A Geometric and Combinatorial Interpretation of Weighted Games

A Geometric and Combinatorial Interpretation of Weighted Games Sarah K. Mason and R. Jason Parsley Winston Salem, NC Clemson Mini-Conference on Discrete Mathematics and Algorithms 17 October 2014

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

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 power.

A weighted voting example Example (Mayor and 5-person city council) A local city council consists of the mayor and 5 other members. Each council member gets 1 vote, while the mayor gets 2 votes. It takes a majority, 4 out of 7 votes, for a measure to pass.

What is weighted voting? Definition A weighted voting system (or game) occurs when you can assign a weight to each voter, and a quota q, so that any combination of voters whose weights add up to q will win; otherwise they lose. We represent such a system as (q : w n,..., w 2, w 1 ). Example 1 has a representation as (4 : 2, 1, 1, 1, 1, 1). Another example: stockholders in a corporation. Each voter gets weight equal to # shares owned.

3 stockholders in the Donut Company The Donut Company is having its annual meeting; its stock-holders will vote yes/no whether to introduce a Halloween donut. Voter 1 Voter 2 Voter 3 25 shares 30 shares 45 shares Question If it takes a majority of shares (say 51 shares) to pass a motion, how do we write this as a weighted voting system? What can we say about the power each voter has?

3 stockholders in the Donut Company Question If it takes a majority of shares (say 51 shares) to pass a motion, how do we write this as a weighted voting system? What can we say about the power each voter has? Answer: As a weighted voting system, (51 : 45, 30, 25) Any 2 of the 3 voters who vote yes will cause the motion to pass. Despite having unequal amounts of stock, each voter possesses the same amount of power.

Let s think more about quotas Back to the Donut Company Voter 1 25 shares Voter 2 30 shares Voter 3 45 shares We can express the quota as either the number of shares required to pass, or as a percentage, or as a decimal. The quota here was 51... or 51%... or as a decimal q = 0.51. We require that 0.5 < q 1 (i.e., both sides can t both win, aka a proper game)

Higher quotas The Donut Company changes its rules it takes more than a simple majority to pass motions. Voter 1 Voter 2 Voter 3 25 shares 30 shares 45 shares Case 2. Quota q = 0.6 = 60% What voters can pair up to win? voters 1 and 2 voters 1 and 3 voters 2 and 3 lose win win

Higher quotas Voter 1 Voter 2 Voter 3 25 shares 30 shares 45 shares Case 3. Quota q = 0.73

Higher quotas Voter 1 Voter 2 Voter 3 25 shares 30 shares 45 shares Case 3. Quota q = 0.73 Now voters 2 and 3 are the only pair who can win. Voter 1 s power has gone down (to 0 in fact!)

Higher quotas Voter 1 Voter 2 Voter 3 25 shares 30 shares 45 shares Case 3. Quota q = 0.73 Now voters 2 and 3 are the only pair who can win. Voter 1 s power has gone down (to 0 in fact!) Case 4. Quota q = 0.8 It takes a consensus of all three voters to win. All three voters again have equal amounts of power.

Coalitions Definition A group of voters who vote the same is called a coalition. If their weights sum to at least q, this is a winning coalition. If no (proper) subset of the coalition s voters forms a winning coalition, then we have a minimal winning coalition or mwc. mwc s tell you everything about a system s behavior Example. Donut majority rule: (51 : 45, 30, 25) {2, 1}, {3, 1}, {3, 2}, {3, 2, 1} are the winning coalitions all but the last one are minimal.

Dictators and Dummies Definition A dictator has weight greater than q. Equivalently, i is a dictator if {i} is a winning coalition. A voter has veto power if she appears in every (minimal) winning coalition. A voter is a dummy if he appears in no minimal winning coalitions. Example. Donut case 3: (73 : 45, 30, 25) {32}, {321} winning coalitions. {32} minimal. 3 and 2 have veto power. 1 is a dummy.

A donut dictator The Donut Company has gone back to requiring a simple majority, and voter 1 decides to sell 10 of his shares to voter 3. Voter 1 Voter 2 Voter 3 15 shares 30 shares 55 shares This is now the system (51 : 55, 30, 15). Q: What are the winning coalitions? The mwc s? winning coalitions: mwc s:

A donut dictator The Donut Company now decides to increase the quota. Voter 1 Voter 2 Voter 3 15 shares 30 shares 55 shares Example. Small quota increase: (60 : 55, 30, 15) {3, 1}, {3, 2}, {3, 2, 1} are the winning coalitions all but the last one are minimal. Example. Larger quota increase: (80 : 55, 30, 15)

Representing weighted games 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 voters.

The 5 different weighted voting games with n = 3 voters For n = 3 voters, there are 5 games. We ve seen these all for the Donut company. winning coals. description # wc s 3, 31, 32, 321 dictator 4 21, 31, 32, 321 majority rule 4 31, 32, 321 3-veto power 3 32, 321 1=dummy 2 321 consenus 1

A weighted game with 3 players Example (Stock-holders in a company) Voter 1 has 25 shares Voter 2 has 30 shares Voter 3 has 45 shares Quota Winning Coalitions 51 12, 13, 23, 123 60 13, 23, 123 74 23, 123 80 123

A weighted game with 3 players Example (Stock-holders in a company) Voter 1 has 25 shares 21 shares Voter 2 has 30 shares 27 shares Voter 3 has 45 shares 52 shares Quota Winning Coalitions WCs 51 12, 13, 23, 123 60 13, 23, 123 74 23, 123 80 123

Increasing quotas winning coals. description # wc s 3, 31, 32, 321 dictator 4 21, 31, 32, 321 majority rule 4 31, 32, 321 3-veto power 3 32, 321 1=dummy 2 321 consenus 1 Weights (45, 30, 25) pass from majority rule to 3-veto power to 1=dummy to consensus as we increase the quota. Weights (52, 27, 21) pass from dictator to 3-veto power to 1=dummy to consensus as we increase the quota.

Ordering coalitions We start by placing an order on coalitions. n = 3: 8 coalitions:, 1, 2, 3, 21, 31, 32, 321 Some coalitions are obviously stronger than others: {1} {2} {31} {321} Definition A partially ordered set or poset has an order (reflexive, anti-symm., transitive) that distinguishes between some of its elements. Let s write the coalitions as a poset.

n = 3: poset M(3) of coalitions 321 w 1 w 2... w n 32 31 3 21 2 1 Ranked, symmetric, SCD, lattice, Sperner (Richard Stanley) If a coalition is winning, so is everything above it. The filter A generated by A consists of all elements above A. Some filters are generated by multiple elements. Each weighted voting game can be described in terms of a filter. A minimal element of a simple game is called a generating winning coalition (gwc) of that game.

Generating winning coalitions Definition A generating winning coalition (or shift minimal coalition) for a weighted voting game is a generator of its order ideal; that is, it is a mwc and not stronger than any other mwc. We classify weighted voting games by their gwc s. All weighted voting systems for n = 3 voters gwc winning coals. description # wc s 3 3, 31, 32, 321 dictator 4 21 21, 31, 32, 321 majority rule 4 31 31, 32, 321 3-veto power 3 32 32, 321 1=dummy 2 321 321 consenus 1

Games poset Σ(M(4)) for n = 4 (Krohn-Sudhölter) 21 Coalitions 4321 432 431 421 43 321 42 32 41 31 4 3 2 1 Games poset Σ(M(4)) 4321 432 431 421 321 {321, 43} {321, 42} 32 {321, 41} 43 {421, 43} 42 41 4

How many games are there? Question How many weighted games are there for n voters? n 1 2 3 4 5 6 7 8 # games 1 Difficulties hard to count order ideals in general harder to count with complement condition unknown for n 10

How many games are there? Question How many weighted games are there for n voters? n 1 2 3 4 5 6 7 8 # games 1 2 Difficulties hard to count order ideals in general harder to count with complement condition unknown for n 10

How many games are there? Question How many weighted games are there for n voters? n 1 2 3 4 5 6 7 8 # games 1 2 5 Difficulties hard to count order ideals in general harder to count with complement condition unknown for n 10

How many games are there? Question How many weighted games are there for n voters? n 1 2 3 4 5 6 7 8 # games 1 2 5 14 Difficulties hard to count order ideals in general harder to count with complement condition unknown for n 10

How many games are there? Question How many weighted games are there for n voters? n 1 2 3 4 5 6 7 8 # games 1 2 5 14 62 566 14755 1366318 Difficulties hard to count order ideals in general harder to count with complement condition unknown for n 10

Number of singly-generated games It is easier to count games with one generating winning coalition. Theorem (M-Parsley) The number of singly-generated games 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

n = 5: poset M(5) of coalitions

n = 5: games poset Σ(M(5)) of 62 possible games 54321 5432 5431 5421 543 5321 5421, 543 4321 5321, 543 542 4321, 543 5321, 542 541 4321, 542 532 5321, 541 54 4321, 532 4321, 541 532, 541 5321, 54 432 531 4321, 532, 541 4321, 54 532, 54 432, 541 521 4321, 532, 54 4321, 531 531, 54 432, 54 4321, 531, 54 432, 531 4321, 521 521, 54 53 431 432, 531, 54 4321, 521, 54 432, 521 4321, 53 521, 53 431, 521 431, 54 432, 521, 54 432, 53 4321, 521, 53 52 421 431, 521, 54 431, 53 432, 521, 53 4321, 52 51 321 43 432, 52 51, 4321 421, 54 431, 521, 53 5

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

n = 6: poset M(6) of coalitions

n = 6: games poset J(M(6)) of 1171 possible games

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

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 (w 3, w 2, q)-space q = 1 ( 1 3, 1 3, 1 3 ) 3 C 3 (1,0,0) q = 0.0

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.

The face p 1 p 3 of C 3 The games poset Σ(M(3)) 1 321 321 32 2/3 1/2 21 31 3 31 21 3 1/3 <3,21> {3,21} 0 p3 1 p1 2 1

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 (w, q)-space represents combinatorics of the weighted games poset.

A vertical line and its corresponding chain The point P = (0.45, 0.3, 0.25) is generic. P Games poset n = 3 21 321 32 31 3 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

Equivalence classes of voters Definition Voters 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 ) 2 21 21, 31, 32, 321 (v 1 v 2 v 3 ) 1 31 31, 32, 321 (v 1 v 2 ), (v 3 ) 2 32 32, 321 (v 1 ), (v 2 v 3 ) 2 321 321 (v 1 v 2 v 3 ) 1

Facets of the polytope associated to a weighted game Theorem (M-Parsley) Let g be an arbitrary weighted game whose n voters 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.

Example: n = 3 321 1 321 32 31 21 3 2/3 1/2 21 31 3 3, 21 1/3 <3,21> 2 1 0 p3 1 p1

But what does it all mean? Probability: volumes of weighted game polytopes Measuring power: geometric power index Relationships: fixed weights, changing quota 1 321 2/3 1/2 21 31 3 1/3 <3,21> 0 p3 1 p1

Ongoing research: Categorize the weighted chains. Let n be the number of voters. Find a formula for the number of weighted games as a function of n. Understand the geometry better. Better understand our geometric power index. Translate the poset properties into voting information.