Bibliography

Selection in branching systems

The closest-looking relatives. In branching Brownian motion with selection, particles branch and a rule retains the rightmost ones; the population size is exogenous. In a budgeted Brownian race nothing branches, the survivor count is endogenous, and the resource is cumulative path-time, not a headcount.

• Brunet, E., and Derrida, B. (1997). “Shift in the velocity of a front due to a cutoff.” Physical Review E 56(3), 2597–2604. The cutoff moves the front; the first quantitative account of what truncating a population does to its leading edge.
• Berestycki, J., Berestycki, N., and Schweinsberg, J. (2013). “The genealogy of branching Brownian motion with absorption.” Annals of Probability 41(2), 527–618. Absorption at a moving barrier as a selection mechanism, worked out in full.
• Maillard, P. (2016). “Speed and fluctuations of N-particle branching Brownian motion with spatial selection.” Probability Theory and Related Fields 166, 1061–1173. The N-BBM benchmark: keep the N rightmost. The fixed-N constraint is exactly what the path-time budget relaxes.

Budgets as fuel

The classification home for the constraint. Finite-fuel control bounds a cumulative control effort; here the fuel is aggregate survivor-time and the control is irreversible killing.

• Beneš, V. E., Shepp, L. A., and Witsenhausen, H. S. (1980). “Some solvable stochastic control problems.” Stochastics 4(1), 39–83. The original finite-fuel problems, solved in closed form. A useful subtitle for the race, not its name.

Search, indices, and optionality

The economics of paying to keep options alive. The race is a continuous-time cousin: each surviving path is a box still open, and the pivotal value prices the option of keeping it.

• Weitzman, M. L. (1979). “Optimal search for the best alternative.” Econometrica 47(3), 641–654. Pandora's rule: open boxes in reservation-value order. The one-inspection-per-box version of the pruning decision.
• Gittins, J. C. (1979). “Bandit processes and dynamic allocation indices.” Journal of the Royal Statistical Society, Series B 41(2), 148–177. The index that decides which arm deserves the next unit of attention; the reason a race of continued-or-killed paths is a bandit at heart, and the name on the repo.
• Audibert, J.-Y., Bubeck, S., and Munos, R. (2010). “Best arm identification in multi-armed bandits.” COLT 2010. Pure exploration under a hard budget of pulls: the discrete, statistical face of the same keep-or-kill question.

Mean-field control

The machinery the solver discretizes: a value function coupled to a density evolution through a shadow price.

• Lasry, J.-M., and Lions, P.-L. (2007). “Mean field games.” Japanese Journal of Mathematics 2(1), 229–260. The coupled HJB/Fokker–Planck template. The race's version swaps the HJB for an obstacle problem, since the only control is stopping.
• Carmona, R., and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications, vols. I–II. Springer. The reference treatment, including control of McKean–Vlasov dynamics, which is the honest label for the population problem here.

The algorithmic tools

• Frank, M., and Wolfe, P. (1956). “An algorithm for quadratic programming.” Naval Research Logistics Quarterly 3, 95–110. The outer loop: linearize the concave objective, call the obstacle problem as the linear-optimization oracle, mix.
• Jaggi, M. (2013). “Revisiting Frank–Wolfe: projection-free sparse convex optimization.” ICML 2013. The modern convergence and duality-gap toolkit; the dual gap reported by the solver and the demo is this one.

The rollout application

Best-of-n sampling from language models is a budgeted race: partial generations drift, compute is metered per token kept alive, and only the best completion is paid. Standard beam search fixes the beam width; wavefront pruning would let the width float under the budget.

• Yao, S., Yu, D., Zhao, J., Shafran, I., Griffiths, T. L., Cao, Y., and Narasimhan, K. (2023). “Tree of Thoughts: deliberate problem solving with large language models.” NeurIPS 2023, arXiv:2305.10601. Explicit keep-or-prune decisions over partial reasoning paths, with a hand-set frontier size.
• Snell, C., Lee, J., Xu, K., and Kumar, A. (2024). “Scaling LLM test-time compute optimally can be more effective than scaling model parameters.” arXiv:2408.03314. Makes the budget explicit: fixed inference compute, allocated across candidate solutions. The allocation rule is the open dial.
• Chen, M., et al. (2021). “Evaluating large language models trained on code.” arXiv:2107.03374. Defines pass@k and its unbiased estimator: the expected-maximum payoff of the rollout race, as the field actually scores it.
• Malladi, S., Jelassi, S., Foster, D., Ash, J. T., and Krishnamurthy, A. (2026). “TailSFT: filtered fine-tuning improves post-training performance.” arXiv:2608.25756. Trains for coverage — shaping the distribution so its maximum is large — and steers by a pass@k extrapolation whose plug-in step the posterior-predictive note on this site replaces.

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