Social Choice Majority Vote Graphs Supermajority Voting Supermajority Vote Graphs Clemson Miniconference on Discrete Mathematics October 00 Craig A. Tovey Georgia Tech
Social Choice HOW should and does (normative) (descriptive) a group of individuals make a collective decision? Typical Voting Problem: select a decision from a finite set given conflicting ordinal preferences of set of agents. No T.U., no transferable good.
Case of Alternatives Majority Rule n voters, alternatives Theorem (Condorcet) If each voter s judgment is independent and equally good (and not worse than random), then majority rule maximizes the probability of the better alternative being chosen.
Social Choice Majority Vote Graphs Supermajority Voting Supermajority Vote Graphs
Notation [m] Π([m]) x..m set of all permutations of [m] Norm of x, default Euclidean A > i A Voter i prefers A to A Social Choice Function (SCF): chooses a winner Social Welfare Ordering (SWO): chooses an ordering
Social Choice What if there are alternatives? Plurality can elect one that would lose to every other (Borda). Alternatives A,,A m Condorcet Principle (Condorcet Winner) IF an alternative is pairwise preferred to every other alternative by a majority 9 t [m] s.t. 8 j [m], j t: i [n]: A t > i A j > n/ THEN the group should select A j.
Condorcet s Voting Paradox Condorcet winner may fail to exist Example: choosing a pizza Craig prefers Onion to Cheese to Mushroom Renu prefers Mushroom to Onion to Cheese Neal prefers Cheese to Mushroom to Onion Each alternative loses to another by / vote
majority preference/vote graph
Pairwise Relationships 8 directed graphs G=(V,E) 9 a population of O( V ) voters with preferences on V alternatives whose pairwise majority preferences are represented by G. Proof: Cover edges of K V with O( V ) ham paths Create voters for each path, each direction
Now the majority preference graph has no edges. Assign to each ordered pair (i,j) a voter with preference ordering { j,i, }. Don t re-use! Flip i and j to create any desired edge.
Now the tournament graph has no edges. Assign to each ordered pair (i,j) a voter with preference ordering { j,i, }. Don t re-use! Flip i and j to create any desired edge. > >
Extremal Problems Given directed graph G=(V,E) what is the least number of voters such that G represents their majority preferences? What graphs require the largest number of voters? Same questions for number of distinct preference orderings.
Social Choice Majority Vote Graphs Supermajority Voting Supermajority Vote Graphs
So, what if there are alternatives and there is no Condorcet winner? some (Cond. consistent) SCFs Copeland: outdegree indegree in majority preference graph. Simpson: min # votes mustered against any opponent Dodgson: minimize the # of pairwise adjacent swaps in voter preferences to make alternative a Condorcet winner Multistage elimination tree (Shepsle & Weingast)
So, what if there are alternatives and there is no Condorcet winner? some (Cond. consistent) SCFs Copeland: outdegree indegree in majority preference graph. Simpson: min # votes mustered against any opponent Leads to idea of supermajority voting in terms of DEFEATING an incumbent If Simpson winner musters > -α of votes, it can t be dislodged by α-majority voting.
Supermajority voting More Motivation: SPATIAL (EUCLIDEAN) MODEL
Definition of Spatial Model Voter i has ideal (bliss) point x i < k Each alternative is represented by a point in < k A i A iff x i -A x i A Can use norms other than Euclidean e.g. ellipsoidal indifference curves
D spatial model Informally used by U.S. press and many others Shockingly effective predictively in current U.S. politics. See Keith Poole s website, e.g. Supreme Court. Similar to single-peaked preferences (a little more restrictive). For polyhedral explanation of nice behavior of singlepeaked prefs, see MOR 00.
Spatial Model Largely descriptive role rather than normative The workhorse of empirical studies in political science k=, are the most popular # of dimensions In U.S. k= gives high accuracy (~90%), k= also very accurate since 980s, and 80s to early 0 th century.
What do the dimensions mean? Different schools of thought Use expert domain knowledge or contextual information to define dimensions and/or place alternatives Fit data (e.g. roll call) to achieve best fit Maximize data fit in st dimension, then nd Impute meaning to fitted model
D is qualitatively richer than D x A A x x A A >A > A > A
Condorcet s voting paradox in Euclidean model x A A x x A Hyperplane normal to and bisecting line segment A A
permitted alternatives, no Condorcet winner exists A x A x x
Chaos theorems McKelvey [979], Schofield [8]. Majority vote can take the agenda anywhere. (not the same meaning of chaos in system dynamics) Supermajority voting may be necessary for stability, finite termination of voting procedures
Social Choice Majority Vote Graphs Supermajority Voting Supermajority Vote Graphs
Supermajority-vote graphs For α>/, are all graphs realizable, and if so how many voters are required?