SUPERMAJORITY RULES (ONE DIMENSION) 1 Extra Credit • Each of you will be assigned an “ideal point” between 1 and 10 inclusive (randomly drawn out of a hat). • We will vote on alternatives between 1 and 10 inclusive using 2/3rds-majority rule. – When we start, there will be no alternative on the floor (no one gets any points if there is no alternative on the floor when we adjourn). – Anyone can propose a number between 1 and 10. If someone seconds that proposal, we will vote on the proposal vs the number on the floor. – The alternative that receives a majority of votes becomes the number on the floor in the next round. – Voting concludes when someone motions adjournment, the motion is seconded, and a majority of players vote to adjourn. That number on the floor wins. • Payoffs. You will receive 10 – (|x – i|) points on homework 2, where x is the winning number and i is your ideal point. • In other words, the closer the final outcome is to your ideal point, the more points you receive. Write your name on your number 2 and give it to me. K-Majority Rule Definition (absolute version): • k-majority rule: a proposal, x, defeats the status quo, q, by k- majority rule if and only if #yeas ≥ k, where 1 ≤ k ≤ N; otherwise, q is chosen. Note: A majority of N can be defined as follows: • For N odd, m = (N+1)/2 • For N even, m = (N/2) + 1 – This lecture focuses on k > m. K-majority rules (with k > m) are used for cloture, to overturn presidential vetoes, to amend the U.S. Constitution, to elect Popes, and for non-procedural decisions among permanent members of the U.N. Security Council, to name a few. 3 Exercise Questions: • Suppose N=7. How many yeas do you need for a 3/4ths- majority of 7? – Note, the number of yeas has to be a whole number, so you round up. • Suppose N=99. How many yeas do you need for a 3/5ths- majority of 99? • Suppose N=11. How many yeas do you need for a 2/3rds- majority of 11? 4 Exercise Suppose N=11 and each individual has single peaked and symmetric preferences, with the ideal points listed below. • Which point represents 2/3rds of the voters from left to right? A B C DE F G H I J K • Which point represents 2/3rds of the voters from right to left? • We call a voter pivotal if his/her vote can determine the outcome of a close vote. The two individuals identified above are the pivots. 5 Exercise A B C DE F G H I J K • Suppose q is at F, can the committee make a proposal that will defeat q under 2/3rds-rule? • Suppose q is at C, can the committee make a proposal that will defeat q under 2/3rds-rule? 6 Exercise A B C DE F G H I J K • Suppose q is at H, can the committee make a proposal that will defeat q under 2/3rds-rule? • Suppose q is at B, can the committee make a proposal that will defeat q under 2/3rds-rule? 7 Equilibrium Recall Definition • Core: Alternative z is an element of the core of an f-voting rule game if there does not exist another alternative y that f individuals prefer to z. – We are currently concerned with f = k > m. 8 K-Majority Voter Theorem Given: 1) n > 2, 2) pairwise k-majority rule, with k > m and everyone voting, 3) alternatives are on a single dimension, 4) preferences are single peaked. Theorem. The core is [M1, M2], where M1 is k individuals from right to left and M2 is k individuals from left to right. 9 Proof of the Theorem Notation M1 = the ideal point of the left pivot (in position N-k+1). M2 = the ideal point of the right pivot (in position k). q = the status quo. L = N-k number of ideal points to the left of M1 R = N-k number of ideal points to the right of M2. Assume and show that q is in the core. Consider an arbitrary x such that x < M1. Note that R ∪ {M2} = N-k+1 individuals prefer q to x; thus even if x gets support from voters within [M1, M2], at most N-(N-k+1) = k-1 individuals prefer x to q, which is less than k. Consider an arbitrary x such that x > M2. Note that L ∪ {M1} = k+1 individuals prefer q to x, which is less than k. 10 Proof of the Theorem Now show that no other point is in the core. Assume q < M1. It follows that q is not in the core because a proposal at M1 will receive at least k yeas (M1 and all voters to the right). Assume q > M2. It follows that q is not in the core because a proposal at M2 will receive at least k yeas (M2 and all voters to the left). 11 Exercise • Suppose N=11 and voting is conducted under 2/3rds rule. Again, what points represent the left and right 2/3rds pivots? q1 q3 q2 A B C DE F G H I J K • What set of alternatives beat q1 under 2/3rds rule? • What set of alternatives beat q2 under 2/3rds rule? • What set of alternatives beat q3 under 2/3rds rule? 12 Corollary of the k-majority Theorem Given: 1) n > 2, 2) pairwise k-majority rule, with k > m and everyone voting, 3) alternatives are on a single dimension, 4) preferences are single peaked, 5) and preference are symmetric. Corollary: for any status quo a proposal x that is closer to both pivots than q is to both pivots will defeat q under pairwise k-majority rule; otherwise q will win. 13 Sketch of proof of the Corollary x q l r M1 M2 q x l r M1 M2 q x l r M1 M2 14 Sketch of proof of the Corollary q x l r M1 M2 15 Win sets Let the k-majority rule “win set” of q be the set of alternatives that k individuals prefer to q: W(q) = {x | xPq} W(q) q M1 M2 • With N=4, k=3, and q at the location shown, the k-majority winset would be all alternatives closer to M1. • Note, k-majority rule winsets are defined by the pivot closest to the status quo. 16 Because of the corollary, alternatives will be drawn toward [M1, M2]. x q l r M1 M2 q y l r M1 M2 q z l r M1 M2 17 Because of the theorem, we get stuck at z l r M1 M2 Note: z cannot be defeated. 18 Practice • Suppose N=11 and voting is conducted under unanimity rule. What points represents left and right pivots? q1 A B C DE F G H I J K • What set of alternatives are in equilibrium under pairwise unanimity rule (i.e.. what is the core)? • What set of alternatives beat q1 under pairwise unanimity rule? 19 Summary • For k > m, k-majority rules behave similar to majority rule (k = m), except the equilibrium is a range rather than a point. • The corollary explains why alternatives a drawn between the two pivots. • For any status quo outside the pivots, the winset is defined by the pivot closest to the status quo. 20.