Median voting rule

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.