Storable Votes and Quadratic Voting are voting systems designed to account for voters' intensity of preferences. We test their performance in two samples of California residents using data on four initiatives prepared for the 2016 California ballot. We bootstrap the original samples and generate two sets of 10,000 multi-elections simulations. As per design, both systems induce minority victories and result in higher expected welfare relative to majority voting. In our parametrization, quadratic voting induces more minority victories and achieves higher average welfare, but causes more frequent inefficient minority victories. The results are robust to different plausible rules-of-thumb in casting votes.

Storable votes allow the minority to win occasionally while treating every voter equally and increasing the efficiency of decision-making, without the need for external knowledge of voters' preferences. This book complements the theoretical discussion with several experiments, showing that the promise of the idea is borne out by the data: the outcomes of the experiments and the payoffs realized match very closely the predictions of the theory.

The storable votes mechanism is a method of voting for committees that meet periodically to consider a series of binary decisions. Each member is allocated a fixed budget of votes to be cast as desired over the multiple decisions. Voters are induced to spend more votes on those decisions that matter to them most, shifting the ex ante probability of winning away from decisions they value less and towards decisions they value more, typically generating welfare gains over standard majority voting with non-storable votes. The equilibrium strategies have a very intuitive feature--the number of votes cast must be monotonic in the voter's intensity of preferences--but are otherwise difficult to calculate, raising questions of practical implementation. In our experiments, realized efficiency levels were remarkably close to theoretical equilibrium predictions, while subjects adopted monotonic but off-equilibrium strategies. We are lead to conclude that concerns about the complexity of the game may have limited practical relevance

Quadratic Voting (QV) is a promising technique for improving group decisionmaking by accounting for preference intensities. QV is a social choice mechanism in which voters buy votes for or against a proposal at a quadratic cost and the outcome with the most votes wins. In some cases, individuals are asymmetrically informed about the effects of legislation and therefore their valuations of legislation. For instance, anti-corruption legislation is the most beneficial to taxpayers and the most detrimental to corrupt officials when corruption opportunities are plentiful, but government officials have better information than taxpayers about how many corruption opportunities exist. I provide an example of a setting in a large population where QV does not achieve approximate efficiency despite majority voting achieving full efficiency. In this example, a society considers an anti-corruption policy that protects taxpayers from corruption by deterring corruption. Officials know whether corruption opportunities exist, but taxpayers are uncertain about whether corruption opportunities exist. I present surprising experimental results showing that in one case where theory predicts QV will perform poorly and majority voting will perform relatively well, QV performs much better than expected and is about as efficient as majority voting.

Democratic systems are built, with good reason, on majoritarian principles, but their legitimacy requires the protection of strongly held minority preferences. The challenge is to do so while treating every voter equally and preserving aggregate welfare. One possible solution is Storable Votes: granting each voter a budget of votes to cast as desired over multiple decisions. During the 2006 student elections at Columbia University, we tested a simple version of this idea: voters were asked to rank the importance of the different contests and to choose where to cast a single extra "bonus vote, " had one been available. We used these responses to construct distributions of intensities and electoral outcomes, both without and with the bonus vote. Bootstrapping techniques provided estimates of the probable impact of the bonus vote. The bonus vote performs well: when minority preferences are particularly intense, the minority wins at least one of the contests with 15--30 percent probability; and, when the minority wins, aggregate welfare increases with 85--95 percent probability. When majority and minority preferences are equally intense, the effect of the bonus vote is smaller and more variable but on balance still positive.

Since their introduction in 1932, Likert and other continuous, independent rating scales have become the de facto toolset for survey research. Scholars have raised significant reliability and validity problems with these types of scales, and alternative methods for capturing perceptions and preferences have gained traction within specific domains. In this paper, we evaluate a new broadly applicable approach to opinion measurement based on quadratic voting (QV), a method in which respondents express preferences by 'buying' votes for options using a fixed budget from which they pay a quadratic price for votes. Comparable QV-based and Likert-based survey instruments designed by Collective Decision Engines LLC were experimentally evaluated by randomly assigning potential respondents to one or the other method. Using a host of metrics, including respondent engagement and process-based metrics, we provide some initial evidence that the QV-based instrument provides a clearer measure of the preferences of the most intense respondents than the Likert-based instrument does. We consider the implications for survey satisfying, a key threat to the continued value of survey research, and reveal the mechanisms by which QV differentiates itself from Likert-based scales, thus establishing QV as a promising alternative survey tool for further political and commercial research. We also explore key design issues within QV-based surveys to extend these promising results.

Voters have strong incentives to increase their influence by trading votes, a practice indeed believed to be common. But is vote trading welfare-improving or welfare-decreasing? We review the theoretical literature and, when available, its related experimental tests. We begin with the analysis of logrolling -- the exchange of votes for votes, considering both explicit vote exchanges and implicit vote trades engineered by bundling issues in a single bill. We then focus on vote markets, where votes can be traded against a numeraire. We cover competitive markets, strategic market games, decentralized bargaining, and more centralized mechanisms, such as quadratic voting, where votes can be bought at a quadratic cost. We conclude with procedures allowing voters to shift votes across decisions -- to trade votes with oneself only -- such as storable votes or a modified form of quadratic voting. We find that vote trading and vote markets are typically inefficient; more encouraging results are obtained by allowing voters to allocate votes across decisions.