Strategy in Range Voting and COAF voting systems

(1) We define “range voting.” (2) More generally we define the wide class of “COAF voting systems,” (3) We reach an understanding of optimum voter strategy in COAF voting systems, at least in a certain probabilistic model, the “Gaussian model,” of how the other voters behave, and in the limit V → ∞ of a large number of voters. (4) This understanding also works for Condorcet and IRV voting (which are not COAF), albeit in a more limited way. (5) We find that range voting is the uniquely best COAF system, in a certain sense: Roughly speaking, range voting is the only COAF system which allows voters to be maximally expressive without being strategically stupid. We also find (for the first time) an infinite number of nontrivial voting systems which satisfy Mike Ossipoff’s “favorite betrayal criterion” (FBC), and show that range voting is the unique FBC-obeying COAF system with maximum voter expressivity. Most of these results were due to me in 1999-2000 but I did not write them up very well at that time. 1 Range and Approval Voting In a C-candidate election conducted with range voting your vote is an assignment of a score in the interval [0, 1] to each candidate. For example a legal vote in a 4-candidate election would be (0, 1, 0.4, 1). The candidate with the greatest score-sum wins. Approval voting [2] is the same, except that only the two endpoints of the [0, 1] interval are allowed as scores. Giving a score of 1 is said to represent your “approval” of that candidate. 2 COAF Voting Systems (Compact set based, One-vote, Additive, Fair) Let there be C candidates and V voters, C fixed, V large. Definition 1. A “COAF Voting System” is a single-winner election that works as follows. Each voter chooses, from a fixed Compact set S ⊂ R of “allowed votes,” One C-vector. The vectors are Added. The maximum entry in the summed C-vector corresponds to the winner. Such a system is “Fair” if S is invariant under the group of C! permutations of the C coordinates of R . Voting system The Compact Set of Allowed Votes Plurality The C permutations of (1, 0, 0, 0, . . . , 0) Approval The 2 vectors of form (±1,±1,±1, . . . ,±1) Dabagh “vote and a half” The (C − 1)C permutations of (2, 1, 0, 0, 0, . . . , 0) Used on Nauru Island The C! permutations of (1, 1/2, 1/3, 1/4, . . . , 1/C) Borda The C! permutations of (C − 1, C − 2, C − 3, . . . , 1, 0) “Vote for-and-against” The (C − 1)C permutations of (+1,−1, 0, 0, 0, . . . , 0) Anti-plurality The C permutations of (−1, 0, 0, 0, . . . , 0) Continuum “cumulative voting” The ~x with ∑C j=1 xj = 1 and xj ≥ 0(∀j) Boehm’s “signed voting” The 2C permutations & sign changes of (±1, 0, 0, 0, . . . , 0) L2-sphere voting The ~x with ∑C j=1 x 2 j = 1 Range Voting The unit C-dimensional hypercube [0, 1] Figure 2.1. Some interesting COAF voting systems. Some well-known voting systems which are not COAF include: Condorcet systems, Instant Runoff (transferable vote), Bucklin, Woodall-DAC [4]; in all four of these cases the votes are preference orderings of the C candidates. N ∗Non-electronic mail to: 21 Shore Oaks Drive, Stony Brook NY 11790. Some people also allow “truncated preference” vote-vectors, such as (3, 2, 12 , 1 2 ), in (their version of) Borda.