text:
Please only ask for at most one
solution at a time, i.e. all passwords except one should be left
blank! (Hit the "reset" button to make all blank.)
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.
(Return to main page.
See puzzle index.
Puzzle answers only available to those who know the password, i.e, CRV members!
Join CRV)
Feb 2007: All three original wide-open math problems have now been (pretty much) knocked off by
the combined efforts of W.D.Smith & Forest W. Simmons! They're being demoted
to ≤[8] and we added some new
Puzzle #0 (Voting power – ultra-easy speed problem):[0]
Three stockholders own 47%, 34%, and 19% of the stock in some corporation, respectively.
What are their relative powers in a 2-way vote?
Type password to get answer:
Puzzle #1: The minimum it takes to elect US president:[1]
What is the smallest percentage of votes needed to elect a US president?
(Assume 2004's population distribution, and a 2-candidate race;
disallow "faithless electors," and assume no need to send
any election to coin-tossing, congress, or the courts for resolution.
Assume state turnouts proportional to populations – or don't [which leads to
a different answer].)
Puzzle #2: Small elections in which many voting systems all disagree:[5]
Find an example election in which
Approval Voting,
Borda Voting,
Plurality Voting,
Plurality Voting with Top-Two Runoff,
Instant Runoff Voting (IRV),
Condorcet methods,
all return different unique-winners.
Election examples with fewer voters and fewer voter-types are preferred
as "more elegant."
Puzzle #3 (interesting – easy once you see it, otherwise hard!):[4]
Three people enter a room and a red or blue mark is made on each person's forehead.
The color of each mark is determined by a coin toss, with the outcome of one coin
toss having no effect on the others. Each person can see the others' marks but not his own.
No communication of any sort is allowed,
except for an initial strategy session before the game begins.
Once they have had a chance to look at the other marks,
they must (simultaneously) guess the color of their own marks, or pass.
The group shares a hypothetical $3 million prize if at least one player guesses
correctly and none guess incorrectly.
The problem is to find a
strategy for the group maximizing its chances of winning the prize.
For example, one obvious strategy for the players would be for one player always to guess
"red" while the others pass. That would give the group a 50% chance of winning
the prize. Can the group do better?
Puzzle #4: Non-monotonicity probabilities in IRV voting:[6]
Consider 17 voters electing one winner from 3 candidates using
IRV (Instant Runoff) voting.
(Each vote is a preference ordering among the three candidates;
the candidate with the fewest top-rank votes is eliminated repeatedly until only
one – the winner – remains.)
Among all the possible elections of this sort: what fraction have the
unhappy "non-monotonic" property
that the winner would have lost to some loser L,
if only some subset of identically-voting voters had
changed their top-rankings away
from L? (Disallow elections involving ties.)
Extra credit:
What if, instead of 17 voters, we allow the number of voters to tend to infinity?
Open: Solved! [8]
What if we then also allow the number of candidates to tend to infinity?
Puzzle #5 (open now solved! – voting systems obeying both ICC and AFB):[7]
Two desirable properties of a voting system – both of which Range Voting has –
are "immunity to candidate-cloning" (ICC) and "avoiding favorite betrayal" (AFB).
AFB: voters should never have strategic incentive to "betray" their favorite
candidate by voting him below some other.
ICC: political parties should be unable to usefully manipulate an election
by running clones of their own, or of an opposed, candidate; voters here are assumed
to vote honestly and to have only tiny preferences (which they may express in their
votes, if they exist) among the clones.
In contrast: the USA's present "plurality" voting system fails both tests:
It was strategically unwise in Florida in 2000 to vote for Nader even if he was your favorite;
and by sponsoring extra "clones" of your top opponent, that can cause their vote
to be split and hence cause both clones to lose, and you to win.
Is range voting (and obvious variants of it) the only nontrivial single-winner
voting system to satisfy both properties?
(A "voting system" inputs a set of votes and [deterministically] outputs the name of a winner.
The votes are numerical scores for some-or-all candidates, or rank-orderings of
some-or-all candidates, optionally with ranking-equalities permitted.)
Note: Many voting systems are known (beyond just variants of range voting) which
satisfy AFB, and many also are known
(such as
[but not BTR-IRV!],
Woodall-DAC, and Heitzig River)
that satisfy ICC.
But I do not know of any systems besides range voting variants
which satisfy both simultaneously.
Partial credit: Must any such system employ continuum votes?
What if the problem is restricted to systems based on pure-rank-order ballots
(with candidate-equalities forbidden)?
Puzzle #6: Dice:[2]
Suppose there are 4 dice: Blue, Green, Red, and White.
These dice have different numbers than usual printed on their six faces.
After observing a long sequence of experiments rolling pairs of these dice,
you conclude that
Puzzle #7: Probabilities of "Condorcet cycles."
Assume there are V voters, V large, each of whom cast any of
the 6 possible votes A>B>C, B>A>C, A>C>B, B>C>A, C>A>B, C>B>A
randomly, independently, and with equal likelihood for all 6 votes,
in a 3-candidate Condorcet election. In other words, all 6V
elections are equally likely.
Then: what is the probability of an A>B>C>A clockwise "Condorcet preference cycle"?
Extra credit: what about if we now permit candidate-equalities in votes
so that twelve votes
Puzzle #8: Probabilities of Favorite-Betrayal Lesser-Evil scenarios with
Condorcet voting in 3-candidate elections with equal rankings permitted:
Again suppose there are V voters and all 12V elections
are equally likely.
We shall only be interested in the V→∞ limit.
Suppose the election proceeds as follows.
If a candidate exists who pairwise-beats (or at least ties)
every other, then he is the winner. (And we disallow elections with more than one winner.)
Otherwise, the candidate X, suffering at most one
pairwise defeat and with that defeat being the "weakest,"
There are two possible ways to define that Y>X is the "weakest defeat":
In either system: what is the probability that this election will include a
"Favorite-Betrayal Lesser-Evil (FBLE) scenario" in which, say, A wins,
but some "C-top and B>A" voters by dishonestly rating their favorite C below top,
can cause the "lesser evil" B to win instead of A (a result they prefer)?
Note: It only counts as FBLE if those voters can only achieve their
B-wins goal by demoting C below top.
Puzzle #9 (open now solved! – voting systems in which semi-honest voting is strategic):
Let a "voting system" mean a map from a set of votes (each a rank-ordering of the candidates
with equality-rankings permitted, or an assignment of a real number score in [0,1] to
each candidate) to the identity of a single "winner" candidate.
Call a voter's vote X "semi-honest" if it, while perhaps differing from his honest opinion,
has the same rank-ordering
as a limit of a sequence of score-assignments that agree in their ordering
with the honest one.
For example, if the voter honestly feels A>B=C>D>E, then a vote A=B=C>D=E
would be semi-honest, a vote A>B=C>D>E would be honest and hence automatically semi-honest,
but both B>A and B>C would be dishonest statements that are not even semi-honest.
Does there exist a nontrivial voting system for 4-candidate
elections in which semi-honest voting is always strategically optimal?
(Range voting has this property for elections with ≤3 candidates.)
Puzzle #10 (cake cutting):
A well known way to cut a cake into two pieces that assures
that each person gets at least half of
the value of the cake (where the two players may value
different cake-parts differently, e.g.
one likes chocolate and another likes cherries) is the "cut and choose" protocol
where player A cuts the cake and then player B chooses his piece.
Can you devise a similar protocol for three cake-eating players assuring each
will regard his/her piece as containing at least as much value as either one
of the other pieces?
Puzzle #11 (pie-style cake cutting):
In the previous problem
completely arbitrary styles of cake-cutting were permitted,
i.e. players could receive arbitrary-shaped (and perhaps not even connected)
pieces. In that case no matter what the number
N of players, there always exists a way to cut the cake into N pieces
so that each player would regard his own piece as at least as valuable (by his own
measure of value) as any of the others.
But now suppose
that the cake (regarded as a circular disk)
must be cut into wedges along radial line segments, with each player getting
exactly one wedge.
does a way exist to
cut the cake into N pieces
so each player regards his own piece as at least as valuable
as any of the others?
Solve the cases N=3 and N=500.
Puzzle #12 (cutting of 2-dimensional cakes):
Given a 2-dimensional disk "cake" and N players each of whom have (perhaps) different
everywhere-positive-and-finite value-densities on that cake.
Is it possible to cut the cake into N disjoint
pieces, one for each player, such that the players
are "envy-free," i.e. regard themselves as owning an at-least-as-valuable piece
of the cake as any rival? We ask this question twice:
Puzzle #13 (range-voting with outlier-discarding):
Suppose each voter's "vote" is an integer
numerical score from 0 to 99 for each candidate, for example a legal vote
in a 4-candidate election might be (99, 54, 0, 99).
Suppose the "final score" for a candidate is the average of all his scores that remain
after you discard the top X% and the bottom X% of them.
If X=50, this
is just the median score, and if X=0 then this is just the usual kind
of range voting. These are both important special cases. But more generally
any fixed X with 0≤X≤50 can be considered.
One of the advantages of ordinary range voting, which appears to be sacrificed in this
more general kind of voting, is the fact that with range voting only a small
amount of information may be sent by each precinct to central tabulating, summarizing
all the ballots (no matter how many) in that precinct. The entire set of
ballots need not be sent.
Puzzle #14 (mostly open – Social science/economics research. Suggested by Aaron Krowne):
Compare macro-economic
metrics, including soft ones like "economic freedom" (a number of indexes for
this exist). One might hypothesize that countries with "better voting
systems" will progress "forward" in these various metrics faster than countries
with worse voting systems.
There may already be enough historical data in the 20-50
most recent years past to reveal some
interesting economic-political correlations.
Colloquially, this seems to be happening. Median incomes in the US
[which has one of the worst voting systems]
continue to fall (since the 70s), the Gini index is rising, class
mobility is falling, the prison population is rising (and accelerating),
the black market sector is growing, and economic freedom is falling.
Meanwhile Ireland and Switzerland [with two of the best, in my opinion,
governmental setups] have impressive economies. (2010: Oops! – Ireland's
banks just blew sky high due to massive risk taking and speculation, then
the government bailed them out sending the ountry into huge debt over 10×GDP,
but Ireland was supposedly a great success when we wrote this, i.e. before that happened...)
Puzzle #15 (open now solved! – multiwinner EP & PR voting systems):
The goal of a "multiwinner voting system" is to, from the "votes,"
determine W winners from C
candidates where 0<W<C.
The system is "proportional" (PR) if,
supposing the voters and candidates each are colored one of
a finite number of colors (i.e. voter Joe is "green," candidate Mary is "black"), and
supposing each voter prefers each candidate of his own color above all others
(and says so in her vote), then the winning slate of candidates will
have the same color composition as the voters (up to unavoidable integer-roundoff effects
and limitations caused by perhaps too few candidates existing of some color).
For example, the famous Hare/Droop-STV system [single transferable vote with
"Droop quotas" and "reweighting"] invented by Hare and Droop in 1800s England,
is PR. That is, if there are 60% red, 30% black, and 10% white voters,
then we will automatically get 6,3, and 1 red, black, and white winners respectively in
a Hare/Droop-STV
election with W=10 assuming enough candidates of each color are available.
The voting system is required to depend upon the votes only, hence
does not know
the candidate- or voter-colors.
"Votes" are either preference orderings or numerical scores
(and perhaps other things would also be permissible – if you can solve this we
are willing to be generous with the conditions).
The system is "efficiently parallelizable" (EP) if an election with an enormous number
of voters can be handled by efficient combination of subresults computed
in each election district from only the votes in that district, and these subresults
consist of a very small amount of data compared
to the large number of votes that they summarize. For EP to hold,
we require all this data
to be transmitted unidirectionally, i.e.
For example the plurality system is EP because each district can simply compute
its total vote counts for each candidate, then pass only these subtotals on to the
central election agency.
Does there exist a nontrivial multiwinner voting system which is both
PR and EP?
(And if "yes," then please construct one!)
Puzzle #16 (Raising the mean of both sets by shuffling them – easy):
Suppose you have two sets A and B of numbers.
Now rearrange the numbers to get two new sets A' and B'
(same numbers, but moved around).
For example we could have
A={1, 3} and B={2, 4}
which after rearrangement becomes
A'={1, 2} and B'={3, 4}.
Note that in this example, the mean of A decreased but the mean of B increased,
with overall mean staying the same.
Question: Is it possible to make the means of both sets increase?
(And if "yes," then please construct example!)
Puzzle #17 – Probability of "Peru scenario":[6]
Consider V voters electing one winner from 3 candidates using
IRV (Instant Runoff) voting.
(Each vote is a preference ordering among the three candidates;
the candidate with the fewest top-rank votes is eliminated repeatedly until only
one – the winner – remains.)
Assume all 6V elections are equally likely.
What is the probability, conditioned on there being a
"Condorcet winner"
W who would pairwise-beat
every other candidate,
that W will be eliminated in the first IRV round?
(Disallow elections involving ties and concern yourself only with
the V→∞ limit.)
We call this the "Peru scenario" since it apparently happened in the
2006 Peru national election.
Extra Credit:
Also evaluate the same probability in the alternative
Dirichlet probabilistic model.
Puzzle #18 – IRV refuses to elect a candidate who beats
every other pairwise by 99:1 margin?[2]
In the Instant Runoff (IRV) voting method,
Prove it is possible for a candidate L to win, even though a candidate W "should have"
won in the sense that, for every candidate X, including X=L,
the ballots indicate that W pairwise-beats X by at least a 99:1 margin?
Puzzle #19 –
A situation with random other voters where your strategically best range vote is honest.
Suppose there are N>0 voters in addition to you in a
range-voting election with 3 candidates.
Suppose you know absolutely nothing about how
the other N voters voted, i.e. regard all
their scores for all candidates as 3N random numbers
independently uniform within the permitted range.
(Equivalently: regard every possible vote by each of them as equally likely.)
Suppose your true utilities for the elections of the candidates
are 0 for the worst candidate,
1000 for the middle candidate, and 2000 for the best candidate.
Prove that for each N>0, including the limit when N→∞, your
uniquely strategically-best
range vote is the "honest" one (0, 0.5, 1) assuming the score-range
is the interval [0,1], and in particular that this
honest vote yields superior expected utility
for you than either "exaggerated" approval-style
vote (0, 0, 1) or (0, 1, 1).
Also, prove this is still true even if your utility for the middle candidate is not
exactly 1000 – i.e. if it is slightly perturbed away from 1000, with the size
of the perturbation being small enough compared to some power of N, then your
strategically best range vote is not approval-style.
Puzzle #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.
Puzzle #21 – (partly open) how many arcs must be deleted from an n-node
directed graph to get rid
of all cycles?[8]
Let F(n) denote the maximum, over all n-node directed graphs G (note: singly-directed arcs only;
bidirected arcs not allowed), of the minimum number of arcs you need to erase from G in order
to cause G to become acyclic. For example, F(2)=0 and F(3)=1.
Find values of F(n), upper and lower bounds on F(n), and asymptotics.
Puzzle #22 – How bad can Condorcet cycles be? [7]
As is well known, voters who provide rank-order ballots can create a
preference cycle in which each candidate
is beaten pairwise by some other (and in which, therefore, it is not easy to
claim there is any clear winner). This can happen with 3 candidates.
We now investigate just how nasty such things can be.
Is it possible for every pair of candidates to both be pairwise-beaten
by some other candidate? If so, find the smallest number of candidates that makes that possible.
Is it possible for every triple of candidates to all three be pairwise-beaten
by some other candidate? If so, find the smallest number of candidates that makes that possible.
Is there any upper limit on K such that
every K-tuple of candidates can all K be pairwise-beaten
by some other candidate?
What is the largest possible number of 3-candidate preference cycles that can
occur simultaneously in an N-candidate election?
What is the largest possible number of 4-candidate preference cycles that can
occur simultaneously in an N-candidate election?
Is it possible for every 4-tuple of candidates (considered on their own with
all others erased from all votes) to contain at least one preference cycle
in an N-candidate election with N≥5?
Puzzle #23 – Avoiding favorite-betrayal (Chris Benham):
Is there a nontrivial
voting system in which the votes are strict rank-orderings of all the candidates,
and in which there is never any strategic incentive
for a voter to "betray his favorite" by ranking him below top?
Puzzle #24 – What is the probability
plurality elects the candidate a majority
least-likes? [6]
Consider 7 voters electing a unique winner from 3 candidates using
plurality voting.
Among all the possible elections of this sort: what fraction have the
unhappy property
that a majority of the voters
agree that the winner is actually the candidate they least-like?
(Assume all 6 possible votes A>B>C,
A>C>B, B>A>C< B>C>A, C>A>B, C>B>A equally likely, all voters independent.)
What if now instead of 7 voters, we consider the
limit as the number of voters tends to infinity, and ask what is the probability
that the plurality winner in a vote for best candidate, coincides with the plurality "winner"
in a vote for worst candidate?
(Hint: your probabilities should range between about 2% and 10%.)
Type password to get answer:
Puzzle #25 – Pancyclicity:[6]
In any N-player
round-robin tournament in which each player has the
same score ("score" is number of other players he defeats pairwise; we only consider
"win" and "loss" game results with "draws" are forbidden), prove that
each pairwise defeat A>B is part of at least one K-player preference cycle
(for example, with K=4, a 4-player cycle including A>B might be "A>B>C>D>A",
but A>C>B>D>A does not count as a "4-cycle including A>B")
and this is simultaneously true for every cycle-length K with 3≤K≤N.
Puzzle #26 – Super-bad tournaments:[5]
Call an N-player
round-robin tournament "super-bad" if every player A loses pairwise to some other
player, who loses to another,... who loses to player B, i.e. there is a path
of pairwise defeats leading from A to B, for every A and B.
This seems about as bad as things possibly could be (from the
standpoint of determining who should be
the "winner") but we shall see that it is an extremely common state of affairs.
Prove a tournament is super-bad if and only if each player is a member of
at least one "cycle" of K defeats, for every value of K from 3 to N
Assuming each pairwise defeat-or-victory is determined by a coin flip
("random tournament"),
find the probability P(N) that an N-node tournament is super-bad.
In particular, compute P(20) accurate to 8 decimals.
Let us say that a player who either pairwise-defeats,
or "defeats" via a 2-long path, each other player,
has "bragging rights."
In the same probabilistic model as in (b), show that,
with probability tending exponentially toward 1 as N→∞,
every player has bragging rights.
Show that not only does every player A "defeat" every player B via a 2-hop defeat-path AXB,
indeed this happens (with probability→1 exponentially as N→∞)
for at least N/5 distinct 2-defeat paths.
Further, there are (again with probability→1 exponentially as N→∞)
at least
Puzzle #27 – A quantity like the "Ramsey numbers" but defined for
directed rather than undirected graphs.[8]
Let R(n) denote the smallest number Q such that every
tournament with ≥Q players
(i.e, complete directed graph on ≥Q nodes)
contains an acyclic n-player subtournament.
Re-expressed in voting-theory language:
R(n) is the smallest number Q of candidates in a ranked-ballot election,
such that it is guaranteed that at least one subset
containing at least n of those candidates can be unambiguously
ranked in relation to one another!
Find values of R(n), upper and lower bounds, and asymptotic behavior.
In particular, prove that R(n) grows exponentially with n.
Puzzle #28 –
How many votes suffice to create any "tournament" configuration?[8]
In a ranked-ballot voting system,
Show that any "tournament" (i.e. configuration of (n-1)n/2 pairwise-beats relationships among
n candidates) is attainable where "A beats B"
when more voters say A>B than B>A in their ballots,
and there are enough voters.
(And let us agree not to allow "ties.")
Define M(n) to be the minimum number of rank-order-ballot
votes required to create any n-player tournament
in the this way (maximized over all such tournaments; elections with "ties" not permitted).
Then show M(2)=1, M(3)=M(4)=M(5)=M(6)=3, and M(n) is always an odd number and M(n+1)≥M(n).
Show M(n)≤2n-1 and M(n)≤n+1.
Show M(n)≥Cn/lg(n)
for all sufficiently large n,
where lg(X)=log2X,
if C is any constant with 0<C<1/2.
Show M(n)≤Kn/lg(n)
for all sufficiently large n,
for some positive constant K.
But what if, to declare that "A beats B," we need, not a mere majority of voters,
but in fact a fraction-f supermajority (for some f=ε+1/2)?
Prove then that unattainable tournaments exist (for each ε>0)
and indeed that an unattainable tournament with O(ε-2) players
exists – and not only exists, but in fact a random tournament of that size
is highly likely to be unattainable.
Puzzle #29 – Team comparisons. [2]
Suppose two "teams" of N numbers are compared by
comparing the largest number on the first team, with the largest on the second;
then the second-largest on the first team, with the second-largest on the second;
and so on, with the team that wins the most comparisons being "the better team."
(This is actually a pretty realistic model of, e.g, chess teams.)
(team mergers)
Suppose A is a better team than B, and C is a better team than D.
Does it (or does it not) follow that the union of A and C, is a better team than
the union of B and D?
Suppose A is a better team than B, and B is a better team than C.
Does it (or does it not) follow that A is a better team than C?
Which possible "betterness structures" are possible among N-man teams?
("different" comparison method)
Suppose we instead compare teams by comparing every number of team #1 with
every number on team #2; whichever team wins the most comparisons, is "better."
For 3-man teams, show that this "different" method of comparing teams, is actually
equivalent to the old method.
Puzzle #30 – Election where Borda and Approval voting maximally disagree. [2]
Find an example election with 4 candidates in which
Approval and
Borda Voting
give reversed results.
(Solutions with fewer voters preferred.
For extra credit, can you generalize your example
to N candidates?)
Puzzle #31 – Election where Borda reverses [2]
Find an example election with N candidates in which
Borda Voting
gives reversed results when the last-place (or first-place) candidate is removed.
Even better: do this for every "weighted positional" voting system.
Puzzle #32 – The "Three Stooges voting problem" (Alan Frieze):
There are 3 candidates for Mayor: A.Moe, B.Larry, and C.Curly.
There are N voters and the winner is a function
F(x1, x2,..., xN)
of the votes of each voter, xj∈{A,B,C}.
Warning: F is
not the simple majority function and it is known only
to the manufacturers of voting machines.
The Stooge Holistic Information Times
claims the voting system is "fair" because if
yi≠xi for all i∈{1,2,...,N},
F(x1, x2,..., xN)
F(y1, y2,..., yN).
But conspiracy theorists claim there exists j so that
the value of F is completely determined by xj, i.e. some voter j
holds total power. Are they right?
Puzzle #33 – (Wisest allocation of vote-weights):
Suppose there are some very large number N of independent
voters who need to make
a yes/no decision.
We suppose this decision has a "right" and "wrong" answer.
Suppose voter K would have (independent) probability PK
of making the "right" decision on his own. (Hopefully all the PK≥1/2,
i.e. hopefully humans are better decision-makers than coin tosses.
The PK may
differ because some voters are wiser or better informed than others, but
we'll assume the PK-1/2 don't vary too tremendously.)
Suppose voter K gets to cast WK votes.
What (approximately)
is the best choice of the weights WK, maximizing the chance that
the collective decision will be the right one?
Puzzle #34 – How much ignoring does Instant Runoff Voting do?
It has often been noted that
Instant Runoff Voting (IRV)
ignores a large portion of a large number of the votes, which is
one reason behind some of the crazy paradoxes IRV suffers.
In this puzzle, we shall prove a theorem showing,
quantitatively, that IRV always ignores asymptotically all
the information in the votes.
Suppose there are V voters and C candidates.
A. Prove
that IRV never examines more than a fraction
of the CV "cell entries" inside the votes,
where Hn=1+1/2+1/3+1/4+...+1/n is the nth "harmonic number."
When C=10, this fraction is 6121/25200≈24%,
when C=20, it is 15%, and when C→∞ it goes to 0.
B. Prove
that IRV usually will not elect Condorcet winners (even when Condorcet Winners exist)
in the random election model
in the limit C→∞ with V fixed,
nor in any limit V,C→∞ with C≥V0.51.
Puzzle #35 – Gerrymandering cancellation theorem
Suppose there are two political parties A & B and two corresponding
kinds of voters (all voters are total party-loyalists).
Suppose the country is gerrymandered by party A.
What is the largest fraction of congress that they can,
by means of that gerrymandering,
cause to become As? (Compare with this: in Ohio in 2006, ≈60% of
voters voted Democratic, but only 39% of the congressmen they
elected were Democratic.)
Now suppose the country instead is optimally-gerrymandered by party B.
Prove this theorem: the "double congress" elected via gerrymandering(A)
[congress #1]
and via gerrymandering(B)
[parallel congress #2] is, in net, of the same composition as the electorate.
Does any analogue of the gerrymandering cancellation theorem
hold if there are three or more parties and voter-types?
Puzzle #36: Additive utilities:
A common model in social choice theory is one in which
each human has a "utility" (real number) for each possible event,
and it is deemed "best" to alter events in order
to maximize the sum (over all humans) of the utility.
In this and the next few puzzles we explore the foundations and limits of that picture.
If the quantity to be maximized is not ∑n Un
but rather
g(∑n f(Un))
where f and g are some 1-to-1 smooth functions,
then argue that this is really just the same thing
as the original additive-utility picture. (This is just
"utilities in disguise" where f and g are the "disguising functions.")
Recognize that many common symmetric
functions in mathematics are just addition in disguise.
In particular, do this for the product function ∏n Un,
the Lp-norm functions
(∑n Unp)1/p with p>0,
and (in a limit sense) the max and min functions.
But not every smooth continuous function invariant under permutations of its N arguments,
and monotonic in each, is just addition in disguise. When N=3 prove that the
"combined addition and multiplication" function abc+a+b+c is not
just addition in disguise.
But when N=2 it is: prove that ab+a+b is just addition in disguise.
An example of a 2-person combining function which is not
just addition in disguise is
It seems reasonable to demand that any method of combining N real utilities to get
a single real "social utility" must
Puzzle #37: Additive utilities II:
Prove that the following three axioms force
any "social utility" to be just the sum of
(perhaps "in disguise" as in the previous puzzle) the individual utilities:
Axiom 1. [invariance under voter reordering]
Social utility does not depend in which order the voters are
Axiom 2. [utility-order-invariance under adding a constant]
After you add an arbitrary constant C to every utility of some particular voter,
then if A was a "better" candidate than B (higher
social utility) then it still is.
Axiom 3. [non-silly]
With random real number input (i.e. each individual utility independently sampled from
some probability density), there will (at least with some
positive probability) exist some unique candidate yielding maximum social utility,
i.e. the social utility does not
merely always say all candidates are equal; and the social utility exists
for generic real number input (personal utilities)
at least within some neighborhood
of the vector with all entries C (for some constant C).
Puzzle #38: NonAdditive utilities:
Criticize the axioms that underlay the preceding two puzzles' proofs
that utility-aggregation must be additive.
Then consider the following non-additive utility-combining method, suggested
by Clay Shentrup:
The "Shentrup Social Utility," as a function
of the personal utilities of each of V people, is:
Puzzle #39: Utility=Log(Wealth)?
Claude Elwood Shannon (and much earlier, Daniel Bernoulli)
had the idea that "utility" and "money" were approximately
related by the formula utility=log(wealth).
Let's explore that.
Show that this formula is consistent with the idea that doubling your wealth causes the same
increase in happiness, regardless of your initial wealth.
Suppose you are offered the opportunity to participate in this wager:
if event E happens, each dollar you bet is replaced by R>1 dollars;
but if E does not happen, you lose all the money you bet.
The probability of E is (you believe) P.
How much money should you bet, in order to maximize your expected Shannon-utility?
Show that, under the Shannon formula, or indeed any formula of the form
utility=ConcaveDownFunction(wealth), removing some money from a random person and giving it
to another random other person, decreases the expected sum of utilities, i.e. "random theft is
socially bad." [Similarly, making any fair monetary bet causes negative expected utility gain,
i.e. such a utility function is "risk averse."]
Also, transferring money from poorer to richer people is socially bad,
but transferring money from richer to poorer people (provided the rich one stays richer)
is socially good.
Criticize the Shannon formula.
How could you try to do a better job than Shannon?
Try to devise a formula that smoothly handles negative wealth (i.e. debt).
What about wagers involving risk of death? How should a bettor optimally handle them?
Puzzle #40: Feel alike ⇒ vote same?
A single-winner voting system satisfies FAVS if the best strategy for any set of "feel alike"
voters is for all members of the set to vote in the same manner.
Show that instant runoff voting (IRV) fails FAVS.
Show that Borda voting fails FAVS.
Show that the Schulze beatpaths Condorcet method (with Simpson-Kramer min max as
a tiebreaker, i.e. of two tied Schulze winners, the one with a milder worst-defeat
is preferred) fails FAVS.
Show that Approval voting satisfies FAVS – but fails it if we allow
incomplete-information scenarios
(i.e. in which the strategic-voter-set does not have complete information about all
the other votes).
Show that range voting always satisfies FAVS in both complete and incomplete information
(It is not clear whether satisfying FAVS is desirable, but this result is of interest
for refuting the vague claim that strategy in range voting is comparatively
"complicated" or "messy.")
Puzzle #41: Invisibly Corrupted Computer Programs.
Explain how it is possible for
an operating system such as LINUX
to be corrupted (so that, e.g, "Mortimer Snerd" can always log in)
in such a way that scrutinizing the full source code for LINUX will fail
to uncover anything special about Mortimer Snerd – even if the scrutinizer is
amazingly incredibly wise and intelligent.
(Thus, inspecting the source code of a program is inadequate for security purposes.)
Hint: LINUX is written in the C language, and the gcc C-compiler is also written in C
and is part of LINUX.
Puzzle #42: Probability of Condorcet cycle:
Suppose there are some odd number V≥3 of voters, each casting
a random rank-ordering of the C candidates as her ballot.
Let P(V,C) be the probability that a Condorcet ("beats all") winner
does not exist.
Prove the surprising and elegant exact equality
Puzzle #43: Approval voting strategy most likely to elect Condorcet winner
Suppose there are C candidates and V voters, and each voter's
honest ranking of the candidates, is random (all C! orderings equally likely).
Suppose we hold a single-winner approval-voting election and every voter
approves her top J candidates (for some J with 0<J<C).
Which value of J leads to the largest chance that the election winner
will be a Condorcet ("beats all") winner (based on the true, honest
voter preference orderings)?
What if we have two approval style elections sequentially,
where voters approve the top J candidates in election #1,
the top K finishers get to run in election #2, and voters approve their top L
candidates in the "runoff" election #2.
What is the "best" (J,K,L) which causes society to have the greatest chance
of electing a Condorcet winner?
Puzzle #44: Shannon Utility Honesty property
Suppose you can buy arbitrary amounts of substances 1, 2, …, n.
If Vk is the amount of substance k that you buy, your total utility is
U = u1V1 + u2V2 + … + unVn
i.e. uk>0 is your utility density for substance k.
Suppose that the total amount of money you spend to buy substance k is xk,
and the total amount of money you can (and must) spend is
Prove that under the following pricing scheme, your best strategy is to provide
an amount of money xk exactly proportional to
Magic pricing scheme:
If you spend $xk to buy substance k, then you get volume
Puzzle #45: (open now solved!) Min-Max matrix product
The min-max product
C = A*B of two n×n matrices A and B (yielding another C)
is defined by
is there a subcubic algorithm to compute min-max matrix products?
[Application to voting:
This yields a subcubic algorithm to find the
candidate-ordering starting from the pairwise matrix.]
Puzzle #46: Probability of unclear election winners
In the random election model, with 3 candidates and
rank-order votes from V→∞ voters,
Puzzle #47: "Vote for N" leading to unclear election winners
Find a rank-order-ballot election with 5 candidates and as few voters as you can,
in which
Puzzle #48: How many votes to count to get confidence?
Suppose there are N voters in a 2-choice election. Each voter is an independent
fair coin flip. We shall be interested only in the limit N→∞.
Then: what is the minimum fraction F of the votes that need
to be counted to obtain probability 1-C or
greater (0<C<1/2) that the winner among the counted votes, is in fact
the winner among all the votes? (Assume the votes are counted in random order.)
Hint: The answer is F = cos(πC)² = cos(π[1-C])².
Puzzle #49: Machievellian Agenda Manipulation
Suppose there are 3 voters and their opinions of the candidates are as follows:
Candidate H seems to be the least-deserving of election.
Construct a "knockout tournament"
sequence of head-to-head elections to cause H to win.
Puzzle #50: How many checks to get confidence?
Suppose you want to know, with correctness probability ("confidence") at least C,
whether a given set of N objects contains B or more "bad" objects.
What is the minimum number S of randomly chosen objects (sampled without replacement)
that you need to examine to answer the question?
In a voting application, the "objects" might be "precincts" or "voting machines."
Puzzle #51: Getting true (not fake) randomness
The Ohio 2004 presidential election was recounted. The recount was supposed to be of
a random subset of precincts. But election administrators arranged a conspiracy to
recount a nonrandom subset of precincts, intentionally selected to make sure that
the recounts would discover little or no discrepancy. (They were later convicted and jailed,
despite their defense that they were just following orders from higher-ups.)
This leads to the question: how can we pick true random choices for such purposes,
that, all players can be confident, were not pre-arranged?
Puzzle #52: How many votes to count to get 100% certainty of winner?
Suppose the candidate with the leading vote share
(in a plurality-style election)
has a fraction V of the votes, 0<V≤1.
How many of the N votes must be counted in order for the counter (who does not
initially know either V or the winner's name) to reach 100% certainty of the identity of this
Puzzle #53: Districting is unavoidably "chaotic"
Suppose it is your job to divide a country (compact connected set in the plane)
into N equipopulous electoral "districts" (ditto) for some given N≥2.
To avoid bias, the only information that is allowed to affect your
districting-drawing is (1) the shape of the country, and (2) the
population-density function. Here "shape" is a EW-mirror-invariant notion,
i.e. two shapes that are the same after mirroring "East" for "West"
are the "same" shape.
Further, to force you to deliver districts which are not completely
asinine, we demand that
The total length of all district-boundaries, is at most twice the minimum
possible length.
Puzzle #54: Better than average
In 1987 John J. Cannell publicized the fact that all 50 US states reported
that their students
were better than the national average on standardized tests.
Puzzle #55: How often is voting honestly worse than not voting with Instant Runoff Voting?
In IRV, voting honestly
can be worse (for a set of co-feeling voters) than not voting at all.
Puzzle #56: How often is voting honestly worse than not voting (with Condorcet)?
[Mostly open]
In every Condorcet voting system, it is a known theorem
that voting honestly
sometimes is worse (for a set of co-feeling voters) than not voting at all.
In this puzzle we investigate how frequent that phenomenon is.
Define a V-voter C-candidate election scenario to be a "participation failure"
if there is some rank-order-ballot vote Q, and some
t>0, such that adding t extra identical Q-type ballots, will cause the
election result to worsen (from the point of view of the t extra voters).
Puzzle #57: How often is dishonesty better strategy than honesty with IRV?
With V voters there are exactly C!V possible different C-candidate elections
employing rank-order ballots (as would be the case with
IRV=instant runoff voting).
Puzzle #58: Gerrymandering
Let N be a number of the form N=2pq with q odd.
(Every number may be written in such a form, of course.)
Suppose there are N small disjoint discs ("equipopulous towns") in a
convex plane region ("country"),
no three collinear, each town either red or blue, with the ratio of red:blue
being ⌈q/2⌉:⌊q/2⌋.
In other words (and oversimplifying slightly), in a slightly-red-majority country,
red always can gerrymander to get 100% of all the congressmen (even if forced to use
convex districts only);
while blue can gerrymander to get almost all the congressmen
(even if forced to use
contiguous districts only).
Puzzle #59: Gerrymandering "Squareland"
Starting at bit #273 after the binary"decimal" point, π in binary first has a string
of 240 bits containing exactly 50% 1-bits and 50% 0-bits.
The only reason we are bringing in π is to get 240 random bits that I clearly did not
choose maliciously; they really are "random."
It is:
Regard this as a 15×16 checkerboard with one person per square,
each person either red (1) or blue (0): "Squareland."
Puzzle #60: Range and Approval as "universal" voting systems
Suppose every voter has a private preference ordering of all candidates, and "approves"
those above some threshold that she chooses.
Depending on how the voters set their thresholds, different
approval-election winners will
result. Call the set of possible approval-voting winners "A."
Puzzle #61: Voting system "stability" (S.J.Brams)
An election outcome is "stable" if it is achieved by some set of (honest) votes
such that no voters of any single type have an incentive to change their votes.
Further, it is "strongly stable" if no subset of colluding (perhaps un-alike)
voters have incentive
to change their votes.
Otherwise it is "unstable."
(Here "election outcomes" are allowed to be "lotteries.")
When considering approval voting it is of
interest to restrict attention to "semi-honest" votes, i.e. in which each voter
genuinely prefers all her approved candidates over all her disapproved ones
(and we forbid all-approved or all-disapproved "null votes").
The idea here is the approval-votes fluctuate with
time in response to opinion polls, random noise,
and voter re-strategizing – but stay semi-honest.
Once the votes reach a
"stable" configuration, they stop fluctuating
because from then on nobody has incentive to change
and repeated opinion polls delivering the
same answer eliminate the random noise.
Puzzle #62: How often is dishonesty better strategy than honesty with Condorcet?
With V voters there are exactly C!V possible different C-candidate elections
employing rank-order ballots.
We shall be considering the "large electorate" limit V→∞ with C held fixed.
Puzzle #63 (ACE): Optimal size for a legislature
Consider the amount of
communication a congressman has to do with both his constituency and
with his fellow legislators.
Puzzle #64: Expected number of votes counted before certain
Suppose you have N voters, V of whom vote for the leading
candidate Larry (Plurality voting).
A counter starts counting the N votes one by one in random order until
first becoming sure that Larry won (and this counter knows N,
but does not a priori know V or Larry's identity).
Puzzle #65: Partitioning squareland into some nice patterns
Puzzle #66: Optimal Districting
Divide an equilateral triangle, square, regular pentagon, regular hexagon, circle, and the surface of a sphere, into N equal-area pieces with the least cutting.
And what about "districting" in three or higher dimensions?)
Puzzle #67: Dice can do damned near anything?
Can Alice construct some number of dice such that,
no matter which two of them Bob selects,
Alice can select a third which "beats" either ("beats" means more than half the time
during any long-enough sequence of rolls of die#1 versus die#2,
will roll a higher number)? And if so, what is the minimum number of dice Alice needs?
More generally: what is the minimum number F(n) of dice Alice needs so that no matter
which n of them Bob selects, Alice can select another from her collection
which pairwise-beats all Bob's dice?
Puzzle #68: A random person
You would like to select a random (uniform) person.
How should you do it?
Pollsters often want to do that, or say that they've done it.
But do they really?
After you think about what it takes to select a random person, you can ponder that
from a position of strength.
Puzzle #69: Optimal way to form a "coalition"
In many (all?) parliamentary governments, if no one political party has
a majority of the seats in parliament, then it is necessary for
some subset of the parties – together holding
a majority of seats –
to form a "coalition" to govern.
Assume each party has a positive "regret" value associated with being in a
coalition with each other party (with the regret values being known functions
of the ordered pair of parties).
Prove that finding the optimum coalition – minimizing
the sum (or the max) of regrets – or even merely approximating
optimality to within a factor of some positive power of N
(where N is the number of parties) is NP-hard.
Hence, if P≠NP, there
is no efficient algorithm to determine
or approximate optimum coalitions.
Puzzle #70: "Marriages" and "Capitalism."
Suppose there are N men and N women. Each man has a preference order
on the women, and vice versa.
We wish to arrange N marriages pairing them off.
But, if the set of marriages has the property that
some man m and woman w would both prefer each other to their current partners,
then we say it is "unstable."
Puzzle #71: The impossibility of districting into squares & triangles
Consider (on a flat Earth) a rectangular "country."
Puzzle #72: Coloring districts
Puzzle #73: Manipulation of most reasonable voting methods is easy
An election using some voting method is "weakly monotone"
with respect to some "manipulator"
co-voting voter subset if,
for every pair of candidates (A,B),
Puzzle #74: How bad can "Shortest Splitline" districts be?
Puzzle #75: Monty Hall problem
This is a famously tricky probability puzzle.
A television game show host asks a contestant to choose one of three closed doors.
Behind each door is a prize to be awarded to the contestant.
One of the prizes is valuable (pot of gold), other two are not (worms).
The contestant chooses a door. But before opening it, the host opens
a different door (revealing worms) and asks: would you like to switch doors?
Should she switch?
Puzzle #76: Monarch family lines
The English Monarchs adopted the rule that the first-born male legitimate child of the Monarch,
would (after the Monarch died) become the new one – but if there were none,
then the oldest female legitimate child would be crowned.
If neither existed, then the line of Monarchs would come to an end.
(In practice, at that point, there would usually be a war. Also, things generally
went very badly for children who were too
young at the time of their enthronement.
But we shall ignore that.)
Suppose each Monarch has N legitimate children who survive him, N≥0,
with probability P(N) –
for some fixed probability distribution P(0), P(1), P(2), ... (all positive).
Puzzle #77: Learning how to play games
"Matrix games" arise all the time in political science, economics,
and of course, gameplaying.
The rules are: there is a rectangular "payoff matrix" M.
There are two players, "White" and "Black."
If White chooses strategy w and Black chooses strategy b
(both announce simultaneously), then
Black pays White M[w,b] dollars. (Which could be negative.)
It is known that there always exists an optimal policy for both players,
which is always a so-called "mixed strategy" in which you employ strategy n
with probability p[n] for some magical fixed probability-vector p,
randomly choosing which pure-strategy to use.
This probability-vector describes your optimal policy completely.
Your problem as a gameplayer, then, is to determine what that optimal vector is.
Puzzle #78: Florida counties' biased enforcement of rules for overseas ballots in 2000
In the 2000 US presidential election in Florida, after Gore challenged the
election there was only one class of ballots which had not yet been
counted – late-arriving overseas ballots. So they became critical.
The NY Times (Front page, 15 July 2001) manually examined the
2490 late-arriving overseas ballots in FL 2000.
They found that 680 violated state election laws and rules
and hence should have been rejected.
Here's how they got counted:
of the 680 uncompliant but accepted ballots 80% were in Bush counties. (Bush-Gore margin: 537.)
(The NY Times also examined the 1824 flaw-free late-overseas ballots, but
there was almost no bias there since only
11 were rejected.)
If the enforcement-levels
for these laws and rules had been unaffected
by whether Gore or Bush happened to be winning in each county, then
show that the chance of an anti-Gore bias this-or-more-severe would happen
by luck, was below 10-50.
Puzzle #79: Biased purging in Ohio 2004
Between 2000 and 2004
(before the November 2004 Bush vs. Kerry US presidential election)
"purging" of lists of registered voters was carried out by
various Ohio counties.
Consider a county to have been "purged" if the number of registered voters
dropped at least 6.5% in this period.
(Note this definition is necessarily somewhat arbitrary, e.g.
some counties might consider themselves to have "purged" even if
their number of registered voters increased from 2000-2004.
For the purposes of this puzzle
we are making an "effective" definition of purging based on results,
not a definition based on alleged intentions.)
The 16 Kerry-Won Counties: The 72 Bush-Won Counties:
4 purges 0 purges
(Athens, Belmont,
Jefferson, Monroe)
[Source: www.feminist.org Ohio registered voter dataset]
Assume that the probability that a county performed a purge
was unaffected by whether they were a Bush- or Kerry-county.
What is the probability that this large an anti-Kerry bias arose by luck?
Puzzle #80: Biased Voter turnout in Lucas County Ohio 2004
In the 2004 US presidential election in Lucas County in the critical state of Ohio:
Assuming that registered voters were equally likely to vote
regardless of whether they supported Bush or Kerry
(and that there was no unethical vote- or turnout manipulation)
show the probability P that an anti-Kerry bias this or more severe would
have happened by luck, obeyed P<10-13.
But perhaps Democratic registered voters (Kerry was Democratic)
are just less likely to vote than Republican ones –
e.g. Democrats are just less-socially-responsible!
Consider that possibility.
Puzzle #81: Hierarchical government
Suppose there are N people in a country.
They wish to elect n congressmen (1≤n<<N).
Suppose the people are of two types "black" and "white." All are
total racists, i.e. only vote for members of their own race.
Analyse how a small racial imbalance in the population (say 0.000001 away from
perfect balance)
is magnified in the congress.
The above could be criticized as a naive, unrealistic model. Here is another. Assume
the blacks, though in the minority, try to dominate the government.
They gerrymander the districts so each is either all-white or has a slight
black majority. They also manipulate the triples in the hierarchical scheme so
that each triple is either 2 blacks versus 1 white, or 3 whites.
How well do these government-takeover plans work?
Suppose the whites "fight back" in the hierarchical scheme by making
all triples be 2:1 or 1:2 (if this is possible). Then what?
Finally suppose our blacks and whites are so racist
they cannot stand being in the same triple.
I.e. all triples are now totally segregated 3:0 or 0:3. Then what?
Puzzle #82: Proxies and cycles
Imagine a government set up as follows. Each voter can either vote on a decision,
or grant her vote to somebody else that she chooses (her "proxy"). That proxy would then have
two votes, or if she were the proxy of two people, would have three, et cetera.
But difficulties arise if a chain of voter-to-proxy arrows is ultimately
In that case everybody in the cycle "passes the buck" and nobody actually votes.
The goal of this puzzle is to model this.
Suppose there are N voters in all (N large) and everybody chooses their proxy
uniformly independently at random from among the other N-1 voters. For each voter
the probability that they vote is P and the probability they "pass the buck" is Q=1-P.
Then what is the number of voters we expect to be disenfranchised
because their vote is transferred into a "buck-passing cycle"?
Puzzle #83: Zero-info honesty
The Zero-Info Honesty criterion (ZIH) for a voting method
"If all the other voters vote randomly independently (all votes equally likely)
then it should be strategically optimal for you to cast an honest vote."
Here "honest" means "rank in honest order of preference" (for a rank-order voting method)
but means "rank with favorite=max, most-hated=min, and other linearly interpolated according
to your utilities for their election" for range voting.
Prove that Borda and Plurality voting both obey ZIH.
It naively seems like Instant Runoff Voting (IRV) and many
(perhaps all) usually-proposed
Condorcet Methods should
obey ZIH. Can you prove or disprove that?
Puzzle #84: Optimum districting longer for larger country?
Consider the "optimum" (i.e. with shortest total length of the inter-district boundaries)
way to divide a country A into N districts.
Now consider a new country B which contains A as a strict subset.
Is the optimum N-district partitioning of B always longer than (or the same length as)
the optimum N-district partitioning of A?
Extra Credit:
What is the probability that a Condorcet Winner exists in a four-candidate
election (vote counts similarly random uniform but now from the 23-dimensional simplex).
Puzzle #86: Copeland is comparatively good with strategic voters
The "Copeland" rank-order-ballot voting system elects whoever
wins the most pairwise contests.
(The "Copeland score," i.e.
total number of pairwise contests you win, must be between 0 and
N-1 inclusive in an N-candidate contest; pairwise-ties count as half.)
One of Copeland's defects is its propensity for having co-winners with
equal-maximal Copeland score.
It is important to add some
tie-breaking method to Copeland to repair that flaw.
In this puzzle we shall show that Copeland enjoys some advantages
related to strategic voters.
Puzzle #87 (Generalized "Monty Hall" probability puzzles, cf. puzzle #75):
Puzzle #88 ("Optimum" voting system maximizing chance of Condorcet winner):[8]
Which weighted positional voting system,
in a 3-candidate election
(in the "Dirichlet" probability model)
maximizes the chance of electing a Condorcet "beats-all" winner?
(Also, for comparison, compute this chance for Plurality, Borda, AntiPlurality, and IRV.)
Puzzle #89 (Probability of Majority-top winner in Dirichlet election):[4]
In the "Dirichlet model" of random 3-candidate elections (rank-order voting: 3!=6 possible
vote types) what is the probability that some candidate will win with a majority of
the top-rank votes (thus avoiding any need for a "runoff")?
Bonus questions:
Also solve for arbitrary number C≥3 of candidates insetad of merely C=3, and
also in the random election model.
Puzzle #90 (Burial works in Condorcet voting):[3]
Prove "burial stategy" works (sometimes) in every Condorcet voting method.
That is, there exists a situation with N candidates
(for each fixed N≥3) in
which A wins, but the B>A>others voters can make B win by strategically
re-voting B>others>A.
Puzzle #91 (Number of count-types to publicize):[3]
(This puzzle suggested by The League of Women Voters of New Mexico.)
Suppose voters in an N-candidate election submit a rank-order ballot
which ranks some subset of the N candidates in order, then leaves the remaining
candidates unranked. For example in a 5-candidate election a legal vote would
be "C>A>B>(all unranked candidates)."
How many different kinds of "total" should the election office in each precinct
publish in order to hold an election
(Give both a formula in terms of N, and compute the numbers when N=10.)
Puzzle #92 (Permit cycles in Condorcet ballots?):[3]
Although it seems senseless to cast a rank-order ballot which includes a
(such as A>B>C>A)... permitting voters to do that would have the advantage that would allow
"dumb totalizing" voting machines to process votes for all Condorcet systems based
on "pairwise matrices." (It is beyond
the capability of such a machine to detect cycles and thus forbid
illegal votes, but if we make cyclic-votes legal, then there is no problem.)
Votes in an N-candidate
election could be cast as (N-1)N/2 pairwise one-bit "A>B versus A<B" choices.
In this puzzle we explore the consequences of permitting cyclic ballots.
Puzzle #93 (Supermajorities):[6]
Suppose, in order for us to say that "society prefers A over B" we demand that
A would beat B in a 2-choice majority vote, by at least a fraction-F supermajority
(for some F with ½≤F<1).
Puzzle #94: The "YN model" – a simple voting model
in which range voting behaves optimally
while many competing voting systems can behave pessimally:[7]
The "YN-model" is a simple model of voting
(with a fair amount of realism) in which range voting
performs optimally with "honest voters."
There are m independent "issues"
and 2m "candidates"; each candidate announces
a clear yes or no stance on each issue, and by some benificent miracle we have a full
democratic marketplace, i.e, every possible issue-stance is
represented. Hence (in the case m=4) we can name the 16 candidates YYYY
("yes" on all 4 issues), YYYN, YYNY,..., NNNN.
Suppose, for each voter, the utility of electing
any given candidate is just the
number of issues
on which that candidate agrees with the voter.
Finally, we suppose without loss of generality
that all m issues each would win if put to a single-issue
yes/no vote, therefore the objectively best candidate is YYYY.
The "without loss of generality" is because we could just
rename the issues to make "yes" become "no" for that issue, e.g. the issue
"raise taxes?" becomes "do not raise taxes," so with such canonical
renaming we can always cause
the best candidate to be named YYYY.
Most analyses of voting system "properties" ignore the fact that "issues" exist and candidates
have stances on them. The YN model is approximately the simplest and most natural
possible voting model which doesn't ignore that. It turns out that honest range voting
is always optimal in the YN model and is "self consistent." But most other commonly
proposed voting methods
turn out to be both "self contradictory" and non-optimal (indeed they can
even behave pessimally)
in the YN model.
Puzzle #95 (The "random YN model"):[8]
The "YN-model" answers to the previous
puzzle have been attacked as
not very relevant to real life. That is, the election examples constructed to
demonstrate that, e.g. Plurality voting can elect the worst winner NNNN,
are (speculate the critics) unlikely to arise.
To address that, now consider random voters. That is, every possible
issue-stance (from among the 2m)
are equally likely and we pick some large number V (far exceeding 2m)
of such random voters. We than canonize them by reversing the signs
of some of the issues so that for each issue considered alone, "Y" is the majority winner.
(This canonical-renaming is purely for convenience and has no genuine effect.)
Puzzle #96 (Human "happiness" and Darwinian evolution):[2]
Here are some nowadays well-accepted
scientific findings about the nature of human happiness (revealed by
survey studies).
Are all of these findings predicted by just one theory – Darwinian evolution?
Puzzle #97 ("Allais Paradox" for utilities):[2]
Maurice Allais conducted the following experiment in the 1950s.
He asked people to preference-order the
following four alternatives:
Puzzle #98 (Random Sampling of a Subset):[4]
Suppose we make n independent tosses of a p-biased coin.
(As is well known) the number h of heads that results is np
in expectation, with
Now suppose we toss some coin (with unknown bias) N times,
with the result we get H
heads. A random subsample of n of the N tosses is then selected
(0≤n≤N, and note this sampling is without replacement).
It contains (say) h heads.
Prove that h's mean=nH/N=np (if we define p=H/N and q=1-p)
and its
Note the "sampling without replacement correction factor"
Puzzle #99 (Efficient generation of correct "random election" pair-margins matrices):[6]
Many election methods for N-candidate elections with rank-order ballots,
as their first step, compute the "pairwise table" giving the "margin" by which A defeats
B (i.e. the number of voters who prefer A>B minus the number who prefer B>A,
for every A,B).
Suppose all V votes are independent random orderings
(all N! orderings equiprobable) with V very large.
Puzzle #100 (Copeland Tie Probability and Behavior):[6]
The "Copeland" voting method for N-candidate elections with rank-order ballots
(1) The number of candidates whom X pairwise-defeats is X's "Copeland score."
(2) The candidate with greatest Copeland score is the winner.
Puzzle #101 (Weighted sampling):[4]
You are a pollster. You want to predict whether the N voters are going to vote
YES or NO in the forthcoming election.
If you pick a sample of n (1≤n<<N)
future voters at random, the variance in your prediction
of the number of YES votes
would be npq where q=1-p and p is the probability a random voter votes YES.
But you want to outperform that –
you want smaller prediction variance using a smaller sample!
Can you do that? And how? Does this really work?
Puzzle #102 (Probability Condorcet Winner exists):
In the random election model (also called the "impartial culture")
Puzzle #103 (Counting rank-order votes in "D-dimensional politics"):[5]
Suppose politicians and voters are points in a D-dimensional "issue space."
Suppose, given a choice between two candidates, voters prefer the one located nearer to them.
Puzzle #104 (Voting system incentivizing voter to reveal honest utilities):[8]
Is there any way to incentivize a voter (by asking her a single question) to state her
sincere normalized utilities for the N candidates, e.g. X=10, Y=3, Z=0?
We know that in this case the voter would consider a 30% chance of getting X
instead of Z to be equal in value to a guarantee of getting Y. But can we use
that kind of knowledge to solve this problem?
Puzzle #105 (Identifying Condorcet Winners by examining few pairs):[8]
A "Condorcet winner" is a candidate who defeats every rival pairwise.
(To avoid annoyance, assume there are no pairwise ties.)
As is well known, if N≥3 then there can
fail to be a Condorcet winner in
an N-candidate election.
Obviously, in an N-candidate election, we can decide whether a
Condorcet winner exists (and if one does exist, identify who)
by examining all (N-1)N/2 candidate-pairs.
Puzzle #106 (Quick elimination of no-hopers in Instant Runoff Voting):[2]
IRV vote-counting software software by "Election Solutions Inc."
often immediately simultaneously eliminates a large number of low-vote-getting candidates
because "his/her defeat was mathematically inevitable."
Puzzle #107 (IRV gets it wrong):[1]
Clay Shentrup wants to know an argument that instant runoff voting (IRV) gets
the winner wrong in at least one of these two elections:
#voters their vote #voters their vote
------- ---------- ------- ----------
(winner = C) 2 B>C>A
(winner = C)
Puzzle #108 (1D political spectrum model):[6]
Here is a simple model of elections:
the "voters" (of which there are an infinite number)
are the uniform distribution on the real interval (-1, 1).
The N candidates each are independent random samples from that distribution, i.e.
points x on the real line with -1<x<+1.
Voters prefer candidates located nearer to them. In this puzzle assume voter honesty.
Puzzle #109 (Information content of ballots):[3]
The number of possible approval-style ballots in an N-candidate election, is 2N;
but 2N-2 if the two "silly" (all-approve or all-disapprove) ballots are excluded.
The number of possible rank-order-style ballots in an N-candidate election, is N!.
Can you draw any "take-home lessons" from this exercise?
Puzzle #110 (Counting rank-order votes):[5]
In an N-candidate election, cast a rank order vote in which equalities in rank are allowed.
For example, if N=4, the vote A>B=C>D would be permitted.
If equalities were not allowed, the number of possible votes would be N! –
but we are allowing them, so how many are there?
Puzzle #111 (CTT voting in legislatures):[3]
Is CTT voting a bad idea for voting in legislatures?
Puzzle #112 (Probabilities voting methods "vulnerable to strategy"):[8]
A single-winner election with honest voters and some given voting method will be called
"vulnerable to strategy" if there exist some subset of the voters, who, by changing
their votes, could cause the winner to change to somebody else whom they (unanimously) prefer.
[This subset is not required to consist of identical voters; their honest
and/or strategic votes both are allowed to be multi-element sets.]
This puzzle investigates the probability that an election is vulnerable to strategy, in
the random election model (also called the "impartial culture")
with V→∞ voters.
Puzzle #113 (What do rank-order-ballot[equalities permitted] systems do if given approval-style votes):[4]
Suppose the ballots are rank-orderings of the N candidates in
which equalities in rank are allowed.
Now suppose that only "approval-style" such ballots are cast, i.e. involving
at most one ">"
and all the others are "=".
Thus the vote A=B>C=D=E would be permitted but A>B=C>D=E would not.
Puzzle #114 (One quantitative measure of clone-immunity [Jack Rudd]):[5]
Suppose you have two camps of supporters, camp A and camp B. You have an
n-candidate election, in which 1 candidate (candidate A) is from camp A, and the
other candidates (B1, B2, B3,..., Bn-1)
are from camp B.
The A-supporters want to elect
A, the B-supporters want to elect anyone but A.
There is, however, a twist to this election: the A-supporters have to vote
first. Their votes are then displayed publicly, and the B-supporters vote in
response to this. Both camps can co-ordinate their votes so as to get an optimal
result from their point of view.
Let c be the least number such that, if the proportion of the electorate in the A
camp is greater than c, they can force the election of candidate A.
Then prove:
Puzzle #115 (How often do "wrong winners" occur in real life with IRV [Clay Shentrup]):[2]
Instant Runoff Voting (for example) can exhibit many paradoxes including
no-show paradox,
and reversal failure.
One form of nonmonotonicity is:
X wins, but increasing X in some votes causes X to lose
(or the time-reverse of this scenario also is non-monotonic).
One form of no-show paradox is:
X loses, but adding some new X-bottom votes causes X to win.
In reversal failure, X wins, but if all the votes are reversed ("trying to select
the worst candidate")
then X still "wins."
Suppose we examine real-life IRV elections
and find some percentage P
of elections in which you can tell that such a paradox existed.
Critics have objected to such discoveries by complaining: "But
this 'paradox' is merely hypothetical. E.g. you hypothesize
that some extra X-bottom voters appear, etc, causing the paradox.
But really they did not.
So your so-called paradox doesn't bother me. I want to see real life elections
where IRV clearly generated the 'wrong winner' otherwise I'm unimpressed."
Puzzle #116 (Inferring political "positions" of politicians in D-space):[6]
At present there are 100 senators, and each votes about 600 times on a yes-no question
during a 2-year senate. Each senator's 600 bits are publicly known. (Actually, some
senators abstain from voting sometimes, so there will be fewer than 600 bits; and
sometimes senators get replaced in the middle of the term, in which case there would be
more than 100 senators.)
Devise an algorithm to place the senators along a line ordered from "leftmost" to "rightmost,"
in such a way that nearby senators tend to usually agree in their votes, whereas far-apart
senators usually disagree in their votes.
More generally, instead of a line, we could have a D-dimensional "issue space."
(D=2 is of interest for pictorial purposes, but we could also consider D=3, etc.)
Given some number D>0,
devise an algorithm to place the senators as points in D-space
in such a way that nearby senators tend to usually agree in their votes, whereas far-apart
ones usually disagree.
Puzzle #117 (Bayes-Laplace-Dirichlet law & "soft quorum" for range voting):[5]
Suppose there is some binary quantity B (i.e. yes=1 or no=0, for example "do you think there
should be a $5/gallon gasoline tax?").
You ask a random sample of S≥0 people for their values of B, and the result is that
Y say "yes"
while N say "no"
where Y+N=S.
Return to main page


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If you've got the capability right now, whether on GameCube, Wii or even PC, go put Wind Waker on. And just look at it. Soak up how a video game released almost a decade ago still looks like it could have come out yesterday.
Once you're done, ask yourself "how did Nintendo do that?". Then come read this.
Indie developer Sönke Christian Seidel, who is one of the folks behind the quietly excellent Pirates of New Horizons, has cracked open the classic Zelda title and broken down, in semi-plain English, exactly how (or in some cases how he's pretty sure) Nintendo made it look so pretty.
The tricks, the tools, the surprises and the silent revolutions, it's all there.
The highlights include a detailed look at Wind Waker's famous ocean and smoke effects, how the game's lighting works and even points out that Nintendo got Link's feet to "stick" to the ground (instead of float all over it) years before other games managed the same feat.
Click the images underneath to expand them; the full set are at the link below.
Zelda: Wind waker Tech and Texture Analysis [Polycount]

subset: commoncrawl


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Chinese Billionaire Killed by Poisoned Cat Meat Stew
A billionaire tycoon from China named Long Liyuan died suddenly and mysteriously two days before Christmas. The only clue: a still-steaming bowl of half-eaten cat stew. Could it be? No, it's simply too easy, too dastardly, altogether too meowtrageous! But it's true: The cat stew was poisoned!
Huang Guang, an agriculture official in Bajia, took Long, the owner of a forestry company, to visit a piece of woodland, according to an official police statement. The two men had a troubled history: Huang had allegedly embezzled money from Long. That should have been reason enough for Long to become suspicious of Huang's later invitation to a local restaurant to indulge in a regional delicacy: slow-boiled cat-meat stew.
Hours after gorging himself on feline goulash, Long began complaining of dizziness; he died of cardiac arrest hours later. Police initially detained the restaurant owner for an extreme case of food poisoning, but Long's family urged them to investigate further, and they were right. They now suspect Huang dosed the dish with the poisonous plant Gelsemium elegans.
The moral here is obvious: Only eat slow-boiled cat-meat when you prepare it yourself. [BBC, Photo via Shutterstock]


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The Charlotte-Mecklenburg Schools in North Carolina are seeking 4 support technicians to support the 3000+ students taking part in the OLPC program as part of Project L.I.F.T. This job will involve day-to-day IT support, maintenance and installations at the school site: including XOs, servers, and networking equipment.
A longer job description follows:
Main Responsibilities:
Prepare and maintain technical infrastructure at 2 of the benefited schools
Laptop maintenance and repair at 2 of the benefited schools
Develop teachers technical skills
Assist during teacher training or any laptop activities at school level
Provide computer support (respond to support calls/emails; research
and solve software and hardware issues; go to the user's workstation
when necessary to assist)
Provide daily break-fix functions for software and hardware supported
Provide software upgrades as necessary
Participate in software testing
Create, maintain and deploy software images
Install and update software applications
Disk backup and recovery
Maintain and update inventory
Knowledge and experience with UNIX or Linux system operator or admin
operating system and software whether it be Linux or Windows
Bachelor's degree in Engineering or related field preferred
Proficient in computer hardware maintenance
Superior problem-solving and analytical skills
Excellent organizational and social skills
Effective time manager
Ability to relate with teachers and non-technical staff as well as children
Willingness to work nontraditional hours to ensure program success
throughout school district
Benefits and Salary:
Salary for this position is competitive and depends on prior
experience. In addition, a comprehensive benefits package is included.


text:
Oprah Winfrey is headed to the supermarket aisle with her own line of refrigerated soups and side dishes.
The media mogul teamed up with food giant Kraft Heinz to launch the line, after creating a joint venture earlier this year.
The new brand, called O, That's Good!, will offer comfort foods with added vegetables, such as mashed potatoes with cauliflower mixed in.
Kraft Heinz Co. said Wednesday the new brand has no connection to Winfrey's deal with Weight Watchers and won't mention the weight loss program on its packaging. Winfrey bought a 10 percent stake in Weight Watchers two years ago and appears in its ads.
The Oprah-branded soups and side dishes will come to some supermarkets this week and be available nationwide by October.
Copyright AP - Associated Press


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Morton's where size matters
Mandarin Oriental
5 Raffles Avenue
Tel: 6338-0066
The steak lineup
For that one night in Morton's I decided that I was going to pretend to be a man because that seemed to be the only appropriate response.
Morton's stays true to the idea of American style over the top indulgence. It is a very masculine restaurant. I was in super-size me land with steak fillets as big as my face and lobster claws as large as my hand. In Morton's land, size matters. Correction, size is everything. Even the tomato was the size of my fist!
The service is winning. It is the integral part of the whole experience from the start to the finish. Part of the whole experience of being pampered is having the waiter literally bring the menu. They recite it off the top of their heads and as show off their platter of meat cuts available and dangle the 1.5 kg lobster in front of you. It is a fun spectacle.
The steaks. They are good, they aren't awe-inspiring but they are good meats. It starts with a good ingredient – a gingerly aged piece of beef – and that's what you are presented with. What I love about it is that, there really is no margin for error here because the steak is presented to you in its naked glory, cooked to your choice of bloodiness.
jumbo lump crab cake
But before we get to the steaks, there are the appetizers to get through. Maybe the colossal shrimp cocktail or a main lobster cocktail? I had neither but I was allocated a colossal shrimp Alexander with beurre blanc sauce which strangely but very amusingly tasted like popcorn. Along with that, the jumbo lump crab cake is outstanding, so if you can't decide between the starters, this is my choice pick.
As for desserts, as far as I can remember this might have been one of the first places to have the ubiquitous molten lava chocolate cake in
Will I come back? Maybe but one condition must be fulfilled - someone else has to picking up the check because I don't feel that the prices are justified. The whole theatric experience was to a certain extent bordering on ludicrous but very entertaining. I can't fault the food it was good but maybe I'm not man enough to stand up to the size of it. A good steak, yes and too much of a good thing, I packed half of mine home and prolonged the pleasure of my NY strip. So maybe this place isn't for me and definitely not for the faint hearted.


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How Much Is Hesitating To Sell Your Covington House Really Costing You?
Hesitating to sell your house in Covington can end up costing you a lot of money! Learn how much in our latest post! Many people don't realize just how much their home is costing them. These costs are even more brutal if you don't like the home you're living or the investment property you own. … Continued


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Whether you're traveling with a group of friends, extended family, or just want a little bit of extra space to stretch out in during your stay, why not book one of Savannah's vacation rentals?
These roomy condos and self-catering properties are designed for visitors who are looking for little homes away from home. Just imagine waking up in your own cozy condo or cottage on Tybee Island, Savannah's beach; having your morning coffee in a funky renovated loft with expansive views of the Savannah River; or luxuriating in an historic property around the corner from Savannah's squares and parks. Whichever you choose, you're sure to have a unique stay in a convenient, comfy place!


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Anti-Americanism is as old (if not older) as the American Revolution itself. Like all nations, America has its flaws. But these defects attract disproportionate attention from the rest of the world. This is partly because of the size and worldwide reach of America's media as well as the United States' superpower status. On a global scale, the choices made by, say, Argentina and Italy just aren't as important for international affairs as decisions made by the United States.
Some of the most insightful analyses of America have been written by non-Americans. The exemplar is Alexis de Tocqueville's Democracy in America (1835/1840). Yet despite the scale and intensity of the attention given to the United States, it's not hard to find articles written by intelligent non-Americans which reflect serious misunderstandings and occasional outright ignorance of the political, economic and cultural currents shaping America.
This brings me to a very odd article that recently appeared in La Civiltà Cattolica: the Italian Jesuit periodical published twice a month and which enjoys a quasi-official status inasmuch as the Vatican's Secretariat of State exercises oversight over the articles it publishes. Entitled "Evangelical Fundamentalism and Catholic Integralism in the USA: A Surprising Ecumenism," its authors Father Antonio Spadaro SJ (Civiltà Cattolica's Editor-in-chief) and Rev. Marcelo Figueroa (a Presbyterian pastor who is Editor-in-chief of L'Osservatore Romano's Argentinean edition), make various assertions about specific political and religious trends in the United States: claims which are, at best, tenuous and certainly badly informed.
Consider, for instance, the authors' analogy between the theological outlook of particular strands of American Evangelicalism and ISIS. As far as I am aware, American self-described fundamentalists are not destroying 2000 year-old architectural treasures, decapitating Muslims, crucifying Middle Eastern Christians, promoting vile anti-Semitic literature, or slaughtering octogenarian French priests. Another questionable contention made in the article is that the Holy Roman Empire was constituted as an effort to realize the Kingdom of God on earth. This particular analysis will come as news to serious historians of that complicated political entity which became, as the saying goes, neither Holy nor Roman nor an Empire.
Various links are also made between climate change skepticism, the faith of white southern Christians (comments which, if applied to other racial groups, would be denounced by some as verging on bigotry), and apocalyptic thinking among some American Evangelicals. Taken together, it is claimed, these things reflect and help fuel a Manichean view of the world on the United States' part. Then there is the article's peculiar association of the heresy of the Prosperity Gospel with recent efforts to protect religious liberty in America.
No doubt, Evangelical scholars and others will highlight the many problems characterizing the article's grasp of the history of Evangelical Christianity and fundamentalism in America. One agnostic friend of mine who happens to be a leading historian of American Evangelicalism at a prestigious secular university described the article's take on this subject to me as "laughably ignorant." I also suspect Rev. Figueroa and Father Spadaro are oblivious, for instance, to many Evangelicals' embrace of natural law thinking in recent decades: something that, by definition, immunizes any serious Christian from fideist tendencies. But two particular claims made by the authors require a more detailed response.
Who's a Manichean?
As noted, the authors assert that Evangelical Fundamentalism has contributed to America adopting a Manichean understanding of international affairs. They argue, however, that Pope Francis rejects any outlook which sees the world in terms of forces of light and forces of darkness. Instead, they maintain, the pope wisely recognizes that at the root of conflicts between nations "there is always a fight for power."
No doubt, the desire for power motivates some international actors. But it is also important to acknowledge that certain ideas—such as Marxism-Leninism, Islamist jihadism, or National Socialism—have driven transnational movements and nation-states to act in ways that are evil because the ideas themselves are evil. For Americans (and anyone else) to recognize this and call these things by their name is not to buy into Manichaeism. It is simply recognition that some ideas are indeed wicked and lead to many people, even nations, engaging in gravely evil acts.
You can't understand, for example, the left-populist regime that's presently destroying Venezuela unless you grasp that its leadership and many of its supporters are partly motivated by a deeply conflictual view of the world. Much of this comes straight from Marx and Lenin (as anyone who has listened to any of the late Hugo Chávez's short three-hour television rants will tell you). It's worth recalling that when President Ronald Reagan called the Soviet Union "the evil empire" in 1983, millions of people behind the Iron Curtain instantly understood what he was talking about. They knew that the systems under which they lived were grounded upon evil ideas about the nature of man and society.
Furthermore, the fact that some Americans describe (often accurately) particular regimes as evil doesn't mean that they view America as an embryonic Kingdom of God on earth. Plenty of American Evangelicals today are deeply distressed, for example, by the state of elite and popular culture in the United States. Nor are they slow to point out these failings, including when these weaknesses manifest themselves in their own ranks. That should make any Western European or Latin American pause before they start attributing Manichaean views of the world to millions of American Christians.
Ecumenism, Evangelicals, and Catholics
A second problematic thesis characterizing the Spadaro-Figueroa article which requires more attention is its characterization of the relationship between many Catholics and Evangelicals in America: a rapport which the priest and the minister plainly have great reservations about.
Father Spadaro and Rev. Figueroa correctly observe that many Catholics and Evangelicals have found common cause in recent decades around issues such as "abortion, same-sex marriage, religious education in schools and other matters generally considered moral or tied to values." They then add that "Both Evangelical and Catholic Integralists condemn traditional ecumenism and yet promote an ecumenism of conflict that unites them in the nostalgic dream of a theocratic type of state."
By "Catholic Integralists," we can safely presume that the authors mean the many American Catholics (routinely labeled as "conservative") who have chosen to ally themselves with Evangelicals to defend things such as the culture of life and religious freedom from the type of doctrinaire secularism which ran rampant under the Obama Administration. But the vast majority of these Catholics aren't "integralists," let alone theocrats-in-waiting. Quite the contrary. Nor are the vast majority of Evangelicals in America pushing theocratic agendas.
If one looks, for example, at statements put together by various scholars and intellectuals involved in movements such as "Evangelicals and Catholics Together," they contain not a shred of theocratic aspiration. The ecumenical discussion between those involved in these endeavors have led over time to genuine fruit in terms of clarification of points in common, removing misconceptions, identifying real doctrinal road-blocks, and identifying areas where practical work to promote the common good can be pursued together. This stands in stark contrast to the bromides and non-sequiturs that characterize ecumenical discussion with the rapidly declining liberal mainstream Protestant confessions who long ago abandoned very basic Christian orthodoxies on faith and morals which most Evangelicals continue to rigorously affirm.
Moreover, when it comes to Evangelical and Catholic Christians in America making the argument that, for example, unborn human beings are entitled to the same protections from the unjust use of lethal force as any other human being, or that religious liberty is more than just freedom of worship, or that parents are entitled to insist that their children not be subjected to the nonsense of "gender theory' at school, these arguments have increasingly been presented in terms of public reason. Catholics have a long tradition of doing this. Yet it is also an approach that many Evangelicals have started embracing in recent years.
This does not add up to the imposition of theocracy or the claiming of special privileges, let alone trying to facilitate quasi-Throne-and-Altar arrangements or some type of Evangelical/Catholic American Nationalism. Contrary to the claims of Father Spadaro and Rev. Figueroa, this is not "a direct virtual challenge to the secularity of the state." It's about maintaining that the truths knowable by all people via their natural reason may be legitimately reflected in the public square of pluralist societies like the United States. Furthermore, the assertion of these truths in this way not only helps facilitate freedom and genuine pluralism (as opposed to the ideology of "diversity") in America; they also help protect non-Christians and non-believers from unjust coercion as much as any other American.
A Credibility Problem
If the Civiltà Cattolica article simply reflected the views of a random Western European Catholic priest and an Argentine Presbyterian minister, few would be concerned about its content. But Civiltà Cattolicaarticles are subject to scrutiny from the Vatican's Secretariat of State. Hence, it's curious that whoever signed off on this article (assuming it was properly vetted) at the Secretariat of State didn't pick up on the authors' conflation of tangentially related matters, or raise questions about the article's emotivist tone, or alert Father Spadaro and Rev. Figueroa to their distinctly amateur grasp of American religious history and the finer points of American politics. If it is the case that red flags were not raised—or were ignored—then all Catholics, American or otherwise, have reason for concern. It is simply not in the universal Church's interests to develop or encourage substantially false understandings of the United States or the Anglosphere more generally.
People—including the pope and his advisors—are free to form views of different nations and the conduct of international affairs. No one expects the bishop of Rome to be uncritical of the United States, or any other country. There is plenty to criticize about America, just as there is to criticize about Argentina (such the economic delusions, systematic envy, and personality-cults encouraged by the poison of Peronism) or Italy (such as the corruption and rampant clientelism in its political and economic culture to which Vatican officials and Italian clerics have not, sadly, proved immune).
Nevertheless, the development of such views should be informed by careful reflection, a command of detail, and an accurate understanding of the history and development of a country. Regrettably, these are lacking in the Spadaro-Figueroa article—and it shows. The greatest damage, however, is to the Holy See's credibility as a serious contributor to international affairs. And that benefits no one, least of all Pope Francis.
Dr. Samuel Gregg is Research Director at the Acton Institute. He has written and spoken extensively on questions of political economy, economic history, ethics in finance, and natural law theory. He is the author of many books, including Becoming Europe (2013) and For God and Profit: How Banking and Finance Can Serve the Common Good (2016).
Crisis Magazine


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Editor's Note: Do you have a weight-loss success story to share? Tell us how you did it and you could be featured on CNN.com.
Story highlights
Rebecca Privitera lost 211 pounds in just over two years
At her peak weight, she was 381 pounds; now she weighs 170
Her husband, Justin, told her she had to lose weight the old-fashioned way
Privitera has lost most of her weight by completing in-home exercise DVDs
Rebecca Privitera desperately wanted to lose the excess weight she had carried around her entire life. In 2010, she had the opportunity to do it when her employer organized a company weight loss challenge.
But when she lined up to weigh in, Privitera realized the scale wouldn't hold her 381-pound frame.
"They wanted me to go down to shipping and receiving and get on the shipping scale," she says.
She refused.
However embarrassing this moment was, it wasn't enough to prompt Privitera to change. A couple of months later, in early 2011, the Cabot, Arkansas, native was put on bed rest because her blood pressure was too high. Doctors gave her three blood pressure medications and an anti-depressant.
Read Rebecca Privitera's story on CNN iReport
"I was out of work for about four weeks," Privitera remembers. "They were afraid I was going to have a stroke."
Before going on bed rest, she had bought a wedding dress. By the time the dress came in, it was too tight to wear. Privitera had to have a second dress specially made overseas.
Instead of a dream night, Privitera's wedding in April 2011 felt like a nightmare.
"I was hot and a sweaty mess," she says. "I was so overweight and so uncomfortable. I was miserable. I couldn't stand it."
A few weeks after her wedding, Privitera's new mother-in-law asked her to join her in a 5-kilometer run. It took her an hour and 14 minutes to complete.
"My body was physically hurting; I didn't want to do it anymore," she says. "I hit rock bottom."
Father's nursing home inspires son to lose weight
After the race, Privitera and her husband, Justin, had an honest conversation. He told her they couldn't afford weight loss surgery or food supplement programs, and she would have to lose weight the old-fashioned way.
She agreed and embarked on a journey to lose more than 200 pounds in just over two years.
Her first step was to get active. Privitera invested in cardio DVDs and started working out at home. She felt too embarrassed to go to a gym and felt that personal training was too expensive.
"The videos showed results that gave me hope, so I gave it a shot," she says.
In the beginning, she was losing 10 to 12 pounds a month. She posted her results on Facebook and the overwhelming response from her friends and family pushed her to continue exercising.
After losing about 65 pounds, Privitera started to modify her eating habits. She started looking up calorie counts for meals before going out to eat in order to make smarter choices at restaurants. She also began to create healthier versions of meals at home using better ingredients.
Privitera was obese her entire life. Her parents divorced when she was 12 years old. Her mother worked two jobs and struggled to make ends meet.
"A lot of our food choices came out of a paper bag – I lived on fast food," she remembers.
There were days during her high school years when she would eat McDonalds for breakfast, lunch and dinner. She recalls being able to eat an entire half-gallon of mint chocolate-chip ice cream in one night.
She was a 16-year-old who weighed more than 300 pounds and developed a blood clot in her leg. Even with this serious medical issue, she still made poor nutrition choices and her weight continued to balloon into adulthood.
'From Fatman to Ironman;' Dad loses 135 pounds
Now, more than two years into her weight-loss journey, Privitera is incorporating new foods into her diet.
"I've followed meal plans with food like quinoa and asparagus," she says. "Until recently, I had never had Brussels sprouts before."
Privitera approaches this journey as exactly that – a journey. She allows herself to indulge once in a while, reminding herself if she wants something she can eat it, just not all the time.
Instead of focusing on calorie restriction, she focuses on eating reasonable portions of healthy foods.
"You have to balance it out," she says. "You can't extreme diet for the rest of your life."
She credits her success to the support of her husband and family, but also to the community of people around her. The majority of her 211-pound loss has come from exercise done right inside her home. She has used programs such as TurboFire and Beachbody, and is now on her fourth round of P90X. She's developed friendships and found accountability partners in the people who use the online message boards for these at-home programs.
Privitera works out six days a week, taking Sunday off, and spends anywhere from 30 minutes to two hours a day exercising. She has been able to avoid any physical setbacks by training safely and not over-exerting herself.
She has continued to use home videos because it keeps her from making excuses to miss workouts. "I hate being pressed for time," she says. "Working out at home, I get up and get started."
She has faced a lot of mental road blocks that nearly made her give up. During these times she has had to rely on people like her mother-in-law to tell her that she has what it takes to keep going and accomplish her goals.
Her motivators have inspired Privitera to become a motivator for others. Privitera says she has helped other women lose a combined 200 pounds by motivating them on Facebook, Instagram and through online communities like the ones she uses. She also regularly teaches a cardio-kickboxing class at a local community center.
About a month ago, she spoke to a class of high school students. She told them that overweight teenagers can make small choices now that will make a world of a difference in their lives.
"One young lady shared that she eats two double cheeseburgers, medium fries, a medium Sprite and three cookies from McDonalds every single night," Privitera recalls. "She had no idea how many calories and how much fat there was in it."
Her weight loss success has inspired the people closest to her. Privitera's husband has started his own program and lost 35 pounds. She says this experience has brought them closer together.
"My husband sees that I have a real passion," she says.
Justin says he's mostly surprised by her new attitude. "She is more positive, upbeat and always looking on the bright side of things," he says. "It inspires me every day to do better. I'm very lucky to have a wife like her."
Privitera set out to complete 13 races in 2013, including three 10Ks. Her 13th race of the year was in late October and it was a big one: her first half-marathon.
She completed the race with her mother-in-law by her side the entire time. Unlike their first race together, this time she wanted to do it and not give up.
"I cried after I finished this half-marathon," she says. "You look for that moment when you cry and think, 'I've done something that some people cannot do.' "


text:
Cory Monteith has taken the first step to getting help.
The Glee star, who has been open about dealing with substance abuse in the past, has checked into rehab, E! News confirms.
"Cory Monteith has voluntarily admitted himself into a treatment facility for substance addiction," his rep tells us. "He graciously asks for your respect and privacy as he takes the necessary steps towards recovery."
In 2011, Monteith told Parade magazine, "I'm lucky on so many counts—I'm lucky to be alive." The TV star opened up about the rollercoaster he went through after dropping out of high school at the age of 16 and the time he realized his life was "out of control."
"Anything and everything, as much as possible," he says about his drug use at the time. "I had a serious problem."
It seems as though girlfriend and costar Lea Michele, who rocks a "Cory" necklace, is supportive of her man's decision to enter rehab, tweeting earlier today, "Feeling so grateful today."
A rep for Glee tells E! News: "Cory is a beloved member of the Glee family and we fully support his decision to seek treatment. Everyone at the show wishes him well and looks forward to his return."
—Reporting by Marc Malkin