# 1 The mathematics of the voting booth

No voting system is perfect. Really, it’s been proven.

The result, called the Gibbard-Satterthwaite Theorem, is an extension of Arrow’s Theorem, named after mathematician and economist Kenneth Arrow.

Arrow showed that in any voting situation with three or more alternative options, no system for ranking those options (apart from dictatorship) is guaranteed to rank the choices of a voting group as a whole in a way that reflects the preferences of that group, even if that preference is unanimous. In geek, any complete, non-dictatorial ranking system for available options {A, B, … n}  greater than two admits cases where it fails to satisfy one or both of the following conditions: unanimity, also called weak Pareto efficiency, (if each member of a group ranks an option A over B, then the group ranks A over B) and independence of irrelevant alternatives (if each individual in a group prefers an option A over B, then one’s preferences concerning an option C should not affect the group preference for A over B). In general, voting systems often fail to respect the pairwise preferences1 of its voting members.

Arrow’s theorem is a result that affects ordinal voting systems, i.e. voting systems that order different available options relative to each other. First-past the-post voting, the method in place in most U. S. elections, is an ordinal voting system. So are ranked-choice systems, where voters rank their preferences first, second, and so forth.

## 1.1 Approval voting and the Gibbard-Satterthwaite Theorem

Approval voting, by contrast, is a cardinal voting system. Cardinal voting systems evaluate options independently of each other, e.g. by rating them on a one-to-ten scale. Approval voting can be thought of as implementing a simpler version of the idea behind the ten-point scale, where the only options are 1 (approval) and 0 (non-approval). Cardinal systems are not affected by Arrow’s impossibility theorem. They do, however, fall within the scope of the Gibbard-Satterthwaite theorem.

The Gibbard-Satterthwaite theorem is named after philosopher Allan Gibbard and economist Mark Satterthwaite, who independently published the result in 1973 and 1975, respectively. According to the theorem, any voting system (apart from dictatorship) for situations involving three or more available candidates admits of cases where a voter can better secure his true preference by not voting for it.2 In plainer language, every voting system allows for situations where a voter can vote strategically, in a way not in accord with his true preferences, to nevertheless secure a preferred outcome.

# 2 Cooperative and non-cooperative, sincere and insincere voting

Let us, then, introduce some definitions.

We say that a vote is strictly cooperative provided that it only votes the absolute preferences of the voter, i.e. that it only votes for candidates that the voter actually approves of. A vote is loosely cooperative provided that it votes for all candidates one approves of. A vote that is both strictly and loosely cooperative, one which votes for exactly those candidates approved, is absolutely cooperative. Conversely, a non-cooperative vote is one that does not vote for all and only unapproved candidates; a non-loosely cooperative vote, one that does not vote for all approved candidates, etc. A vote is absolutely uncooperative provided it votes for all and only unapproved candidates; strictly uncooperative, provided it only votes for candidates the voter does not approve of, and so forth.

A vote is sincere provided that for a ranking of options, if a voter votes for option n, he also votes for every option preferred to n. A vote is insincere if it is not sincere.

# 3 Strategies in approval voting

As one might expect, the most strategic vote in any given situation depends on how other voters are casting their ballots. As a general rule, the best result for the group in approval voting is achieved through large-scale cooperative voting. However, a small group may secure a better outcome for themselves, at the expense of the group, by voting non-cooperatively. Lastly, as more people begin to vote non-cooperatively, standard non-cooperative voting strategies become less and less likely to secure favorable outcomes.

Certain voting strategies will always fail to advance one’s preferences: strictly uncooperative voting, and voting for all or no candidates, for instance, cannot serve as winning strategies under any circumstances. However, various sincere and insincere strategies make sense in various situations. If there is a popular candidate that one sufficiently dislikes, for example, one might vote for all other candidates, included those outside of one’s absolute preferences, in order to maximize opposition to that candidate. Likewise, if one’s other choices on a board of directors are sufficiently secure given the likely choices of others, one might ‘bullet vote’ for one, less popular candidate to increase that candidate’s chances of securing a position.

# 4 An example

Let us consider an example. Haima is voting for the members of the board of a political action committee, which has five seats open for the coming year. The members of the board are to be decided via a simple extension of approval voting to multi-winner elections: the top five vote-getters receive seats. However, multiple evenly-matched blocs within the electorate intend to pursue non-cooperative strategies, with various groups planning to leave off candidates they would approve of under a cooperative strategy in order to vote for their preferred slate of five. The voting environment as a whole leans toward being non-cooperative. Haima has eleven candidates that she approves of, both within and outside of the aforementioned groups, but is not herself party to any of them. How should Haima vote?

Answer: leaving aside Haima’s vote and that of other like-minded voters, the most likely situation appears to be one where various blocs elect their most popular members, leading to a board that may be bogged down by the tensions of its conflicting constituencies. While a restricted vote focusing on her most preferred candidates might make sense in a more cooperative environment, it makes less sense here. Haima should focus on providing more support to a broad set of candidates – either all eleven of her picks, or a subset that excludes only those likely to be elected without her help.

Next, assume that a second voter in the same situation, Pam, is a partisan to one of the groups. Assuming that a conflicting board isn’t desirable to anybody, Pam will want to hedge her bets. To maximize a desirable outcome, she votes to maximize opposition to the faction most hostile to hers. Hence, perhaps excepting those whose election is completely secure, she votes for all candidates she approves of outside of that faction, and might even extend her vote to candidates on whom she is neutral if she feels the situation urgent enough.

# 5 Conclusion

Every voting system allows for some element of strategy. The best strategy for a given situation will depend on the voting system being used, one’s desired aims, and on factors such as whether the broader voting environment is a cooperative one. While approval voting avoids certain problems associated with ordinal voting systems, it requires a broad cooperative voter base to function well. When voters restrict their votes to their most preferred candidate(s), approval voting threatens to replicate the same problems as winner-take-all elections.  When used judiciously, however, it can lead to the election of acceptable options broadly reflecting the preferences of its electorate.

1 I. e. if a majority of the group rank A over B, then the group ranks A over B.

2 This result was later extend to multi-winner elections by John Duggan and Thomas Schwartz.