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Preference Approval Voting (PAV) and Fallback Voting (FV) are two voting rules that combine approval and preferences. They were first introduced by Brams and Sanver (2009). Under PAV, voters rank the candidates and indicate which ones they approve of; with FV, they rank only those candidates they approve of. In this paper, we supplement the work of Brams and Sanver (2009) by exploring some other normative properties of FV and PAV. We show among other that FV and PAV satisfy and fail the same criteria; they possess two properties that AV does not: Pareto optimality and the fact of always electing the absolute Condorcet winner when he exists. For threecandidate elections and a very large electorate, we compare FV and PAV to other voting rules by evaluating the probabilities of satisfying the Condorcet majority criteria. We find that PAV performs better than the Borda rule. We also find that in terms of agreement, FV and PAV are closer to scoring rules than to Approval voting. Our analysis is performed under the Impartial Anonymous Culture assumption.

International audience ; A voting rule that permits some voters to favor a candidate by providing only the initial segment of their sincere rankings is said to be vulnerable to the truncation paradox. In this paper, we consider four models for counting truncated ballots, optimistic, pessimistic (the most common), averaged, and round-down. Under the impartial anonymous culture assumption, the choice of model generally has a real impact on truncation-paradox vulnerability, but there are exceptions. When the election is decided by a one-shot scoring rule, the optimistic model is invulnerable to the truncation paradox, but all other models are vulnerable. We identify new voting rules immune to the truncation paradox, such as the Modified Borda Count. To obtain a more complete picture of the impact of processing model, we assess the likelihood of the truncation paradox in three-candidate elections with large electorates, focusing not only on oneshot scoring rules but also scoring rules with one-by-one or below-average elimination. Our assessment confirms that the processing model for truncated ballots may really matter.

Preference Approval Voting (PAV) and Fallback Voting (FV) are two voting rules that combine approval and preferences. They were first introduced by Brams and Sanver (2009). Under PAV, voters rank the candidates and indicate which ones they approve of; with FV, they rank only those candidates they approve of. In this paper, we supplement the work of Brams and Sanver (2009) by exploring some other normative properties of FV and PAV. We show among other that FV and PAV satisfy and fail the same criteria; they possess two properties that AV does not: Pareto optimality and the fact of always electing the absolute Condorcet winner when he exists. For threecandidate elections and a very large electorate, we compare FV and PAV to other voting rules by evaluating the probabilities of satisfying the Condorcet majority criteria. We find that PAV performs better than the Borda rule. We also find that in terms of agreement, FV and PAV are closer to scoring rules than to Approval voting. Our analysis is performed under the Impartial Anonymous Culture assumption.

International audience ; A voting rule that permits some voters to favor a candidate by providing only the initial segment of their sincere rankings is said to be vulnerable to the truncation paradox. In this paper, we consider four models for counting truncated ballots, optimistic, pessimistic (the most common), averaged, and round-down. Under the impartial anonymous culture assumption, the choice of model generally has a real impact on truncation-paradox vulnerability, but there are exceptions. When the election is decided by a one-shot scoring rule, the optimistic model is invulnerable to the truncation paradox, but all other models are vulnerable. We identify new voting rules immune to the truncation paradox, such as the Modified Borda Count. To obtain a more complete picture of the impact of processing model, we assess the likelihood of the truncation paradox in three-candidate elections with large electorates, focusing not only on oneshot scoring rules but also scoring rules with one-by-one or below-average elimination. Our assessment confirms that the processing model for truncated ballots may really matter.

For a given voting rule, if voters can favor a preferred outcome by providing only a part of their sincere rankings on the competing candidates, rather than listing their entire preference rankings on all the competing candidates, this rule is said to be vulnerable to the truncation paradox. In this paper, we show that the way of dealing with the truncated ballot can really impact the occurrence of the paradox: this paradox never occurs with any one-shot scoring rules when truncated ballot a treated according the optimistic model. The optimistic model is, along with the pessimistic model and the averaged model, the three most common ways of dealing with truncated preferences. The few papers that assess the likelihood of the occurrence of this paradox implicitly assume the pessimistic model. In this paper, we assess the likelihood of the truncation paradox under the two other models for three-candidate elections and large electorates. We focus on whole families of one-shot scoring rules, iterative scoring rules both with one-by-one eliminations and with elimination by the average. This assessment confirms that the choice of model may really matter: under the optimistic model, all the one-shot scoring rules are immune to the truncation paradox, whereas it is is more likely to occur under the pessimistic model than under the averaged model; for each of the scoring runoff rules, we find that the likelihood of the truncation paradox is higher under the pessimistic model, and lower under the optimistic model. Our analysis is performed under the Impartial Anonymous Culture assumption.

For a given voting rule, if voters can favor a preferred outcome by providing only a part of their sincere rankings on the competing candidates, rather than listing their entire preference rankings on all the competing candidates, this rule is said to be vulnerable to the truncation paradox. In this paper, we show that the way of dealing with the truncated ballot can really impact the occurrence of the paradox: this paradox never occurs with any one-shot scoring rules when truncated ballot a treated according the optimistic model. The optimistic model is, along with the pessimistic model and the averaged model, the three most common ways of dealing with truncated preferences. The few papers that assess the likelihood of the occurrence of this paradox implicitly assume the pessimistic model. In this paper, we assess the likelihood of the truncation paradox under the two other models for three-candidate elections and large electorates. We focus on whole families of one-shot scoring rules, iterative scoring rules both with one-by-one eliminations and with elimination by the average. This assessment confirms that the choice of model may really matter: under the optimistic model, all the one-shot scoring rules are immune to the truncation paradox, whereas it is is more likely to occur under the pessimistic model than under the averaged model; for each of the scoring runoff rules, we find that the likelihood of the truncation paradox is higher under the pessimistic model, and lower under the optimistic model. Our analysis is performed under the Impartial Anonymous Culture assumption.

For a given voting rule, if voters can favor a preferred outcome by providing only a part of their sincere rankings on the competing candidates, rather than listing their entire preference rankings on all the competing candidates, this rule is said to be vulnerable to the truncation paradox. In this paper, we show that the way of dealing with the truncated ballot can really impact the occurrence of the paradox: this paradox never occurs with any one-shot scoring rules when truncated ballot a treated according the optimistic model. The optimistic model is, along with the pessimistic model and the averaged model, the three most common ways of dealing with truncated preferences. The few papers that assess the likelihood of the occurrence of this paradox implicitly assume the pessimistic model. In this paper, we assess the likelihood of the truncation paradox under the two other models for three-candidate elections and large electorates. We focus on whole families of one-shot scoring rules, iterative scoring rules both with one-by-one eliminations and with elimination by the average. This assessment confirms that the choice of model may really matter: under the optimistic model, all the one-shot scoring rules are immune to the truncation paradox, whereas it is is more likely to occur under the pessimistic model than under the averaged model; for each of the scoring runoff rules, we find that the likelihood of the truncation paradox is higher under the pessimistic model, and lower under the optimistic model. Our analysis is performed under the Impartial Anonymous Culture assumption.

A voting rule is said to be vulnerable to the truncation paradox if some voters may want to favor a more preferable outcome by providing only a part of their sincere rankings on the competing candidates rather than listing their entire preference rankings on all the competing candidates. This voting paradox was first introduced by Brams (1982). This paper provides for three-candidate elections and for large electorates, a characterization and an evaluation of the likelihood of the truncation paradox for the whole family of one-shot scoring rules and runoff scoring rules. We assume three scoring models for dealing with incomplete rankings: the pessimistic, the optimistic and the averaged scoring models. We find that under the optimistic model, all the one-shot scoring rules are immune to the truncation paradox and this paradox is more likely to occur under the pessimistic scoring model than under the averaged scoring model. For each of the scoring runoff rules, we find that the likelihood of the truncation paradox is higher under the pessimistic scoring model and it is lower under the optimistic scoring model. Our analysis is performed under the Impartial Anonymous Culture assumption.

International audience ; Under Approval Voting (AV), each voter just distinguishes the candidates he approves of from those appearing as unacceptable. The Preference Approval Voting (PAV) is a hybrid version of the approval voting first introduced by Brams and Sanver (2009). Under PAV, each voter ranks all the candidates and then indicates the ones he approves. In this paper, we provide analytical representations for the probability that PAV elects the Condorcet winner when she exists in three-candidate elections with large electorates. We also provide analytical representations for the probability that PAV elects the Condorcet loser. We perform our analysis by assuming the assumption of the Extended Impartial Culture. This analysis allows us to measure at which extend, PAV performs better than AV both on the propensity of electing the Condorcet loser and on that of the non-election of the Condorcet loser.

International audience ; The Borda Effect, first introduced by Colman and Poutney (1978), occurs in a preference aggregation process using the Plurality rule if given the (unique) winner there is at least one loser that is preferred to the winner by a majority of the electorate. Colman and Poutney (1978) distinguished two forms of the Borda Effect:-the Weak Borda Effect describing a situation under which the unique winner of the Plurality rule is majority dominated by only one loser; and-the Strong Borda Effect under which the Plurality winner is majority dominated by each of the losers. The Strong Borda Effect is well documented in the literature as the Strong Borda Paradox. Colman and Poutney (1978) showed that the probability of the Weak Borda Effect is not negligible; they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing in three-candidate elections, the representations for the limiting probabilities of the (Weak) Borda Effect for the whole family of the scoring rules and scoring runoff rules. We highlight that there is a relation between the (Weak) Borda Effect and the Condorcet efficiency. We perform our analysis under the Impartial Culture and the Impartial Anonymous Culture which are two well-known assumptions often used for such a study.

International audience The Borda Effect, first introduced by Colman and Poutney (1978), occurs in a preference aggregation process using the Plurality rule if given the (unique) winner there is at least one loser that is preferred to the winner by a majority of the electorate. Colman and Poutney (1978) distinguished two forms of the Borda Effect:-the Weak Borda Effect describing a situation under which the unique winner of the Plurality rule is majority dominated by only one loser; and-the Strong Borda Effect under which the Plurality winner is majority dominated by each of the losers. The Strong Borda Effect is well documented in the literature as the Strong Borda Paradox. Colman and Poutney (1978) showed that the probability of the Weak Borda Effect is not negligible; they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing in three-candidate elections, the representations for the limiting probabilities of the (Weak) Borda Effect for the whole family of the scoring rules and scoring runoff rules. We highlight that there is a relation between the (Weak) Borda Effect and the Condorcet efficiency. We perform our analysis under the Impartial Culture and the Impartial Anonymous Culture which are two well-known assumptions often used for such a study.

International audience ; Under Approval Voting (AV), each voter just distinguishes the candidates he approves of from those appearing as unacceptable. The Preference Approval Voting (PAV) is a hybrid version of the approval voting first introduced by Brams and Sanver (2009). Under PAV, each voter ranks all the candidates and then indicates the ones he approves. In this paper, we provide analytical representations for the probability that PAV elects the Condorcet winner when she exists in three-candidate elections with large electorates. We also provide analytical representations for the probability that PAV elects the Condorcet loser. We perform our analysis by assuming the assumption of the Extended Impartial Culture. This analysis allows us to measure at which extend, PAV performs better than AV both on the propensity of electing the Condorcet loser and on that of the non-election of the Condorcet loser.