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 number of candidates is a whole number between 2 and 5 ×4
• the number of judges is a whole number between 3 and 6 ×4
• 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 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 largest jury is a whole number between 21 and 1001 ×2
• the manipulating voter's true ranking ranks all 3 of them exactly once ×2
• there are 12 majority patterns on 5 candidates up to relabelling ×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 beats B by exactly two
• a cyclic electorate has its first two margins of one sign
• 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 mixed jury is no worse than a uniform one at the same average
• 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 score is never smaller than the votes still to be won
• a single judge finds for the conclusion in a quarter of profiles
• a single vote moves at most twice the yolk's radius further out
• a single voter is right with the average competence
• A wins all 4 of its pairs
• above one half, every two voters added make the majority likelier to be right
• after the swaps the Dodgson winner beats every rival
• an average competence above one half
• an odd electorate
• an odd electorate of 21 to 101
• an odd jury, so a majority always exists
• 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
• an odd number of voters
• an odd number of voters leaves no tie
• an odd panel of three to nine voters, each better than a coin
• and 2 of the 56 ways of counting the ballots
• and becomes reliable
• and every one is a dictator or an oligarchy
• and it is silent on more profiles than the majority quota is
• and it is unanimity on all three
• and it rises towards that ceiling with every two voters added
• 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 stays below Guilbaud's limit
• and the aligned rule gives up universal domain
• and the drawn profile really is one of them
• and the more voters share it, the rarer cycles are
• and the premise vote better
• at 51 voters each rule is within three points of its limit
• at 51 voters runoff beats Borda beats plurality beats anti-plurality
• at every size the majority's accuracy is under the chance that the shared competence exceeds one half
• averaged over the worlds, enough correlation turns the comparison round
• 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 C by exactly two
• B beats D by exactly two
• 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
• below one half, every two added make it likelier to be wrong
• 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
• Borda's agreement is a share of the electorates counted
• 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 beats A by exactly two
• C beats D by exactly two
• C does not beat everybody
• C really beats A
• C really beats D
• chain: the pruned search finds exactly what the plain one does
• conj: the pruned search finds exactly what the plain one does
• 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
• Copeland's agreement is a share of the electorates counted
• correlated errors help the verdict vote
• counting the three-way cycles gives the same answer as looking for a winner
• cycles are commoner among close contests than among lopsided ones
• 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 A by exactly two
• 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 candidate's Borda score is the number of head-to-head votes it gets
• each covariance entry is the exact covariance of one random ranking's verdicts
• each extra candidate makes a missing winner likelier
• each judge is consistent
• each jury size is the smallest odd one that reaches the target
• each size has a crossover, and larger courts cross earlier
• each voter group carries a count and an order
• each voter group's size is a whole number between 1 and 500
• electorates at each size is a whole number between 50 and 800
• enough electorates without a Condorcet winner at every size
• enough electorates without a Condorcet winner to count
• 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 member of the family shows Anscombe's paradox
• every pattern is produced by one or three voters
• every profile in the space was decided one way or the other
• every ranking the voter could submit was built
• every rule misses the Condorcet winner in some electorate of at most 21
• every survivor gives a consistent verdict here
• every survivor is consistent on every profile
• every systematic anonymous rule on this agenda is considered
• every troubled profile lands in one of the four outcomes
• every voter is counted once
• every voter prefers the start to the target
• exactly one pattern — the transitive order — needs a single voter
• for a large court the conclusion vote switches at √½
• for three judges and two premises the crossover is the square root of a half
• for three judges the average difference is (3/2)p²(1 − p)²(2p² − 1)
• for three judges the crossover is exactly √½, where (3/2)p²(1 − p)²(2p² − 1) vanishes
• four of the six rankings are single-peaked on the axis A, B, C
• free: the pruned search finds exactly what the plain one does
• in the liable world the premise vote is never the worse
• 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
• its average majority is 3m/(4m + 1), below three quarters
• larger sets find for the conclusion no more often
• 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 survives on the chain and the free pair and not on the conjunction
• 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
• more correlation, a less reliable large majority
• more premises never need a smaller court
• no candidate beats every other, which is what having no Condorcet winner means
• no electorate showing Anscombe's paradox has an average majority of three quarters or more
• no misreport helps the voter once only two candidates remain
• not every profile is aligned, which is what makes the restriction a restriction
• of the 216 profiles of three voters, 204 have a Condorcet winner
• on the chain a pair of quotas is safe exactly when the stricter threshold has the stricter quota
• on the conjunction exactly one triple of quotas is safe
• on the conjunction, one survivor for each non-empty set of judges
• on the ranking, the survivors are the dictators and nothing else
• 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
• one for each judge
• one to eight thousand sampled elections for each number of candidates
• one to six competences between 0.3 and 0.9, none exactly one half
• one to three odd electorates up to 45
• one to three target accuracies
• party X wins three groups of five and loses every issue three groups to two
• 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 loses ground as candidates are added and Borda stays ahead of it
• plurality puts A above B in the first profile
• plurality puts B above A in the second
• Plurality's agreement is a share of the electorates counted
• 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
• rank: the pruned search finds exactly what the plain one does
• reachability is computed
• rules that need not respond monotonically are searched for three judges only
• so it is at least as accurate as the simple majority and the best voter
• some profiles make the two procedures disagree
• some profiles produce an inconsistent majority
• some rules are complete
• some set of swaps always makes the candidate a Condorcet winner, with enough voters
• spreading the same product unevenly lowers the premise vote
• the agenda is one of chain, conj, rank
• the agenda is one this family knows
• the average voter is better than a coin
• the conclusion vote never convicts wrongly more often than the premise vote
• the Coombs winner ends with a strict majority
• the correlation is one the two competences allow
• the counting floor sits below Stearns's n + 2, which sits below McGarvey's n(n − 1)
• the court has an odd number of judges from 3 to 31
• the crossover falls as the court grows
• the dictatorship gives up anonymity
• the distinct winners counted by walking the list agree with the set
• the Dodgson winner is unique here
• the electorate is a whole number between 2 and 2000
• the electorate is odd, so the two-candidate control cannot end level
• the enumerated profiles' probabilities add to one
• 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 is a whole number between 5 and 51
• the largest electorate is odd, so no majority is level
• the largest electorate searched is a whole number between 3 and 6
• the largest number of candidates is a whole number between 50 and 2000
• the log-odds weighted majority achieves the best accuracy any rule reading the votes can
• 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 on this court contradicts itself
• 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 margin covariance is positive definite
• the mixed jury improves as it grows
• 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, cycleodds, cyclescatter, cyclecands, cyclemodel, jury, jurycorr, juryweights, jurysize, juryhetero, mcgarvey, mcgarveymin, dodgson, dodgsonfreq, mcgarveybound, dodgsonmargins, dodgsontideman, survivors, blocked, oligarchy, vetoprice, quotagrid, truthworlds, truthcurve, truthliable, truthcross, truthprior, trutherrors, premthreshold, premliable, premaverage, premdisagree, premdisjunct, premsize, uchero, ucworlds, ucmap, ucspread, uccorr, ucsize, oshero, osfreq, osanscombe, oswagner, osspread, effhero, effminimal, effthree, effcandidates, effborda
• the most candidates is a whole number between 4 and 12
• the number of distinct winners lies between one and the number of rules
• the number of electorates is a whole number between 200 and 6000
• 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 number of voters is a whole number between 3 and 15
• the option is one this view reads
• 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 ranking is totally blocked and the conjunction is not
• the rule being manipulated is one of plurality, borda, irv, coombs
• the rule to violate is one of plurality, borda, irv, coombs
• the seed is a whole number between 1 and 999
• the shortcut does not fall behind as electorates grow
• the simulated electorates is a whole number between 200 and 4000
• the simulated electorates per count is a whole number between 5000 and 100000
• the simulated share of cycles is within four standard errors of the exact limit
• 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
• the verdict vote depends only on the product of the competences
• the voters in each electorate is a whole number between 11 and 401
• the walk reaches the target
• the weighted vote is never level on this panel
• the yolk touches or crosses every median line
• there is a share of liable cases at which the two procedures tie
• this profile has no Condorcet winner, so every candidate needs some swaps
• three candidates reproduce Guilbaud's exact limit within four standard errors
• three judges with three yes-or-no answers each
• three to five judges, each holding a consistent judgement set
• three voters at 0.6: 0.648, by hand
• three voters can never produce a party that loses every issue and wins; five can
• three voters cannot produce Anscombe's paradox on three issues; five can
• three voters: 12 of the 216 profiles cycle
• two groups whose shares add to one
• two judges overrule least and all three most
• two to five correlations below 0.9
• two voters for each of the n(n − 1)/2 contests
• unanimity gives up at least one condition
• unanimity gives up completeness
• unanimity is consistent wherever it is defined
• unanimity is defined somewhere
• under the first model the chance rises with every two voters added
• under the second it stays below one in sixteen
• when every voter shares the axis, no electorate cycles at all
• when only the weak element fails the verdict vote is the more reliable
• where the majority's set is consistent the rule returns exactly it
• while the premise vote is already reliable just above a half
• with five alternatives the verdict vote convicts the innocent and the premise vote does not
• with five premises the conclusion vote almost never finds liability and the premise vote almost always does
• with many premises the procedures disagree on most liable defendants
• with one premise the two procedures are the same
• with perfectly correlated errors the two procedures coincide
• with these competences a larger court makes the verdict vote worse in the liable world
• with three issues the election and the issue majorities disagree in between a tenth and three tenths of electorates

• at a quota of 9 Luxembourg is powerless exactly when the quota is even ×9
• 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
• 300 small holders are close to the limit for the first block
• A and B are interchangeable
• a block smaller than the quota and its complement holds a/(1 − a)
• a division outside the core is refused by some group
• a family demanding more than there is rules the core out
• a game that is not convex has an arrival-order vector the core refuses
• a member who can never swing a vote has no power by either count
• a merger helps under majority and hurts under unanimity and under a quota short of it
• a network view's game is one of its own
• 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
• a whole-number seed
• an even number of players from 6 to 60
• an input the model never reads gets nothing when the unknowns are filled independently
• an ocean of 200 holders is already close to the oceanic limit
• and at least one of the other rules fails one
• and D adds nothing
• and every corner of the core is one of them
• and in 1981 more than in 1973
• and it falls at about the square-root rate over the whole sweep
• and neither end gains from losing it
• 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
• below one half the line beats every split off it
• between one and four assemblies
• between three and eight increasing sample sizes
• Bird's division is one no group refuses
• Bird's payments add to the cost of the whole tree
• both cheapest trees cost the same
• C is not interchangeable with them, so symmetry is a real condition here
• cut is read only by the linkcut view
• cutting a link changes both of its ends by the same amount
• donations that raised the donor's share of orders
• 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 arrival-order vector shares out exactly the whole
• every coalition's value is a whole number
• every estimate is a share
• every family in the table really is balanced
• every game has a core while no coalition is worth more than 2/n of the whole
• every held share below one half gives a stable set
• every marked split really attains the smallest imbalance
• every merger lost, gained or broke even
• every order adds exactly the number of pairs, one arrival at a time
• every order has exactly one pivotal member
• every ordering ends with everybody in
• every other split on the lattice is beaten by one of the three
• every player's weights add up to one
• every price is in the core when the sides are equal
• every split hands out the whole
• every split outside the core is beaten from inside it
• every ε lies inside the drawing
• fill is read only by the attribtable view
• filled from matching houses, the unread input takes credit
• fills is read only by the attrib view
• four players have forty-one minimal balanced families
• graph is read only by the network and linkcut views
• graphs is read only by the networks view
• in 1958 Luxembourg is never pivotal
• in 1973, with more members to share with, Luxembourg has power
• in a convex game every arrival-order vector is a corner of the core
• in the drawn case Banzhaf's share rises and the share of orders does not
• mergers go both ways
• model is read only by the attribution views
• net is read only by the network-bill views
• no coalition objects to a split none of them can beat
• no coalition of the other members totals exactly one short of the quota
• no group pays more under Bird's rule than its own cheapest tree would cost
• no half-and-half split beats another
• no other split on the table has a quieter list of complaints
• no point of the line beats another
• no split in the core beats another
• no split survives when some balanced family over-demands
• nothing both beats the split and is beaten by it
• on this game the nucleolus and the average split are different points
• one A and two B's block A's share 2.15, and one of each does not
• one more vote on the quota and there is such a coalition
• one to three networks
• relaxing by less does not
• relaxing by the computed amount lets a split through
• short in proportion, the scarce side's share rises towards the whole pair
• so their average lies in the core
• some coalition holds one player and not the other
• some coalition objects to every split drawn here
• some donation raises the donor's normalised Banzhaf share
• some family attains the maximum
• some house in the data matches the known inputs
• 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 allocation uses exactly the market's goods
• the average split is the average of the arrival-order vectors
• the B's gain and the A is held exactly level
• the balance verdict is computed for every split
• the best quota lies near the estimate
• the bill is drawn on a triangle for three users
• the capacities the users need is a list of 2 to 5 numbers
• the census finds enlargements in which an old member gains
• 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 convex game with values the square of the coalition's share has a core
• the core is drawn for three players
• the core of the convex game is not empty
• 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 core's corners are found for three players
• the count of minimal balanced families matches the known values
• the credits add up to the prediction less the starting value
• the cut is one of the network's links
• 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 excess over a half lies between 0 and 0.45/√n
• the family chosen is a whole number between 1 and 5
• the fill is one of baseline, marginal, conditional
• the fills are baseline, marginal or conditional
• the game is big enough that listing the orders is the point
• the game is one of majority, pairs
• the game is one of partnership, majority, gloves, shops, capacity, dummy, square
• the game names a value for every coalition
• the go-between takes the largest share
• 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 held share lies on the lattice
• the imbalance separates the splits, so the condition bites
• the integral of the slope along the diagonal is the average over orders
• the kernel is drawn for three players
• the lattice holds every value exactly
• 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 marginal view is drawn for three players
• the marginal view is written for a game whose value is shared out
• the model is one of proxy, and
• the network is one of line3, triangle, vee, star4, bridge
• the network is one of remote, far, tie
• the network's whole earning is shared out
• the nucleolus survives every objection exactly when some split does
• the number of users is a whole number between 4 and 11
• the objection is drawn for three players
• the option is one this view reads
• the other tree's Bird division is in the core too
• the others fail
• the pairwise surpluses are drawn for three players
• the picture is symmetric between B and C
• the power shares add to one
• the proportional rule needs somebody to be worth something alone
• the proposed split is beaten
• the proposed split shares out the three pairs
• the quota is more than half the weight and no more than all of it
• the rarer input of the product takes the larger share
• 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 scarce side takes everything
• the scarce side's share falls as the market grows
• 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 split into two pairs is the certificate that fails most often
• the split is three shares adding to one
• the split lies on the lattice
• 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 subset formula and the average over orders agree
• the sweep and the balancedness condition agree about whether any split survives
• the symmetric set and the low line are stable
• the table is drawn for three players
• the test is drawn for three players
• the three lefts and three of the rights earn more alone than they are paid
• 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, attrib, attribtable, attribdata, network, networks, linkcut, marginals, convexity, mst, mstcore, msttable, mstbig, stabledom, stablesym, stablediscrim, stablecheck, stableconvex, eec, parity, quotasweep, newmembers, merge, donate, glovestep, glovepath, glovereplica, gloveblock, gloveequal, repbox, repshrink, reprate, repwho, repcoalition, repequal
• the weights are not all nought
• the weights are positive
• the whole group is worth a whole number
• the worst-treated A and B both prefer splitting their joint endowment evenly
• there is more than one layer, or the figure has nothing to show
• three players: the simulated share matches the exact formula
• two games on the same number of players
• two hundred lefts against three hundred rights take nearly everything
• two to four games this family knows
• users is read only by the large network view
• 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
• width times r settles from twenty replicas
• with 400 small holders the block's power is within a hundredth of the oceanic limit
• with equal sides the average over orders splits each pair evenly
• with every input known the value is the prediction itself
• with fewer links than some of the others
• with more players the typical game is further from having a core
• with values up to one, three players have a core with chance five-sixths

• 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
• every random instance with 2 people and 4 items had an allocation envy-free up to any item ×5
• 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 cap at an equal share almost never works; no cap always does
• 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
• an allocation envy-free up to any item exists
• an envy-free split exists on the quarter-unit grid
• an odd number of fully labelled triangles at every fineness
• an option below the bargaining share changes nothing; above it, it is paid exactly
• 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
• and one envy-free up to one item
• and up to any item implies up to one
• at each end the free room is chosen
• at the maximin split all three are equally well off
• at the product's maximum neither person values the other's bundle more than their own
• each corner is the best or the worst envy-free split for someone
• each generated valuation totals exactly 100
• envy-free implies envy-free up to any item
• every corner of it is envy-free
• every envy-free allocation is also envy-free up to one item
• every person still waiting gets a calling mark
• every small triangle has one corner for each housemate
• every way of giving each item to one of the people was formed
• halving asks at most n⌈log₂ n⌉ questions
• in a typical house some room's envy-free rent spans a fifth to a third of the total
• in nearly every house someone can gain by misreporting
• no corner does better for the worst-off than maximin
• nobody left behind values the slice at more than their share of what remained
• on the boundary every choice is a free room
• 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
• reporting the truth gives the truthful outcome
• rescaling one person's values leaves the product's maximiser unchanged
• some allocation is envy-free up to one item but not up to any
• somewhere two neighbouring points choose different rooms
• the best posted price is a half, realising 1/8
• the budget balances exactly at a threshold of a quarter
• the buyer's best bid against the linear ask is 2b/3 + 1/12
• 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 closed form for the budget agrees with a direct integration
• 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 is content with either bundle up to any item
• 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, efx, efxcensus, efxcut, rent, rent2, rentrefine, rentexact, rentregion, rentcorners, rentwidth, rentlie, rentlies, rentcap
• the division is proportional and not envy-free: somebody prefers another's piece
• the envy left is at most twice the grid step
• the envy-free splits form a hexagon
• the exact arithmetic stays inside the safe integer range
• the exact value of the left piece and the decimal one agree
• the fineness of the triangulation is a whole number between 3 and 24
• the finite-horizon share is 1/(1 + δ) plus an alternating term ±δᵀ/(1 + δ)
• the fully labelled small triangles are odd in number
• the instance is one of three, noef, two
• 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 offers match the closed form (1 − δ)/(1 − δ^(1 + 1/α))
• 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 rents add up to the whole
• the rescaling factor is between 1 and 10
• the result is envy-free up to any item for both
• the round-robin allocation is envy-free up to one item
• the rule asks n(n+1)/2 − 1 questions
• the seller's best ask against the linear bid is 2s/3 + 1/4
• the solver gives 1/(1 + δ) for linear values
• the split the grid found, and the equal split, are both inside
• 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 two offers bracket the Nash split and close in on it
• the value asked for is still ahead of the knife
• the value matrix gives a row of values for two, three or four people
• the whole surplus is 1/6 and the double auction realises 9/64
• trading whenever b > s leaves a budget short by 1/6
• two calling marks never print on top of one another
• two or three people
• two people
• under the cutter's own measure the two pieces are exactly equal
• understating every value pays
• with one buyer and seller McAfee's rule trades nothing; with sixty-four of each it loses under one per cent
• with very short delays the split is r₂/(r₁ + r₂)
• with δ near one the split for value √s is the Nash split, 1/3

• 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
• 3 to 12 soldiers ×3
• 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
• 4 to 10 soldiers for the weaker colonel
• 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 four-by-four game with at least five equilibria is drawn
• a game where one player has a single strategy always has a pure equilibrium
• a pure equilibrium has index +1
• a random game has an odd number of equilibria (Wilson)
• 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
• after two million rounds the mixture is still far from the equilibrium at a third each
• 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 all-pay revenue all average (n − 1)/(n + 1)
• 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 does second-price
• and so is the closed form for breaking it
• and stays there from the tenth round on
• 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 every round the best reply against the history bounds the value from above and the guarantee from below
• 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
• at the two ends each seller charges t and earns t/2
• 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 mixtures are probability distributions
• 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
• common-interest games without a pure equilibrium
• each step moves the share the way the flow points
• equilibria of index +1 minus those of index −1
• even the calmest arrangement found leaves some seller a gain
• evenly spaced sellers give an average walk of 1/(4n)
• every allocation is beaten by some other
• 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 run in Shapley's game is the previous run plus the run three back plus two
• every traveller is strictly slower after the zero-cost link is added
• first-price
• first-price revenue averages (n − 1)/(n + 1)
• first-price revenue is always less spread out
• first-price variance matches the formula
• five sellers have an equilibrium at 1/6, 1/6, 1/2, 5/6, 5/6
• four sellers have an equilibrium in pairs at a quarter and three quarters
• from every start, play ends on one of the three diagonal equilibria
• games found with none, one and two pure equilibria
• games with an even number of equilibria
• in a zero-sum game committing is worth exactly the value, and nothing more
• in rock–paper–scissors each run is two rounds longer than the last
• in Shapley's game each run is longer than the last by the real root of x³ = x² + 1
• in this game committing is worth strictly more than any equilibrium
• independent payoffs agree with the exact chance
• more risk aversion, more first-price revenue
• moving both sellers in from the ends lowers both profits
• no commitment on a lattice of 121 does better than the best candidate
• no equilibrium pays the row chooser less than the minmax
• no game found with more than 2ⁿ − 1 equilibria at these sizes
• 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
• no triangle in the family is an equilibrium
• nothing an equilibrium of the stage game pays is outside what repetition sustains
• on a street of 101 stands no arrangement of three sellers is an equilibrium
• one to four rivals inside the square
• play passes through the same six profiles in the same order, round after round
• pure equilibria average one per game
• sampled best-reply cycles agree with the exact expectation
• sampled games agree with the exact chance
• second-price
• second-price variance matches the formula
• six sellers have many equilibria
• some pair of columns achieves the column chooser's best mixture
• taking turns, the two sellers end within a stand of the middle
• 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 reply bids (n − 1)/(n − 1 + α) of the value
• the best reply to rivals bidding (n − 1)/n of their values is to bid (n − 1)/n of one's own
• the best reserve is one half for every number of bidders
• 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 commonest support size is in the middle
• 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 cycle visits six different profiles
• the distance to the equilibrium falls like one over the square root of the rounds
• the drawn distribution is a correlated equilibrium
• the drawn distribution is whole numerators
• the drawn distribution sums to one
• the equilibria of two and four sellers make customers walk 1/4 and 1/8 on average
• 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 expected number of pure equilibria is exactly one
• the first colonel's mixture earns at least the value against every allocation
• 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 mean number of equilibria rises with the size of the game
• the median and the mean are different places
• the mixed rest point attracts exactly when no invading mixture can do as well against it as it does against itself
• the mixture approaches (1/4, 1/2, 1/4), not a third each
• the mode of the game family is one of payoff, mixed, dominance, network, correlated, potential, select, replicator, cycle, commit, folk, discount, global, iterate, aucthero, auctbest, auctmean, auctspread, auctrisk, auctreserve
• the option is one this view reads
• 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 programme is bounded
• 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 colonel's mixture concedes at most the value to every allocation
• the second colonel's mixture uses only allocations the first colonel's mixture holds to the value exactly
• 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 shortfall from 1/e is close to 1/(en) at a hundred strategies
• the simplex method terminates
• 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 sellers' equilibrium is the middle stand
• 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
• three or four sellers
• three sellers have no equilibrium at all
• two parties taking turns end at the median voter
• two sellers at the centre are an equilibrium
• two sellers have exactly one equilibrium, back to back in the middle
• 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 no reserve the revenue is (n − 1)/(n + 1)
• with the rival at the far end, moving towards it always lowers a seller's profit
• with three sellers someone wants to move at almost every turn
• with two bidders the first-price standard deviation is 1/√72
• with two columns
• zero-sum games have a saddle point with chance n! n! / (2n − 1)!

• 1's ranking of side one ranks all 4 of them exactly once ×20
• A's ranking of side two ranks all 4 of them exactly once ×5
• B's ranking of side two ranks all 4 of them exactly once ×5
• column 1 gives every member of side two exactly one partner ×5
• column 1 is itself a stable matching ×5
• hospital 1's list names distinct residents ×5
• side two's 5-th column is side one's 1-th ×5
• the 1-th generalised median is one of the stable matchings ×5
• with one extra the share stays a small minority at 10 a side ×5
• C's ranking of side two ranks all 4 of them exactly once ×4
• enough instances with an odd number of stable matchings at 6 a side ×4
• the matching drawn as unstable ranks all 4 of them exactly once ×4
• the median is stable in every instance at 6 a side ×4
• a sampled market of 4 has a stable grouping ×3
• D's ranking of side two ranks all 4 of them exactly once ×3
• the size of each side is a whole number between 2 and 6 ×3
• E's ranking of side two ranks all 5 of them exactly once ×2
• the member of side two whose lists are searched is a whole number between 0 and 3 ×2
• the number of markets is a whole number between 4 and 60 ×2
• the size of side one is a whole number between 20 and 2000 ×2
• 100 to 1,000 markets a size
• 100 to 1,000 markets at each size
• a grouping best for every capital at once exists in exactly two markets in nine
• a hospital left short holds the same residents in each
• a market with 12 to 30 stable matchings is found
• 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 third or more of the receivers are matched beyond the fortieth name
• A's list names distinct hospitals
• allowing longer cycles never covers fewer pairs
• allowing three-way exchanges helps substantially at moderate compatibility
• an even number of people, four to ten
• an odd number of stable matchings, at least five
• an odd number of stable matchings, at least three
• and is one of the market's stable matchings
• and matches the same residents
• at balance the share with two stable partners climbs towards all
• at balance, side one does far worse when side two proposes
• 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 least three stable matchings
• 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
• 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
• eight to sixteen a side
• every grouping of the drawn market is broken
• every hospital has between one and four places
• every market of three has a stable grouping
• every matching found is stable
• 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
• F's ranking of side two ranks all 6 of them exactly once
• far more proposers than receivers get their first choice
• fifty or a hundred a side, 500 to 4,000 markets
• following the pointers always leads into a cycle
• from 40 a side to 1,000 the measured mean stays between 0.28 and 0.45 of e⁻¹ n ln n
• G ranks every other person once
• G's list names distinct hospitals
• H ranks every other person once
• I ranks every other person once
• in every market of three each capital gets its first choice in some stable grouping
• in every market the strict core is a single allocation, and it is the cycles' allocation
• instances counted per size is a whole number between 20 and 1000
• J ranks every other person once
• most members have two stable partners at balance and few do with one extra on either side
• no member of the proposing side has a profitable misreport
• no pair blocks the constructed matching
• no sampled pair blocks the constructed matching
• no stable matching beats an extreme for the side it favours
• no stable matching is larger than the largest matching
• nobody ends with a house worse than their own
• 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 census supplies a market with the fewest stable groupings
• the construction returns a one-to-one pairing
• the count of pairs put the blocking question
• the first round's traders number about √(πn/2)
• the grouping drawn on the right is stable, checked triple by triple
• the histogram fits inside the figure with room for its labels
• 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 mean lies above the median: the tail runs to the right
• the median is one of the stable matchings
• 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, market, marketflip, marketimb, marketmulti, marketspread, mediantable, mediangen, medianscatter, mediancensus, trimarket, tricensus, trishrink, trioptimal, tripairs
• the option is one this view reads
• 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 proposers' average rank is near ln n
• the receivers' average rank is near n / ln n
• the roommates instance has exactly as many stable pairings as the market has stable matchings
• the rotation search finds exactly the stable matchings that trying every matching finds
• the rounds cannot outnumber the proposals available
• the search finds a market with no stable grouping
• the seed is a whole number between 0 and 1000000
• the settled matching has no blocking pair
• the share of side one with a single stable partner falls as the market grows
• 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
• three to six increasing market sizes up to 2,000
• two to eight residents
• two to four sizes between 4 and 14
• two to six hospitals
• under the smallest-total-rank rule some agent often gains
• under top trading cycles no agent in any market gains by lying
• with one extra receiver, side one does about as well either way
• with two of each kind no grouping can be broken

• 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
• at k = 1 the ratio is (4k + 2)/(3k + 3) ×5
• constraint row 1 constrains at least one variable ×5
• constraint row 1 has one coefficient per variable ×5
• 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
• the right-hand side is a list of 2 to 2 numbers ×4
• Dantzig's rule makes 2^3 − 1 moves on the 3-dimensional cube ×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
• √n times the spread is close to √(6ζ(4) − 4ζ(3))
• a coarse auction falls short by at most nε
• a fine one never falls short
• 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
• a smaller increment takes more bids
• a worst-case choice matches half, hindsight matches all
• after the cut, exactly two units cross
• an even number of odd vertices, at most twenty
• 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 last corner is the optimum, 5^(n−1)
• and the objective grows along the ray
• and the weight taken out is positive
• and the x₂ terms
• at every size the sample mean is within three and a half standard errors of 1 + 1/4 + … + 1/n²
• at n = 100 the optimum uses a row's (k+1)-th cheapest entry with chance about 2^(−k−1)
• at three thousand points the components per point follow the limit to within three hundredths, the largest gap near c = 1
• at two thousand points the mean is within sampling error of ζ(3)
• at weight limit 10 the dual bound is strictly above the best packing
• at ε = 0 the bidding returns to a state it has been in, and never ends
• below 1/n the result is optimal
• between three and six edges
• Bland's counts obey a(n) = a(n−1) + a(n−2) + 1
• both reach the optimum
• both the program and its dual are empty
• buyers who want one item each always have clearing prices
• buyers whose values add always have clearing prices
• Christofides' tour is at most 3/2 of the relaxation's optimum (Wolsey)
• Christofides' tour is within half as long again as the shortest
• Dantzig's rule on the cube makes 2^n − 1 moves
• dimensions up to between 4 and 12
• 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 auction ends
• every bidder holds an object within ε of their best at the final prices
• every complementary product is exactly zero
• every constraint coefficient is a whole number of size at most 40
• every corner has whole coordinates
• every corner is feasible
• every corner the walk reaches is feasible
• every dual certificate at this optimum gives the same price for the swept constraint
• every edge is covered by halves summing to at least one
• every feasible primal value is at most every feasible dual value
• every fractional value is a half
• every move climbs at a positive rate and gains
• every move strictly improves the objective
• 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 positive increment ends the war
• 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
• every share in the relaxation's best point is one half
• every vertex has even degree
• every whole-number core point lies between the two corners
• here the relaxation with its cut is exact
• inside the sector the optimum is that vertex's line
• leaving nought on the left and a negative number on the right
• markets with pairs wanted whole sometimes have none
• more noise, fewer moves
• n times the variance approaches 4ζ(2) − 4ζ(3)
• 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 point carries more than 1 of the fractional matching
• 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
• on every six-point graph some cheapest cover agrees with the corner's 0s and 1s
• on random instances both rules beat the worst-case guarantee
• on the triangular instance the two kinds of chance give the same expectation
• paying every buyer their marginal contribution is a core outcome
• paying every seller theirs is one too
• prices fail exactly for pair values strictly between 0 and 4
• Prim's and Kruskal's methods find the same cheapest tree
• random programs of the same size take a handful of moves
• six to sixteen workers fit the table
• 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 bases at the corner certify it and some do not
• 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
• taking the cheapest remaining pair each time ends up far above the optimum
• taking the settled 1s, leaving the settled 0s and covering the rest exactly always gives a cheapest cover
• the area under the limiting component curve is ζ(3)
• the auction ends
• the auction's total is within nε of the best
• 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 best-of-both and worst-of-both of two core points are in the core
• the bundle's value is a whole number between 0 and 8
• the buyers' corner gives each buyer what they add to the market
• the constraint whose right-hand side is swept is a whole number between 1 and 2
• the core has a corner best for both buyers and a corner best for both sellers
• the core is not empty
• the corners come in the order of the reflected Gray code
• the corners number the permutations, and no more
• the cost table is square
• the cube has dimension 3 to 5
• the cube is drawn in two or three dimensions
• the double cover's cheapest cover has the size of its largest matching
• 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 prices and surpluses cover every bid
• 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 edge the negative price points along improves the objective
• the exact arithmetic stays inside the safe integer range
• the exact mean for two workers is 5/4
• the exact optimum and the decimal one agree
• the exact optimum at the top of the sweep and the decimal one agree
• the exact tour is computed for at most eighteen cities
• the feasible vertices are in convex position
• the first move takes the steepest edge, which is not the edge to the optimum
• the first order decomposes the table completely
• the fixed second right-hand side is a whole number between 1 and 60
• the fractional matching equals the relaxation's value
• the fractional table needs at least two whole assignments
• the gap between the prices' bound and the value won is between 0 and nε
• 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 limit is ζ(3) divided by the density at nought
• 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 objective never falls along the walk
• the optimal corner uses all three values
• 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 option is one this view reads
• the pairing of the odd cities costs at most half the shortest tour
• the pentagon's double cover needs five points, half of each pair
• the pivot rule is one the figure knows
• the plane of the core is two matched buyers' payoffs
• 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, auction, auctionnet, auctioneps, pricewar, auctionrandom, core2, coreextremes, coreprices, bundleplane, bundlesweep, bundlehalf, bundleascend, bundlecensus, bundletriangle, walk, rules, prices, degenerate, kleeminty, kmgray, kmcount, kmsmooth, kmrates, tspfamily, tspratio, tsprandom, tspcuts, tsphalves, tspdouble, tspchristofides, tspalgs, tspstrip, tspwolsey
• the prices certify the marked assignment as the cheapest
• 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 bounded along the edge
• the program is bounded in the entering direction
• the program is empty
• the program is unbounded
• the program walked has at least one feasible vertex
• the program walked is bounded — an objective that increases without limit has no optimal vertex to draw
• 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 relaxation is bounded
• the relaxation is bounded, since every variable is at most one
• the relaxation is feasible
• the relaxation is never worse than the best whole allocation
• the relaxation never exceeds the tour
• the relaxation never exceeds the whole cover
• the relaxation's optimum is 3k + 3
• the relaxation's value equals its dual's
• the round empties at least one more cell than it found
• the row and column prices certify every assignment checked as cheapest
• the share table is one of thirds, quarters, sparse
• the share tends to 1 − 1/e
• the shortcut tour visits every city once
• the shortest tour is 4k + 2
• the shortest tree is the zigzag
• 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 table lists the walk on a cube of dimension 2 to 5
• the table's size is a whole number between 3 and 6
• the top of the sweep is a whole number between 10 and 80
• the tree costs about ζ(3) and the assignment about π²/6
• the two agree at some right-hand sides
• the two optima agree as decimals too
• the two rows of a basis are not parallel
• the two rules take different numbers of steps on this program
• the valuation table is one of four, war
• the walk ends at a corner whose prices are all non-negative
• the walk ends at the optimum found by trying every corner
• the walk starts at a feasible corner
• the weights add to one
• the weights cancel the x₁ terms
• the whole cover lies between the relaxation and twice it
• 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
• this market has a gap, so its relaxation's best point is fractional
• three buyers on the three pairs leave a gap
• three or more constraints meet at the optimum
• three to nine workers fit the picture
• tree ≤ shortest tour, and the doubled tour ≤ twice the tree
• trying every one of the 5,040 assignments finds the same cheapest cost
• two different assignments differ somewhere
• two to four corners of the walk
• two to four noise levels between 0 and 0.5
• two to six items, each a whole-number weight and value up to 20
• uniform costs climb towards the same limit from further below
• 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
• with a gap, no grid of prices clears the market
• with a pair wanted whole, the rising prices stop with an item unsold at a positive price, and no allocation clears there
• with no density at nought the cost grows with n
• with no gap, the relaxation's dual prices clear the market
• with no noise the cube still takes every corner
• with only the degree conditions, no edge crosses between the clusters
• with values that add, the rising prices stop at prices that clear the market
• with whole-number values and ε below 1/n the auction's assignment is optimal

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