Single-crossing Property Research Articles

Sorting out single-crossing preferences on networks

We extend the single-crossing property to tree networks to facilitate its application in network games. It is equivalent to intermediate preferences and order restriction (also extended to networks). Moreover, to facilitate broader applications in real world cases and simulations, we develop algorithms that answer the following two questions. Given a preference profile, can we construct a tree graph that supports single-crossing? Given a set of alternatives, can we generate single-crossing preference profiles with associated tree graphs?

Nonexclusive competition under adverse selection

Consider a seller of a divisible good, facing several identical buyers. The quality of the good may be low or high, and is the seller’s private information. The seller has strictly convex preferences that satisfy a single-crossing property. Buyers compete by posting arbitrary menus of contracts. Competition is non-exclusive in that the seller can simultaneously and secretly trade with several buyers. We fully characterize conditions for the existence of an equilibrium. Equilibrium aggregate allocations are unique. Any traded contract must yield zero proflt. If a quality is indeed traded, then it is traded e‐ciently. Depending on parameters, both qualities may be traded, or only one of them, or the market may break down completely to a no-trade equilibrium.

Surplus Maximization and Optimality

Expected consumer's surplus rarely represents preferences over price lotteries. Still, I give sufficient conditions for policies which maximize aggregate expected surplus to be interim Pareto Optimal. Besides two standard partial equilibrium conditions, I assume that feasible prices satisfy a single-crossing property; and each consumer's indirect utility satisfies increasing differences in the price and income. I use the result to extend well-known welfare conclusions beyond the knife-edge quasilinear utility case. Since increasing differences puts no upper bound on risk aversion, the result is useful for applications in which risk aversion is important. (JEL D11, D24, D42, D81, D83, L42)

Multi-task agency with unawareness

The paper introduces the problem of unawareness into multi-dimensional Principal–Agent theory. We introduce two key parameters to describe the problem, the extent and the effect of unawareness, show under what conditions it is optimal for the Principal to propose an incomplete or a complete contract, and characterize the incentive power of optimal linear contracts. If Agents differ in their unawareness, optimal incentive schemes can be distorted for both aware and unaware Agents, because, different from standard contract theory, the single-crossing property fails to hold. In this case, even aware Agents can be subject to inefficiently high or low incentives.

Sorting Out Single-Crossing Preferences

We extends the single-crossing property of preferences to tree graphs, and show that it is equivalent to intermediate preferences and order restriction, also extended to tree graphs. Moreover, to facilitate the use of single-crossing in network games, we develop algorithms to answer the following questions. Given a preference profile, can we construct a tree graph that supports single-crossing, if it is? Given a set of alternatives, can we generate single-crossing preference profiles with associated tree graphs?

Complexity of solution structures in nonlinear pricing

This paper characterizes and enumerates the possible solution structures in nonlinear pricing problem when the number of buyer types is given. It is shown that the single-crossing property, which is a standard assumption in the literature, reduces the complexity of solving the problem dramatically. The number of possible solution structures is important when the pricing problem is solved under limited information.

Majority voting and the single-crossing property when voters belong to separate groups

We provide conditions under which a Condorcet winner exists when voters are exogenously distributed in groups, with preferences satisfying the single-crossing property separately inside each group. We also show that the majority voting social preference is acyclic.

Conditions for incentive compatibility in models with multidimensional allocation functions and one-dimensional types

We extend the conditions for incentive compatibility in mechanism design problems to a more general structure of preferences than that found in the literature, for the case where type is one-dimensional but the outcome function becomes multidimensional. This is so, at least, as long as preferences can be represented by means of sub-utility functions, it is adopted a weak single-crossing property and direct mechanisms turn out to be differentiable. When direct mechanisms are not differentiable, local incentive conditions still remain fully incentive compatible, provided utility is weakly separable in the outcome function, or else, it exhibits linearity in the type.

Multi-Task Agency with Unawareness

The paper introduces the problem of unawareness into multi-dimensional Principal-Agent theory. We introduce two key parameters to describe the problem, the extent and the e ffect of unawareness, show under what conditions it is optimal for the Principal to propose an incomplete or a complete contract, and characterize the incentive power of optimal linear contracts. If Agents differ in their unawareness, optimal incentive schemes can be distorted for both aware and unaware Agents, because, di fferent from standard contract theory, the Single-Crossing Property fails to hold. In this case, even aware Agents can be subject to ineciently high or low incentives.

Pre-sale information

In markets as diverse as that for specialized industrial equipment or that for retail financial services, sellers or intermediaries may earn profits both from the sale of products and from the provision of pre-sale consultation services. We study how a seller optimally chooses the costly quality of pre-sale information, next to the price of information and the product price, and obtain clear-cut predictions on when information is over- and when it is underprovided, even though we find that information quality does not satisfy a standard single-crossing property. Buyers who are a priori more optimistic about their valuation end up paying a higher margin for information but a lower margin for the product when they subsequently exercise their option to purchase at a pre-specified price.

Complementarity with complete but P-acyclic preferences

This paper extends the formulation of complementarity in Milgrom and Shannon (1994) to the case of complete but P -acyclic preferences. In such a case, quasi-supermodularity and the single-crossing property on their own do not guarantee monotone comparative statics or equilibrium existence: an additional condition, monotone closure, is required.

Non-Exclusive Competition Under Adverse Selection

Consider a seller of a divisible good, facing several identical buyers. The quality of the good may be low or high, and is the seller's private information. The seller has strictly convex preferences that satisfy a single-crossing property. Buyers compete by posting arbitrary menus of contracts. Competition is non-exclusive in that the seller can simultaneously and secretly trade with several buyers. We fully characterize conditions for the existence of an equilibrium. Equilibrium aggregate allocations are unique. Any traded contract must yield zero pro t. If a quality is indeed traded, then it is traded efficiently. Depending on parameters, both qualities may be traded, or only one of them, or the market may break down completely to a no-trade equilibrium.

On the Existence of Monotone Pure-Strategy Equilibria in Bayesian Games

We extend and strengthen both Athey’s (2001) and McAdams’(2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilatttices and can handle both a weaker form of quasisupermodularity than is employed by McAdams and a weaker single-crossing property than is required by both Athey and McAdams. Our proof — which is based upon contractibility rather than convexity of best reply sets — demonstrates that the only role of single-crossing is to help ensure the existence of monotone best replies. Finally, we do not require the Milgrom-Weber (1985) absolute continuity condition on the joint distribution of types.

Do dividend changes signal future earnings?

Abstract Signaling models contributed to the corporate finance literature by formalizing “the informational content of dividends” hypothesis. However, these models are under criticism as the empirical literature found weak evidences supporting a central prediction: the positive relationship between changes in dividends and changes in earnings. We claim that the failure to verify this prediction does not invalidate the signaling approach. The models developed up to now assume or derive utility functions with the single-crossing property. We show that, in the absence of this property, signaling is possible, and changes in dividends and changes in earnings can be positively or negatively related.

Asymmetric common-value auctions with applications to private-value auctions with resale

We study a model of common-value auctions with two bidders in which bidders’ private information are independently and asymmetrically distributed. We present sufficient and necessary conditions, respectively, under which the expected revenues from first-price and second-price auctions can be ranked. Using these conditions and a bid-equivalence between common-value auctions and private-value auctions with resale, we extend the revenue-ranking result of Hafalir and Krishna [Am Econ Rev 98, 2008a] and provide necessary conditions for their ranking to hold. In addition, we provide sufficient and necessary conditions for the opposite ranking of revenues, respectively. Our analysis helps clarify the roles of two forms of regularity assumptions (buyer-regularity and seller-regularity) in ranking revenues and illustrates how revenue ranking is linked to submodularity and supermodularity of the common-value function and to a single-crossing property of a function derived from the monopoly or monopsony pricing function in the resale stage.

Contests with a stochastic number of players

We study Tullock's (1980) n-player contest when each player has an independent probability 0 < p ⩽ 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n > 2 individual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed.

Communication with Two-Sided Asymmetric Information

Even though people routinely ask experts for advice, they often have private information as well. I study strategic communication when both the expert and the decision maker have private information. In one-way communication, non-monotone equilibria may arise (i.e., the expert conveys whether the state is extreme or moderate instead of low or high), even if preferences satisfy the single-crossing property. In two-way communication, the decision maker cannot credibly reveal her information when communicating first to the expert and hence benefits little from two-way sequential communication. This result provides another explanation for the bottom up arrangement of information flow in organizations.

Dynamic pricing of inventory/capacity with infrequent price changes

We consider a problem of dynamically pricing a single product sold by a monopolist over a short time period. If demand characteristics change throughout the period, it becomes attractive for the company to adjust price continuously to respond to such changes (i.e., price-discriminate intertemporally). However, in practice there is typically a limit on the number of times the price can be adjusted due to the high costs associated with frequent price changes. If that is the case, instead of a continuous pricing rule the company might want to establish a piece-wise constant pricing policy in order to limit the number of price adjustments. Such a pricing policy, which involves optimal choice of prices and timing of price changes, is the focus of this paper. We analyze the pricing problem with a limited number of price changes in a dynamic, deterministic environment in which demand depends on the current price and time, and there is a capacity/inventory constraint that may be set optimally ahead of the selling season. The arrival rate can evolve in time arbitrarily, allowing us to model situations in which prices decrease, increase, or neither. We consider several plausible scenarios where pricing and/or timing of price changes are endogenized. Various notions of complementarity (single-crossing property, supermodularity and total positivity) are explored to derive structural results: conditions sufficient for the uniqueness of the solution and the monotonicity of prices throughout the sales period. Furthermore, we characterize the impact of the capacity constraint on the optimal prices and the timing of price changes and provide several other comparative statics results. Additional insights are obtained directly from the solutions of various special cases.

Contracting with imperfect commitment and noisy communication

This paper provides new analytical tools for studying principal-agent problems with adverse selection and limited commitment. By allowing the principal to use general communication devices we overcome the literature's common, but overly restrictive focus on one-shot, direct communication. In addition, general communication devices solve two fundamental problems of contracting with imperfect commitment: First, they allow us to identify the ‘local downward’ incentive constraints as the relevant ones if the agent's preferences satisfy a single-crossing property. Second, we show how one may restrict the cardinality of the message spaces of the communication device. An example illustrates our arguments and the suboptimality of one-shot, direct communication.

Endogenous Public Expenditures on Education

We construct a model of the determination of public funding of education through majority voting. Households have the option of privately supplementing public education. Alternatively, they can opt out of public education completely and choose private education. We find that in general the single-crossing property cannot be used to establish existence of a majority voting equilibrium. Numerical solutions of the model reveal (i) when public education inputs and private supplements are substitutes, private school enrollment is often zero; and (ii) the funding level for public education is very sensitive to the productivity of private supplements and the elasticity of substitution between public inputs and private supplements.