For mathematicians

On this page we discuss some mathematical properties of range voting, and related areas, for those who really want to know.

• Puzzle page (puzzles - answers reachable by CRV members). Note: some of these problems are open ones that would be excellent research topics.
• A completely idiotic Instant Runoff Voting (IRV) election.
• Crude Axiomatic characterization of range voting and a precise writeup (pdf, 10 pages) and (ps) which proves many key theorems about range voting. See also Marcus Pivato's beautiful axiomatic characterization of range voting (2011).
• Why Arrow's "impossibility" theorem does not apply to range voting.
• 3-candidate instant runoff voting paradox probabilities.
• About voter honesty vs. strategy and the Gibbard-Satterthwaite theorem.
• "Properties" of voting systems – a bad yardstick
• Theorem (& proof) explaining why Instant Runoff Voting (IRV) leads to 2-party domination (just like under the flawed plurality system it was supposed to "fix.") Also, shows, remarkably, that the probability of a FBLE (favorite-betrayal lesser-evil) situation in a random large IRV election is arccot(√2)/π≈0.195913, albeit arguably 100%, and also does a similar FBLE analysis for Condorcet voting. More on this.
• Some quibbles over the definition of "Condorcet" method
• What if the Approval and Condorcet winners conflict? Amazing "no-conflict" theorem! (Simplified)
• A mathematical way to abolish Gerrymandering
• Definition of "Bayesian Regret" and measurements indicating Range Voting is best by this yardstick
• Argument that strategic range voting approximately maximizes number of "pleasantly surprised" voters
• Venzke proof that Condorcet voting methods exhibit "favorite betrayal." Another Venzke Proof of a related result. Another.
• Unintuitive examples where approval-style range voting is not strategically optimal. But these examples do not occur (i.e. the departure of the best approval-style vote from optimality is negligible) in large realistic elections in which 3-or-more-way lead-ties (and near-ties) are regarded as negligibly unlikelier than 2-way lead-ties.
• Unintuitive examples where dishonest approval-style range voting is strategically optimal. But this cannot happen in 3-candidate elections, nor can the dishonesty ever involve your favorite (or your most-hated) candidate. Also it in practice in 4- and 5-candidate elections seems rare and has small probabilistic effect.
• Range voting strategy experiments showing that honesty is actually a pretty good strategy.
• Range voting problems caused by strategic voters.
• Survey of voting methods obeying Mike Ossipoff's "Favorite Betrayal Criterion" (FBC). We conjecture Continuum Range Voting is the unique voting method (aside from obvious variants) obeying both FBC and immunity to candidate-cloning (assuming voters express slight preferences among clones).
• Theorem and proofs showing (essentially) that no pure-rank-order-ballot voting system can obey both the favorite betrayal criterion and immunity to candidate cloning. But range voting obeys both. More precisely, we give numerous sets of criteria that range voting obeys but no pure-rank-order-ballot voting system can obey.
• Computer simulations of "random normal elections."
• Jan Kok's brilliantly simple idea showing how to run range voting elections on ordinary voting machines designed for plurality elections.
• Useful index of election-pathology examples.
• Quick proof of Duncan Black's singlepeakedness theorem.
• Collection of skew Hadamard matrices of all orders≤100, including exhaustive list when n≤28.
• Survey of directed graph Ramsey Numbers.
• Crude discussion of theoreticians' attempts to get optimality theorems for voting methods.
• Numerical simulations about apportionment methods; they support Webster's = Sainte-Laguë's method.
• New and superior apportionment method(s). And an introduction to apportionment.
• Jobst Heitzig thoughts on utilities, efficiency, equality. An essay on the mathematical foundations of utility and its history.
• Theorems about point runoff systems: all of them are non-monotonic, exhibit "no show paradoxes," exhibit "favorite betrayal," and disobey consistency with respect to partitioning into districts.
• Every weighted positional rank-order ballot system (except for anti-plurality voting) exhibits favorite betrayal, and every weighted positional system is vulnerable to candidate-cloning.
• Collections of election data.
• In 2006 Balinski & Laraki proposed a voting system very similar to range voting. But it is different because based on medians not averaging. Here is a critique.
• Some simple voting optimality theorems; pretty trivial and presumably previously known, although I have not seen such clean statements before. But I believe Bayesian Regret is a better optimality measure than any of these measures.
• Proof that "participation failure" is forced in Condorcet methods with at least 4 candidates.
• An interesting and simple model of issues and voting the YN model in which range votng performs optimally (with honest voters) while most rival voting methods can perform pessimally or near-pessimally badly.
• Criticism of the voting-related work of mathematician Don Saari.
• Two research grant proposals we made in June 2007: ElectionTracker, IEVS.
• Stay away from Voting Matters; it appears to be a biased pseudo-journal.
• Combinatorics of halving lines.
• Analysis of different voting systems vis-a-vis Los Angeles Mayoral Elections.
• Analysis of true versus instant runoffs based on real-life data – true runoff winners differ from plain-plurality winners over three times as often.
• The Romania 2009 presidential election featured one or more high Condorcet cycles, and the apparent approval-voting and range-voting winners officially finished 3rd and/or 6th.

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