Table Of ContentVoting, Arbitration, and Fair Division
The mathematics of social choice
Marcus Pivato
Trent University
March 10, 2007
ii
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(cid:13)
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Contents
I Voting and Social Choice 1
1 Binary Voting Procedures 3
1A Simple Majority Voting: May’s Theorem . . . . . . . . . . . . . . . . . . . . . . 3
1B Weighted Majority Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6
1C Vector-Weighted Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1D Voting Power Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Multi-option Voting Systems 17
2A Plurality voting & Borda’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . 17
2B Other voting schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2B.1 Pairwise elections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2B.2 The Condorcet Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2B.3 Borda Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2B.4 Approval voting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2C Abstract Voting Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2C.1 Preferences and Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2C.2 Voting procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2C.3 Desiderata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2D Sen and (Minimal) Liberalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2E Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2F Strategic Voting: Gibbard & Satterthwaite . . . . . . . . . . . . . . . . . . . . 46
II Social Welfare Functions 53
3 Utility and Utilitarianism 55
3A Utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3B The problem of interpersonal comparisons . . . . . . . . . . . . . . . . . . . . . 60
3C Relative Utilitarianism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3D The Groves-Clarke Pivotal Mechanism . . . . . . . . . . . . . . . . . . . . . . . 62
3E Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
iii
iv CONTENTS
III Bargaining and Arbitration 69
4 Bargaining Theory 71
4A The von Neumann-Morgenstern Model . . . . . . . . . . . . . . . . . . . . . . . 71
4B The Nash Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4C Hausdorff Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 The Nash Program 89
5A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5B Normal-form games and Nash equilibria . . . . . . . . . . . . . . . . . . . . . . 93
5C The Nash demand game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5D The Harsanyi-Zeuthen concession model . . . . . . . . . . . . . . . . . . . . . . 103
5E Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5F The Rubinstein-St˚ahl Alternating Offers model . . . . . . . . . . . . . . . . . . 109
5G Proof of Rubinstein’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Interpersonal Comparison Models 131
6A The Utilitarian Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6B The Proportional Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6C Solution Syzygy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6D Contractarian Political Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Renormalized Solutions 155
7A Kalai & Smorodinsky’s Relative Egalitarianism . . . . . . . . . . . . . . . . . . 155
7B Relative Utilitarianism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
IV Fair Division 161
8 Partitions, Procedures, and Games 163
8A Utility Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8B Partition Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8C Partition Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9 Proportional Partitions 171
9A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9B Banach and Knaster’s ‘Last Diminisher’ game . . . . . . . . . . . . . . . . . . . 171
9C The Latecomer Problem: Fink’s ‘Lone Chooser’ game . . . . . . . . . . . . . . 175
9D Symmetry: Dubins and Spanier’s ‘Moving Knife’ game . . . . . . . . . . . . . . 178
9E Connectivity: Hill and Beck’s Fair Border Procedure . . . . . . . . . . . . . . . 180
9E.1 Appendix on Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
CONTENTS v
10 Pareto Optimality 189
10A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10B Mutually Beneficial Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10C Utility Ratio Threshold Partitions . . . . . . . . . . . . . . . . . . . . . . . . . 192
10D Bentham Optimality & ‘Highest Bidder’ . . . . . . . . . . . . . . . . . . . . . . 196
11 Envy and Equitability 199
11A Envy-freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11B Equitable Partitions & ‘Adjusted Winner’ . . . . . . . . . . . . . . . . . . . . . 203
11C Other issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11C.1 Entitlements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11C.2 Indivisible Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11C.3 Chores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11C.4 Nonmanipulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Bibliography 215
vi CONTENTS
Part I
Voting and Social Choice
1
Chapter 1
Binary Voting Procedures
Democracy is the worst form of government except all those other forms that have been tried
from time to time. —Winston Churchill
The word ‘democracy’ has been used to describe many different political systems, which
often yield wildly different outcomes. The simple rule of ‘decision by majority’ can be made
complicated in several ways:
Granting veto power to some participants (e.g. the permanent members of the UN Se-
•
curity Council or the President of the United States), possibly subject to ‘override’ by a
sufficiently large majority of another body (e.g. the United States Senate). .
Requiring a majority by two different measures (e.g. in a federal system, a constitutional
•
ammendment might require the support of a majority of the population and a majority
of states/provinces).
Giving different ‘weight’ to different voters (e.g. different shareholders in a publically
•
traded corporation, or different states in the European Union).
Forcing voters to vote in ‘blocs’ (e.g. political parties)
•
In this chapter we will consider the simplest kind of democratic decision-making: that between
two alternatives. Nevertheless, we will see that aforementioned complications engender many
surprising phenomena.
1A Simple Majority Voting: May’s Theorem
Prerequisites: 2C.1 Recommended: 2C.3
§ §
The most obvious social choice function for two alternatives is the simple majority vote.
The more elaborate voting procedures (Borda count, pairwise votes, approval voting, etc.) all
reduce to the majority vote when = 2. Indeed, the conventional wisdom says that majority
|A|
3
4 CHAPTER 1. BINARY VOTING PROCEDURES
vote is the ‘only’ sensible democratic procedure for choosing between two alternatives. The
good news is that, for once, the conventional wisdom is right.
Suppose that = A,B . In 2C.3 we introduced three desiderata which any ‘reasonable’
A { } §
voting procedure should satisfy:
ρ
(M) (Monotonicity) Let ρ be a profile such that A B. Let v be some voter such that
⊑ ∈ V
ρ
B A, and let δ be the profile obtained from ρ by giving v a new preference ordering
(cid:22)
v
δ δ δ
, such that A B (all other voters keep the same preferences). Then A B.
(cid:23) (cid:23) ⊑
v v
(A) (Anonymity) Let σ : be a permutation of the voters. Let ρ be a profile, and let
V−→V
δ be the profile obtained from ρ by permuting the voters with σ. In other words, for any
ρ δ
v , δ(v) = ρ σ(v) . Then A B A B .
∈ V ⊒ ⇐⇒ ⊒
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(N) (Neutrality) Let ρ be a profile, and let δ be the profile obtained from ρ by reversing the
positions of A and B for each voter. In other words, for any v ,
∈ V
ρ δ
A B B A .
(cid:23) ⇐⇒ (cid:23)
(cid:18) v (cid:19) (cid:18) v (cid:19)
Then the outcome of δ is the reverse of the outcome of ρ. That, is, for any B,C ,
∈ A
ρ δ
A B B A .
⊒ ⇐⇒ ⊒
(cid:16) (cid:17) (cid:16) (cid:17)
A semistrict voting procedure is a function Π : R∗( , ) ( ). Thus, the voters must
V A −→P A
provide strict preferences as input, but the output might have ties. Let V := #( ). We say Π
V
is a quota system if there is some Q [0..V] so that, for any ρ R( , ),
∈ ∈ V A
ρ ρ
(Qa) If # v ; A B > Q, then A = B.
∈ V ≻
v
(cid:26) (cid:27)
ρ ρ
(Qb) If # v ; B A > Q, then B = A.
∈ V ≻
v
(cid:26) (cid:27)
ρ
(Qc) If neither of these is true, then A B.
≈
For example:
The simple majority vote is a quota system where Q = V/2.
•
The two thirds majority vote is a quota system where Q = 2V. If an alternative does
• 3
not obtain at least two thirds support from the populace, then it is not chosen. If neither
alternative gets two thirds support, then neither is chosen; the result is a ‘tie’.
