# Luck, skill, and depth of competition in games and social hierarchies

arXiv: https://arxiv.org/abs/2312.04711 · HTML: https://mlsanity.com/paper/2312.04711/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 7 Dec 2023.

## Builds on (most cited first, 5 of 5)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Efficient computation of rankings from pairwise comparisons](https://mlsanity.com/paper/2207.00076/) (2022)
- [A physical model for efficient ranking in networks](https://mlsanity.com/paper/1709.09002/) (2017)
- [Ranking with multiple types of pairwise comparisons](https://mlsanity.com/paper/2206.13580/) (2022)
- [Depth, balancing, and limits of the Elo model](https://mlsanity.com/paper/1511.02006/) (2015)

## Built on by (most cited first, 6 of 6)

- [Estimation of partial rankings from sparse, noisy comparisons](https://mlsanity.com/paper/2501.02505/) (2025)
- [RALS: Resources and Baselines for Romanian Automatic Lexical Simplification](https://mlsanity.com/paper/2607.20078/) (2026)
- [Model inference for ranking from pairwise comparisons](https://mlsanity.com/paper/2512.15269/) (2025)
- [On rankings in multiplayer games with an application to the game of Whist](https://mlsanity.com/paper/2604.00641/) (2026)
- [Multiple latent orderings better predict language model preferences](https://mlsanity.com/paper/2609.22170/) (2026)
- [All against the machine: the Solo score for rating skill in variable environments](https://mlsanity.com/paper/2610.04523/) (2026)

## Abstract (arXiv)

Patterns of wins and losses in pairwise contests, such as occur in sports and games, consumer research and paired comparison studies, and human and animal social hierarchies, are commonly analyzed using probabilistic models that allow one to quantify the strength of competitors or predict the outcome of future contests. Here we generalize this approach to incorporate two additional features: an element of randomness or luck that leads to upset wins, and a "depth of competition" variable that measures the complexity of a game or hierarchy. Fitting the resulting model to a large collection of data sets we estimate depth and luck in a range of games, sports, and social situations. In general, we find that social competition tends to be "deep," meaning it has a pronounced hierarchy with many distinct levels, but also that there is often a nonzero chance of an upset victory, meaning that dominance challenges can be won even by significant underdogs. Competition in sports and games, by contrast, tends to be shallow and in most cases there is little evidence of upset wins, beyond those already implied by the shallowness of the hierarchy.