Monotone Allocations without Single-Crossing: When to Bunch and When to Jump

In the world of economics, there is a classic puzzle about how a seller can get the most value from a buyer when the seller does not know exactly what the buyer wants. Imagine a government trying to buy electricity from a power company, or a regulator setting prices for a utility. The buyer knows their own costs and capabilities, but the seller does not. To get the best deal, the seller must design a menu of options that encourages the buyer to reveal their true nature. For decades, economists believed that if the buyer's preferences followed a simple, consistent rule—where a more capable buyer always values an extra unit of service more than a less capable one—the solution was straightforward. The seller would simply offer a smooth, continuous menu where better types get more service, and the only complication would be to slightly reduce the service for the less capable types to save money. This smoothness was considered a fundamental law of the trade.

However, real-world technology often breaks this simple rule. Sometimes, a technology has a "minimum efficient scale," a specific size where it becomes most efficient. Below this size, a highly efficient firm might actually value an extra unit of output less than a less efficient firm because it is not yet big enough to use it well. Above this size, the ranking flips, and the efficient firm suddenly values the extra unit much more. This creates a situation where the standard rules of the game fail. The relationship between a buyer's type and their desire for more service is no longer a straight line; it twists and turns. When this happens, the old assumption that the best contract must be smooth and continuous collapses. The question then becomes: what does the best contract look like when the rules of the game are broken?

A team of researchers has now mapped out exactly how to solve this problem, revealing that the answer depends entirely on how the "twist" in the buyer's preferences interacts with the seller's ideal plan. They found that there are only three possible outcomes, determined by a single geometric fact: whether the seller's ideal plan ever crosses the line where the buyer's preferences flip. If the ideal plan never crosses this line, the best contract is smooth and continuous, just like in the old theory, but with a twist: it groups the least capable buyers together into a single pool, giving them the exact same service level, before separating the rest. This bunching happens not because the ideal plan is messy, but because the value of information evaporates as buyers get closer to the point where their preferences flip.

If the ideal plan crosses a flat line where the preference flip happens, the smooth contract is no longer just suboptimal; it is strictly worse than a contract that jumps. In this scenario, the best deal involves a sudden, discontinuous leap in the amount of service offered. There is a specific type of buyer for whom the seller offers a whole range of quantities, from a low amount to a high amount, and the buyer is indifferent between any of them. The seller effectively skips a whole middle ground of service levels, offering only a small package and a large package, with nothing in between. This jump is not a choice; it is forced by the geometry of the problem. The researchers proved that any attempt to smooth out this jump would strictly reduce the seller's profit.

The third and most complex scenario occurs when the ideal plan crosses a line that is itself rising or falling. Here, the jump is not forced, but it is also not forbidden. The seller has a choice. They can stick with a smooth contract, or they can introduce a jump. If they choose to jump, the contract follows a very specific path: it pools the low types, jumps to a new level, and then follows a "mirror" path that reflects the initial pool level across the preference-flip line, until the seller's ideal plan eventually overtakes it. The researchers showed that this mirror path is the only way to maintain the necessary incentives after the jump. Whether the seller chooses the jump or the smooth path depends on a precise calculation of profits, which they solved in closed form for a wide range of economic situations.

The most significant achievement of this work is not just finding these shapes, but proving that they are truly the best possible solutions among all conceivable contracts, including random ones. In many complex economic problems, it is difficult to be certain that a proposed solution is the absolute best, because there are so many ways to mix and match options. The researchers developed a new method to certify their solutions. They constructed a mathematical "weight" that acts like a certificate of optimality. This weight proves that no other contract, whether smooth, jagged, or random, can beat the one they found. They applied this method to forty different geometric configurations of the problem, covering every possible way the curves could interact. In every case, they confirmed that their proposed contract was the global optimum.

One surprising finding is that random contracts, or lotteries, do not help the seller in any of these scenarios. Even though the problem involves non-standard preferences, the best strategy is always a fixed, deterministic menu. A lottery, where a buyer might get a small package or a large package with some probability, is strictly worse than a fixed menu that offers the average of those two options. This holds true even at the point where the contract jumps; the seller does not need to offer a coin flip to the marginal buyer. The jump itself is sufficient to satisfy the incentives. The researchers also showed that in some cases, the best contract involves excluding the least capable buyers entirely, while in others, the most capable buyers receive the perfect, undistorted service.

The paper also revisited a famous numerical example from previous literature that had been left unresolved. By applying their new method, the researchers were able to identify the true optimal contract for that problem, correcting a long-standing error in the field. They found that the previously proposed solution was not the best, and their new contract, which they could prove was optimal, earned a slightly higher profit. This demonstrates the power of their approach: it does not rely on guessing or simulation, but on a rigorous proof that works even when the numbers are messy and the curves are nonlinear.

Ultimately, this work provides a complete taxonomy for a class of problems that had been considered too difficult to solve generally. It shows that when the standard rules of economics break down, the solution is not chaos, but a structured set of possibilities. The contract will either be smooth, it will jump, or it will follow a mirror path. The specific outcome is determined by the shape of the technology and the distribution of buyer types. The researchers have provided the tools to calculate exactly which of these three paths is the right one for any given situation, turning a previously intractable problem into a solvable one with clear, certifiable answers.