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Abstract We consider the Bernoulli bandit problem where one of the arms has win probability α and the others β, with the identity of the α arm specified by initial probabilities. With u = max(α, β), v = min(α, β), call an arm with win probability u a good arm. Whereas it is known that the strategy of always playing the arm with the largest probability of being a good arm maximizes the expected number of wins in the first n games for all n , we conjecture that it also stochastically maximizes the number of wins. That is, we conjecture that this strategy maximizes the probability of at least k wins in the first n games for all k , n . The conjecture is proven when k = 1, and k = n , and when there are only two arms and k = n - 1.
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This content will become publicly available on November 8, 2025
Luck, skill, and depth of competition in games and social hierarchies
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, 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. Competition in sports and games, by contrast, tends to be shallow, and in most cases, there is little evidence of upset wins.
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- Award ID(s):
- 2404617
- PAR ID:
- 10614244
- Publisher / Repository:
- American Association for the Advancement of Science
- Date Published:
- Journal Name:
- Science Advances
- Volume:
- 10
- Issue:
- 45
- ISSN:
- 2375-2548
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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