Dealing with Randomization in Economic Design

Economic design has seen the emergence of a number of attractive randomized rules for allocation, matching, and even voting. In this essay, I would like to raise three fundamental questions in the context of randomization in economic design, which I think have not been sufficiently addressed in the literature. I will take the perspective of social choice theory and refer to randomized collective choice rules, which map a collection of individual preference relations to a socially most-preferred, representative, or otherwise adequate lottery over the alternatives. However, since alternatives may represent allocations (with or without payments), matchings, coalition structures, or any other type of economic outcome, the following observations should be equally relevant for mechanism design, market design, auction theory, matching markets, random assignment, fair division, and so forth. When aggregating the preferences of multiple agents into a single collective choice, it is easily seen that certain cases call for randomization or other means of tie-breaking. For example, if there are two alternatives, a and b, and two agents such that one prefers a and the other one b, there is no deterministic way of selecting a single alternative without violating basic fairness conditions (referred to as anonymity and neutrality). However, ex ante fairness can easily be restored by returning an even chance lottery over a and b. When allowing for more randomization than is necessary to break ties, classic impossibilities such as Arrow’s theorem, the Gibbard-Satterthwaite theorem, the no-show paradox, or the incompatibility of Condorcetconsistency with population-consistency can be circumvented under suitable assumptions about the agents’ preferences over lotteries (e.g., Gibbard, 1977; Brandl et al., 2016, 2017; Brandl and Brandt, 2017). This is reminiscent of game theory, where the availability of randomized strategies is a crucial requirement of fundamental results like the minimax theorem and the Nash equilibrium existence theorem. In the context of voting, two rules that turned out to be particularly desirable from an axiomatic perspective are random (serial) dictatorship and a rule that returns maximal lotteries (whose existence follows from the minimax theorem) (see, e.g., Fishburn, 1984a; Brandl et al., 2016; Aziz et al., 2017). These rules have also been considered in matching and allocation subdomains of the general voting domain where random serial dictatorship is known as random priority and maximal lotteries as mixed popular