Market-Based Explanations of Collective Decisions

We consider approval-based committee elections, in which a size-k subset of available candidates must be selected given approval sets for each voter, indicating the candidates approved by the voter. A number of axioms capturing ideas of fairness and proportionality have been proposed for this framework. We argue that even the strongest of them, such as priceability and the core, only rule out certain undesirable committees, but fail to ensure that the selected committee is fair in all cases. We propose two new solution concepts, stable priceability and balanced stable priceability, and show that they select arguably fair committees. Our solution concepts come with a non-trivial-to-construct but easy-tounderstand market-based explanation for why the chosen committee is fair. We show that stable priceability is closely related to the notion of Lindahl equilibrium from economics. number of weaker properties, such as extended justified representation (EJR) [2], proportional justified representation (PJR) [19] , and justified representation (JR) [2] . In fact, the core is so strong that, for the time being, it is not known whether there always exists a committee in the core for approval-based elections. However, even this strong property can sometimes allow dramatically unfair committees, as the example below illustrates. Example 1. Fix an integer L. Consider the following instance with n = kL voters. Voters 1, . . . L approve candidates c 1 , . . . , c k . For each i ∈ [k -1] voters iL + 1, . . . iL + L -1 approve candidate c k+i . The remaining k -1 voters approve candidates c 2k , . . . , c 3k-2 , each voter approving a different candidate. This instance is depicted below. The candidates correspond to the boxes; each candidate is approved by the voters who are below the corresponding box. For this instance, the committee W 1 = c 1 , . . . , c k (the committee marked green) is in the core. In fact this committee would be uniquely selected by the following natural rule: "among all committees in the core (assuming it is non-empty), select the one that maximizes the total number of approvals". This committee gives zero satisfaction to a majority of the voters. One can argue that a committee that consists of the blue candidates, along with one green candidate, e.g., W 2 = c 1 , c k+1 , . . . , c 2k-1 , is a much fairer choice. Similar observations have been made by Bredereck et al. [4] for the axiom of extended justified representation (EJR). They observe that in practice many very different committees satisfy EJR, and concluded that EJR on its own does not guarantee a sensible selection of committees.