From Arrow’s Impossibility to Schwartz’s Tournament Equilibrium Set
Perhaps the most influential result in social choice theory is Arrow’s impossibility theorem, which states that a seemingly modest set of desiderata cannot be satisfied when aggregating preferences . While Arrow’s theorem might appear rather negative, it can also be interpreted in a positive way by identifying what can be achieved in preference aggregation.
In this talk, I present a number of variations of Arrow’s theorem–such as those due to Mas-Colell and Sonnenschein  and Blau and Deb –in their choice-theoretic version. The critical condition in all these theorems is the assumption of a rationalizing binary relation or equivalent notions of choice-consistency. The bulk of my presentation contains three escape routes from these results. The first one is to ignore consistency with respect to a variable set of alternatives altogether and require consistency with respect to a variable electorate instead. As Smith  and Young  have famously shown, this essentially characterizes the class of scoring rules, which contains plurality and Borda’s rule. For the second escape route, we factorize choice-consistency into two parts, contraction-consistency and expansions-consistency . While even the mildest dose of the former has severe consequences on the possibility of choice, varying degrees of the latter characterize a number of appealing social choice functions, namely the top cycle, the uncovered set, and the Banks set [3,9,4]. Finally, I suggest to redefine choice-consistency by making reference to the set of chosen alternatives rather than individual chosen alternatives . It turns out that the resulting condition is a weakening of transitive rationalizability and can be used to characterize the minimal covering set and the bipartisan set. Based on a two decades-old conjecture due to Schwartz , the tournament equilibrium set can be characterized by the same condition or, alternatively, by a weak expansion-consistency condition from the second escape route. Whether Schwartz’s conjecture actually holds remains a challenging combinatorial problem as well as one of the enigmatic open problems of social choice theory.
Throughout the presentation I will discuss the algorithmic aspects of all considered social choice functions. While some of the mentioned functions can be easily computed, other ones do not admit an efficient algorithm unless P equals NP [13,5,7].