of bandwidth on each link in their route. Buyers can specify multiple routes (corresponding to a source–destination pair). The total flow can then be split among these multiple routes. We first propose a one-sided VCG-type mechanism. Players do not report a full valuation function but only a two-dimensional bid signal: the maximum quantity that they want and the per-unit price they are willing to pay. The proposed mechanism is a weak Nash implementation, i.e., it has a non-unique Nash equilibrium that implements the social-welfare maximizing allocation. We show the existence of an efficient Nash equilibrium in the corresponding auction game, though there may exist other Nash equilibria that are not efficient. We then generalize this to arbitrary bundles of various goods. Each buyer submits a bid separately for each good but their utility function is a general function of allocations of bundles of various divisible goods. We then present a double-sided auction mechanism for multiple divisible goods. We show that there exists a Nash equilibrium of this auction game which yields the efficient allocation with strong budget balance.

Core equivalence and welfare properties without divisible goods

We study an economy where all goods entering preferences or production processes are indivisible. Fiat money not entering consumers’ preferences is an additional perfectly divisible parameter. We establish a First and Second Welfare Theorem and a core equivalence result for the rationing equilibrium concept introduced in Florig and Rivera (2005a). The rationing equilibrium can be considered as a natural extension of the Walrasian notion when all goods are indivisible at the individual level but perfectly divisible at the level of the entire economy. As a Walras equilibrium with money is a special case of a rationing equilibrium, our results also hold for Walras equilibria with money.

Competitive Equilibria in Economies With Multiple Divisible and Indivisible Commodities and No Money

A general equilibrium model is considered with multiple divisible and multiple indivisible commodities. In models with indivisibles it is always assumed that an indivisible commodity, called money, is present that is used to transfer the value of certain amounts of indivisible goods. For these economies with a finite number of divisible and indivisible goods and money and without producers it is well understood that a general equilibrium exists if the individual demands and supplies for the indivisible goods belong to a same class of discrete convexity. In this paper we a model with multiple divisible and multiple indivisible commodities, in which none of the divisible goods may serve as money. Moreover, there are a finite number of producers owning a non-increasing returns to scale technology. One of the producers is assumed to have a linear production technology in order to produce divisible goods. Individual endowments being sufficiently large for production and discrete convexity guarantees the existence of a competitive equilibrium.

THE EFFECT OF AUCTION FORMAT ON EFFICIENCY AND REVENUE IN DIVISIBLE GOODS AUCTIONS: A TEST USING KOREAN TREASURY AUCTIONS*

This paper measures the efficiency and revenue properties of the two most popular formats for divisible goods auctions: the uniform‐price and discriminatory auction. We analyze bids into the Korean Treasury auctions which have used both formats. We find that the discriminatory auction yields statistically higher revenue. Unlike previous work that uses data from only one format, we are able to compare the efficiency properties of the two formats. We find that the discriminatory auction better allocates treasury bills to the highest value financial institutions. However, the differences in revenue and efficiency are not large because the auctions are very competitive.

Optimal procurement mechanisms for divisible goods with capacitated suppliers

Procurement auction literature typically assumes that the suppliers are uncapacitated [see, e.g. Dasgupta and Spulber in Inf Econ Policy 4:5–29, 1990 and Che in Rand J Econ 24(4):668–680, 1993]. Consequently, the auction mechanisms award the contract to a single supplier. We study mechanism design in a model where suppliers have limited production capacity, and both the marginal costs and the production capacities are private information. We provide a closed-form solution for the revenue maximizing direct mechanism when the distribution of the cost and production capacities satisfies a modified regularity condition [Myerson in Math Oper Res 6(1):58–73, 1981]. We also present a sealed low bid implementation of the optimal direct mechanism for the special case of identical suppliers. The results in this paper extend to other principle-agent mechanism design problems where the agents have a privately known upper bound on allocation.

VCG-Kelly Mechanisms for Allocation of Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals

rdquoThe VCG-Kelly mechanism is proposed, which is obtained by composing the communication efficient, one- dimensional signaling idea of Kelly with the VCG mechanism, providing efficient allocation for strategic buyers at Nash equilibrium points. It is shown that the revenue to the seller can be maximized or minimized using a particular one-dimensional family of surrogate valuation functions.

Bidding for money

We analyze monetary exchange in a model that allows for directed search and multilateral matches. We consider environments with divisible goods and indivisible money, and compare the results with those in models that use random matching and bilateral bargaining. Two different pricing mechanisms are used: ex ante price posting, and ex post bidding (auctions). Also, we consider settings both with and without lotteries. We find that the model generates very simple and intuitive equilibrium allocations that are similar to those with random matching and bargaining, but with different comparative static and welfare properties.

A Search-Theoretic Monetary Business Cycle Model with Capital Formation

Search-theory has become the main paradigm for the micro-foundation of money. But no comprehensive business cycle analysis has been undertaken yet with a search-based monetary model. This paper extends the model with divisible goods and divisible money of Shi (JET, 1998) to allow for capital formation, analyses the monetary propagation mechanism and contrasts the model's implications with US business cycle stylized facts. The propagation mechanism based on a feedback between increased search intensity and depleted inventories only survives in the presence of non-negligible capital adjustment costs. With intermediate adjustment costs the model is able to replicate fairly well the volatility and cross-correlation with output of key US time series, including sales and inventory investment.

More on Ausubel's Dynamic Double Auction

This paper examines the incentive-compatibility of a close variant of Ausubel's (2006) dynamic auction for divisible heterogeneous goods. Assuming only quasi-linear, (weakly) concave utilities with private values, this paper removes Ausubel's assumptions that value functions are strictly concave and non-satiated, that demand functions are measurable, and that the price tâtonnement is continuous and reaches an equilibrium in finite time. Strict concavity is a problematic assumption when applying the auction design, for instance, to commodity exchanges between firms motivated by locally linear value functions. Since merely concave value functions may induce non-unique levels of optimal consumption, bidders in the design modification proposed here submit (set-valued) demand correspondences rather than (single-valued) demand functions. The auctioneer then selects a suitable vector of demand levels from these bids and feeds them into a discrete tatonnement process based Shor's (1985) subgradient algorithm. For quasi-linear, concave utilities, this algorithm converges to equilibrium prices for any starting point. This paper shows that misrepresentations of demand that lead the tâtonnement process away from the true price equilibrium are strictly inferior to truth-telling as the run-time of the process tends to infinity and as its step-size tends to zero. This result, based on only elementary techniques from convex analysis, is closely related to Ausubel's conclusion that truthful revelation of demand is a weakly dominating strategy.

Competitive equilibria in economies with multiple indivisible and multiple divisible commodities

In this paper we consider a general equilibrium model with a finite number of divisible and a finite number of indivisible commodities. In models with indivisibilities it is typically assumed that there is only one divisible good, which serves as money. The presence of money in the model is used to transfer the value of certain amounts of indivisible goods. For such economies with only one divisible commodity Danilov et al. showed the existence of a general equilibrium if the individual demands and supplies belong to a same class of discrete convexity. For economies with multiple divisible goods and money van der Laan et al. proved existence of a general equilibrium if the divisible goods are produced out of money using a linear production technology and no other producers are present in the model. In the model to be presented in this paper we consider an economy without production and with multiple divisible and multiple indivisible commodities. Convexity is replaced by pseudoconvexity, while the indivisible parts of individual demands should belong to some class of discrete convexity. One of the divisible commodities serves as money. Money is strictly desired by all consumers and there should be enough money present in the economy. To guarantee existence of a general equilibrium individual demands should be products of divisible and indivisible parts. In case there is no money, at least one linear production technology besides possibly other producers should be present in order to produce divisible goods.

Public Goods Provision: On Generalizing the Jackson-Moulin Mechanism

Jackson and Moulin (1992) proposed a simple mechanism to implement a wide class of cost-sharing rules for the provision of a binary public good. Bag (1997) generalized this mechanism to divisible public goods under the assumption of “constant marginal benefit.” This paper generalizes the Jackson–Moulin mechanism to a broader setting, including the cases of both decreasing and constant marginal benefit. Moreover, this paper shows the impossibility of generalizing the mechanism further.

When Less (Potential Demand) Is More (Revenue): Asymmetric Bidding Capacities in Divisible Good Auctions

We show that asymmetry in bidders' capacity constraints plays an important role in inhibiting collusion and promoting competitive outcomes in multi-unit auctions in which the final value of the good is common knowledge. This effect appears to be related to the increased difficulty of coordination when there are significant differences between bidders. Due to its impact on collusive outcomes, asymmetry in bidding capacities has a more powerful impact on the seller's revenue than does the auction type. Consistent with the finding in Sade et al. (2006) that the discriminatory auction is more susceptible to collusion than the uniform-price auction, asymmetry in capacity constraints has a greater impact in discriminatory auctions.

The Ex Ante Incentive Compatible Core in Exchange Economies With and Without Indivisibilities

The ex ante incentive compatible core of an exchange economy with private information is the (standard) core of a socially designed characteristic function, which expresses the fact that coalitions allocate goods by means of random incentive compatible mechanisms. We first survey some results in the case of perfectly divisible goods. Examples then show that the ex ante incentive compatible core can be empty, even if utility functions are quasi-linear. If, in addition to quasi-linearity, further assumptions are made (like independent private values), the non-emptiness of the core follows nevertheless from d'Aspremont and Gerard-Varet's construction of incentive compatible, ex post efficient mechanisms. We also introduce a private information version of Shapley and Scarf's economies with indivisible goods, and prove that the ex ante incentive compatible core is always non-empty in this framework.