There are two major impossibility theorems in social choice:

- Arrow’s theorem (1951)
^{1}: the only system to aggregate individual preferences into social preferences (e.g., a voting mechanism) that satisfies the properties of transitivity, monotonicity and independence of irrelevant alternatives for all possible individual preferences is the dictatorial system. - Gibbard–Satterthwaite’ theorem (1973)
^{2}: any voting system that is not dictatorial and that does not veto any option is susceptible to strategic voting by some voter, meaning that he will not vote according to his true preferences.

A couple clarifications are in order. First, let us briefly explain the properties. Transitivity means that if, according to some preferences, A is preferred to B and B is preferred to C, then A should be preferred to C. Monotonicity means that if A is socially preferred to B and now a person who voted for B changes her mind to vote for A, this can never result in B being elected. Independence of irrelevant alternatives (IIA) is more subtle. Imagine that, given the choice between A and B, A is elected, but that if a third choice, C, is added, then B is elected. This inconsistency is avoided if IIA is satisfied. A dictatorial system considers the preferences of only one individual.

Secondly, both theorems have an unrestricted preference domain, which means that individuals can have any preferences, and any number of alternatives. For instance, when the Arrow’s theorem says the only system that satisfies the three properties is the dictatorial, it means that for any system other than the dictatorial, one can find individual preferences so that at least one of the properties is not satisfied. This remark opens the door for a non-dictatorial voting system to satisfy the three properties as long as the preferences of the individuals are restricted in some way (for instance, if individuals are not very different or if some “extreme” preferences are not allowed) or if the number of alternatives is also restricted in some way (for instance, if there are only two of them). A similar thing can be said about the second theorem.

After these negative results, the social choice literature opened two complementary research lines (see Barberà, 2011 ^{3} for a survey). One seeks to find out the properties that different choice mechanisms satisfy, and a second one to detect which restrictions in the individual preferences can reasonably be assumed to find election rules satisfying the desired properties. One important restriction is considering only “single peaked” preferences. This means that for some particular way of ordering the alternatives (closedness to a given point, alphabetical order or any other arbitrary way of ordering them), any voter has a best alternative and that the other alternatives are worse as they are further from the best one in that ordering. The ordering of the alternatives must be the same for all the individuals. For instance, suppose there are five alternatives ordered alphabetically as A, B, C, D and E. Voter 1 may have B as his preferred alternative. If he has single peaked preferences, this implies that both A and C are worse than B, and also that D is worse than C, and that E is even worse.

Some positive results involving this kind of preferences were already known before Arrow’s theorem or shortly after it. For instance, Black’s theorem (1948) ^{4} states that, if individual preferences are single-peaked, then majority rule (the election of the alternative with more than half of the votes) is a non-dictatorial preference aggregation rule that satisfies monotonicity, transitivity and IIA, and, further, is strategy-proof (voting sincerely is always the best thing to do).

It is within these lines of research that Alcalde and Vorsatz (2018) ^{5} present a new result. The motivation for their theorems is the problem of locating a public facility in a city taking into account the preferences of the inhabitants. Some individuals may like to have the facility as close to them as possible (these individuals show single-peaked preferences), while others may want just the opposite (and show single-dipped preferences). For instance, if the facility is a dog park, dog owners probably want it near their homes, and not owners, far. Other examples are a football stadium or a school.

The authors’ goal is to find a characterization of all strategy-proof rules on that preference domain. In other words, they want to know all the rules that, applied to those preferences, have the property that individuals want to declare their true preferences. First, they show that, without further restricting the preferences of the individuals, the goal is impossible to achieve. Then, the next natural question is what further minimal restriction on the preferences one needs to assume to escape from the Gibbard-Satterthwatite impossibility theorem. They find that, if for any individual the peak or the dip corresponds to their location, then one can indeed solve the problem. See that this means that the location of individuals is known and that they cannot report preferences indicating otherwise. What the social planner does not know is whether the location corresponds to a peak or a dip, and also, the planner does not know how rapidly the utility decreases or increases as the public facility is located further from the peal or dip.

Finally, the authors find that all strategy-proof rules on this new preference domain share the following structure (in their words):

- agents are only asked about their types of preferences (single-peaked or single-dipped), and the answers determine one or two preselected locations;

- if only one location has been preselected, then it is finally chosen; and

- if two locations have been preselected, then all agents that are situated strictly between the preselected alternatives have to indicate their ordinal preferences between them (and, in case of indifference, their entire preferences) in order to choose the winning location.

- The particular ways the preselected locations that pass to the second phase are picked and the form in which the final location is chosen have to satisfy some monotonicity conditions.

One such a rule is the following: ask the question in (i). If there are agents that declare to have single-peaked preferences, the mean of the locations of these agents is selected (we are in case ii above); otherwise, go to a second round and ask agents to vote by simple majority between the two extremes (case iii). Here is an example: Suppose we have a city spread on the interval [0, 4], and that there are three voters located at points 1, 2 and 3. Consider the social choice rule to select the point to locate the public facility defined through the following procedure: if the set of agents with single-peaked preferences is non-empty, choose the mean location of these agents; otherwise, choose by simple majority (in case or a tie, choose location at point 4) between the extremes 0 and 4. This rule is strategy-proof, Pareto efficient, non-dictatorial.

## References

- Arrow, K. J. 1951.
*Social Choice and Individual Values*. John Wiley and Sons, New York, NY. ↩ - Black, D. 1948. On the Rationale of Group Decision-making.
*Journal of Political Economy*, 56: 23–34,. ↩ - Barberà, S. 2011. Strategy-proof social choice. In: Arrow, K.J., Sen, A.K., Suzumura, K. (Eds.),
*Handbook of Social Choice and Welfare*, vol. 2. North-Holland, Netherlands, 731–831. ↩ - May, K.O. 1952. A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision.
*Econometrica*, 20(4): 680–684. ↩ - Alcalde-Unzu, J., and Vorsatz, M. 2018. Strategy-proof location of public facilities.
*Games and Economic Behavior*, 112: 21–48. ↩