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Graphs of d-dimensional polytopes are d-connected
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex
Balinski's_theorem
American and French mathematician
disconnect the graph of the remaining vertices and edges; it is known as Balinski's theorem. He also proved the Hirsch conjecture for several different classes
Michel_Balinski
Proof all ranked voting rules have spoilers
Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no procedure for group
Arrow's_impossibility_theorem
Pathological behavior by an apportionment rule
methodology can resolve observed paradoxes. However, as shown by the Balinski–Young theorem, it is not always possible to provide a perfectly fair resolution
Apportionment_paradox
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Basic concept of graph theory
if and only if it has an orientation that is strongly connected. Balinski's theorem states that the polytopal graph (1-skeleton) of a k-dimensional convex
Connectivity_(graph_theory)
Line segment joining two adjacent vertices in a polygon or polytope
In a polygon, two edges meet at each vertex; more generally, by Balinski's theorem, at least d edges meet at every vertex of a d-dimensional convex polytope
Edge_(geometry)
Graph-theoretic description of polyhedra
segments. The 3-connectivity of a polyhedral graph is a special case of Balinski's theorem that the graph of any k {\displaystyle k} -dimensional convex polytope
Steinitz's_theorem
Graph which remains connected when k or fewer nodes removed
polytope forms a k-vertex-connected graph (Balinski's theorem). As a partial converse, Steinitz's theorem states that any 3-vertex-connected planar graph
Vertex_connectivity
Graph made from vertices and edges of a convex polyhedron
planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties
Polyhedral_graph
Combinitorics of Polyhedra
obtained from the vertices and edges of polytopes (their 1-skeleta). Balinski's theorem states that the graph obtained in this way from any d-dimensional
Polyhedral_combinatorics
Graphs formed by a hypercube's edges and vertices
vectors in S {\displaystyle S} . is a n-vertex-connected graph, by Balinski's theorem. is planar (can be drawn with no crossings) if and only if n ≤ 3 {\displaystyle
Hypercube_graph
graph is d {\displaystyle d} -vertex connected. This is known as Balinski's theorem. the edge graph contains a subdivision of the complete graph K d +
Graph_of_a_polytope
Single-winner cardinal voting system
judgment (MJ) is a single-winner voting system proposed in 2010 by Michel Balinski and Rida Laraki. It is a kind of highest median rule, a cardinal voting
Majority_judgment
Mathematical principles
method of rounding suffers one or more paradoxes, as proven by the Balinski–Young theorem. The mathematical theory of apportionment identifies what properties
Mathematics_of_apportionment
Axiom of decision theory and social sciences
many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski–Young theorem, and the money pump arguments
Independence of irrelevant alternatives
Independence_of_irrelevant_alternatives
historian and professor at Harvard University Michel Balinski 1954, known for Balinski's theorem; mathematician and economist, winner of the John von
List of Williams College people
List_of_Williams_College_people
Election result affecting losing candidate
situations, called cyclic ties. Rated voting systems are not subject to Arrow's theorem, allowing them to be spoilerproof so long as voters' ratings are consistent
Spoiler_effect
Property of apportionment methods
allowing the house size to vary, satisfies strong monotonicity in this sense. Balinski and Young proved that an apportionment method is VRM if-and-only-if it
Vote-ratio_monotonicity
Electoral systems with independent candidate ratings
impossibility theorem, a theorem on the limitations of ranked-choice voting Gibbard's theorem, a generalization of the Gibbard-Satterthwaite theorem applicable
Rated_voting
Type of voting rules for an election
median voter theorem. They note that as with other cardinal voting rules, highest medians are not subject to Arrow's impossibility theorem. However, critics
Highest_median_voting_rules
Rule in math and political science
than 2 or 3, or giving party C any other than 0 or 1 seat. The Balinski–Young theorem proved in 1980 that if an apportionment method satisfies the quota
Quota_rule
Spoiler effect in RCV and two-round systems
votes needed to survive until the final round. By Black's median-voter theorem, the candidate who appeals most to the median voter will be the majority-preferred
Center_squeeze
Choosing a candidate other than preferred to undercut a less desired one
to maximize one's satisfaction with the election's results. Gibbard's theorem shows that no voting system has a single "always-best" strategy, i.e. one
Strategic_voting
Computational problem in graph theory
is minimized when the weight of C is maximized. By the max-flow min-cut theorem, a minimum cut, and the optimal closure derived from it, can be found by
Closure_problem
Voting system
Social Choice and Welfare. 59 (2): 305–333. doi:10.1007/s00355-022-01397-4. Balinski, Michel; Laraki, Rida (2020-03-01). "Majority judgment vs. majority rule"
Two-round_system
American mathematician
includes constructing polytopes to prove the sufficiency condition for the g-theorem (with Carl Lee), discovering fiber polytopes (with Bernd Sturmfels), and
Louis_Billera
Class of apportionment methods
Webster's method and Huntington-Hill perform well even without quota-caps. Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal
Rank-index_method
Number of votes a candidate needs to win
1007/978-3-319-64707-4_7, ISBN 978-3-319-64707-4, retrieved 2024-05-10 Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal
Electoral_quota
Single-winner electoral system
1007/s00355-020-01269-9. S2CID 226196615 – via Springer Link. Smith, Warren D. "On Balinski & Laraki's "majority judgment" median-based range-like voting scheme".
Graduated_majority_judgment
Vote supporting only a single candidate
vulnerability to bullet voting, due to its use of ranked ballots; however, Balinski and Laraki showed in their study of highest median rules that this can
Bullet_voting
Proportional-representation electoral system
84 (2): 481–496. doi:10.2307/1963530. JSTOR 1963530. S2CID 146438586. Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal
Sainte-Laguë_method
Proportional-representation voting system
1007/978-3-319-64707-4_9, ISBN 978-3-319-64707-4, retrieved 2024-05-10 Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal
Largest_remainder_method
Voting system that makes outcomes proportional to vote totals
fitting (IPF). It was proposed for elections by the mathematician Michel Balinski in 1989, and first used by the city of Zürich for its council elections
Proportional_representation
Way to distribute seats in a legislative body
methods". Electoral Studies. 17: 3–19. doi:10.1016/S0261-3794(97)00052-8. Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal
Apportionment_(politics)
Index of articles associated with the same name
many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski–Young theorem, and the money pump arguments
Voting_criteria
Single-winner ranked vote system
voters vote according to preferences along a spectrum, the median voter theorem guarantees the absence of Condorcet cycles. Consequently such cycles can
Copeland's_method
Family of voting systems
Meeting of the Public Choice Society, New Orleans, March 8-10, 2013 (PDF). Balinski, Young (2001). Fair Representation. Brookings Institution Press. ISBN 0-8157-0090-3
Party-list proportional representation
Party-list_proportional_representation
Criterion for evaluating rules for fair division
Victoriano (2016-09-01). "The whole and its parts: On the coherence theorem of Balinski and Young". Mathematical Social Sciences. 83: 11–19. doi:10.1016/j
Coherence_(fairness)
Multi-winner electoral system
method with single-member districts. It was proposed in 2008 by Michel Balinski (who also invented the single-winner voting system called majority judgment)
Biproportional_apportionment
Emerging field of applied ethics
practice, even as some theorems bear the name of the person making the conjecture rather than finding the proof. Folk theorems, or mathematical folklore
Ethics_in_mathematics
American game theorist, economist, and professor
receive noisy signals of this true preference order (cf. Condorcet's jury theorem). Using a simple probabilistic model for these noisy signals, Young showed
Peyton_Young
Form of political manipulation
(November 2007). "Flagrant Gerrymandering: Help from the Isoperimetric Theorem?". SIAM News. 40 (9). Archived from the original on 29 October 2021. Retrieved
Gerrymandering
Single-winner ranked-choice electoral system
that instant runoff rewards strategic withdrawal by candidates. Gibbard's theorem demonstrates that no (deterministic, non-dictatorial) voting method can
Instant-runoff_voting
Rule for proportional allocation
1007/978-3-319-64707-4_3. ISBN 978-3-319-64707-4. Retrieved 2021-09-01. Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal
Highest_averages_method
Property of electoral systems
35 (2): 285–300. doi:10.1137/0135023. ISSN 0036-1399. JSTOR 2100667. Balinski, Michel; Laraki, Rida (2011-01-28). Majority Judgment. The MIT Press. doi:10
Consistency_criterion
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