The median voting rule is a rule for group decision-making along a one-dimesional spectrum. An example is members of a city-council who have to decide on the total amount of annual city budget. Another example is several people working in the same office who have to decide on the air-conditioning temperature. Each member has in mind an ideal amount (called a "peak"), and prefers the actual amount to be as close as possible to his peak.

A simple way to decide is the average voting rule: ask each member what his peak is, and take the average of all peaks. But this rule easily manipulable. For example, suppse Alice's peak is 30, George's peak is 40, and Chana's peak is 50. If all voters report their true peaks, the actual amount will be 40. But Alice may manipulate and say that her peak is actually 0; then the average will be 30, which is Alice's actual peak. Thus, Alice has gained from the manipulation. Similarly, any agent whose peak is different than the outcome has an incentive to manipulate and report a false peak.

In contrast, the median rule determines the actual budget at the median of all votes. This simple change makes the rule strategyproof: no voter can gain by reporting a false peak. In the above example, the median is 40, and it remains 40 even if Alice reports 0. In fact, as Alice's true peak is below the median, no false report by Alice can potentially decrease the median; Alice can only increase the median, but this will make her worse-off.

Preconditions

The median voting rule holds in any setting in which the agents have single peaked preferences. This means that there exists some linear ordering > of the alternatives, such that for each agent i with peak pi:

  • If pi > a > b, then agent i prefers a to b;
  • If b > a > pi, then agent i prefers a to b.

Note that single-peakedness does not imply any particular distance-measure between the alternatives. In particular, if a > pi > b, then the agent may prefer either a to b or b to a.

Once such a linear order exists, the median of any set of peaks can be computed by ordering the peaks along this linear order.

Proof of strategyproofness

Here is a proof that the median rule is strategyproof:

  • Consider first a voter whose peak is below the median. Reporting a lower peak will not change the median; reporting a higher peak will either keep the median unchanged or increase the median. In all cases, the voter does not gain.
  • Similarly, consider a voter whose peak is above the median. Reporting a higher peak will not change the median; reporting a lower peak will either keep the median unchanged or decrease the median. In all cases, the voter does not gain.

Using similar reasoning, one can prove that the median rule is also Group-strategyproof, that is: no coalition has a coordinated manipulation that improves the utility of one of them without harming the others.

Generalized median rules

Median with phantoms

The median rule is not the only strategyproof rule. One can construct alternative rules by adding fixed votes, that do not depend on the citizen votes. These fixed votes are called "phantoms". For every set of phantoms, the rule that chooses the median of the set of real votes + phantoms is group-strategyproof.

For example, suppose the votes are 30,40,50. Without phantoms, the median rule selects 40. If we add two phantoms at 0, then the median rule selects 30; if we add two phantoms at 100, then the meidan rule select 50; if we add medians at 20 and 35, the median rule selects 35.

Here are some special cases of phantom-median rules, assuming all the votes are between 0 and 100:

  • If there are n-1 phantoms at 0, then the median rule returns the minimum of all real votes.
  • If there are n-1 phantoms at 100, then the median rule returns the maximum of all real votes.
  • If there are n-1 phantoms at 50, then the median rule returns 50 if some ideal points are above and some are below 50; otherwise, it returns the vote closest to 50.

Moulin[1] proved the following characterizations:

  • A rule is anonymous, strategyproof and Pareto-efficient for all single-peaked preferences iff it is equivalent to a median rule with at most n-1 phantoms.
  • A rule is anonymous and strategyproof for all single-peaked preferences iff it is equivalent to a median rule with at most n+1 phantoms.
  • A rule is strategyproof for all single-peaked preferences iff it is equivalent to a minmax rule of the following form. There are 2n parameters, bS for any subset S of voters. The rule returns the minimum over all subsets S, of the maximum of (all peaks of voters in S, and bS).

Moulin's characterizations consider only rules that are "peak only", that is, the rule depends only on the n peaks. Ching[2] proved that all rules that are strategyproof and continuous, even if they are not "peak only", are augmented median rules, that is, can be described by a variant of the median rule with some 2n parameters.

Berga and Serizawa[3] characterized generalized median rules as the only strategyproof rules on "minimally-rich domains". They proved that the unique maximal domain that includes a minimally-rich domain, which allows for the existence of strategyproof rules satisfying the "no vetoer" condition, is the domain of convex preferences.

Border and Jordan[4] and Barbera, Gul and Stacchetti[5] generalized the notions of single-peaked preferences and median voting rules to multidimensional settings.

The median voter theorem says that any voting rule that satisfies the Condorcet Criterion will necessarily choose the candidate closest to the voter whose peak is the median of all peaks.[6]

Highest median voting rules are different voting rules, that work for general preferences (not only single-peaked). Each voter is asked to rank each of the candidates; the median score of each candidate is computed; and a candidate with the highest median score is elected.

The Gibbard–Satterthwaite theorem says that every strategyproof rule on three or more alternatives must be a dictatorship. The median rule apparently contradicts this theorem, because it is strategyproof and it is not a dictatorship. In fact there is no contradiction: the Gibbard-Satterthwaite theorem applies only to rules that operate on the entire preference domain (that is, only to voting rules that can handle any set of preference rankings). In contrast, the median rule applices only to a restricted preference domain - the domain of single-peaked preferences.

Dummet and Farquharson[7] present a sufficient condition for stability in voting games.

References

  1. ↑ Moulin, H. (1980-01-01). "On strategy-proofness and single peakedness". Public Choice. 35 (4): 437–455. doi:10.1007/BF00128122. ISSN 1573-7101. S2CID 154508892.
  2. ↑ Ching, Stephen (1997-12-01). "Strategy-proofness and "median voters"". International Journal of Game Theory. 26 (4): 473–490. doi:10.1007/BF01813886. hdl:10722/177668. ISSN 1432-1270. S2CID 42830689.
  3. ↑ Berga, Dolors; Serizawa, Shigehiro (2000-01-01). "Maximal Domain for Strategy-Proof Rules with One Public Good". Journal of Economic Theory. 90 (1): 39–61. doi:10.1006/jeth.1999.2579. ISSN 0022-0531.
  4. ↑ https://academic.oup.com/restud/article-abstract/50/1/153/1568395. Retrieved 2023-11-15. {{cite web}}: Missing or empty |title= (help)
  5. ↑ Barberà, Salvador; Gul, Faruk; Stacchetti, Ennio (1993-12-01). "Generalized Median Voter Schemes and Committees". Journal of Economic Theory. 61 (2): 262–289. doi:10.1006/jeth.1993.1069. ISSN 0022-0531.
  6. ↑ Black, Duncan (1948-02-01). "On the Rationale of Group Decision-making". Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. ISSN 0022-3808. S2CID 153953456.
  7. ↑ Dummett, Michael; Farquharson, Robin (1961). "Stability in Voting". Econometrica. 29 (1): 33–43. doi:10.2307/1907685. ISSN 0012-9682. JSTOR 1907685.
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