The average voting rule is a rule for

group decision-making Group decision-making (also known as collaborative decision-making or collective decision-making) is a situation faced when individuals collectively make a choice from the alternatives before them. The decision is then no longer attributable to ...

when the decision is a ''distribution'' (e.g. the allocation of a budget among different issues), and each of the voters reports his ideal distribution. This is a special case of

budget-proposal aggregation Budget-proposal aggregation (BPA) is a problem in social choice theory. A group has to decide on how to distribute its budget among several issues. Each group-member has a different idea about what the ideal budget-distribution should be. The probl ...

. It is a simple aggregation rule, that returns the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of all individual ideal distributions. The average rule was first studied formally by Michael Intrilligator. This rule and its variants are commonly used in economics and sports.

Characterization

Intrilligator proved that the average rule is the unique rule that satisfies the following three axioms: * ''Completeness:'' for every ''n'' distributions, the rule returns a distribution. * ''Unanimity for losers'': if an issue receives 0 in all individual distributions, then it receives 0 in the collective distribution. * ''Strict and equal sensitivity to individual allocations'': if one voter increases his allocation to one issue while all other allocations remain the same, then the collective allocation to this issue strictly increases; moreover, the rate of increase is the same for all voters (that is, it depends only on the issue). Elkind, Greger, Lederer, Suksompong and Teh present two other characterizations of the average rule under the assumption that agent's utilities are based on L1 distance. They also show that the average rule is fairer than all other rules they considered.

Manipulation

An important disadvantage of the average rule is that it is not strategyproof – it is easy to manipulate. For example, suppose there are two issues, the ideal distribution of Alice is (80%, 20%), and the average of the ideal distributions of the other voters is (60%, 40%). Then Alice would be better off if she reports that her ideal distribution is (100%, 0%), since this will pull the average distribution closer to her ideal distribution. If all voters try to manipulate simultaneously, the computed average may be substantially different than the "real" average: in a two-issue setting with true average close to (50%, 50%), the computed average may vary by up to 20 percentage points when there are many voters, and the effect can be more extreme when the true average is more lopsided.

Variants

The weighted average rule gives different weights to different voters (for example, based on their level of expertise). The trimmed average rule discards some of the extreme bids, and returns the average of the remaining bids. Renault and Trannoy study the combined use of the average rule and the majority rule, and their effect on minority protection.

Other rules

Rosar compares the average voting rule to the median voting rule, when the voters have diverse

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and interdependent preferences. For uniformly distributed information, the average report dominates the median report from a utilitarian perspective, when the set of admissible reports is designed optimally. For general distributions, the results still hold when there are many agents.

References

{{improve categories, date=November 2023 Electoral systems Participatory budgeting