From root: How can simulations help me take better actions?

When should a rugby manager make substitutions?

A rugby manager picks when to substitute each position group: bring on the front-row replacements early, or save them for the closing minutes?

The simulation below is fitted to thousands of real match events and learns the rate at which scoring events happen, minute by minute, with an intensity that depends on which players are on the pitch and how long they have been there.

Drag the substitution-timing sliders for each position group and watch how the predicted win probability shifts.

Rugby manager decision support: pick when each home position group comes off the pitch; the simulator (fitted to thousands of real match events) shows how the win-margin distribution shifts. This is a research model fitted to match events, not a rugby manager decision tool.

Simulation

Score margin (home − away) for the in-progress match

−50−250+25+50

Win-margin distribution across completed matches

 

 

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The link to this project can be found here.

Data and licence: This is an entirely non-commercial research and learning model, built purely to demonstrate that a simulation can be fit to realistic match-event data. None of the underlying data is redistributed here; the dashboard ships only the fitted model coefficients, not any match records or source data. Unlike the other dashboards in this series there is no open-data licence to cite.

References and further reading

Dixon, Mark J., and Stuart G. Coles. 1997. “Modelling Association Football Scores and Inefficiencies in the Football Betting Market.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 46 (2): 265–80. https://doi.org/10.1111/1467-9876.00065.
Hardwick, Robert J. 2022. “Stochadex: A Simulation Engine Written in Go.” GitHub. https://stochadex.github.io/.
Hardwick, Robert J. 2023. “trywizard: An Event-Based Rugby Match Simulator Written in Go.” GitHub. https://github.com/umbralcalc/trywizard.
Hardwick, Robert J. 2026a. “How can simulations help me take better actions?” umbralcalculations, May. https://umbralcalc.github.io/posts/how_can_simulations_help_me_take_better_actions.html.
Hardwick, Robert J. 2026b. “How does a simulation learn its parameters from data?” umbralcalculations, November. https://umbralcalc.github.io/posts/how_does_a_simulation_learn_its_parameters_from_data.html.
Hardwick, Robert J. 2026c. “What is a state partition?” umbralcalculations, November. https://umbralcalc.github.io/posts/what_is_a_state_partition.html.
Mauboussin, Michael J. 2012. The Success Equation: Untangling Skill and Luck in Business, Sports, and Investing. Harvard Business Review Press.