pith. sign in

arxiv: 1906.10968 · v1 · pith:ADR7XKEOnew · submitted 2019-06-26 · 🧮 math.NA · cs.NA· math.OC

Stochastic hybrid differential games and match race problems

Simone Cacace , Roberto Ferretti , Adriano Festa This is my paper

Pith reviewed 2026-05-25 15:28 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC
keywords stochastic differential gameshybrid systemsmatch racesailingvalue functionnumerical approximationtwo-player games
0
0 comments X

The pith

A stochastic two-player hybrid differential game models match races between sailing boats, with a convergent numerical scheme to compute its value function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up a general framework for stochastic two-player hybrid differential games and applies it to a match race in which each sailing boat tries to advance windward while impeding the opponent. It constructs an approximation scheme that converges to the value function of this game. The scheme is tested on representative racing scenarios to produce strategies under varying wind conditions. If the modeling holds, the computed value function yields explicit optimal controls for each boat as functions of position, velocity, and wind state.

Core claim

The match-race problem between two sailing boats is formalized as a stochastic hybrid differential game whose value function can be approximated by a convergent numerical scheme that yields usable strategies on typical racing courses.

What carries the argument

Convergent approximation scheme for the value function of the stochastic hybrid differential game.

If this is right

  • Explicit position-dependent strategies become available for each boat as functions of the current state and wind.
  • The value function quantifies the probability that one boat finishes ahead under optimal play.
  • The same scheme extends to other hybrid games with stochastic switching dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be adapted to other pursuit-evasion problems with discrete mode switches, such as vehicle lane changes or robotic navigation.
  • If the wind model is replaced by real-time sensor data, the scheme might support on-the-water tactical decisions.
  • The hybrid structure suggests that similar games arise whenever continuous motion is interrupted by discrete events like tacking or rounding marks.

Load-bearing premise

The chosen hybrid boat dynamics, stochastic wind model, and payoff structure are close enough to real sailing that the computed value function produces useful strategies.

What would settle it

Direct comparison of the computed optimal trajectories and win probabilities against recorded outcomes from actual match races under similar wind conditions.

Figures

Figures reproduced from arXiv: 1906.10968 by Adriano Festa, Roberto Ferretti, Simone Cacace.

Figure 1
Figure 1. Figure 1: Model of the boat speed. Geometric setting (a), one of the two [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Level sets of the speed profile. We remark that we can also incorporate in the maximum speeds ¯s A and ¯s B an additional dependency on the states (x, q, r). This can be useful to model the rules about the right of way in match race competitions. For instance, with a little abuse of notation, we can choose s¯ A(x, θ, q) = ¯s A  1 − ν1e − x 2 ν2 q−1 (and similarly ¯s B(x, θ, r)) to introduce a penalizatio… view at source ↗
Figure 3
Figure 3. Figure 3: One-dimensional problem. Value functions (a), zoom of their differ [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test 1a. Optimal strategy for both players in symmetric conditions, [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Test 1b. Optimal strategy for both players in symmetric conditions, [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Test 2. Symmetric conditions, A (red) plays the optimal strategy for [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Test 3. Asymmetric conditions, player B (black) is ahead at the start, [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
read the original abstract

We discuss the general framework of a stochastic two-player, hybrid differential game, and we apply it to the modelling of a "match race" between two sailing boats, namely a competition in which the goal of both players is to proceed in the windward direction, while trying to slow down the other player. We provide a convergent approximation scheme for the computation of the value function of the game, and we validate the approach on some typical racing scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a general framework for stochastic two-player hybrid differential games and specializes it to a match-race model between two sailing boats whose objective is windward progress while impeding the opponent. It constructs a convergent approximation scheme for the value function of the resulting game and illustrates the scheme on several representative racing scenarios.

Significance. If the convergence analysis holds, the work supplies a concrete numerical method for a class of hybrid stochastic games that had previously lacked convergent discretizations. The sailing application serves as a non-trivial test case that exercises the hybrid and stochastic components simultaneously. The manuscript does not claim or demonstrate empirical fidelity to observed race data; its contribution is therefore primarily methodological rather than predictive for real sailing.

minor comments (3)
  1. §3 (or the section defining the hybrid dynamics): the precise form of the stochastic wind process and the payoff functional (windward progress minus opponent slowing) should be stated with explicit parameter values or ranges used in the numerical examples so that the scenarios are fully reproducible.
  2. The convergence theorem (presumably in §4 or §5) would benefit from a short remark clarifying whether the scheme remains convergent when the hybrid switching times are themselves stochastic rather than deterministic.
  3. Figure captions for the racing scenarios should indicate the discretization parameters (time step, spatial grid size, number of wind states) employed, to allow readers to assess the resolution at which the reported value functions were obtained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points requiring substantive response or revision at this stage. The contribution remains methodological, as correctly observed.

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper presents a mathematical framework for stochastic hybrid differential games with a claimed convergent approximation scheme for the value function, applied to a match-race model. No equations, fitted parameters, self-citations, or ansatzes are referenced in the abstract or summary that reduce any prediction or result to its inputs by construction. The convergence claim and numerical validation on scenarios remain independent of any self-referential fitting or renaming; modeling assumptions about dynamics and wind are stated as premises rather than derived quantities. This is the standard case of a self-contained mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the modeling choices (hybrid dynamics, stochastic wind, payoff) are implicit but not enumerated.

pith-pipeline@v0.9.0 · 5592 in / 1015 out tokens · 17827 ms · 2026-05-25T15:28:25.361157+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    M. S. Branicky, V. S. Borkar, S. K. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE transactions on automatic control 43 (1) (1998) 31–45. 18 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 100 200 300 400 500 600 700 800 900 1000 0 5 10 15 20 25 30 35 40 45 -10 -8 -6 -4 -2 0 2 4 6 wind direction trajectories -0.01 0 0.01 0.02 0.03 0....

    work page 1998

  2. [2]

    Bensoussan, J

    A. Bensoussan, J. Menaldi, Hybrid control and dynamic programming, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Appli- cation and Algorithm 3 (4) (1997) 395–442

    work page 1997

  3. [3]

    Dharmatti, M

    S. Dharmatti, M. Ramaswamy, Hybrid control systems and viscosity solu- tions, SIAM Journal on Control and Optimization 44 (4) (2005) 1259–1288

    work page 2005

  4. [4]

    Ferretti, H

    R. Ferretti, H. Zidani, Monotone numerical schemes and feedback con- struction for hybrid control systems, Journal of Optimization Theory and Applications 165 (2) (2014) 507–531

    work page 2014

  5. [5]

    Barles, P

    G. Barles, P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis 4 (3) (1991) 271–283

    work page 1991

  6. [6]

    Yong, Differential games with switching strategies, Journal of Mathe- matical Analysis and Applications 145 (2) (1990) 455–469

    J. Yong, Differential games with switching strategies, Journal of Mathe- matical Analysis and Applications 145 (2) (1990) 455–469. 19 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 35 40 45 50 -10 -8 -6 -4 -2 0 2 4 6 8 wind direction trajectories -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 200 400 600 800 10...

    work page 1990

  7. [7]

    R. J. Elliott, N. J. Kalton, The existence of value in differential games of pursuit and evasion, Journal of Differential Equations 12 (3) (1972) 504– 523

    work page 1972

  8. [8]

    Shaiju, S

    A. Shaiju, S. Dharmatti, Differential games with continuous, switching and impulse controls, Nonlinear Analysis: Theory, Methods & Applications 63 (1) (2005) 23–41

    work page 2005

  9. [9]

    B. E. Asri, S. Mazid, Stochastic differential switching game in infinite hori- zon, Journal of Mathematical Analysis and Applications 474 (2) (2019) 793 – 813

    work page 2019

  10. [10]

    Ishii, S

    H. Ishii, S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic pdes arising in switching games, Funkcial. Ekvac 34 (1) (1991) 143– 155. 20 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 50 100 150 200 250 300 350 400 450 -2 0 2 4 6 8 10 12 14 16 18 -5 -4 -3 -2 -1 0 1 2 wind direction trajectories -0.02 -0.01 0 0.01 0.02 0.03 0.04 0....

    work page 1991

  11. [11]

    Y. Hu, S. Tang, Switching game of backward stochastic differential equa- tions and associated system of obliquely reflected backward stochastic dif- ferential equations, Discrete and Continuous Dynamical Systems-Series A 35 (11) (2015) 5447–5465

    work page 2015

  12. [12]

    Y. Hu, S. Tang, Multi-dimensional bsde with oblique reflection and optimal switching, Probability Theory and Related Fields 147 (1-2) (2010) 89–121

    work page 2010

  13. [13]

    Hamadene, J

    S. Hamadene, J. Zhang, Switching problem and related system of reflected backward sdes, Stochastic Processes and their applications 120 (4) (2010) 403–426

    work page 2010

  14. [14]

    Hamadene, M.-A

    S. Hamadene, M.-A. Morlais, Viscosity solutions of systems of pdes with interconnected obstacles and switching problem, Applied Mathematics & Optimization 67 (2) (2013) 163–196. 21

    work page 2013

  15. [15]

    Spenkuch, S

    T. Spenkuch, S. R. Turnock, M. Scarponi, R. A. Shenoi, Modelling multiple yacht sailing interactions between upwind sailing yachts, Journal of Marine Science and Technology 16 (2) (2011) 115–128

    work page 2011

  16. [16]

    R. C. Dalang, F. Dumas, S. Sardy, S. Morgenthaler, J. Vila, Stochastic optimization of sailing trajectories in an upwind regatta, Journal of the Operational Research Society 66 (5) (2015) 807–821

    work page 2015

  17. [17]

    Ferretti, A

    R. Ferretti, A. Festa, Optimal route planning for sailing boats: A hy- brid formulation, Journal of Optimization Theory and Applications 181 (3) (2019) 1015–1032

    work page 2019

  18. [18]

    A. B. Philpott, S. G. Henderson, D. Teirney, A simulation model for pre- dicting yacht match race outcomes, Operations Research 52 (1) (2004) 1–16

    work page 2004

  19. [19]

    Tagliaferri, A

    F. Tagliaferri, A. Philpott, I. Viola, R. Flay, On risk attitude and optimal yacht racing tactics, Ocean Engineering 90 (2014) 149 – 154

    work page 2014

  20. [20]

    Tagliaferri, I

    F. Tagliaferri, I. Viola, A real-time strategy-decision program for sailing yacht races, Ocean Engineering 134 (2017) 129 – 139

    work page 2017

  21. [21]

    Ferretti, Choosing between two fluctuating options: a hybrid control approach, Applied Mathematical Sciences 8 (3) (2014) 139–146

    R. Ferretti, Choosing between two fluctuating options: a hybrid control approach, Applied Mathematical Sciences 8 (3) (2014) 139–146

    work page 2014

  22. [22]

    URL http://www.sailing.org/documents/racingrules/

    WS, 2017 - 2020 racing rules of sailing (2017). URL http://www.sailing.org/documents/racingrules/

    work page 2017

  23. [23]

    Vinckenbosch, Stochastic control and free boundary problems for sail- boat trajectory optimization, PhD thesis, EPFL, Lausanne, 2012

    L. Vinckenbosch, Stochastic control and free boundary problems for sail- boat trajectory optimization, PhD thesis, EPFL, Lausanne, 2012

    work page 2012

  24. [24]

    Frehse, On the regularity of the solution of a second order variational inequality, Boll

    J. Frehse, On the regularity of the solution of a second order variational inequality, Boll. Un. Mat. Ital. (4) 6 (1972) 312–315. 22

    work page 1972