Figure: Democracy Figure: Democracy Figure: Democracy Is Democracy Possible?

Minhyong Kim

WMI Masterclass June, 2020 Is Democracy Possible?

Our approach today:

Toy Models

The hope is that they still cast light on the complicated phenomena of the world. I. Methods Faculty Senate (elect one)

Candidates: A,B,C,D,E 55 votes altogether.

Rank Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

This is called a preferential ballot.

Who should win? Plurality Method

Election of candidate with most first place votes

In our example, using the plurality method, A wins. Plurality Method

Is the plurality method reasonable?

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2 Countries using Plurality Vote for Parliamentary

Antigua Azerbaijan Bahamas Bangladesh Barbados Belize Bermuda Bhutan Botswana Brazzaville Canada Comoros Cook Islands Cote d’Ivoire Dominica Eritrea Ethiopia Gabon Gambia Ghana Grenada India Iran Jamaica Kenya Kuwait Laos Liberia Malawi Malaysia Maldives Marshall Islands Micronesia Myanmar Nigeria Niue Oman Palau Saint Kitts and Nevis Saint Lucia Saint Vincent and the Grenadines Samoa Seychelles Sierra Leone Singapore Solomon Islands Swaziland Tanzania Tonga Trinidad and Tobago Tuvalu Uganda United Kingdom Yemen Zambia Two-Round System

Count how many first place votes each candidate receives. If some candidate receives a majority, then that person wins. If no candidate receives a majority, eliminate all candidates except those two who have received the largest number of first place votes. Now, conduct a new election based on the preferences of the voters for these top two candidates. Two-Round System

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

C, D, E are eliminated. Two-Round System

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B 2 B B 3 4 B B 5 B A A A A A Votes 18 12 10 9 4 2

B wins by a large majority. Countries using Two-round System for Presidential Elections

Afghanistan, Argentina, Austria, Benin, Brazil, Bulgaria, Burkina Faso, Cape Verde, Chile, Colombia, Costa Rica, Croatia, Czech Republic, Cyprus, Djibouti, Dominican Republic, East Timor, Ecuador, Egypt, El Salvador, Finland, France, Ghana, Guatemala, Haiti, India, Iran, Indonesia, Kyrgyzstan, Liberia, Malawi, North Macedonia, Peru, Poland, Portugal, Romania, Russia, Senegal, Serbia, Slovakia, Slovenia, Turkey, Ukraine, Uruguay and Zimbabwe. Exhaustive Ballot

If no candidate gets a majority based on first place votes, eliminate the candidate with the fewest first place votes and hold a new election based on voting only for the smaller collection of candidates. Repeat the process until some candidate receives a majority of the first place votes. Exhaustive Ballot

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

E is eliminated. Exhaustive Ballot

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D 2 D B C B C 3 D D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

D is eliminated. Exhaustive Ballot

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C 2 B C B C 3 4 C C B C B 5 B A A A A A Votes 18 12 10 9 4 2

B is eliminated. Exhaustive Ballot

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A C 2 C C 3 4 C C C 5 A A A A A Votes 18 12 10 9 4 2

C wins! Use of Exhaustive Ballot

Scottish government, Host City of the Olympic Games, President of the European Parliament, Speaker of the British House of Commons, Speaker of the House of Commons of Canada, Leader of UK Conservative Party Method

Figure: Jean-Charles de Borda (1733-1799): French mathematician, physicist, political scientist, and sailor. Borda Count Method

Assign points

In an election with n candidates, a first place vote earns a candidate n − 1 points, second place vote n − 2 points, third place vote n − 3 points, and so on. Borda Count Method

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

A: 18 × 4+0+0+0+0=72 points B: 12 × 4 + 14 × 3 + 11 × 1 =48+42+11=101 points C: 10 × 4 + 11 × 3 + 34 × 1=40+33+34=107 points D: 9 × 4 + 18 × 3 + 18 × 2 + 10 × 1=36+54+ 36+10=136 points E: 6 × 4 + 12 × 3 + 37 × 2=24+36+ 74=134 points In our example, using the Borda count method, D wins. Summary

Method Winner Plurality A Two-Round System B Exhaustive Ballot C Borda Count D Condorcet Criterion

Figure: Nicola de Condorcet (1743-1794): French philosopher, mathematician, and political scientist

His ideas, including support for a liberal economy, free and equal public instruction, constitutional government, and equal rights for women and people of all races, have been said to embody the ideals of the Age of Enlightenment and Enlightenment rationalism. He died in prison after a period of flight from French Revolutionary authorities. (Entry from Wikipedia) Condorcet Criterion

Pairwise Comparison Condorcet Criterion

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

A versus B 18 vs. 37 B wins. Condorcet Criterion

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

B versus C 16 vs. 39 C wins. Condorcet Criterion

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

C versus D 12 vs. 43 D wins. Condorcet Criterion

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

D versus E 27 vs. 28 E wins. Condorcet Criterion

Rank Ballot Ballot Ballot Ballot Ballot Ballot 1 A B C D E E 2 D E B C B C 3 E D E E D D 4 C C D B C B 5 B A A A A A Votes 18 12 10 9 4 2

E versus A, 37:18 E versus B, 33:22 E versus C, 36:19 Condorcet Criterion

E is the Condorcet candidate, i.e., preferred by the voters to all other candidates in pairwise comparison. Condorcet candidate should win, right?

Method Winner Plurality A Two-Round B Exhaustive Ballot C Borda Count D Condorcet Criterion E

What should we do? II. Social Choice Theory

Stanford Encyclopaedia of Philosophy:

Social choice theory is the study of collective decision processes and procedures. It is not a single theory, but a cluster of models and results concerning the aggregation of individual inputs into collective outputs. Social Choice Theory

(Complicated) collection of individual preferences

Method ? Social Preference

Election, legislation, constitution, ... Modern social choice theory has its roots in the enlightenment and its emphasis on reason. Social Choice Theory

Continued from the Stanford Encyclopaedia Pioneered in the 18th century by Nicolas de Condorcet and Jean-Charles de Borda and in the 19th century by Charles Dodgson (also known as Lewis Carroll), social choice theory took off in the 20th century with the works of Kenneth Arrow, Amartya Sen, and Duncan Black. Its influence extends across economics, political science, philosophy, mathematics, and recently computer science and biology. Apart from contributing to our understanding of collective decision procedures, social choice theory has applications in the areas of institutional design, welfare economics, and social epistemology. Social Choice Theory

Figure: Immanuel Kant (1724–1804) Social Choice Theory

”Of the three forms of the state, that of democracy is, properly speaking, necessarily a despotism, because it establishes an executive power in which ”all” decide for or even against one who does not agree; that is, ”all,” who are not quite all, decide, and this is a contradiction of the general will with itself and with freedom..” from ’Perpetual Peace: A Philosophical Sketch (1795)’. For Kant:

Social choice must be grounded in a metaphysics of morals. III. Social Choice in the 20th Century Social Choice Theory: Mathematical Approach

What are some desirable features of a social choice method? For example, when you said some system is better than another, what were your reasons?

Figure: Kenneth Arrow (1921–): American economist, writer, and political theorist Social Choice Theory Social Choice Machine:

(Complicated) collection of individual preferences

?

? Social Preference Social Choice Theory: Mathematical Approach

What are some desirable features of a social choice method? Arrow listed three he viewed as essential. A1. Consensus: Suppose everyone ranks A above B. Then the method should rank A above B. A2. Independence: The relative ranking of A and B should be independent of the presence of other candidates. A3. No dictator: There is no voter who always determines the social preference. By focusing carefully on these three features, would like to build a social choice machine. Social Choice Theory: Arrow’s Theorem

Arrow’s Impossibility Theorem:

There is no social choice method having features A1, A2, A3.

A Perfect Social Choice Machine is impossible. Social Choice Theory: Arrow’s Theorem

Stanford Encyclopaedia of Philosophy: The technical framework in which Arrow gave the question of social orderings a precise sense and its rigorous answer is now widely used for studying problems in welfare economics. The impossibility theorem itself set much of the agenda for contemporary social choice theory. Arrow accomplished this while still a graduate student. In 1972, he received the Nobel Prize in economics for his contributions. Social Choice Theory: Arrow’s Theorem

Difficulty is already clear with one example.

Rank Ballot Ballot Ballot Ballot 1 A B C D 2 B C D A 3 C D A B 4 D A B C Votes 1 1 1 1 Social Choice Theory: Arrow’s Theorem

For proof, see ’Amartya Sen, Arrow and the Impossibility theorem.’

Basic idea: If A1 and A2 are satisfied, then the machine is necessarily a dictatorship. So the theorem is sometimes called Arrow’s dictator theorem. Say a set D of voters is deciding if their choices determine the outcome. Note that there is a deciding set. Given a deciding set D, use A1 and A2 to remove a voter from D and still have it be deciding. Eventually, end up with one deciding voter - dictator. Arrow’s Theorem: Basic Framework

Ck = {1, 2, 3,..., k}: Set of candidates.

Pk : Orderings of Ck . Examples:

C2 = {1, 2}. Then P2 = {12, 21}.

C3 = {1, 2, 3}. Then

P3 = {123, 132, 213, 231, 312, 321} Arrow’s Theorem: Basic Framework

A social choice method is a collection of functions

n S : Pk → Pk

Sk,n {p1, p2,..., pn} −→ p for every k and n satisfying properties given by Arrow’s principles. For example, S is a dictatorship if

S(p1, p2,..., pn) = pi

for some i. Theorem says that if A1 and A2 are satisfied, then S is a dictatorship. Arrow’s Theorem: An Analogy Find a solution to the equation

x + y = 2.

Find a solution to the system

x + y = 2; x − y = 2.

x = 2, y = 0.

Find a solution to the system

x + y = 2; x − y = 2; 3x + 4y = 5.

None. Arrow’s Theorem: A Comparison Interesting to compare with Newton’s three laws of motion.

Figure: Isaac Newton’s Mathematical Principles of Natural Philosophy Arrow’s Theorem: A Comparison

First law In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. Second law In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. Third law When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. Arrow’s Theorem: An Analogy

Figure: ‘Perpetual Motion’ Machine Arrow’s Theorem: An Analogy

Impossibility of Perpetual Motion - Modern Thermondynamics

Impossibility of Perfect Social Choice - Modern Voting Theory Arrow’s theorem: Circumvention

Relax A2, independence: The relative ranking of A and B should be independent of the presence of other candidates. Weaken the type of ranking: what we have been discussing is called a transitive total ordering. Relax requirement on possible input: Only allow ‘reasonable (p1, p2,..., pn) as input. That is, allow for the possibility that sometimes the machine will give ‘wrong’ answers.

A real world machine need only work with high probability.