# On the structure of stable tournament solutions

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## Abstract

A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (1) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (2) we completely characterize which simple choice functions give rise to stable choice functions, and (3) we prove a strong relationship between stability and a new property of tournament solutions called *local reversal symmetry*. Based on these findings, we provide the first concrete tournament—consisting of 24 alternatives—in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.

### Keywords

Choice consistency Tournament solutions Bipartisan set Tournament equilibrium set### JEL Classification

D7 C6## Notes

### Acknowledgements

This material is based on work supported by Deutsche Forschungsgemeinschaft under Grants BR 2312/7-1 and BR 2312/7-2, by a Feodor Lynen Research Fellowship of the Alexander von Humboldt Foundation, by ERC Starting Grant 639945, by a Stanford Graduate Fellowship, and by the MIT-Germany program. The authors thank Christian Geist for insightful computer experiments and Paul Harrenstein for helpful discussions and preparing Fig. 1.

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