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.
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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.
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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
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