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We investigate the performance of alternative q-majority rules when voters can engage in logrolling agreements. The environment we consider involves a group of n voters who make a number of binary decisions, e.g. whether to undertake some `projects'. Each voter obtains a payoff (positive or negative) from each project. When voters vote sincerely, and if payoffs are independently drawn from a symmetric distribution, simple majority rule maximizes the expected sum of payoffs. We propose an algorithm to identify the likely outcomes when voters can instead form logrolling agreements - i.e. mutually beneficial agreements among two or more voters to vote insincerely on some set of projects. In a simulation exercise, this algorithm is applied to a large number of randomly generated payoff matrices. The simulation results suggest that the possibility to logroll significantly and systematically alters the relative performance of alternative q-majority rules: As the number of potential projects (as well as the ability of voters to construct logrolling agreements) increases, rules requiring larger-than-simple majorities (including unanimity rule) `catch up' to and eventually outperform simple majority rule. We conduct laboratory experiments to investigate whether human subjects engage in the types of agreements assumed by our algorithm, as well as how their payoffs are affected relative to sincere voting. The set of situations we consider is such that simple majority rule would outperform unanimity rule under sincere voting, but both rules are predicted to perform equally well when logrolling is introduced. We find that subjects do engage in the agreements we predict, though less than we predicted. Compared to sincere voting, the payoffs achieved under majority rule are slightly smaller, and those achieved under unanimity rule are substantially larger than they would be under sincere voting. As predicted, both rules perform roughly equally in an aggregate payoff sense and the possibility of logrolling under unanimity rule allows Pareto-improvement.
