## Puzzles about or related to Voting & Democracy

Puzzles range from "easy" to "so hard nobody can currently solve them" (i.e. "open"); we've taken pity on you by giving a very-subjective "difficulty level" from [0] to [9]. If you can solve one of the open [9] problems please let warren.wds at gmail.com know! [You can also contribute new puzzles to him...]

For terminology, it may help to consult our glossary. "Range Voting" will generally mean "continuum range voting" and not range voting with scores restricted to the discrete set {0,1,...,99} for mathematical analysis purposes.

#6: Dice[2]

#7: Probabilities of "Condorcet cycles."

#8: Probabilities of Favorite-Betrayal Lesser-Evil scenarios with Condorcet voting in 3-candidate elections with equal rankings permitted

#9 (open Now solved! – voting systems in which semi-honest voting is strategic)

#10 (cake cutting)

#11 (pie-style cake cutting)

#12 (cutting of 2-dimensional cakes)

#14 (mostly open – Social science/economics research. Suggested by Aaron Krowne)

#15 (open Now solved! – multiwinner EP & PR voting systems)

#16 (Raising the mean of both sets by shuffling them – easy)

#18 – IRV refuses to elect a candidate who beats every other pairwise by 99:1 margin?

#19 – A situation with random other voters where your strategically best range vote is honest.

#20 – (partly open) How often do "Condorcet cycles" arise in real elections with rank-order ballots? Give some prominent real historical examples of such elections

#21 – (partly open) how many arcs must be deleted from an n-node directed graph to get rid of all its cycles?

#23 – Avoiding favorite-betrayal (Chris Benham)

#24 – What is the probability plurality elects the candidate a majority least-likes?

#27 – A quantity like the "Ramsey numbers" but defined for directed rather than undirected graphs.

#32 – The "Three Stooges voting problem" (Alan Frieze)

#33 – Wisest allocation of vote-weights

#34 – How much ignoring does Instant Runoff Voting do?

#35 – Gerrymandering cancellation theorem

#39: Utility=Log(Wealth)?

#40: Feel alike ⇒ vote same?

#41: Invisibly Corrupted Computer Programs.

#42: Probability of Condorcet cycle

#43: Approval voting strategy most likely to elect Condorcet winner

#44: Shannon Utility Honesty property

#45: (Open) Min-Max matrix product

#46: Probability of unclear election winners

#47: "Vote for N" leading to unclear election winners

#48: How many votes to count to get confidence?

#49: Machievellian Agenda Manipulation

#50: How many checks to get confidence?

#51: Getting true (not fake) randomness

#52: How many votes to count to get 100% certainty of winner?

#53: Districting is unavoidably "chaotic"

#54: Better than average

#55: How often is voting honestly worse than not voting with Instant Runoff Voting?

#56: How often is voting honestly worse than not voting (with Condorcet)?

#57: How often is dishonesty better strategy than honesty with IRV?

#58: Gerrymandering

#59: Gerrymandering "Squareland"

#60: Range and Approval as "universal" voting systems

#61: Voting system "stability" (S.J.Brams)

#62: How often is dishonesty better strategy than honesty with Condorcet?