Efficient mechanisms for bilateral trading

We consider bargaining problems between one buyer and one seller for a single object. The seller's valuation and the buyer's valuation for the object are assumed to be independent random variables, and each individual's valuation is unknown to the other. We characterize the set of allocation mechanisms that are Bayesian incentive compatible and individually rational, and show the general impossibility of ex post efficient mechanisms without outside subsidies. For a wide class of problems we show how to compute mechanisms that maximize expected total gains from trade, and mechanisms that can maximize a broker's expected profit.     

Multidimensional Bargaining Under Asymmetric Information

A buyer with private information regarding marginal valuation bargains with a seller to determine price and quantity of trade. Depending on parameter values, a high‐valuation buyer wants either to reveal information to create value or to conceal it to capture value. In the first case, equilibrium trades are efficient. In the second case, the low‐valuation buyer purchases less than her efficient quantity, and there can be a one‐period delay in trade. The quantity distortion is the only inefficiency that persists when time between offers approaches zero. There exist equilibria that are independent of the seller's prior beliefs.     

Weak monotonicity and Bayes–Nash incentive compatibility

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.     

Weak monotonicity and Bayes–Nash incentive compatibility

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.     

Selling an asset to a competitor

A seller decides whether to allocate an item among two potential buyers. The seller and buyer 1 interact ex post in such a way that each of them suffers a negative externality if the other possesses the item. We show that the optimal allocation rule favors buyer 2, who does not interact ex post with the seller, and in particular bidder 1 may not obtain the good even if his valuation is highest. The auction is therefore subject to resale. When resale is possible, the seller must distort the original auction. We show that the mechanism depends crucially on the way resale is organized ex post. The seller may decide to always allocate the good to the agent with the highest valuation when rents are fully extracted by an intermediary on the resale market. However, she may resort to a stochastic mechanism when the winner of the primary auction has full bargaining power in the resale stage.     

Optimal selling strategies when buyers may have hard information

Consider a revenue-maximizing seller who can sell an object to one of n potential buyers. Each buyer either has hard information about his valuation (i.e., evidence that cannot be forged) or is ignorant. The optimal mechanism is characterized. It turns out that more ignorance can increase the expected total surplus. Even when the buyers are ex ante symmetric, the object may be sold to a buyer who does not have the largest willingness-to-pay. Nevertheless, an additional buyer increases the expected total surplus in the symmetric case, whereas more competition can be harmful if there are ex ante asymmetries.     

Ascending price Vickrey auctions

We show that an ascending price auction for multiple units of a homogeneous object proposed by Ausubel (i) raises prices for packages until they reach those nonlinear and non-anonymous market clearing prices at which bidders get their marginal products and (ii) the auction is a primal–dual algorithm applied to an appropriate linear programming formulation in which the dual solution yields those same market clearing prices. We emphasize the similarities with efficient incentive compatible ascending price auctions to implement Vickrey payments when there is a single object or when objects are heterogeneous but each buyer does not desire more than one unit. A potential benefit of these common threads is that it helps to establish the principles upon which Vickrey payments may be implemented through decentralized, incentive compatible procedures.     

Single Object Auctions With Externalities: A Tractable Model

We consider the single object auction model with allocative externalities in a private valuation and quasi‐linear setting. We model externalities by assuming that every agent has a private valuation (for the object) and a strict ranking of other agents. The utility for an agent when another agent receives the object is the product of his own valuation and a real number that depends on the rank of this agent in his ranking. When the only private information is the valuation of the agents, we characterise the implementable allocation rules and use these to derive the optimal auction. The optimal auction collects payments from agents who do not receive the object.     

Asymmetric first-price menu auctions under intricate uncertainty

This paper studies asymmetric first-price menu auctions in the procurement environment where the buyer does not commit to a decision rule and asymmetric sellers have interdependent costs and statistically affiliated signals. Sellers compete in bidding a menu of contracts, where a contract specifies a vector of characteristics and a payment required from the buyer for delivering these characteristics. The buyer does not commit ex-ante to a decision rule but rather upon observing all the menus offered by sellers chooses the best contract. This paper establishes the existence of a continuum of separating monotone equilibria in this game bounded above by the jointly ex-post efficient outcome and below by the jointly interim efficient outcome. It shows that the jointly ex-post efficient equilibrium outcome is the only ex-post renegotiation-proof outcome and it is also ex-ante robust to all continuation equilibria.     

Convergence to perfect competition of a dynamic matching and bargaining market with two-sided incomplete information and exogenous exit rate

Consider a decentralized, dynamic market with an infinite horizon and incomplete information in which buyers and sellers' values for the traded good are private and independently drawn. Time is discrete, each period has length δ, and each unit of time a large number of new buyers and sellers enter the market. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, the seller and winning buyer exit with their realized utility. Traders who fail to trade either continue in the market to be rematched or exit at an exogenous rate. We show that in all steady state, perfect Bayesian equilibria, as δ approaches zero, equilibrium prices converge to the Walrasian price and realized allocations converge to the competitive allocation.     

Efficiency versus optimality in procurement

We study procurement procedures that simultaneously determine the specification and price of a good. Suppliers can offer and produce the good in either of two possible specifications, both of which are equally good for the buyer. Production costs are interdependent and unknown at the time of bidding. Each supplier receives two signals about production cost, one per specification. Our model is a special case of the interdependent value settings with multidimensional types in Jehiel and Moldovanu (Econometrica 69:1237–1259, 2001) where an efficient and incentive compatible mechanism exists. We characterize equilibrium bidding behavior if the winning supplier is selected purely on the basis of price, regardless of the specification offered. While there is a positive chance of obtaining an inefficient specification, this procurement mechanism involves lower information rents than efficient mechanisms, suggesting that there is a trade-off between minimizing expected expenditure for the good, and ensuring that the efficient specification is chosen.     

Learning to play games in extensive form by valuation

Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.     

Efficient performance in two agent bargaining

This paper concerns ex ante incentive efficient performance in the bargaining of one buyer and one seller over a single item. For any element in a specified class of distributions of the buyer's and the seller's valuations of the item, all of the (interim) individually rational and efficient allocation rules are characterized. It is also shown that the “buyer's bid” and the “seller's price” double auctions are efficient for all distributions in the specified class. These results are obtained by studying efficiency as a linear programming problem.     

Efficient assignment respecting priorities

A widespread practice in assignment of heterogeneous indivisible objects is to prioritize some recipients over others depending on the type of the object. Leading examples include assignment of public school seats, and allocation of houses, courses, or offices. Each object comes with a coarse priority ranking over recipients. Respecting such priorities constrains the set of feasible assignments, and therefore might lead to inefficiency, highlighting a tension between respecting priorities and Pareto efficiency. Via an easily verifiable criterion, we fully characterize priority structures under which the constrained efficient assignments do not suffer from such welfare loss, and the constrained efficient rule (CER) is indeed efficient. We also identify the priority structures for which the CER is singleton-valued and group strategy-proof.     

Equilibrium Partner Switching in a Bargaining Model With Asymmetric Information

We study a model in which the seller of an indivisible object faces two potential buyers and makes an offer to either of them in each period. We find that the seller's ability to extract surplus from them depends crucially on the value of the cost of switching from one buyer to the next. If the seller is pessimistic about the buyers' valuations and there is a switching cost, however small, then the market is a natural bilateral monopoly; the second buyer is never called on. If the switching cost is zero, or if the seller is optimistic, then switching, and possibly recall of the original buyer, may occur.     

On the Interplay of Hidden Action and Hidden Information in Simple Bilateral Trading Problems

A buyer and a seller can exchange one unit of an indivisible good. While producing the good, the seller can exert unobservable effort (hidden action). Then the buyer realizes whether his or her valuation is high or low, which stochastically depends upon the seller's effort level (hidden information). The parties are risk neutral—they can rule out renegotiation and write complete contracts. It is shown that the first best cannot be achieved whenever the ex post efficient trade decision is trivial. The second-best contract is characterized and an application of the model to the choice of risky projects is briefly discussed. Journal of Economic Literature Classification Numbers: C72, C78, D23, D82.     

Intertemporal pricing in laboratory posted offer markets with differential information

Summary. This paper analyzes intertemporal seller pricing and buyer purchasing behavior in a laboratory retail market with differential information. A seller posts one price each period that a buyer either accepts or rejects. Trade occurs over a sequence of "market periods" with a random termination date. The buyer and seller are differentially informed: The seller's cost of producing a unit of a fictitious good is known and constant in all periods, but the buyer's value for the good (demand) is a random variable governed by a Markov Process whose structure is common knowledge. At the beginning of each period the unit's value is determined by "nature" and is privately revealed only to the buyer. The market termination rule is a binary random variable. We conduct 32 laboratory experiments designed to study intertemporal pricing by human subjects in the Posted Offer Institution when demand follows a stochastic process. There are four series of experiments: 8 with simulated buyers, 8 with inexperienced subjects, 8 with once experienced subjects, and 8 with twice experienced subjects.     

Bargaining with a possibly committed seller

We consider negotiations with an open time horizon where a buyer has private information about his valuation and does not know whether the seller is committed to the advertised price. This setting combines two common specifications made in the non-cooperative bargaining literature: one side is privately informed about its valuation, which is drawn from a continuum, and the other side is possibly committed to a fixed offer. We analyze the game both in discrete and in continuous time and show convergence of the two settings, which extends results from Abreu and Gul [2000. Bargaining and reputation. Econometrica 68, 85–117]. One interesting result is that as time proceeds, the non-committed seller becomes less likely to concede in a given period, i.e., it appears as if he becomes more “stubborn.” We further show that a seller may prefer to negotiate with a “worse” buyer as this enhances the value of his possible commitment.     

Exclusive contracts when the incumbent can establish a direct retailer

This paper constructs a model of anticompetitive exclusive dealings with potential downstream competition. Unlike in previous studies, the incumbent can establish a direct retailer with some fixed payment and can offer an exclusive contract to a downstream buyer twice. We show that the existence of these two options helps the incumbent deter socially efficient entry and earn almost monopoly profits even in the absence of scale economies and downstream competition.     

Sequential selling mechanisms

Summary In this paper we attempt to formalize the idea that a mechanism that involves multilateral communication between buyers and sellers may be dominated by one that involves simple bilateral communication. To do this we consider the well known problem in which a seller tries to sell a single unit of output to a group ofN buyers who have independently distributed private valuations. Our arguments hinge on two considerations. First, buyers communicate their willingness to negotiate with the seller sequentially, and second, buyers have the option of purchasing the good from some alternative supplier. It is shown that the seller cannot improve upon a procedure in which she offers the good to each buyer in turn at a fixed price. The seller reverts to multilateral communication if possible, only when no buyer is willing to pay the fixed price. In reasonable environments buyers will be too impatient to wait for the outcome of a multilateral negotiation and all communications will be bilateral.In many problems in mechanism design, informed traders have no alternative to participation in the mechanism that is offered by its designer. The best mechanism from the designer's point of view is then the one that is most efficient at extracting informational rents, that is, a simple auction. In a competitive environment it is likely to be costly for buyers to participate in an auction or any other multilateral selling scheme in which the seller must process information from many different buyers because alternative trading opportunities will be disappearing during the time that the seller is collecting this information. Buyers might be willing to participate in an auction, but only if they could be guaranteed that the competition that they face will not eliminate too much of their surplus.At the other extreme to the auction is a simple fixed price selling scheme 1. The seller simply waits until he meets a buyer whose valuation is high enough, given the opportunities that exist in the rest of the market, for him to be willing to pay this price. The seller extracts the minimum of the buyer's informational rents since the price that a buyer pays is independent of his valuation. Yet the seller might like this scheme if adding a second bidder to the process makes it very difficult for him to find a buyer with a valuation high enough to want to participate.In the presence of opportunity costs, the seller faces a trade-off between his ability to extract buyers informational rents and his ability to find buyers who are willing to participate in any competitive process. In practice this trade-off will impose structure on the method that is used to determine a price. In markets where there are auctions, limits are put on buyer participation. In tobacco auctions bids are submitted at a distinct point in time from buyers who are present at that time. In real estate auctions time limits are put on the amount of time the seller will wait before making a decision. These restrictions on participation are presumably endogenously selected by the seller (possibly in competition with other mechanism designers) with this trade-off in mind.On the other hand, markets in which objects appear to trade at a fixed price are rarely so simple. A baker with a fixed supply of fresh bagels is unlikely to collect bids from buyers and award the bagels to the high bidder at the end of the day. Buyers are unlikely to be willing to participate in such a scheme since they can buy fresh bagels from a competitor down the street. Yet despite the fact that bagels sell at a fixed price throughout the day, most bakers are more than willing to let it be known that they will discount price at the end of the day on any bagels that they have not yet sold. Selling used cars presents a similar problem. Each potential buyer for the used car is likely to have inspected a number of alternatives, and is likely to know the prices at which these alternative can be obtained. A seller who suggests that buyers submit a bid, then wait until the seller is sure that no higher offer will be submitted is asking buyers to forgo these alternative opportunities with no gain to themselves. To avoid the rigidity of the pure fixed price scheme most used cars are sold for a fixed price or best offer. These examples suggest that the best selling mechanism may involve a complex interplay between participation and surplus extraction considerations.The purpose of this paper is to provide a simple formalism within which the factors that determine the best contract can be evaluated. We consider the best known environment from the point of view of auction design in which there are a large number of buyers with independent private valuations for a unit of an indivisible commodity that is being sold by a single supplier who acts as the mechanism designer. We modify this standard problem in two critical ways. First, we assume that the seller meets the potential buyers sequentially rather than all at once. Secondly we assume that buyers have a valuable alternative that yields them a sure surplus. This creates a simple bidding cost that is effectively the expected loss in surplus (created by the disappearance of outside alternatives) that the buyer faces during the time that he spends negotiating with the seller.These simple assumptions allow us to calculate the impact of competition and communication costs using completely standard arguments from the mechanism design literature. We are able to show that with these assumptions the seller's expected surplus will be highest if the object is sold according to the following modified fixed price scheme: the seller contacts each of the potential buyers in turn and either offers to negotiate or announces that he no longer wishes to trade. If he offers to negotiate and the buyer agrees, the buyer immediately has the option of trading for sure with the seller at a fixed price set ex ante. If the buyer does not wish to pay this fixed price, he may submit an alternative bid. The seller will then continue to contact new buyers, returning to trade with the buyer only if no buyer wishes to pay the fixed price and no higher bid is submitted.It will be clear that in our environment, both the simple fixed price scheme and the simple auction are feasible. The simple auction prevails when the fixed price is set equal to the maximum possible valuation, while the simple fixed price scheme occurs when the fixed price is set so that buyers are willing to participate if and only if they are willing to pay the fixed price. Our results will show that a simple auction in never optimal for the seller. The seller can always strictly improve his payoff by moving to a scheme in which there is some strictly positive probability that trade will occur at the fixed price. On the other hand, there are reasonable circumstances in which the seller cannot achieve a higher payoff than the one she gets by selling at a fixed price. It is shown that for any positive participation cost, there is a large, but finite, number of potential buyers so that the seller cannot achieve a higher payoff than what she gets by selling at a fixed price. Two simple, but important continuity results are also illustrated. As the cost of participation in the mechanism increases (decreases), the probability with which the seller's unit of output is sold at a fixed price goes to one (zero) in the best modified fixed price mechanism for the seller.Our paper is not the first to generate such a modified fixed price scheme. Both McAfee and McMillan (1988) and Riley and Zeckhauser (1983) come up with similar schemes for the case in which the seller must bear a fixed cost for each new buyer that she contacts. There are two essential differences between our model and theirs. First, as the cost is interpreted as the opportunity cost of participation in the mechanism, it is reasonable to imagine that the seller advertises the mechanism ex ante. Another way of putting this is that the seller pays a fixed rather than a variable cost to communicate the mechanism to buyers. This makes it possible to assume that the mechanism is common knowledge to the seller and all the buyers at the beginning of the communication process. For this reason we can make our case using completely standard arguments. Secondly, the mechanism in the opportunity cost case plays a different allocative role than it does in the case when the seller bears a cost. The mechanism must decide whether buyers should communicate with the seller or pursue their alternative activities, as well as who should trade and at what price. It is this allocative role that makes bilateral communication superior to multilateral communication in a competitive environment. These differences allow us to show, for example, that a simple fixed price scheme is undominated for the seller when the number of buyers is finite. As shown by McAfee and McMillan, this is only possible when the number of potential buyers is infinite when the seller bears the cost of communication.Remarkably, the existence of opportunity costs to buyer participation is not, by itself, sufficient to explain why sellers might prefer bilateral communications mechanisms. Samuelson (1983) and McAfee and McMillan (1987) show that when buyers must pay a fixed cost to submit a bid, which is equivalent to giving up a valuable alternative, a seller cannot expect to earn more than she does in a second price auction (though Samuelson shows that the reserve price may depend on the number of potential buyers). One of the contributions of this paper is to show that the assumption that buyers make their participation decisions simultaneously is critical to this result. Simultaneous entry decisions means that whether or not any particular buyer is assigned to the alternative activity is independent of any other buyer's valuation. With sequential communication the seller is able to relax this constraint. It is precisely the enlargement of the class of feasible mechanism that breaks down the optimality of the simple auction.The second author acknowledges the support of the Social Sciences and Humanities Research Council of Canada and the CRDE at the Université de Montreal.