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.     

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.     

Price and information disclosure in the private art market: A signalling game

In this paper, we model private art market agents’ strategic interactions in presence of two types of asymmetric information, about artwork quality and buyer’s knowledge, assuming the seller does not know how informed is the buyer while the buyer does not know the quality of the artwork before purchase. If the seller can choose either a high or a low price and the buyer can signal his type to the seller, we identify the conditions for both equilibria with pooling buyer signalling strategy and with separating strategy, as well as conditions for equilibria where the seller fixes the price according to the actual quality and where he posts prices trying to take advantage of buyer’s limited information. Finally, we identify the condition for the emergence of a “counter-lemon” result, where low-quality artworks and uninformed collectors exit the market, suggesting that seller uncertainty does not directly benefit the buyers, but it can impact the quality traded in the market.     

Optimal Information Disclosure in Auctions and the Handicap Auction

We analyse a situation where a monopolist is selling an indivisible good to risk-neutral buyers who only have an estimate of their private valuations. The seller can release, without observing, certain additional signals that affect the buyers' valuations. Our main result is that in the expected revenue-maximizing mechanism, the seller makes available all the information that she can, and her expected revenue is the same as it would be if she could observe the part of the information that is "new" to the buyers. We also show that this mechanism can be implemented by what we call a handicap auction in interesting applications. In the first round of this auction, each buyer picks a price premium from a menu offered by the seller (a smaller premium costs more). Then the seller releases the additional signals. In the second round, the buyers bid in a second-price auction where the winner pays the sum of his premium and the second highest non-negative bid. In the case of a single buyer, this mechanism simplifies to a menu of European call options.     

SEARCH,ADVERSE SELECTION,AND MARKET CLEARING

We examine a dynamic decentralized trading model with infinitesimal sellers and buyers to investigate whether or not the market fails to clear in the limit of search friction vanishing. A seller, who has private information about product quality, and a buyer are matched to bargain over price. They form a long‐term relationship if they reach agreement. They return to the matching pool if they fail to agree or the existing relationship is dissolved. The market fails to clear if and only if the ratio of agents' patience over the dissolution rate exceeds a threshold.     

SELLING A LEMON UNDER DEMAND UNCERTAINTY

We consider a seller who has private information about the quality of her good but is uncertain about buyer arrivals. Assuming that the high‐quality seller insists on a price, we show that the low‐quality seller's surplus and pricing strategy crucially depend on buyers' knowledge about the demand state. If they are also uncertain about demand, then demand uncertainty increases the low‐quality seller's expected payoff, and her optimal strategy is to lower the price after some time. If buyers know the demand state, then demand uncertainty does not affect the low‐quality seller's payoff, but she must employ a sophisticated pricing strategy.     

Monopoly pricing in the binary herding model

How should a monopolist price when selling to buyers who learn from each other’s decisions? Focusing on the case in which the common value of the good is binary and each buyer receives a binary private signal about that value, we completely answer this question for all values of the production cost, the precision of the buyers’ signals, and the seller’s discount factor. Unexpectedly, we find that there is a region of parameters for which learning stops at intermediate and at extreme beliefs, but not at beliefs that lie between those intermediate and extreme beliefs.     

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.     

An optimal auction with correlated values and risk aversion

We consider an auction setting where the buyers are risk averse with correlated private valuations (CARA preferences, binary types), and characterize the optimal mechanism for a risk-neutral seller. We show that the optimal auction extracts all buyer surplus whenever the correlation is sufficiently strong (greater than 1/3 in absolute value), no matter how risk averse the buyers are. In contrast, we note that a sufficiently risk-averse seller would not use a full rent extracting mechanism for any positive correlation of the valuations even if the buyers were risk neutral.     

Buyer groups as strategic commitments

Buyer cooperatives, buyer alliances, and horizontal mergers are often perceived as attempts to increase buyer power. In contrast to prior research emphasizing group size, I show that even small buyer groups composed of buyers with heterogeneous preferences can increase price competition among rival sellers by committing to purchase exclusively from one seller. Without transfer payments, at least one buyer group exists for each pair of sellers and buyer groups membership is chosen to achieve indifference between the two sellers. With transfer payments, and just two sellers, the grand coalition is a coalition-proof subgame perfect equilibrium (CP-SPNE), though equilibria with arbitrarily many buyer groups also exist. With three sellers (and with more sellers when the distribution of buyers is symmetric), a CP-SPNE always exists, all coalition-proof equilibria are payoff equivalent and have at least one buyer group for each pair of firms, so the grand coalition is not an equilibrium.     

A Walrasian Rubinstein and Wolinsky model

We provide a full dynamic analysis of a continuous-time variant of Rubinstein and Wolinsky (1985) matching and bargaining model with unbalanced flows of buyers and sellers. The focus is on the price limit as the frictions of search are removed. It is found that a necessary and sufficient condition for the limit price to be Walrasian at all times is the alignment of the initial buyer and seller stocks with the flows.     

A choice of auction format in seller cheating: a signaling game analysis

In this paper, we examine which auction format, first-price or second-price, a seller will choose when he can profitably cheat in a second price auction by observing all bids by possible buyers and submitting a shill bid as pretending to be a buyer. We model this choice of auction format in seller cheating as a signaling game in which the buyers may regard the selection of a second price auction by the seller as a signal that he is a shill bidder. By introducing trembling-hand perfectness as a refinement of signaling equilibrium, we find two possible strictly perfect signaling equilibria. One is a separating equilibrium in which a noncheating honest seller selects a first price auction and a cheating seller does a second price auction. In another pooling equilibrium, however, both cheating and non-cheating sellers select a second price auction. The conclusion that a seller chooses a second price auction even if he cannot cheat is in contrast to the previous literature, which focused on the case of independent values. We thank an anonymous referee for useful comments that have improved the paper. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 15310023 and (C) 18530139.     

Price formation in a matching market with targeted offers

We model a market where the surpluses from seller–buyer matches are heterogeneous but common knowledge. Price setting is synchronous with search: buyers simultaneously make one personalized offer each to the seller of their choice. With impatient players efficient coordination is not possible, and both temporary and permanent mismatches occur. Nonetheless, for patient players efficient matching (with monopsony wages) is an equilibrium. The setting is inspired by a labor market for highly skilled workers, such as the academic job market, but it can be easily adapted to, for example, the housing market or Internet advertising auctions.     

How private is private information? The ability to spot deception in an economic game

We provide experimental evidence on the ability to detect deceit in a buyer–seller game with asymmetric information. Sellers have private information about the value of a good and sometimes have incentives to mislead buyers. We examine if buyers can spot deception in face-to-face encounters. We vary whether buyers can interrogate the seller and the contextual richness. The buyers’ prediction accuracy is above chance, and is substantial for confident buyers. There is no evidence that the option to interrogate is important and only weak support that contextual richness matters. These results show that the information asymmetry is partly eliminated by people’s ability to spot deception.     

Asymmetric information in bilateral trade and in markets: An inversion result

I consider bilateral trade between a seller and a buyer with private valuations. The seller makes a take-it-or-leave-it price offer. If the seller observes the buyer?s valuation (symmetric information), bilateral trade is trivially efficient. If the seller cannot observe the valuation (asymmetric information), bilateral trade is inefficient. This bilateral trading game is embedded into a large matching market. In the steady-state equilibrium of the market game, the relation between the informational regime and efficiency is inverted: With small frictions efficiency obtains if information is asymmetric. If information is symmetric, however, the trading outcome can be very inefficient—even if frictions vanish.     

Prices for cash and cash for prices? Theory and evidence on convenient pricing

A transaction between a seller and a buyer incurs a payment cost. The payment cost is borne by the seller, depending on the payment instrument the buyer chooses, cash or card. Card payment is more costly than cash payment, so the seller prefers that the buyer pays cash. In this article, we study the strategy of the seller setting a convenient price, which simplifies transactions and pushes the buyer to pay cash. The theoretical analysis, which models both the seller and the buyer in a game setting, derives two propositions: (1) the seller is more likely to set a more convenient price and (2) the buyer is more likely to pay cash a more convenient price. The empirical analysis supports both propositions. Thus, sellers adopt a convenience pricing strategy – prices for cash – and this strategy pushes buyers to pay cash – cash for prices.     

Two-part tariffs set by a risk-averse monopolist

This paper revisits the classical issues of two-part tariffs by considering risk aversion of a monopolistic seller. Under demand uncertainty, equilibrium unit price declines and approaches towards marginal cost as the seller becomes more risk averse. Marginal-cost pricing prevails, irrespective of the seller’s risk attitude, if clients are homogenous. Under cost uncertainty, unit price is higher than marginal cost and monotonically increases in risk aversion. The model is then extended to accommodate buyers’ risk aversion and it is found that demand uncertainty makes unit price decline in the seller’s risk aversion again but increase in buyers’ risk aversion.     

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.     

Monopolistic screening and uninformed buyers

In the standard monopolistic screening problem, buyers obtain information rent as a result of possessing private information; if a contract can be offered before the buyer knows his valuation, the seller can extract the full (expected) surplus. I consider a situation where the buyer may or may not have private information about his valuation at the time the contract is offered. Is the seller (strictly) better off as compared to the standard situation? The answer depends crucially on the specific model. In the 2-types model, unless the probability (that the buyer is uninformed) reaches a critical threshold, the seller is unable to benefit from the buyer's ignorance. In the continuum-types model, on the other hand, optimal expected profit is strictly higher than in the standard model whenever this probability is positive.     

DIRECTED SEARCH,COORDINATION FAILURE,AND SELLER PROFITS: AN EXPERIMENTAL COMPARISON OF POSTED PRICING WITH SINGLE AND MULTIPLE PRICES

We experimentally examine posted pricing and directed search. In one treatment, capacity‐constrained sellers post fixed prices, which buyers observe before choosing whom to visit. In the other, firms post both “single‐buyer” (applied when one buyer visits) and “multibuyer” (when multiple buyers visit) prices. We find, based on a 2 × 2 (two buyers and two sellers) market and a follow‐up experiment with 3 and 2 × 3 markets, that multibuyer prices can be lower than single‐buyer prices or prices in the one‐price treatment. Also, allowing the multibuyer price does not affect seller profits and increases market frictions.