arXiv 13 Feb 2026 · Econometrics
arXiv:2602.13450 · PDF · DOI · OpenAlex · Extracted main text
Algorithms for computing equilibria, optima, and fixed points in nonconvex problems often depend sensitively on practitioner-chosen initial conditions. When uniqueness of a solution is of interest, a common heuristic is to run such algorithms from many randomly selected initial conditions and to interpret repeated convergence to the same output as evidence of a unique solution or a dominant basin of attraction. Despite its widespread use, this practice lacks a formal inferential foundation. We provide a simple probabilistic framework for interpreting such numerical evidence. First, we give sufficient conditions under which an algorithm's terminal output is a measurable function of its initial condition, allowing probabilistic reasoning over outcomes. Second, we provide sufficient conditions ensuring that an algorithm admits only finitely many possible terminal outcomes. While these conditions may be difficult to verify on a case-by-case basis, we give simple sufficient conditions for broad classes of problems under which almost all instances admit only finitely many outcomes (in the sense of prevalence). Standard algorithms such as gradient descent and damped fixed-point iteration applied to sufficiently smooth functions satisfy these conditions. Within this framework, repeated solver runs correspond to independent samples from the induced distribution over outcomes. We adopt a Bayesian approach to infer basin sizes and the probability of solution uniqueness from repeated identical outputs, and we establish convergence rates for the resulting posterior beliefs. Finally, we apply our framework to settings in the existing industrial organization literature, where random-restart heuristics are used. Our results formalize and qualify these arguments, clarifying when repeated convergence provides meaningful evidence for uniqueness and when it does not.
appendix boundary found by appendix_command · 81% of the source is main text. Read the extracted text to check this.
The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Berry, Steven and Levinsohn, James and Pakes, Ariel (1995) Automobile Prices in Market Equilibrium | 0.874 | 5 | 2 | 100% |
| 2 | Pakes, Ariel and McGuire, Paul (1994) Computing Markov-Perfect Nash Equilibria: Numerical Implications of a Dynamic Differentiated Product Model | 0.644 | 2 | 2 | 100% |
| 3 | Besanko, David and Doraszelski, Ulrich and Kryukov, Yaroslav and Sat… (2010) Learning-by-doing, organizational forgetting, and industry dynamics | 0.644 | 2 | 2 | 100% |
| 4 | Conlon, Christopher and Gortmaker, Jeff (2020) Best practices for differentiated products demand estimation with pyblp | 0.585 | 3 | 1 | 100% |
| 5 | Crawford, Gregory S and Yurukoglu, Ali (2012) The Welfare Effects of Bundling in Multichannel Television Markets | 0.511 | 2 | 1 | 100% |
| 6 | Morrow, W Ross and Skerlos, Steven J (2011) Fixed-point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand | 0.511 | 2 | 1 | 100% |
| 7 | Morrow, W Ross and Skerlos, Steven J (2010) Fixed-point Approaches to Computing Bertrand-Nash Equilibrium Prices under Mixed Logit Demand: A Technical Framework for Analysi… | 0.511 | 2 | 1 | 100% |
| 8 | Boender, C. Guus E. and Romeijn, H. Edwin (1995) Stochastic Methods | 0.405 | 1 | 1 | 100% |
| 9 | Bajari, Patrick and Benkard, C Lanier and Levin, Jonathan (2007) Estimating Dynamic Models of Imperfect Competition | 0.405 | 1 | 1 | 100% |
| 10 | Bhojanapalli, Srinadh and Neyshabur, Behnam and Srebro, Nati (2016) Global optimality of local search for low rank matrix recovery | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 29 scored citations.