Exchanging heterogeneous goods via sealed bid auctions and transportation systems

A primary commodity such as wheat, rice, coffee, oil, etc., is shipped fromm locations where it was grown or pumped ton manufacturers. Each manufacturer processes, packages, advertises, and distributes the commodity under a consumer product brand name. The resulting heterogeneous good is sold at a sealed bid auction, in competition with the other manufacturers of the consumer product, tok final customers. The problem to be considered in this paper is to find a way of determining prices for the goods produced and the physical exchanges between seller and buyer which satisfy flow conditions and which take into account the evaluations of the goods by both sellers and buyers. The first model for doing this is given in section 1, which combines the idea of a sealed bid auction due to Shapley, Shubik and Thompson, with a conventional transportation system. The sealed bid auction is used to determine the exchange prices, and the transportation system is used to calculate the production and transportation costs. It is suggested that the resulting model type can also be applied in a wide range of problems that arise in the marketing of goods sold under brand names (i.e., heterogeneous goods) regardless of whether they are actually exchanged at formal auctions. We show in section 6 that our model is a generalization of the transshipment model in a recent paper by Dubey and Shapley [1]. In their model they considered a number of oligopolists engaged in transshipping and trading goods. Their oligopolists set their prices in order to maximize profits, rather than having them determined by an auction process as is done in our model. In section 7, we extend the model to one in which the wholesalers are permitted to make positive profits. We show how to calculate the values of coalitions of the various players in the model.The work of the first named author was prepared as part of the activities of the Management Sciences Research Group, Carnegie Mellon University, under Contract No. N00014-85-K-0198 NR 047-048 with the Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Manipulation games in economies with indivisible goods

In this paper we study the strategic aspects of the No-Envy solution for the problem of allocating a finite set of indivisible goods among a group of agents when monetary compensations are possible. In the first part of the paper we consider the case where each agent receives, at most, one indivisible good. We prove that the set of equilibrium allocations of any direct revelation game associated with a subsolution of the No-Envy solution coincides with the set of envy-free allocations for the true preferences. Under manipulation all the subsolutions of the No-Envy solution are equivalent. In the second part of the paper, we allow each agent to receive more than one indivisible good. In this situation the above characterization does not hold any more. We prove that any Equal Income Walrasian allocation for the true preferences can be supported as an equilibrium allocation of any direct revelation game associated with subsolutions of the No-Envy solution, but also non-efficient allocations can be supported.     

Warm-glow giving in networks with multiple public goods

This paper explores a voluntary contribution game in the presence of warm-glow effects. There are many public goods and each public good benefits a different group of players. The structure of the game induces a bipartite network structure, where players are listed on one side and the public good groups they form are listed on the other side. The main result of the paper shows the existence and uniqueness of a Nash equilibrium. The unique Nash equilibrium is also shown to be asymptotically stable. Then the paper provides some comparative statics analysis regarding pure redistribution, taxation and subsidies. It appears that small redistributions of wealth may sometimes be neutral, but generally, the effects of redistributive policies depend on how public good groups are related in the contribution network structure.     

House allocation with fractional endowments

This paper studies a generalization of the well known house allocation problem in which agents may own fractions of different houses summing to an arbitrary quantity, but have use for only the equivalent of one unit of a house. It departs from the classical model by assuming that arbitrary quantities of each house may be available to the market. Justified envy considerations arise when two agents have the same initial endowment, or when an agent is in some sense disproportionately rewarded in comparison to her peers. For this general model, an algorithm is designed to find a fractional allocation of houses to agents that satisfies ordinal efficiency, individual rationality, and no justified envy. The analysis extends to the full preference domain. Individual rationality, ordinal efficiency, and no justified envy conflict with weak strategyproofness. Moreover, individual rationality, ordinal efficiency and strategyproofness are shown to be incompatible. Finally, two reasonable notions of envy-freeness, no justified envy and equal-endowment no envy, conflict in the presence of ordinal efficiency and individual rationality. All of the impossibility results hold in the strict preference domain.     

Hedging pure endowments with mortality derivatives

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. (2009). The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.     

Hedging life insurance with pure endowments

We extend the work of Milevsky et al., [Milevsky, M.A., Promislow, S.D., Young, V.R., 2005. Financial valuation of mortality risk via the instantaneous Sharpe ratio (preprint)] and Young, [Young, V.R., 2006. Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio (preprint)] by pricing life insurance and pure endowments together. We assume that the company issuing the life insurance and pure endowment contracts requires compensation for their mortality risk in the form of a pre-specified instantaneous Sharpe ratio. We show that the price P^m,n for m life insurances and n pure endowments is less than the sum of the price P^m,0 for m life insurances and the price P^0,n for n pure endowments. Thereby, pure endowment contracts serve as a hedge against the (stochastic) mortality risk inherent in life insurance, and vice versa.     

Multi-unit auctions with complementarities: Issues of efficiency in electricity auctions

In this paper, we analyze the ability of different auction structures to induce the efficient dispatch in a one-shot framework where generators know their own and competitors' costs with certainty. In particular, we are interested in identifying which, if any, rules in an auction structure yield only the efficient dispatch in equilibrium. We find that a critical component to a successful auction design is the way in which demand is bundled and hence the way bids are defined. While an auction mechanism which allows for more than one winner in an auction may support inefficient dispatches in equilibrium, we find that an auction where there is exactly one winner per lot, where the lots are formed to capture the cost structure of generation plants, and all lots are auctioned simultaneously, supports only efficient dispatches in equilibrium.     

Technological endowments in entrepreneurial partnerships

This paper discusses a novel argument to interpret the importance of thinking of collaborative partnerships in pre-competitive agreements. To do so, we adopt a dynamic iterative process to model technology diffusion between the partners of an agreement. We find that the success of an agreement of a given length hinges around identifying the suitable efficient combinations of the initial technological endowments of partners. As the time horizon of the agreement expands, the probability of identifying a suitable partner decreases, thus justifying the prevalence of short-horizon R&D agreements.     

An impossibility result for housing markets with fractional endowments

The housing market setting constitutes a fundamental model of exchange economies of goods. House allocation with fractional endowments of houses was considered by Athanassoglou and Sethuraman (2011) who posed the open problem whether individual rationality, efficiency, and weak strategyproofness are compatible for the setting. We show that the three axioms are incompatible.     

Large auctions with risk-averse bidders

We study private-value auctions with n risk-averse bidders, where n is large. We first use asymptotic analysis techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . .). These explicit approximations show that in all large k-price auctions the effect of risk-aversion is O(1/n ²) small. Hence, all large k-price auctions with risk-averse bidders are O(1/n ²) revenue equivalent. The generalization, that all large auctions are O(1/n ²) revenue equivalent, is false. Indeed, we show that there exist auction mechanisms for which the limiting revenue as ${n\longrightarrow \infty }We study private-value auctions with n risk-averse bidders, where n is large. We first use asymptotic analysis techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . .). These explicit approximations show that in all large k-price auctions the effect of risk-aversion is O(1/n ²) small. Hence, all large k-price auctions with risk-averse bidders are O(1/n ²) revenue equivalent. The generalization, that all large auctions are O(1/n ²) revenue equivalent, is false. Indeed, we show that there exist auction mechanisms for which the limiting revenue as n? ￥{n\longrightarrow \infty } with risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even to leading-order as n? ￥{n\longrightarrow \infty }.     

All-pay auctions with risk-averse players

We study independent private-value all-pay auctions with risk-averse players. We show that: (1) Players with low values bid lower and players with high values bid higher than they would bid in the risk neutral case. (2) Players with low values bid lower and players with high values bid higher than they would bid in a first-price auction. (3) Players’ expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-averse players and the seller’s expected revenue. In particular, we show that in all-pay auctions the seller’s expected payoff in the risk-averse case may be either higher or lower than in the risk neutral case.     

Additive habit formation: consumption in incomplete markets with random endowments

We provide a detailed characterization of the optimal consumption stream for the additive habit-forming utility maximization problem, in a framework of general discrete-time incomplete markets and random endowments. This characterization allows us to derive the monotonicity and concavity of the optimal consumption as a function of wealth, for several important classes of incomplete markets and preferences. These results yield a deeper understanding of the fine structure of the optimal consumption and provide a further theoretical support for the classical conjectures of Keynes (The general theory of employment, interest and money. Cambridge University Press, Cambridge, 1936).     

Monotonicity in asymmetric first-price auctions with affiliation

I study monotonicity of equilibrium strategies in first-price auctions with asymmetric bidders, risk aversion, affiliated types, and interdependent values. Every mixed-strategy equilibrium is shown to be outcome-equivalent to a monotone pure-strategy equilibrium under the “priority rule” for breaking ties. This provides a missing link to establish uniqueness in the “general symmetric model” of Milgrom and Weber (Econometrica 50:1089–1122, 1982). Non-monotone equilibria can exist under the “coin-flip rule” but they are distinguishable: all non-monotone equilibria have positive probability of ties whereas all monotone equilibria have zero probability of ties. This provides a justification for the standard empirical practice of restricting attention to monotone strategies. Hendricks et al. (2003) provide an overview of recent empirical work. For a survey of experimental work, see Kagel and Levin (2002).     

Choice of item pricing feedback schemes for multiple unit reverse combinatorial auctions

Recently, interest in combinatorial auctions has extended to include trade in multiple units of heterogeneous items. Combinatorial bidding is complex and iterative auctions are used to allow bidders to sequentially express their preferences with the aid of auction market information provided in the form of price feedbacks. There are different competing designs for the provision of item price feedbacks; however, most of these have not been thoroughly studied for multiple unit combinatorial auctions. This paper focuses on addressing this gap by evaluating several feedback schemes or algorithms in the context of multiple unit auctions. We numerically evaluate these algorithms under different scenarios that vary in bidder package selection strategies and in the degree of competition. We observe that auction outcomes are best when bidders use a naïve bidding strategy and competition is strong. Performance deteriorates significantly when bidders strategically select packages to maximize their profit. Finally, the performances of some algorithms are more sensitive to strategic bidding than others.     

Common-value all-pay auctions with asymmetric information

We study two-player common-value all-pay auctions in which the players have ex-ante asymmetric information represented by finite connected partitions of the set of states of nature. Our focus is on a family of such auctions in which no player has an information advantage over his opponent. We find sufficient conditions for the existence of equilibrium with monotone strategies, and show that such an equilibrium is unique. We further show that the ex-ante distribution of equilibrium effort is the same for every player (and hence the players’ expected efforts are equal), although their expected payoffs are different and they do not have the same ex-ante probability of winning.     

Optimization problem solving with auctions in Cultural Algorithms

In this paper we investigate the role that the competition among knowledge sources plays in the problem solving behavior of a population as the complexity of the optimization problem solving landscapes increases. We employ a type of game theoretic mechanism, auctions, in our study. Our main goal is to determine whether it matters if knowledge sources competed for individuals using auction mechanisms or weighted majority win mechanisms as the landscape complexity increased. The weighted majority win situation allows the Knowledge Sources to make predictions about future success whereas the Auction mechanism allows them to invest in their future by using tokens earned from recent performance. This latter approach allows contextual Knowledge Sources such as Situational Knowledge to play a larger role. Although the results are preliminary, it appears that the auction mechanism is more efficient when solving a problem in situations where contextual information is available. In that case, it is easier for the knowledge sources to make judicial bids or investments. However, once the landscapes become chaotic there is less contextual information available and correspondingly little advantage to the auction mechanism over the weighted majority win situation in terms of the number of generations needed to achieve a solution.     

The buy price in auctions with discrete type distributions

This paper considers second-price, sealed-bid auctions with a buy price where bidders’ types are discretely distributed. We characterize all equilibria in which bidders whose types are less than the buy price bid their own valuations. Budish and Takeyama (2001) analyze the two-bidder, two-type framework. They show that if bidders are risk-averse, then the seller can obtain a higher expected revenue from the auction with a certain buy price than from the auction without a buy price. We extend their revenue improvement result to the n-bidder, two-type framework. In case of three or more types, however, bidders’ risk aversion is not a sufficient condition for a revenue improvement. We point out that even if bidders are risk-averse, the seller cannot always obtain a higher expected revenue from the auctions with a buy price.     

Revenue-maximizing Dutch auctions with discrete bid levels

This paper is concerned with setting a predetermined number of bid levels in a Dutch auction to maximize the auctioneer’s expected revenue. As a departure from the traditional methods used by applied economists and game-theorists, a novel approach is taken in this study to tackle the problem by formulating the auctioning process as a constrained nonlinear program and applying standard optimization techniques to solve it. Aside from proposing respective closed-form formulae for computing the optimal bid levels and the auctioneer’s maximum expected revenue, we also show that the bid decrements should be increasing if there are two or more bidders in the Dutch auction. Additionally, the auctioneer’s maximum expected revenue increases with the number of bidders as well as the number of bid levels. Finally, managerial implications of the key findings as well as limitations of this research work are discussed.     

Ascending bid auctions with behaviorally consistent bidders

Decision makers whose preferences do not satisfy the independence axiom of expected utility theory, when faced with sequential decisions will act in a dynamically inconsistent manner. In order to avoid this inconsistency and maintain nonexpected utility, we suggest the idea of behavioral consistency. We implement this notion by regarding the same decision maker at different decision nodes as different agents, and then taking the Bayesian — Nash equilibrium of this game. This idea is applied to a finite ascending bid auction game. We show the condition for the existence of an equilibrium of this game, and we also characterize the equilibrium in those cases when it exists. In particular, when the utility functionals are both quasi-concave and quasi-convex, then there is an equilibrium in dominant strategies where each bidder continues to bid if and only if the prevailing price is smaller than his value. In the case of quasi-concavity it is shown that, in equilibrium, each bidder has a value such that he continues with positive probability up to it, and withdraws after that.This research was supported by the NSF under Grant No. SES87-08360. We would like to thank Professor Irving H. LaValle for his helpful suggestions.     

Equilibrium strategies in random-demand procurement auctions with sunk costs

^** Email: hinz{at}ifor.math.ethz.ch We address an auction model which captures basic features ofbalancing markets for electricity. The existence and uniquenessof equilibrium are examined and a method for explicit calculationof bid strategies is presented.