Why a Hard Problem Can Make an Old Idea Worth Revisiting

An approach that seemed exhausted can become worth another look after a different approach also fails. In Solving Problems of Unknown Difficulty, Nicholas Wu develops a mathematical explanation: unsuccessful attempts teach a solver about both individual ideas and the difficulty shared by the whole problem.

The November 3, 2026 version of this preprint presents theoretical results, not experimental evidence about how researchers or entrepreneurs actually behave. Its central finding is conditional on the model: revisiting earlier approaches can be part of an optimal strategy rather than evidence of indecision.

Two explanations for no breakthrough

The baseline model gives a solver a fixed amount of effort to allocate at each moment. Generating a new approach requires paying a cost. Each approach is either valid or flawed, but the solver cannot observe which. A flawed approach never succeeds; a valid one can produce a breakthrough unpredictably as effort accumulates.

The problem itself is either easy or hard, also unobserved. Valid approaches produce breakthroughs faster on an easy problem than on a hard one. The solver values earlier success more than later success and chooses a strategy to maximize the expected discounted reward after brainstorming costs.

This creates two interpretations of an unsuccessful attempt. Perhaps the approach is flawed, making a switch attractive. Alternatively, perhaps the problem is hard, making further work on existing approaches more defensible.

Wu first analyzes a benchmark where difficulty is known. With sufficiently low brainstorming costs, the optimal strategy is sequential: generate an approach, work on it until a fixed effort threshold, then move to a new one. Along this path, abandoned approaches are never revisited.

Why another failure changes an old idea

Unknown difficulty changes the information structure. Although approaches’ validity is independent at the outset, their prospects share an unknown factor: the speed at which valid approaches can deliver success. Work on one approach therefore changes assessments of others.

The optimal search pattern starts with concentrated effort on a new approach. If it does not succeed, the solver eventually generates another. Once the newer approach has received as much effort as the older ones, the solver splits effort equally among them before expanding the search again. The effort threshold for introducing another approach increases as more approaches are generated.

The intuition is subtle. Failure on the newer approach can make the solver more convinced that the problem is hard. That can increase the assessed probability that an older approach was valid after all: its disappointing performance becomes less diagnostic of a bad idea.

This does not mean the old approach’s expected breakthrough rate increases. The paper explicitly distinguishes a higher belief in validity from a higher expected success rate. Returning to old approaches becomes worthwhile relative to the alternative of paying for another new one.

The model also distinguishes difficulty from impossibility. If valid approaches retain a positive success rate even in the hard state, the optimal policy keeps generating approaches along a history without success. If the hard state makes success impossible, the solver eventually stops generating new approaches, while continuing to work on existing ones because that work is free in the model.

Incentives concern how to work

To analyze contracts, Wu develops a smoother model with a continuum of approaches, characterized by search breadth and depth. He establishes a mathematical limiting connection to the discrete model under specified parameter scaling.

An investor then offers an entrepreneur a share of the value of success. The entrepreneur privately bears the cost of broadening search, and the investor cannot observe precisely what the entrepreneur works on. The disagreement concerns how to work, not whether to work.

Under a fixed success share, the entrepreneur searches less broadly than a planner maximizing total surplus. A larger share encourages broader search but leaves less of the breakthrough’s value for the investor.

Time-varying contracts introduce opposing forces. With known difficulty, the characterized optimal interior contract offers a declining success share. Early exploration has persistent benefits, giving an investor able to commit to future terms a reason to reward early success more generously.

Learning can push the other way. As unsuccessful work makes the problem appear harder, larger later rewards can encourage continued exploration. The paper illustrates contracts whose rewards fall, rise, and then fall again.

Where the conclusions stop

The dynamic-contract optimality result is restricted to contracts inducing admissible interior search responses; it is not a characterization of every possible contract. Deadlines, for example, fall outside that optimization class.

The baseline also assumes initially identical approaches and a fixed underlying difficulty. Heterogeneous approaches complicate the search state, and difficulty that changes with the solver’s actions remains a proposed extension. The preprint therefore offers a conditional explanation of why revisiting ideas and tolerating early failure can make economic sense—not a universal prescription for managing innovation.