What the figures prove — page 6

• district 1 gets exactly the seats it is due ×4
• district 1's total survives the rounding ×3
• in a 2 × 2 × 2 box every table of halves rounds inside quota ×3
• party 1 gets exactly the seats it is due ×3
• the quotient in district 1, party A rounds to its seats without a tie ×3
• the quotient in district 1, party B rounds to its seats without a tie ×3
• the quotient in district 1, party C rounds to its seats without a tie ×3
• the quotient in district 1, party D rounds to its seats without a tie ×3
• the dial at δ = 0/20 is Adams's own answer ×2
• the dial at δ = 10/20 is Webster's own answer ×2
• the dial at δ = 20/20 is Jefferson's own answer ×2
• the house size is a whole number between 2 and 200 ×2
• 4 to 16 half cells
• a column with a fractional cell has a second one
• a divisor above the interval awards fewer than the house size
• a divisor strictly inside the interval awards exactly the house size
• a quota violation is a whole number of seats away from the band
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a region outside quota does not sit exactly on its quota
• a row with a fractional cell has a second one
• a table inside every cell's quota exists
• a table meeting every district's target and every party's target exists
• a table of 2 to 4 districts by 2 to 4 parties with positive whole vote counts
• a trial divisor awards no more seats than the search allows for
• Adams does the opposite
• Adams is found outside quota
• Adams is never found failing house or population monotonicity, which is a theorem about every divisor method
• Adams's awarded seats sum to the house size
• almost every instance drawn is usable
• alternate halves give every row exactly one seat
• and every column exactly one
• and exactly one table does it at least cost
• and Hamilton is found losing a seat as the house grows
• and Hill's is the one no transfer improves for the relative difference
• and losing a seat to a region that grew more slowly
• and the best of them puts exactly one seat where the fair share is 0
• and the smallest never gains one
• and Webster's bias for the largest region is smaller than either of theirs
• apportioning each district on its own gets at least one party's total wrong
• both ends of the interval land inside the drawn divisor axis
• each entry of the populations is a whole number between 1 and 10000000
• every cell ends at the floor or the ceiling of its share
• every column holds none or two halves
• every half shares one line in each direction with another half
• every house size swept has a seat for every region
• every line holds no halves or exactly two
• every method searched fails at least one of the three properties
• every row holds none or two halves
• floor division is taken of a whole number by a positive whole number
• Hamilton gives every region the floor or the ceiling of its own quota
• Hamilton is never found outside quota, which is a theorem about it
• Hamilton's awarded seats sum to the house size
• Hamilton's seats sum to the house at every size swept
• Hill seats every region before any second seat
• Hill's awarded seats sum to the house size
• in a 3 × 3 × 3 box some cannot, the smallest with sixteen halves
• Jefferson gives the largest region more than its quota on average and the smallest less
• Jefferson is found outside quota
• Jefferson is never found failing house or population monotonicity, which is a theorem about every divisor method
• Jefferson's apportionment is improvable under every one of the three
• Jefferson's awarded seats sum to the house size
• multipliers exist that round the votes to exactly these seats
• no fair share is a whole number, so its quota is two seats wide
• no region's seat count falls anywhere in the swept range
• one seat target per district
• one seat target per party
• one seat total per district
• one seat total per party
• party A's total survives the rounding
• party B's total survives the rounding
• party C's total survives the rounding
• party D's total survives the rounding
• some allocation is stable for the district measure
• some allocation is stable for the relative measure
• some allocation is stable for the share measure
• some table meets every district's seats and every party's seats
• some whole-number table meets every line total
• the allocations stable for the district measure are all the same allocation
• the allocations stable for the relative measure are all the same allocation
• the allocations stable for the share measure are all the same allocation
• the awarded seats sum to exactly the house size
• the biproportional table is among the within-quota tables exactly when it breaks no quota
• the closed end of the interval awards exactly the house size
• the cycle alternates rows and columns and has an even number of cells
• the cycle through the halves has an even number of cells
• the dial changes the answer at all
• the dial has a step exactly at one half
• the districts' and parties' seats fill one house
• the districts' seats and the parties' seats add to the same house
• the divisor and the priority ranking award the same seats
• the divisor method is one of jefferson, webster, adams
• the exact arithmetic stays inside the safe integer range
• the exact floor of a quota and its decimal floor agree
• the exact quotas sum to exactly the house size
• the exact quotas sum to the house size in both censuses
• the fall the sweep found is by a single seat
• the family contains a table whose biproportional seats break a quota
• the family of second censuses is small enough to walk in full
• the faster-growing region holds exactly one seat fewer after the census
• the fitted shares meet every district's and party's total
• the floors leave between none and one seat per region over
• the growth step is a whole number between 1 and 100000
• the halves contain a cycle of seven, and none shorter that is odd
• the house has at least one seat for every region
• the house has at most 24 seats, so every table can be visited
• the house is large enough for every seat the method gives away for free
• the house is large enough for the seats the method gives away
• the interval of divisors is not empty
• the largest house swept is a whole number between 2 and 200
• the largest region never loses a seat as the dial turns toward Jefferson
• the marked region really holds fewer seats in the larger house
• the method is one of hamilton
• the method is one of hamilton, jefferson, webster, hill, adams
• the method is one of jefferson, webster, adams
• the mode is one of quota, alabama, divisor, population, family, bias, impossible, unfair, biproportional, quota2d, fit2d, certificate2d, cycle2d, rarity2d, halves3d, graph3d, slices2d, nearest3d, search3d
• the named quota violation is a real comparison of the seats against the exact quota
• the number of growth steps is a whole number between 1 and 12
• the number of instances generated is a whole number between 50 and 1200
• the number of instances searched is a whole number between 20 and 400
• the number of random tables is a whole number between 50 and 400
• the number of regions is a whole number between 3 and 6
• the number of steps along the dial is a whole number between 8 and 40
• the open end of the interval awards more than the house size
• the populations is a list of 2 to 8 numbers
• the populations together stay inside the exact range
• the priority list reaches past the house size, so the interval has both ends
• the region that lost a seat grew by the larger exact ratio
• the rounds drawn bring the wrong totals down by at least a factor of a million
• the seats awarded along the dial sum to the house size
• the second census fills exactly the same house
• the second census is larger than the first
• the second census totals what it says it does
• the seed is a whole number between 1 and 1000000
• the signpost offset is a fraction between zero and one
• the slower-growing region holds exactly one seat more after the census
• the smallest house swept is a whole number between 2 and 200
• the sweep is short enough to draw one house size at a time
• the sweep runs over at least two house sizes
• trying to mark alternate halves as a seat and not a seat runs into two neighbours marked alike
• Webster is found outside quota
• Webster is never found failing house or population monotonicity, which is a theorem about every divisor method
• Webster's apportionment is the one no transfer improves for the difference in seats per person
• Webster's awarded seats sum to the house size

• voter 1 ranks A and B the same way in both profiles ×5
• the total for 000 is the minorities plus the margins it overrides ×4
• each ballot ranks all 3 of them exactly once ×3
• the 3-voter profiles fall into one class per pattern of the pair ×3
• the number of candidates is a whole number between 2 and 5 ×3
• the number of judges is a whole number between 3 and 6 ×3
• the space of 3-voter profiles was built entire ×3
• B wins all 4 of its pairs ×2
• each candidate of the pair is a whole number between 0 and 2 ×2
• every one of the 64 profiles is examined ×2
• the manipulating voter's true ranking ranks all 3 of them exactly once ×2
• A ≻ B ≻ D ≻ C is not an improvement and is not marked as one
• A ≻ C ≻ B ≻ D is not an improvement and is not marked as one
• A ≻ C ≻ B is not an improvement and is not marked as one
• A ≻ C ≻ D ≻ B is not an improvement and is not marked as one
• A ≻ D ≻ B ≻ C is not an improvement and is not marked as one
• A ≻ D ≻ C ≻ B is not an improvement and is not marked as one
• A does not beat everybody
• A has more Borda points than B
• A has more Borda points than C
• A has more Borda points than D
• A has more Borda points than E
• A has more first places than B
• A has more first places than C
• A has more first places than D
• A has more first places than E
• a profile is between one and eight voter groups
• a rational is a whole numerator over a non-zero whole denominator
• A really beats B
• A wins all 4 of its pairs
• an odd number of judges
• an odd number of judges, so majorities are decisive
• an odd number of judges, three to seven, each with a yes or no on three questions
• and it is silent on more profiles than the majority quota is
• and most do not
• and none is both, which is the impossibility this agenda exhibits
• and on some that are not aligned, so alignment is sufficient and not necessary
• and some are consistent
• and some do not, so neither procedure is always wrong
• and the aligned rule gives up universal domain
• and the drawn profile really is one of them
• B ≻ A ≻ C ≻ D elects B, which the voter ranks above D
• B ≻ A ≻ C elects B, which the voter ranks above C
• B ≻ A ≻ C is not an improvement and is not marked as one
• B ≻ A ≻ D ≻ C elects B, which the voter ranks above D
• B ≻ C ≻ A ≻ D elects B, which the voter ranks above D
• B ≻ C ≻ A elects B, which the voter ranks above C
• B ≻ C ≻ A is not an improvement and is not marked as one
• B ≻ C ≻ D ≻ A elects B, which the voter ranks above D
• B ≻ D ≻ A ≻ C elects B, which the voter ranks above D
• B ≻ D ≻ C ≻ A elects B, which the voter ranks above D
• B beats every other candidate head to head
• B does not beat everybody
• B has more Borda points than A
• B has more Borda points than C
• B has more Borda points than D
• B has more Borda points than E
• B really beats C
• Borda can be flipped by moving a third candidate somewhere in this range
• Borda puts A above B in the first profile
• Borda puts B above A in the second
• C ≻ A ≻ B ≻ D is not an improvement and is not marked as one
• C ≻ A ≻ B is not an improvement and is not marked as one
• C ≻ A ≻ D ≻ B is not an improvement and is not marked as one
• C ≻ B ≻ A ≻ D elects B, which the voter ranks above D
• C ≻ B ≻ A is not an improvement and is not marked as one
• C ≻ B ≻ D ≻ A elects B, which the voter ranks above D
• C ≻ D ≻ A ≻ B is not an improvement and is not marked as one
• C ≻ D ≻ B ≻ A is not an improvement and is not marked as one
• C does not beat everybody
• C really beats A
• C really beats D
• Coombs can be flipped by moving a third candidate somewhere in this range
• Coombs eliminated exactly one candidate in every round but the last
• Coombs puts A above B in the first profile
• Coombs puts B above A in the second
• Coombs reaches a winner without a level elimination
• counting the three-way cycles gives the same answer as looking for a winner
• D ≻ A ≻ B ≻ C is not an improvement and is not marked as one
• D ≻ A ≻ C ≻ B is not an improvement and is not marked as one
• D ≻ B ≻ A ≻ C is not an improvement and is not marked as one
• D ≻ B ≻ C ≻ A is not an improvement and is not marked as one
• D ≻ C ≻ A ≻ B is not an improvement and is not marked as one
• D ≻ C ≻ B ≻ A is not an improvement and is not marked as one
• D beats every other candidate head to head
• D does not beat everybody
• D really beats A
• D wins all 4 of its pairs
• dictatorship of judge 1 gives up at least one condition
• dictatorship of judge 1 is consistent wherever it is defined
• dictatorship of judge 1 is defined somewhere
• E does not beat everybody
• each judge is consistent
• each voter group carries a count and an order
• each voter group's size is a whole number between 1 and 500
• every assignment of positions to judges is considered
• every ballot the voter could submit produces a single winner
• every class holds the same number of profiles
• every consistent rule leaves some proposition undecided
• every inconsistent majority on this agenda has the same shape
• every judge's own three answers hang together
• every profile in the space was decided one way or the other
• every ranking the voter could submit was built
• every systematic anonymous rule on this agenda is considered
• every troubled profile lands in one of the four outcomes
• instant runoff can be flipped by moving a third candidate somewhere in this range
• instant runoff eliminated exactly one candidate in every round but the last
• instant runoff puts A above B in the first profile
• instant runoff puts B above A in the second
• instant runoff reaches a winner without a level elimination
• majority is consistent on every aligned profile
• majority is defined somewhere
• majority never fails exactly when no minimally inconsistent set has more than two members
• majority, aligned profiles only gives up at least one condition
• majority, aligned profiles only is consistent wherever it is defined
• majority, aligned profiles only is defined somewhere
• no candidate beats every other, which is what having no Condorcet winner means
• no misreport helps the voter once only two candidates remain
• not every profile is aligned, which is what makes the restriction a restriction
• on this profile the majority corner is not a consistent one
• one candidate has strictly the most Borda points
• one candidate has strictly the most first places
• one disagreeing profile is drawn
• plain majority on every proposition is the rule the impossibility rules out
• plurality can be flipped by moving a third candidate somewhere in this range
• plurality puts A above B in the first profile
• plurality puts B above A in the second
• premise-based gives up at least one condition
• premise-based is consistent wherever it is defined
• premise-based is defined somewhere
• raising the quota never reduces how often the rule says nothing
• some profiles make the two procedures disagree
• some profiles produce an inconsistent majority
• some rules are complete
• the agenda is one this family knows
• the Coombs winner ends with a strict majority
• the dictatorship gives up anonymity
• the distinct winners counted by walking the list agree with the set
• the electorate is a whole number between 2 and 2000
• the electorate is odd, so the two-candidate control cannot end level
• the exact arithmetic stays inside the safe integer range
• the exhausted profile space is small enough to give every profile its own cell
• the exhausted profile space stays small enough to walk
• the honest ballot appears exactly once among the submissions
• the honest winner does not depend on the order the ballots are listed in
• the instant-runoff winner ends with a strict majority
• the largest electorate searched is a whole number between 3 and 6
• the majority between A and B is decided rather than level
• the majority between A and C is decided rather than level
• the majority between A and D is decided rather than level
• the majority between B and C is decided rather than level
• the majority between B and D is decided rather than level
• the majority between C and D is decided rather than level
• the majority quota is inconsistent on some profiles
• the majority tournament runs in a cycle through every candidate
• the majority's three answers do not hang together, which is the whole point
• the majority's verdicts are the median judge's
• the margin between A and B is antisymmetric
• the margin between A and C is antisymmetric
• the margin between A and D is antisymmetric
• the margin between A and E is antisymmetric
• the margin between B and C is antisymmetric
• the margin between B and D is antisymmetric
• the margin between B and E is antisymmetric
• the margin between C and D is antisymmetric
• the margin between C and E is antisymmetric
• the margin between D and E is antisymmetric
• the mode is one of profile, cycle, rules, iia, manipulate, judge, agenda, escape, doctrinal, impossible, escapes, quota, aligned, distance, cube, distancesweep, kemeny, agendas, mis, quotak, median
• the number of distinct winners lies between one and the number of rules
• the number of judges is odd, so no question ties
• the number of voters in the exhausted space is a whole number between 2 and 6
• the ordered pairs inside a class counted two ways agree
• the pair is two different candidates
• the premise-based answer is consistent
• the premise-based rule gives up systematicity
• the profile space has one entry for every assignment of ballots
• the quota is safe exactly when it exceeds n(k − 1)/k
• the rule being manipulated is one of plurality, borda, irv, coombs
• the rule to violate is one of plurality, borda, irv, coombs
• the two opposed counts for A against B sum to the electorate
• the two opposed counts for A against C sum to the electorate
• the two opposed counts for A against D sum to the electorate
• the two opposed counts for A against E sum to the electorate
• the two opposed counts for B against C sum to the electorate
• the two opposed counts for B against D sum to the electorate
• the two opposed counts for B against E sum to the electorate
• the two opposed counts for C against D sum to the electorate
• the two opposed counts for C against E sum to the electorate
• the two opposed counts for D against E sum to the electorate
• the two profiles really differ on some ballot
• the two rules disagree on this profile, which is what makes it a choice
• the two-candidate contest has a winner whichever ballot the voter submits
• the unanimity quota is inconsistent on none
• three judges with three yes-or-no answers each
• three to five judges, each holding a consistent judgement set
• unanimity gives up at least one condition
• unanimity gives up completeness
• unanimity is consistent wherever it is defined
• unanimity is defined somewhere
• where the majority's set is consistent the rule returns exactly it

• the layer formula and the average over all 3! orders agree for A ×2
• the layer formula and the average over all 3! orders agree for B ×2
• the layer formula and the average over all 3! orders agree for C ×2
• the quota is a whole number between 1 and 120 ×2
• the weights is a list of 3 to 3 numbers ×2
• A and B are interchangeable
• a family demanding more than there is rules the core out
• a member who can never swing a vote has no power by either count
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a satisfied set of certificates finds splits, unless the core is a single point off the lattice
• a split some coalition can beat is objected to
• a stable split is one every objection to which can be answered
• and at least one of the other rules fails one
• and D adds nothing
• and it falls at about the square-root rate over the whole sweep
• and satisfies the other three
• and the additivity column's rule gives a different answer here
• and the efficiency column's rule gives a different answer here
• and the symmetry column's rule gives a different answer here
• and the the null player column's rule gives a different answer here
• and their heights add to it
• at least one assembly drawn has power and weight ordered differently
• at least one family is satisfied, so the table is separating them
• averaging what each adds satisfies all four on this game
• averaging what each player adds passes every condition on every game tested
• between one and four assemblies
• between three and eight increasing sample sizes
• C is not interchangeable with them, so symmetry is a real condition here
• each entry of the capacities the users need is a whole number between 1 and 200
• each entry of the split argued about, as three weights is a whole number between 0 and 100
• each entry of the weights is a whole number between 1 and 40
• each share lands on the grid the search used
• every coalition's value is a whole number
• every estimate is a share
• every family in the table really is balanced
• every marked split really attains the smallest imbalance
• every ordering ends with everybody in
• every split hands out the whole
• no coalition objects to a split none of them can beat
• no other split on the table has a quieter list of complaints
• no split survives when some balanced family over-demands
• on this game the nucleolus and the average split are different points
• relaxing by less does not
• relaxing by the computed amount lets a split through
• some coalition holds one player and not the other
• some coalition objects to every split drawn here
• some family attains the maximum
• some split is individually rational
• some split on the lattice pays every player at least what they earn alone
• somebody can swing something, or the assembly decides nothing
• the balance verdict is computed for every split
• the capacities the users need is a list of 2 to 5 numbers
• the certificate is drawn for three players
• the closed-form shares pay the whole bill
• the coalitions together demand more than there is to give
• the core is drawn for three players
• the core, nucleolus and excess views are written for a game whose value is shared out, and this game's value is a bill
• the denominator the splits are searched over is a whole number between 12 and 180
• the empty coalition is worth nothing
• the error at the largest sample is a third of the error at the smallest
• the error is a distance
• the exact arithmetic stays inside the safe integer range
• the exact shares add to one
• the family chosen is a whole number between 1 and 5
• the game is big enough that listing the orders is the point
• the game is one of partnership, majority, gloves, shops, capacity, dummy
• the game names a value for every coalition
• the grid divides the whole group's value evenly
• the group A pays no more together than it would alone
• the group AB pays no more together than it would alone
• the group ABC pays no more together than it would alone
• the group ABCD pays no more together than it would alone
• the group ABD pays no more together than it would alone
• the group AC pays no more together than it would alone
• the group ACD pays no more together than it would alone
• the group AD pays no more together than it would alone
• the group B pays no more together than it would alone
• the group BC pays no more together than it would alone
• the group BCD pays no more together than it would alone
• the group BD pays no more together than it would alone
• the group C pays no more together than it would alone
• the group CD pays no more together than it would alone
• the group D pays no more together than it would alone
• the imbalance separates the splits, so the condition bites
• the kernel is drawn for three players
• the layer formula and the average over all 4! orders agree for D
• the layers stack up to the largest requirement
• the least core is drawn for three players
• the marginal contributions along one ordering add up to the whole
• the nucleolus survives every objection exactly when some split does
• the objection is drawn for three players
• the pairwise surpluses are drawn for three players
• the power shares add to one
• the proportional rule needs somebody to be worth something alone
• the quota is more than half the weight and no more than all of it
• the requirements are given smallest first
• the resolution of the exhaustive sweep is a whole number between 12 and 120
• the resolution of the relaxed sweep is a whole number between 12 and 120
• the resolution of the sweep the verdict is checked against is a whole number between 12 and 120
• the resolution the proposals are searched over is a whole number between 8 and 48
• the rule in the additivity column fails additive
• the rule in the efficiency column fails efficient
• the rule in the symmetry column fails symmetric
• the rule in the the null player column fails null player
• the search finds no balanced family of at most three coalitions outside the table
• the search is written for three players
• the second game names a whole value for each of the eight coalitions of three, starting at nothing
• the seed is a whole number between 0 and 10000
• the shares add to what the whole group is worth
• the shares add up to what the whole group is worth
• the split argued about, as three weights is a list of 3 to 3 numbers
• the split hands out the whole
• the splits achieving the minimum are one point to within a grid step
• the splits give different surpluses, so the table is comparing them
• the sweep and the balancedness condition agree about whether any split survives
• the test is drawn for three players
• the users do not all need the same thing, or there is nothing to divide unevenly
• the view is one of axioms, core, nucleolus, excess, unique, power, sample, layers, balanced, certificate, leastcore, surplus, kernel, counter
• the weights are not all nought
• the weights are positive
• the whole group is worth a whole number
• there is more than one layer, or the figure has nothing to show
• two games on the same number of players
• two to four games this family knows
• what all four players are worth together is a whole number between 4 and 40
• when a stable split exists, every balanced split is one

• person 1's valuation is a list of 6 to 6 numbers ×11
• person 2 receives at least a 1/4 share by their own measure ×11
• person 1 values their own share at least as highly as person 1's ×9
• each entry of person 1's valuation is a whole number between 0 and 100 ×7
• person 1's cake totals 100 ×7
• the value matrix, person 1's row is a list of 3 to 3 numbers ×7
• trimming piece 1 is worth exactly a third of the trimming to person 3 ×6
• person 1's values is a list of 2 to 6 numbers ×5
• the control matrix, person 1's row is a list of 3 to 3 numbers ×5
• no allocation of these goods is envy-free — all 8 were tested and each failed ×4
• person 1 strictly prefers person 3's piece to their own ×4
• person 1's own piece is worth at least 25 to person 1 ×4
• person 1's valuation of the whole cake is 100 ×4
• person 1's values of the 4 pieces account for the whole cake ×4
• person 2 receives at least a 1/4 share ×4
• person 2's calling mark stays over the bar it belongs to ×4
• each entry of person 1's values is a whole number between 0 and 100 ×3
• each entry of the control matrix, person 1's row is a whole number between 0 and 100 ×3
• each entry of the value matrix, person 1's row is a whole number between 0 and 100 ×3
• person 1's items total 100 ×3
• person 1's values of the three shares account for the whole cake ×3
• person 2 called the knife at exactly 25 ×3
• piece 1 is worth exactly a third of the cake to person 1 ×3
• the control matrix: person 1 values the whole set at 100 ×3
• the valuations are given for 2 to 2 people ×3
• the value matrix: person 1 values the whole set at 100 ×3
• person 1 buys only goods of the best value for money to them ×2
• person 1 spends exactly their budget ×2
• person 1's goods total 100 ×2
• the number of people is a whole number between 2 and 6 ×2
• a knife is asked to pass a non-negative amount of value
• a person sent left values the left part at their share or more
• a person sent right values the right part at their share or more
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• and each gets at least half by their own measure
• and fewer than the last diminisher from four people on
• and moves the sum's maximiser
• at the product's maximum neither person values the other's bundle more than their own
• each generated valuation totals exactly 100
• every envy-free allocation is also envy-free up to one item
• every person still waiting gets a calling mark
• every way of giving each item to one of the people was formed
• halving asks at most n⌈log₂ n⌉ questions
• nobody left behind values the slice at more than their share of what remained
• person 2 does not envy person 1's piece before the trimming is divided
• person 4, who never called, is left with at least 25
• rescaling one person's values leaves the product's maximiser unchanged
• the chooser envies nobody, having taken the piece it valued higher
• the chooser gains strictly more than half — the promise is not tight
• the chooser's piece is worth at least half the cake to the chooser
• the chooser's values of the two pieces sum to the whole cake
• the claim that somebody is left envious is either made or not made
• the claim that the chooser strictly gains is either made or not made
• the control matrix gives a row of values for two, three or four people
• the control was searched over every allocation too
• the control: a chooser with the cutter's own measure gains exactly nothing
• the control: the same search finds an envy-free allocation where one exists
• the cutter's envy is exactly zero: the piece left behind is worth what the other is
• the division being shown is one of cutchoose, trim, items, moving, evenpaz, lastdim, cutcount, nash, scale, prices, mnwitems, mnwsearch
• the division is proportional and not envy-free: somebody prefers another's piece
• the exact arithmetic stays inside the safe integer range
• the exact value of the left piece and the decimal one agree
• the largest number of people is a whole number between 16 and 128
• the marks made match the count the recursion predicts
• the number of instances is a whole number between 50 and 600
• the number of items in the control matrix is a whole number between 2 and 6
• the number of items in the value matrix is a whole number between 2 and 6
• the number of segments the cake is cut into is a whole number between 2 and 12
• the product-maximising allocation is envy-free up to one item
• the product's maximiser is envy-free up to one item on every instance
• the rescaling factor is between 1 and 10
• the round-robin allocation is envy-free up to one item
• the rule asks n(n+1)/2 − 1 questions
• the sum's maximiser is not
• the three pieces go to three different people
• the trimmed piece ties person 2's second largest exactly
• the value asked for is still ahead of the knife
• the value matrix gives a row of values for two, three or four people
• two calling marks never print on top of one another
• two or three people
• under the cutter's own measure the two pieces are exactly equal

• moving participant 1 from state 0 lowers the potential by exactly what they save ×12
• the run from 0.05 has closed on the rest point ×5
• the run from 0.05 has moved away from the rest point ×5
• nobody improves by moving alone out of state 1 ×4
• the column chooser's options names all 2 of them in one to three characters ×2
• the row chooser's options names all 2 of them in one to three characters ×2
• the time drawn is between 2 and 40 ×2
• a caption for the matrix is a short description, or not given
• a caption renames the game only when the placement supplies its own payoffs
• a cell carries both marks exactly when neither chooser can gain by moving alone
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a win and a loss are whole numbers between one and four
• above one, investing pays even if the other certainly does not
• all the traffic arrives at T
• all the traffic leaves S after the link is added
• all the traffic leaves S before the link is added
• and a move that does not improve raises it by exactly what it costs
• and above it, keeping it does
• and at that belief the two actions are worth exactly the same
• and below nothing, staying out pays even if the other certainly invests
• and it is a region rather than a segment
• and none pays the column chooser less
• and so is the closed form for breaking it
• and the bracket has closed to a small fraction of what it started at
• and the optimistic one never falls
• and the two never cross
• and when a win and a loss are equal no invader does worse there than the centre does
• at least one column holds the row chooser exactly to the value
• at least one order of elimination was run
• at least one state has no improving move, so an equilibrium exists
• at the centre every pure strategy earns the same, so a third each is an equilibrium
• at the critical discount factor the two streams are worth exactly the same
• at the cutoff a chooser believes the other is above it with probability exactly a half
• at the mixed equilibrium the leader earns the same from either row
• at the rest point both strategies earn the same, so it is the mixed equilibrium
• at the symmetric cutoff the other is believed above it with probability exactly a half
• below it, breaking the agreement pays more
• between four and forty rounds are drawn
• between one and four starting mixtures, each three positive shares adding to one
• between one and six starting shares, each strictly between nought and one
• between two and four participants
• both of them sit on the diagonal, which is what makes the two tie-breakers comparable
• committing is worth at least as much as every equilibrium of the simultaneous game
• each step moves the share the way the flow points
• every cell of the matrix was put to both tests
• every distribution on the lattice was tested
• every order of elimination reaches the same surviving set
• every payoff is a whole number no larger than 20 in size
• every traveller is strictly slower after the zero-cost link is added
• in a zero-sum game committing is worth exactly the value, and nothing more
• in this game committing is worth strictly more than any equilibrium
• no commitment on a lattice of 121 does better than the best candidate
• no equilibrium pays the row chooser less than the minmax
• no mixing weight on the sweep guarantees the row chooser more than the peak
• no mixture of the punisher's holds the other below the minmax
• no mixture on the lattice holds the row chooser below the value
• no route is cheaper than the one everybody is on, so the flow after is an equilibrium
• no split of the traffic on the lattice costs less in total than the computed optimum
• nothing an equilibrium of the stage game pays is outside what repetition sustains
• some pair of columns achieves the column chooser's best mixture
• the agreement is worth more than the punishment and less than breaking it
• the best correlated equilibrium is at least as good as the best pure one
• the best use of the added link is somewhere between none of the traffic and all of it
• the bimatrix carries two payoffs in every cell
• the bimatrix drawn by the payoff mode is one of coordination, none, dominant, three
• the bimatrix has between 2 and 4 columns
• the bimatrix has between 2 and 4 rows
• the bimatrix is rectangular — every row offers the same columns
• the centre resists every invader exactly when a win is worth more than a loss
• the closed form for keeping the agreement is the limit of its own terms
• the closed form for the least possible total agrees with pricing its own flow edge by edge
• the column chooser's mixture holds the row chooser to at most the value against every row
• the computed optimum is itself a point of the lattice
• the correlated mode draws a two-by-two game
• the cost of a fixed-cost link is a whole number between 1 and 400
• the cost per unit of flow on a congestible link is a whole number between 1 and 10
• the critical discount factor is a real fraction of the way between nothing and certainty
• the crossing sits strictly inside the unit interval
• the drawn distribution is a correlated equilibrium
• the drawn distribution is whole numerators
• the drawn distribution sums to one
• the equilibrium costs strictly more in total than the optimum
• the exact arithmetic stays inside the safe integer range
• the exact value and its decimal recomputation agree
• the first option is the better reply above the crossing and the worse below it
• the first resource never gets cheaper as more use it
• the first resource prices every load it can carry, as whole numbers no larger than forty
• the fixed-cost link is cheap enough that the old routes beat the new one
• the fixed-cost link is dear enough that the added link is taken by everybody
• the four payoff pairs are not all on one line
• the game drawn has exactly two pure equilibria
• the game the commitment mode draws is one of chicken, stag, battle, stengel, pennies
• the game the correlated mode draws is one of chicken, stag, battle
• the game the replicator mode draws is one of hawk, stag
• the game the selection mode draws is one of chicken, stag, battle
• the lattice contains at least one correlated equilibrium
• the lattice of distributions has a denominator between 4 and 24
• the mixed rest point attracts exactly when no invading mixture can do as well against it as it does against itself
• the mode of the game family is one of payoff, mixed, dominance, network, correlated, potential, select, replicator, cycle, commit, folk, discount, global, iterate
• the payoff gap is not constant, so the two strategies can tie at most once
• the pessimistic cutoff never rises
• the product of the shares grows at every sampled time
• the product of the shares shrinks at every sampled time
• the pure equilibrium AA is a correlated equilibrium too
• the pure equilibrium BB is a correlated equilibrium too
• the pure equilibrium CD is a correlated equilibrium too
• the pure equilibrium DC is a correlated equilibrium too
• the pure equilibrium HH is a correlated equilibrium too
• the pure equilibrium SS is a correlated equilibrium too
• the ratio of the equilibrium's total to the least possible total is exact
• the reading is off by between two and forty hundredths
• the region above both minmax values has an interior
• the row chooser's maximin and the column chooser's minimax are the same number
• the row chooser's mixture is worth at least the value against every single column
• the second never gets cheaper as more use it
• the second prices every load it can carry, as whole numbers no larger than forty
• the selection mode draws a two-by-two game
• the shares still add to one at the end of the run, so the integration stayed on the triangle
• the simultaneous game has an equilibrium to compare against
• the stage game carries two payoffs in every cell
• the stage game has between 2 and 4 columns
• the stage game has between 2 and 4 rows
• the stage game is rectangular — every row offers the same columns
• the stage game the discount mode repeats is one of prisoner, stag, chicken
• the stage game the folk-theorem mode repeats is one of prisoner, stag, chicken
• the state of least potential is one of them
• the surviving column is a best reply to the surviving row
• the surviving row is a best reply to the surviving column
• the tie falls strictly inside the population, so there is a mixed rest point
• the traffic entering the network is a whole number between 3 and 40
• the two lines are not parallel, or there is no crossing to draw
• the two lines meet at the mixed equilibrium
• the two routes cost the same before the link is added, so nobody moves
• the two sequences bracket that half at every stage, this one included
• the zero-sum matrix has between 2 and 4 columns
• the zero-sum matrix has exactly two rows — the mixing probability is one number
• the zero-sum matrix is rectangular — every row offers the same columns
• what arrives at A leaves A
• what arrives at B leaves B
• with a win worth a loss the product of the shares is conserved along the orbit
• with two columns

• 1's ranking of side one ranks all 4 of them exactly once ×14
• hospital 1's list names distinct residents ×5
• A's ranking of side two ranks all 4 of them exactly once ×4
• B's ranking of side two ranks all 4 of them exactly once ×4
• the matching drawn as unstable ranks all 4 of them exactly once ×4
• C's ranking of side two ranks all 4 of them exactly once ×3
• D's ranking of side two ranks all 4 of them exactly once ×2
• the member of side two whose lists are searched is a whole number between 0 and 3 ×2
• a hospital left short holds the same residents in each
• a profitable misreport is profitable under the ranking actually held
• a proposer still has a name left to propose to
• A ranks every other person once
• a stable pairing exists exactly when there is no odd ring
• a stable partition exists
• A's list names distinct hospitals
• an even number of people, four to ten
• and is one of the market's stable matchings
• and matches the same residents
• at every size most instances, but not all, have a stable pairing
• at least one of the matchings has no blocking pair at all
• at the top, every member of side two has its worst stable partner
• B ranks every other person once
• B's list names distinct hospitals
• between one and fifteen pairings to draw
• C ranks every other person once
• C's list names distinct hospitals
• D ranks every other person once
• D's list names distinct hospitals
• E ranks every other person once
• E's list names distinct hospitals
• E's ranking of side two ranks all 5 of them exactly once
• each annotation fits inside its own cell with a gap to the cell beside it
• each member of side one settles on the last name it proposed to
• every hospital has between one and four places
• every matching of the two sides was formed
• every ordered pair of stable matchings was joined and met
• every pair of the two sides is put the blocking question
• every proposer ends the construction matched
• every random market has a stable assignment
• every ranking the participant could submit was formed
• every stable assignment fills each hospital to the same count
• every stable matching lies between the two the construction can reach
• every stable pairing pairs across the two sides
• every stable partition has the same odd rings
• F ranks every other person once
• F's list names distinct hospitals
• G ranks every other person once
• G's list names distinct hospitals
• H ranks every other person once
• I ranks every other person once
• J ranks every other person once
• no member of the proposing side has a profitable misreport
• no stable matching is larger than the largest matching
• nobody proposes to the same name twice
• nothing with a blocking pair is anything the construction returns
• side one has one ranking per member
• side two has one ranking per member
• submitting the true ranking returns the truthful matching
• the bottom of the order is what the construction returns with side two proposing
• the construction returns a one-to-one pairing
• the count of pairs put the blocking question
• the instance has a matching that stability excludes, or there is nothing to draw
• the instance has at least one stable matching
• the join of two stable matchings is stable
• the market has a stable assignment
• the matching side one proposing returns is in the zero band of the census
• the matching side two proposing returns is in the zero band of the census
• the matching this mode draws is one stability actually excludes
• the meet of two stable matchings is stable
• the mode of the match family is one of run, blocking, lattice, strategy, seats, rural, partial, roommates, roomcensus, partition, twosided
• the pairings number (n − 1)!!
• the pointwise better of two stable matchings is a matching
• the pointwise worse of two stable matchings is a matching
• the proposals counted and the proposals drawn are the same proposals
• the roommates instance has exactly as many stable pairings as the market has stable matchings
• the rounds cannot outnumber the proposals available
• the settled matching has no blocking pair
• the size of each side is a whole number between 2 and 6
• the tally accounts for every matching exactly once
• the top is at least as good as every stable matching for side one
• the top of the order is what the construction returns with side one proposing
• the true ranking is one of the rankings searched
• the whole strategy space of both sides was searched
• the zero column of the tally is the stable set
• two to eight residents
• two to six hospitals

• at b₁ = 0 the optimum is the lowest dual line ×41
• at b₁ = 1/2 the optimum is the lowest dual line ×40
• the program at b₂ = 2 has at least one feasible vertex ×33
• the program at b₂ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×33
• the program at b₂ = 5/2 has at least one feasible vertex ×32
• the program at b₂ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×32
• the program at b₁ = 2 has at least one feasible vertex ×31
• the program at b₁ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×31
• the program at b₁ = 5/2 has at least one feasible vertex ×30
• the program at b₁ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×30
• the weights on person 1's task 1 add back to the share ×16
• objective (0, 4): runs away exactly when the dual is empty ×9
• objective (-4, 0): runs away exactly when the dual is empty ×8
• objective (0, -4): runs away exactly when the dual is empty ×8
• objective (-4, -4): runs away exactly when the dual is empty ×7
• objective (-4, -4): the two optima agree ×7
• objective (-4, 0): the two optima agree ×5
• round 1: what is left still has a whole assignment inside it ×5
• person 1's shares add to a whole task ×4
• task 1 is exactly covered ×4
• constraint row 1 constrains at least one variable ×3
• constraint row 1 has one coefficient per variable ×3
• objective (0, -4): the two optima agree ×3
• row 1 of A times the ray is not positive ×2
• the decomposition needs at most n² − 2n + 2 = 5 whole assignments ×2
• the right-hand side is a list of 2 to 2 numbers ×2
• a program here has between two and five constraint rows
• a rational is a whole numerator over a non-zero whole denominator
• a rational is never divided by zero
• a set of prices whose total equals the cheapest assignment exists
• an intersection is kept exactly when it satisfies every constraint
• and doubling b doubles the optimum, as a line through the origin must
• and every corner is a whole assignment
• and every pair the assignment uses is tight
• and gives the same value at the corner
• and its dual is empty
• and its dual is unbounded
• and its weights add to one
• and the fractional corner is a half on every edge
• and the objective grows along the ray
• and the weight taken out is positive
• and the x₂ terms
• at weight limit 10 the dual bound is strictly above the best packing
• between three and six edges
• both the program and its dual are empty
• each decomposition rebuilds every entry exactly
• each entry of the right-hand side is a whole number between 0 and 400
• each fan line lies on or above the optimum everywhere
• each piece's slope is the dual variable on that piece
• every complementary product is exactly zero
• every constraint coefficient is a whole number of size at most 40
• every dual certificate at this optimum gives the same price for the swept constraint
• every feasible primal value is at most every feasible dual value
• every objective coefficient is a whole number of size at most 40
• every pair of constraints was formed, not a selection of them
• every point between the two tied vertices is dual feasible
• every program with coefficients in {−1, 0, 1} is classified
• every set of people is willing to take at least as many tasks as there are of them
• inside the sector the optimum is that vertex's line
• leaving nought on the left and a negative number on the right
• no dual variable and no reduced cost is negative
• no feasible lattice point beats the best vertex
• no pair's two prices exceed its cost
• no program is infeasible with a dual that is optimal
• no program is optimal with a dual that is infeasible
• no program is optimal with a dual that is unbounded
• no program is unbounded with a dual that is optimal
• no program is unbounded with a dual that is unbounded
• no share is negative
• no slack and no variable is negative at the optimum
• so a mixture containing it would give a share to a pairing the corner refuses
• so its total is half the number of edges
• so the prices add to the cheapest assignment's cost, which certifies it
• some order gives a genuinely different decomposition of the same table
• some program is infeasible with a dual that is infeasible
• some program is infeasible with a dual that is unbounded
• some program is optimal with a dual that is optimal
• some program is unbounded with a dual that is infeasible
• the best dual bound equals the linear optimum, not the whole-number one
• the best whole-number point is strictly worse than the linear optimum
• the constraint whose right-hand side is swept is a whole number between 1 and 2
• the corners number the permutations, and no more
• the cost table is square
• the drawn dual region is convex
• the drawn primal region is convex
• the dual bound is never below the best value
• the dual certificate is worth exactly what the primal optimum is worth
• the dual is drawn only for a program with two constraints — with more, the dual polytope has more than two variables and is not a polygon in the plane
• the dual optimum is unique, so each constraint has one price — a degenerate optimum, with more than two constraints through one point, has a whole set of them and no single table of products
• the dual program has at least one feasible vertex
• the dual program is bounded — an objective that increases without limit has no optimal vertex to draw
• the dual region has at least two vertices
• the exact arithmetic stays inside the safe integer range
• the exact optimum and the decimal one agree
• the exact optimum at the top of the sweep and the decimal one agree
• the feasible vertices are in convex position
• the first order decomposes the table completely
• the fixed second right-hand side is a whole number between 1 and 60
• the fractional table needs at least two whole assignments
• the grid holds more matrices than there are permutations
• the grid the search runs on is a whole number between 3 and 8
• the high end of the sweep is a whole number between 1 and 400
• the low end of the sweep is a whole number between 0 and 400
• the multipliers reproduce the objective exactly
• the number of people is a whole number between 2 and 4
• the objective has one coefficient per variable
• the objective is not identically zero
• the optimal vertex has a dual certificate — a non-negative multiplier on each binding constraint
• the optimal vertex's coordinates are labelled clear of every dot, every value and the axis ticks
• the optimum has a dual certificate
• the optimum is linear across this interval, so the drawn segment is exact
• the polytope figure's mode is one of primal, dual, slack, shadow, birkhoff, extreme, support, assign, hungarian, lottery, fractional, fourcases, unbounded, infeasible, bothempty, directions, envelope, chambers, subgradient, intgap, intsweep, knapsack
• the primal maximum and the dual minimum are the same number
• the primal program has at least one feasible vertex
• the primal program is bounded — an objective that increases without limit has no optimal vertex to draw
• the program has at least one feasible vertex
• the program is bounded — an objective that increases without limit has no optimal vertex to draw
• the program is empty
• the program is unbounded
• the reduced cost is the multiplier on the variable's own non-negativity
• the region itself is never empty
• the relaxation has a corner that is not whole
• the round empties at least one more cell than it found
• the share table is one of thirds, quarters, sparse
• the side of the square of right-hand sides is a whole number between 10 and 60
• the size of the assignment is a whole number between 2 and 4
• the slope of the optimum equals the dual variable for the swept constraint
• the support contains a whole assignment
• the sweep meets a corner of the optimum where two dual vertices tie
• the sweep produced at least one linear piece
• the sweep runs over between two and sixty units of the right-hand side
• the sweep solved the program once at every whole right-hand side
• the table is one the family knows
• the table is square
• the table is used up exactly
• the top of the sweep is a whole number between 10 and 80
• the two agree at some right-hand sides
• the two optima agree as decimals too
• the weights add to one
• the weights cancel the x₁ terms
• the whole-number optimum never exceeds the linear one
• there are n! = 6 whole assignments
• there is one breakpoint between consecutive pieces
• there is one product per constraint and one per variable
• thirty-two objective directions on the ring
• two different assignments differ somewhere
• two to six items, each a whole-number weight and value up to 20
• weak duality was checked on the whole cross product of the two vertex lists
• whenever both are optimal their optima are equal
• which beats every whole matching, so the relaxation is genuinely loose

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