## Abstract

While traditional social choice models assume that the set of candidates is known and fixed in advance, recently several researchers [2, 5, 7, 15, 18] have proposed to reject this hypothesis. In particular, the unavailable candidate model of Lu and Boutilier [15] considers voting situations in which some candidates may not be available and focuses on minimising the number of binary disagreements between the voters and the consensus ranking. In this paper, we extend this model and present two new voting rules based on a finer notion of disagreement, called dissatisfaction. The dissatisfaction of a voter is defined as the disutility gap between its preferred available candidate and the candidate elected by the consensus ranking. In the first approach, called *ex ante dissatisfaction* rule, the disutility is independent of the set of available candidates whereas the second approach, called *ex post dissatisfaction* rule, assumes that the disutility depends on which candidates are actually available. We provide several results for the two rules. On the one hand, we show that the ex ante rule actually coincides with standard positional scoring rules; therefore, a consensus ranking can be computed in polynomial time. On the other hand, we exhibit strong links between ex post rule and Kemeny rule and we provide a polynomial-time approximation scheme (PTAS) for the ex post problem.

This work was mainly conducted while at LIP6, Sorbonne Université.

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

## Notes

- 1.
With this point of view, the candidate in the

*i*-th position can be seen as the preimage of*i*by*r*, i.e. \(r_i = r^{-1}(i)\). - 2.
These conventions are aimed at simplifying the proofs and do not interfere with the search of optimal rankings since \(a_{\emptyset }\) is not in \(C\).

- 3.
The atomic elements being the subsets of \(C\),

*P*should actually be defined on \(\mathscr {P}(\mathscr {P}(C))\) and the probability that the set of available candidates is equal to*S*would be \(P(\{S\})\). We nevertheless write*P*(*S*) for the sake of readability. - 4.
The notation \(\tilde{P}(k)\) must not be confused with the probability that the set of available candidates is of cardinality

*k*, which is actually equal to \(\tilde{P}(k) \times \left( {\begin{array}{c}m\\ k\end{array}}\right) \). - 5.
Referring to the definition of \(\hat{\varDelta }_P\), one can note that the first argument of \(\hat{\varDelta }_P\) here is voter

*v*itself while the second argument is its preference order. - 6.
Studying other aggregators is a perspective that would allow us to give more focus on fairness in the consensus production.

- 7.
This remark motivates our choice of considering disutilities instead of utilities because, when \(\rho _v(1) = 0\) for all \(v\in V\), the ex post dissatisfaction measure can be seen as a simple sum of expectations of \(\rho _v(v_S({{\,\mathrm{top}\,}}_r(S)))\).

- 8.
\(\epsilon > 0\) so both \((1-\epsilon )^{\frac{1}{m-1}}\) and \(\frac{q - (1+\epsilon )^{\frac{1}{m-1}}}{q-1}\) are strictly lower than 1.

- 9.
It is unclear whether there exists a PTAS if the normalised DF are not bounded.

## References

Apesteguia, J., Ballester, M.A.: A measure of rationality and welfare. J. Polit. Econ.

**123**(6), 1278–1310 (2015)Baldiga, K.A., Green, J.R.: Assent-maximizing social choice. Soc. Choice Welfare

**40**(2), 439–460 (2013)Bartholdi, J., Tovey, C.A., Trick, M.A.: Voting schemes for which it can be difficult to tell who won the election. Soc. Choice welfare

**6**(2), 157–165 (1989)Berg, S., Lepelley, D.: On probability models in voting theory. Statistica Neerlandica

**48**(2), 133–146 (1994)Boutilier, C., Lang, J., Oren, J., Palacios, H.: Robust winners and winner determination policies under candidate uncertainty. In: Twenty-Eighth AAAI Conference on Artificial Intelligence. AAAI, Québec (2014)

Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D.: Handbook of Computational Social Choice. Cambridge University Press, Cambridge (2016)

Chevaleyre, Y., Lang, J., Maudet, N., Monnot, J., Xia, L.: New candidates welcome! possible winners with respect to the addition of new candidates. Math. Soc. Sci.

**64**(1), 74–88 (2012)Dutta, B., Jackson, M.O., Le Breton, M.: Strategic candidacy and voting procedures. Econometrica

**69**(4), 1013–1037 (2001)Dutta, B., Jackson, M.O., Le Breton, M.: Voting by successive elimination and strategic candidacy. J. Econ. Theory

**103**(1), 190–218 (2002)García-Lapresta, J.L., Pérez-Román, D.: Consensus measures generated by weighted kemeny distances on weak orders. In: 2010 10th International Conference on Intelligent Systems Design and Applications, pp. 463–468. IEEE Computer Society, Cairo (2010)

Gilbert, H., Portoleau, T., Spanjaard, O.: Beyond pairwise comparisons in social choice: a setwise kemeny aggregation problem. In: Thirty-Fourth AAAI Conference on Artificial Intelligence, pp. 1982–1989 (2020)

Klamler, C.: A distance measure for choice functions. Soc. Choice Welfare

**30**(3), 419–425 (2008)Lang, J., Maudet, N., Polukarov, M.: New results on equilibria in strategic candidacy. In: Vö, B. (ed.) SAGT 2013. LNCS, vol. 8146cience. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41392-6_2

Laslier, J.F.: And the loser is\(\ldots \) plurality voting. In: Felsenthal, D., Machover, M. (eds.) Electoral Systems. SCW, pp. 327–351. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-20441-8_13

Lu, T., Boutilier, C.E.: The unavailable candidate model: a decision-theoretic view of social choice. In: Proceedings of the 11th ACM Conference on Electronic Commerce, pp. 263–274 (2010)

Lumet, C., Bouveret, S., Lemaître, M.: Fair division of indivisible goods under risk (2012)

Naamani-Dery, L., Kalech, M., Rokach, L., Shapira, B.: Reducing preference elicitation in group decision making. Expert Syst. Appl.

**61**, 246–261 (2016)Oren, J., Filmus, Y., Boutilier, C.: Efficient vote elicitation under candidate uncertainty. In: IJCAI, pp. 309–316 (2013)

Shieh, G.S.: A weighted kendall’s tau statistic. Stat. Prob. Lett.

**39**(1), 17–24 (1998)Viappiani, P.: Characterization of scoring rules with distances: application to the clustering of rankings. In: Twenty-Fourth International Joint Conference on Artificial Intelligence (2015)

## Author information

### Authors and Affiliations

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2021 Springer Nature Switzerland AG

## About this paper

### Cite this paper

Grivet Sébert, A., Maudet, N., Perny, P., Viappiani, P. (2021). Preference Aggregation in the Generalised Unavailable Candidate Model. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham. https://doi.org/10.1007/978-3-030-87756-9_3

### Download citation

DOI: https://doi.org/10.1007/978-3-030-87756-9_3

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-030-87755-2

Online ISBN: 978-3-030-87756-9

eBook Packages: Computer ScienceComputer Science (R0)