Geometric and Combinatorial Weighted Voting: Some Open Problems

Transcription

1 Geometric and Combinatorial Weighted Voting: Some Open Problems Sarah K. Mason and R. Jason Parsley Winston Salem, NC Encuentro Colombiano de Combinatoria 2016

2 Types of voting In voting for most political offices, e.g., congressman, senator, governor but not president(!), the election runs on the idea of Every voter has the same role. one person, one vote However there are plenty of situations where different people have different numbers of votes: governments (city council, UN) corporations (stockholders) groups of friends Different voters have different roles, different amounts of weight.

3 Weighted games A weighted game is a situation in which n players, each with a certain weight, vote on a yes or no motion. For one coalition to win, the weights of its players must reach a certain fixed quota q. Example (A company with 3 stockholders) Voter 1 has 25 shares Voter 2 has 30 shares Voter 3 has 45 shares Quota Winning Coalitions 50 12, 13, 23, , 23, ,

4 Weighted games A weighted game is a situation in which n players, each with a certain weight, vote on a yes or no motion. For one coalition to win, the weights of its players must reach a certain fixed quota q. Example (A company with 3 stockholders) Voter 1 has 25 shares 21 shares Voter 2 has 30 shares 28 shares Voter 3 has 45 shares 51 shares Quota Winning Coalitions Winning Coalitions 50 12, 13, 23, 123 3, 13, 23, , 23, , 23, , ,

5 The right definition of a weighted game Definition Two sets of quotas & weights are isomorphic (represent the same weighted game) iff they have the same winning coalitions. Example. These games (q : w 1, w 2, w 3, w 4 ) are all isomorphic. (8 : 5, 3, 3, 1) (4 : 3, 2.1, 1, 0.5) (65 : 40, 30, 29, 1) (1002 : 1000, 998, 2, 1) (7.4 : 5.8, 3.2, 2.3, 1.4) (11 : 10, 1, 1, 0) Winning Coalitions: {4, 2}, {4, 3}, {4, 3, 2}, {4, 3, 1}, {4, 2, 1} {4, 3, 2, 1}. Definition (The right definition of a weighted game.) We should really only discuss isomorphism classes of weighted games. There are only 5 games for n = 3 players.

6 Simple games Definition A simple game g is a pair ([n], W g ) in which [n] = {1, 2,..., n} is a finite set of players, W g is the set of winning coalitions for g, and if S W g and S R [n], then R W g. (monotonicity) A simple game is proper if A W g A c / W g. Example (All proper simple games with 3 players) winning coals. description # wc s 3, 31, 32, 321 dictator 4 21, 31, 32, 321 majority rule 4 31, 32, veto power 3 32, 321 1=dummy consenus 1

7 n = 3: poset M(3) of coalitions {a 1, a 2,... a j } {b 1, b 2,..., b k } j k and a i b i ( i j) Ranked, symmetric, SCD, lattice, Sperner (Richard Stanley) If a coalition is winning, so is everything above it. The filter A 1, A 2,..., A k generated by A 1, A 2,..., A k consists of all elements above and including A 1, A 2,..., A k. Every simple game can be described in terms of a filter. (Note that not all such games are weighted.) The minimal elements of a simple game are called the generating winning coalitions (gwc s) of that game.

8 Games poset J(M(4)) for n = 4 (Krohn-Sudhölter) 21 Coalitions Games (filters) poset J(M(4)) {321, 43} {321, 42} 32 {321, 41} 43 {421, 43}

9 Games poset J(M(4)) for n = 4 (Krohn-Sudhölter) 21 Coalitions Games (filters) poset J(M(4)) {321, 43} {321, 42} 32 {321, 41} 43 {421, 43}

10 Counting games on n players Theorem (M-Parsley) The number of games with one generating winning coalition is s(2j + 1) = j k=0 4j k C k, s(2j) = 2 s(2j 1) where C k is the k th Catalan number. Equivalently, ( ) n s(n) = 2 n. n/2 Open Problem # 1 What is the number of simple games on n players with two generators? (with k generators?)

11 n = 5: Poset M(5) of coalitions

12 n = 5: Games poset J(M(5)) of 62 possible games , , , , , , , , , , , 532, , , , , 532, , , , , 531, , , , , 531, , 521, , , , , , , 521, , , 521, , 521, , , 521, , , 52 51, , , 521, 53 5

13 n = 5: Games poset J(M(5)) of 119 possible games

14 n = 6: Poset M(6) of coalitions

15 n = 6: Games poset J(M(6)) of 1171 possible games Open Problem # 2 How many simple games on n players exist? How many weighted games on n players exist? Open Problem # 3 Which poset properties does J(M(n)) posess? What do these properties tell us about real-world voting situations?

16 Geometry of weighted games Idea: using coords (q : w n,... w 2, w 1 ), graph these games in R n+1 weighted voting is scale invariant normalize so that total weight W = 1 call the configuration region C n := (0, 1] n Definition. The region of allowable weights n consists of w n w n 1 w 2 w 1 0 w n + w n w 2 + w 1 = 1 n an (n 1)-dimensional simplex in R n, with vertices at (1, 0, 0,...) ( 1 2, 1 2, 0, 0,...) ( 1 3, 1 3, 1 3, 0, 0,...)..., ( 1 n, 1 n,..., 1 n ) p 1 p 2 p 3... p n

17 3 and C 3 3 lies in the plane w 3 + w 2 + w 1 = 1 intersection of 3 half-planes in (w 3, w 2, w 1 ) ( 1 2, 1 2, 0) in (q : w 3, w 2 )-space q = 1 ( 1 3, 1 3, 1 3 ) 3 C 3 (1,0,0) q = 0.0

18 Each weighted game is a polytope The set of points where a coalition A s weight equals the quota forms a hyperplane h A which intersects C n. Hyperplane h A : q = w A := v i A w i Each weighted game g is a polytope P g : for each winning coalition A, take all points below or on h A for each losing coalition B, take all points above h B The polytopes P g are convex and n-dimensional. Open Problem # 4 An n-dimensional polytope is said to be simple if n facets meet at each vertex. For which games g is P g simple?

19 Example 1 The face p 1 p 3 of C The games poset J(M(3)) /3 1/ /3 <3,21> {3,21} 0 p3 1 p1 2 1

20 Geometry Games poset Call a point (w n,... w 2, w 1 ) generic if none of its partial sums equal each other. Theorem (M-Parsley) Above any generic point, the line from q = 0.5 to q = 1 passes through precisely the polyhedra representing voting games as along some maximal saturated chain in the weighted games poset. i.e., The geometry of weighted games in (q : w)-space represents combinatorics of the weighted games poset. Open Problem # 5 What do the poset properties tell us about the geometry?

21 A vertical line and its corresponding chain The point P = (0.45, 0.3, 0.25) is generic. P Games poset n = Example results q (0.75, 1] 321 q (0.7, 0.75] 32 q (0.55, 0.7] 31 q (0.5, 0.55] 21

22 Equivalence classes of players Definition Players v i and v j are equivalent (written v i v j ) in a game g iff switching v i and v j fixes the winning coalitions in g. Example: gwc winning coals. equivalence classes # of equiv classes 3 3, 31, 32, 321 (v 1 v 2 ), (v 3 ) , 31, 32, 321 (v 1 v 2 v 3 ) , 32, 321 (v 1 v 2 ), (v 3 ) , 321 (v 1 ), (v 2 v 3 ) (v 1 v 2 v 3 ) 1

23 Facets of the polytope associated to a weighted game Theorem (M-Parsley) Let g be an arbitrary weighted game whose n players form k equivalence classes. If the degree of g in the Hasse diagram of the poset of weighted games is d, then the polytope corresponding to g has n k + d facets. * equivalence classes, covers, and # covered gives # of facets* Corollary Every saturated chain in the weighted games poset may be achieved by some piecewise linear motion through C n. Open Problem # 6 How many vertices do the polytopes P g have? (Upper bound?)

24 Example: n = /3 1/ , 21 1/3 <3,21> p3 1 p1

25 Summary of open problems: 1 What is the number of simple games on n players with two generators? With k generators? 2 How many simple/weighted games on n players exist? 3 Which poset properties does J(M(n)) posess? 4 An n-dimensional polytope is said to be simple if n facets meet at each vertex. For which games g is P g simple? 5 What do the poset properties tell us about the geometry? 6 How many vertices do the polytopes P g have? Are these polytopes with few ( n 4) vertices? 7 Choose your own favorite polytope/poset question! (f -vectors, shellability, etc) 8 Translate the poset properties (and geometric properties) into real-world information about voting situations. Muchas Gracias!