Abstract
An important and surprising phenomenon in voting theory is the No-Show Paradox (NSP), which occurs if a voter is better off by abstaining from an election. While it is known that certain voting rules suffer from this paradox in principle, the extent to which it is of practical concern is not well understood. We aim at filling this gap by analyzing the likelihood of the NSP for six Condorcet extensions (Black’s rule, Baldwin’s rule, Nanson’s rule, Max-Min, Tideman’s rule, and Copeland’s rule) under various preference models using Ehrhart theory as well as extensive computer simulations. We find that, for few alternatives, the probability of the NSP is rather small (less than 4% for four alternatives and all considered preference models, except for Copeland’s rule). As the number of alternatives increases, the NSP becomes much more likely and which rule is most susceptible to abstention strongly depends on the underlying distribution of preferences.
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Notes
- 1.
- 2.
Note that other discriminating Condorcet extensions such as Kemeny’s rule, Dodgson’s rule, and Young’s rule are NP-hard to compute see, e.g., Brandl et al. (2016a).
- 3.
Tideman’s rule is arguably the least well-known voting rule presented here. It was proposed to efficiently approximate Dodgson’s rule and is not to be confused with ranked pairs which is sometimes also called Tideman’s rule. Also note that the ‘dual’ rule returning alternatives for which the sum of weighted pairwise majority wins is maximal is not a Condorcet extension.
- 4.
For both MaxiMin and Tideman’s rule, this holds by the observation that a weak Condorcet winner does not lose any pairwise majority comparison. Black’s rule fails to be a weak Condorcet extension by definition; a counterexample for Baldwin’s, Nanson’s, and Copeland’s rule is given by Fishburn (1977).
- 5.
More precisely, \(P_n\) is a dilated polytope depending on n, \({P_n = nP = \{n\mathbf {x}:\mathbf {x}\in P\}}\).
- 6.
Theoretically, we only require \(g_{ad}-1 \le g_{bc} \le g_{ad}\). As either all \(g_{xy}\) are even or all \(g_{xy}\) are odd, this collapses to \(g_{ad} = g_{bc}\).
- 7.
Some inequalities are omitted to remove redundancies when taken together with later constraints.
- 8.
We choose this informal notation for the sake of readability. It is to be understood in a way that \(P_1\) is the polytope described by (in)equalities labelled (basis) as well as (A). We additionally assume for all polytopes that the sum of voters per type adds up to n and each type consists of a nonnegative number of voters.
- 9.
This effect is only relevant when there are at least four alternatives.
- 10.
For Black’s rule, we find that the polynomial would be of period \(q \approx 2.7\times 10^7\) corresponding to a mid two-digit GB file size.
- 11.
Felsenthal and Nurmi (2018) also show that none of the two rules fares strictly better than the other. Indeed, there are profiles where a manipulation is possible according to Baldwin’s rule but not using Nanson’s rule and vice versa.
- 12.
For Black’s rule, manipulation is only possible either towards or away from a Condorcet winner since Borda’s rule is immune to strategic abstention and manipulation is impossible from Condorcet winner to Condorcet winner.
- 13.
For increasing m the computations quickly become very demanding. The values for \({m=30}\) and \({n\ge 99}\) are determined with 50 000 runs each only. The size of all 95% confidence intervals is, however, still within \(0.5\%\).
- 14.
For instance the profile allowing for a manipulation under Copeland’s rule is immune to the NSP for all other rules. It features 10 alternatives and 30 voters. Baldwin’s and Nanson’s rule exhibit the NSP for the same profile.
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Acknowledgements
Preliminary results of this chapter were presented at the 14th Meeting of the Society for Social Choice and Welfare (Seoul, June 2018), the 7th International Workshop on Computational Social Choice (Troy, June 2018), the AAMAS-IJCAI Workshop on Agents and Incentives in Artificial Intelligence (Stockholm, July 2018), and the 18th International Conference on Autonomous Agents and Multiagent Systems (Montréal, May 2019).
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Brandt, F., Hofbauer, J., Strobel, M. (2021). Exploring the No-Show Paradox for Condorcet Extensions. In: Diss, M., Merlin, V. (eds) Evaluating Voting Systems with Probability Models. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-030-48598-6_11
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