Constant scoring rules for choosing one among many alternatives

Conclusions

When a group is to select a decision alternative from a finite set of m ⩾ 3 feasible alternatives, it is often desirable to choose the Condorect (majority) alternative when there is one. Hence Condorcet efficiency —the likelihood that a selection procedure will choose the Condorcet alternative, given that one exits — is an important measure for comparing selection procedures.

This study compared Condorcet efficiencies for constant voting rules C _k, where C _k has each member of the group vote for his k most preferred alternatives. Three different probabilistic procedures for generating voter preference profiles were discussed. Previous results, based largely on simulation data, were summarized, and new analytical results that corroborated our previous findings were proved. In particular, for the IC procedure, as the number of voters tends to infinity, Condorcet efficiencies are single-peaked and symmetric about m/2. And, for the MC procedure, the Condorcet efficiency of rule C _k equals the Condorcet efficiency of rule C _m−k for k = 1, 2,; ..., m − 1. This symmetry property of equal efficiencies for C _k whose k are equidistant from m/2 does not hold, however, for the IAC procedure, even in the limit for large numbers of voters.

The general evidence collected thus far on C _k-rules strongly suggests that a value of k that maximizes Condorcet efficiency for m alternatives never exceeds m/2 and will sometimes be less than m/2. Moreover, it appears most likely that Condorcet efficiency drop off as k moves away in either direction from the efficiency-maximizing value.