Social Choice Theory By Shiyan Li

History The theory of social choice and voting has had a long history in the social sciences, dating back to early work of Marquis de Condorcet (the 1st rigorous mathematical treatment of voting) and others in the 18th century. The theory of social choice and voting has had a long history in the social sciences, dating back to early work of Marquis de Condorcet (the 1st rigorous mathematical treatment of voting) and others in the 18th century. Now it is a branch of discrete mathematics. Now it is a branch of discrete mathematics.

Purpose Social Choice Theory is the study of systems and institution for making collective choice, choices that affect a group of people. Social Choice Theory is the study of systems and institution for making collective choice, choices that affect a group of people. Be used in multi-agent planning, collective decision, computerized election and so on. Be used in multi-agent planning, collective decision, computerized election and so on. Voters Alternatives

Simple Majority Voting Choose one from two possible alternatives by a group of participants. Choose one from two possible alternatives by a group of participants. Consider a democratic voting situation. Consider a democratic voting situation.

Preferences and Outcome Alternatives: x or y Alternatives: x or y Every voter has a preferences. Every voter has a preferences. Three possible situations of each voter ’ s preference: i) x is strictly better than y: +1 ii) y is strictly better than x: -1 iii) x and y are equivalent: 0 Three possible situations of each voter ’ s preference: i) x is strictly better than y: +1 ii) y is strictly better than x: -1 iii) x and y are equivalent: 0 After the voting: i) x is winner: +1 ii) y is winner: -1 iii) x and y tie: 0 After the voting: i) x is winner: +1 ii) y is winner: -1 iii) x and y tie: 0

General List Use a list to describe a collection of n voters ’ preferences e.g. (-1, +1, 0, 0, -1, …, +1, -1) Use a list to describe a collection of n voters ’ preferences e.g. (-1, +1, 0, 0, -1, …, +1, -1) General List: D = (d 1, d 2, d 3, …, d n-1, d n ) d i is +1, -1 or 0 depending on whether individual i strictly prefers x to y, y to x or is indifferent between them. General List: D = (d 1, d 2, d 3, …, d n-1, d n ) d i is +1, -1 or 0 depending on whether individual i strictly prefers x to y, y to x or is indifferent between them. n entries

General List Consider the sum of list D: When d 1 +d 2 +d 3 + … +d n-1 +d n > 0, x is to be chosen, simple majority voting assigns +1. When d 1 +d 2 +d 3 + … +d n-1 +d n 0, x and y tie, simple majority voting assigns 0. Consider the sum of list D: When d 1 +d 2 +d 3 + … +d n-1 +d n > 0, x is to be chosen, simple majority voting assigns +1. When d 1 +d 2 +d 3 + … +d n-1 +d n 0, x and y tie, simple majority voting assigns 0.

Formal Definition of Simple Majority Vote Use the sign function to formally define the simple majority vote: (d 1, d 2, …, d n ) sgn(d 1 +d 2 + … +d n ) Use the sign function to formally define the simple majority vote: (d 1, d 2, …, d n ) sgn(d 1 +d 2 + … +d n ) Function N +1 and N -1 : N +1 : associates with a list D the number of d i ‘ s that are strictly positive N -1 : associates with a list D the number of d i ‘ s that are strictly positive Function N +1 and N -1 : N +1 : associates with a list D the number of d i ‘ s that are strictly positive N -1 : associates with a list D the number of d i ‘ s that are strictly positive

Formal Definition of Simple Majority Vote E.g.: for list D = (+1, -1, -1,0, +1, +1), ∵ n = 6, n/2 = 3, N +1 (+1, -1, -1,0, +1, +1) = 3 > n/2 N -1 (+1, -1, -1,0, +1, +1) = 2 n/2 N -1 (+1, -1, -1,0, +1, +1) = 2 <n/2 ∴ g(+1, -1, -1,0, +1, +1) = +1 g (d 1, d 2, d 3, …, d n ) = +1 if N +1 (d 1, d 2, d 3, …, d n ) > n/2 -1 if N -1 (d 1, d 2, d 3, …, d n ) > n/2 0 otherwise

Rule of Simple Majority Voting Social Choice Rule: is a function f(d 1, d 2, …, d n ), the domain of the function is the set of all list to which f assigns some unambiguous outcome: +1, -1 or 0. Social Choice Rule: is a function f(d 1, d 2, …, d n ), the domain of the function is the set of all list to which f assigns some unambiguous outcome: +1, -1 or 0. A social choice rule of simple majority voting can be characterized by 4 properties (Kenneth O. May, 1952). A social choice rule of simple majority voting can be characterized by 4 properties (Kenneth O. May, 1952).

Property 1 of Rule f Property 1 – Universal Domain: f satisfies universal domain if it has a domain equal to all logically possible lists (i.e. any combination of the individual voters ’ preferences) of n entries of +1, -1 or 0. Property 1 – Universal Domain: f satisfies universal domain if it has a domain equal to all logically possible lists (i.e. any combination of the individual voters ’ preferences) of n entries of +1, -1 or 0.

Property 2 of Rule f One-to-one Correspondence: is a function s from the set {1, 2, …, n} to itself such that s is defined on every integer from 1 to n and no outcome is assigned to two different integers: s(i) = s(j) implies i = j. One-to-one Correspondence: is a function s from the set {1, 2, …, n} to itself such that s is defined on every integer from 1 to n and no outcome is assigned to two different integers: s(i) = s(j) implies i = j. one-to-one correspondence S(i)i not one-to-one correspondence S(i)i i i

Property 2 of Rule f Permutation: Given two lists D = (d 1, d 2, …, d n ) and D ’ = (d 1 ’, d 2 ’, …, d n ’ ) say that D and D ’ are permutation of one another if there is a one-to-one correspondence s on {1, 2, …, n} such that d s(i) ’ = d i. Permutation: Given two lists D = (d 1, d 2, …, d n ) and D ’ = (d 1 ’, d 2 ’, …, d n ’ ) say that D and D ’ are permutation of one another if there is a one-to-one correspondence s on {1, 2, …, n} such that d s(i) ’ = d i. E.g.: voter: (+1, +1, +1, 0, 0, -1, -1) and voter: (-1, 0, +1, +1, 0, -1, +1) are permutation of one another via the one-to-one correspondence: 1->3, 2->4, 3->7, 4->2, 5->5, 6->1, 7->6. E.g.: voter: (+1, +1, +1, 0, 0, -1, -1) and voter: (-1, 0, +1, +1, 0, -1, +1) are permutation of one another via the one-to-one correspondence: 1->3, 2->4, 3->7, 4->2, 5->5, 6->1, 7->6.

Property 2 of Rule f Property 2 – Anonymity: A social choice rule will satisfy this property if it does not make any difference who votes in which way as long as the numbers of each type are the same (i.e. equal treatment of each voter). Formal Definition: A social choice rule f satisfies anonymity if whenever (d 1, d 2, …, d n ) and (d 1 ’, d 2 ’, …, d n ’) in the domain of f are permutations of one another then f(d 1, d 2, …, d n ) = f(d 1 ’, d 2 ’, …, d n ’) E.g.: if D = (+1, +1, +1, 0, 0, -1, -1) and D ’ = (-1, 0, +1, +1, 0, -1, +1) so D and D ’ are permutations of each other, and if f(d 1, d 2, …, d n ) = f(d 1 ’, d 2 ’, …, d n ’) then social choice rule f satisfies anonymity. Property 2 – Anonymity: A social choice rule will satisfy this property if it does not make any difference who votes in which way as long as the numbers of each type are the same (i.e. equal treatment of each voter). Formal Definition: A social choice rule f satisfies anonymity if whenever (d 1, d 2, …, d n ) and (d 1 ’, d 2 ’, …, d n ’) in the domain of f are permutations of one another then f(d 1, d 2, …, d n ) = f(d 1 ’, d 2 ’, …, d n ’) E.g.: if D = (+1, +1, +1, 0, 0, -1, -1) and D ’ = (-1, 0, +1, +1, 0, -1, +1) so D and D ’ are permutations of each other, and if f(d 1, d 2, …, d n ) = f(d 1 ’, d 2 ’, …, d n ’) then social choice rule f satisfies anonymity.

Property 3 of Rule f Property 3 – Neutrality: A social choice rule satisifies neutrality if whenever (d 1, d 2, …, d n ) and (-d 1, -d 2, …, -d n ) are both the domain of f then f(d 1, d 2, …, d n )=-f(-d 1, -d 2, …, -d n ) Property 3 – Neutrality: A social choice rule satisifies neutrality if whenever (d 1, d 2, …, d n ) and (-d 1, -d 2, …, -d n ) are both the domain of f then f(d 1, d 2, …, d n )=-f(-d 1, -d 2, …, -d n ) Note: The condition of anonymity is a way of treating individuals equally, the condition of neutrality is a way of treating alternatives x and y equally. Note: The condition of anonymity is a way of treating individuals equally, the condition of neutrality is a way of treating alternatives x and y equally.

Property 4 of Rule f i-Variants: Suppose there are D = (d 1, d 2, …, d n ) and D ’ = (d 1 ’, d 2 ’, …, d n ’ ); D and D ’ are i-variants if for all j≠i, d j =d j ’. Thus two i-variants differ in at most the ith entry. (Note: It has not strictly stipulated the relationship of d i and d i ’, i.e., it is possible that d i =d i ’, d i >d i ’, or d i d i ’, or d i <d i ’.) E.g.: Two lists D = (+1, -1, -1, 0, +1, -1, +1) and D ’ = (+1, -1, 0, 0, +1, -1, +1) are 3-variants since they differ only at the third place E.g.: Two lists D = (+1, -1, -1, 0, +1, -1, +1) and D ’ = (+1, -1, 0, 0, +1, -1, +1) are 3-variants since they differ only at the third place

Property 4 of Rule f Purpose: Simple majority voting can not be strictly characterized by property 1~3 yet (unresponsive). Purpose: Simple majority voting can not be strictly characterized by property 1~3 yet (unresponsive). E.g.: Assume a constant rule (function) const 0 (D) that always generates result 0 for any point in its domain. i.e. const 0 (D) 0 This constant rule satisfies all 3 properties mentioned above. D contains all logically possible lists. – Property 1 For all permutations D ’, const 0 (D) = const 0 (D) = 0. – Property 2 For all lists in D, const 0 (D) = -const 0 (-D) = 0. – Property 3 So, we still need a property to constrain rule f to simple majority more strictly. E.g.: Assume a constant rule (function) const 0 (D) that always generates result 0 for any point in its domain. i.e. const 0 (D) 0 This constant rule satisfies all 3 properties mentioned above. D contains all logically possible lists. – Property 1 For all permutations D ’, const 0 (D) = const 0 (D) = 0. – Property 2 For all lists in D, const 0 (D) = -const 0 (-D) = 0. – Property 3 So, we still need a property to constrain rule f to simple majority more strictly.

Property 4 of Rule f Property 4 – Positive Responsiveness: f satisfies positive responsiveness if for all i, whenever (d 1, d 2, …, d n ) and (d 1 ’, d 2 ’, …, d n ’ ) are i-variants with di ’ > di, then f(d 1, d 2, …, d n ) ≥ 0 implies f(d 1 ’, d 2 ’, …, d n ’ ) = +1. Property 4 – Positive Responsiveness: f satisfies positive responsiveness if for all i, whenever (d 1, d 2, …, d n ) and (d 1 ’, d 2 ’, …, d n ’ ) are i-variants with di ’ > di, then f(d 1, d 2, …, d n ) ≥ 0 implies f(d 1 ’, d 2 ’, …, d n ’ ) = +1.

Property 4 of Rule f Positive responsiveness can be inferred by multi i-variants. E.g.: Suppose to apply lists #1 below to f which is a rule satisfies positive responsiveness: f(+1, 0, -1, 0, 0, +1, -1) = 0. First find a 3-variant list #2 of #1: (+1, 0, 0, 0, 0, +1, -1), so f(+1, 0, 0, 0, 0, +1, -1) = +1. Second find a 4-variant list #3 of #2: (+1, 0, 0, +1, 0, +1, -1), so f(+1, 0, 0, +1, 0, +1, -1) = +1. Then it can be concluded that f(+1, 0, -1, 0, 0, +1, -1) = 0 implies f(+1, 0, 0, +1, 0, +1, -1) = +1, although list #1 and #3 are not i-variants. Positive responsiveness can be inferred by multi i-variants. E.g.: Suppose to apply lists #1 below to f which is a rule satisfies positive responsiveness: f(+1, 0, -1, 0, 0, +1, -1) = 0. First find a 3-variant list #2 of #1: (+1, 0, 0, 0, 0, +1, -1), so f(+1, 0, 0, 0, 0, +1, -1) = +1. Second find a 4-variant list #3 of #2: (+1, 0, 0, +1, 0, +1, -1), so f(+1, 0, 0, +1, 0, +1, -1) = +1. Then it can be concluded that f(+1, 0, -1, 0, 0, +1, -1) = 0 implies f(+1, 0, 0, +1, 0, +1, -1) = +1, although list #1 and #3 are not i-variants.

Property 4 of Rule f “ Negative Responsiveness ” : Suppose rule f satisfies property 1~4. For all i, whenever D = (d 1, d 2, …, d n ) and D ’ = (d 1 ‘, d 2 ‘, …, d n ‘ ) are i- variants with d i ‘ -d i ). If f(D) ≤ 0 then f(-D) = -f(D) ≥ 0 by neutrality. So f(-D) ≥ 0. There is a list -D’ which together with –D are i-variants with -d i ‘ > -d i. Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness. So f(D’) = -f(-D’) = -1 For summary: If f satisfies positive responsiveness and neutrality then for all i, whenever D = (d 1, d 2, …, d n ) and D ’ = (d 1 ‘, d 2 ‘, …, d n ‘ ) are i-variants with d i ‘ -d i ). If f(D) ≤ 0 then f(-D) = -f(D) ≥ 0 by neutrality. So f(-D) ≥ 0. There is a list -D’ which together with –D are i-variants with -d i ‘ > -d i. Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness. So f(D’) = -f(-D’) = -1 For summary: If f satisfies positive responsiveness and neutrality then for all i, whenever D = (d 1, d 2, …, d n ) and D ’ = (d 1 ‘, d 2 ‘, …, d n ‘ ) are i-variants with d i ‘ < d i, such that f(D) ≤ 0 implies f(D’) = -1

May’s Theorem Simple majority voting is the only rule that satisfies all four properties (or conditions) simultaneously. Simple majority voting is the only rule that satisfies all four properties (or conditions) simultaneously.

May’s Theorem May ’ s Theorem: If a social choice rule f satisfies all of i) universal domain ii) anonymity iii) neutrality iv) positive responsiveness then f is simple majority voting. May ’ s Theorem: If a social choice rule f satisfies all of i) universal domain ii) anonymity iii) neutrality iv) positive responsiveness then f is simple majority voting.

Proof of May’s Theory Step 1: If rule f satisfies conditions i), ii), iii) and iv). So the value of f(D) only depends on the number of +1 ’ s, 0 ’ s and -1 ’ s by anonymity. Suppose there are n elements in D, N +1 (D) and N -1 (D) is the number of +1 ’ s and -1 ’ s in D correspondingly. So the number of 0 ’ s is n - N +1 (D) - N -1 (D). Therefore, f(D) is entirely determined by N +1 (D) and N -1 (D) by anonymity. Step 1: If rule f satisfies conditions i), ii), iii) and iv). So the value of f(D) only depends on the number of +1 ’ s, 0 ’ s and -1 ’ s by anonymity. Suppose there are n elements in D, N +1 (D) and N -1 (D) is the number of +1 ’ s and -1 ’ s in D correspondingly. So the number of 0 ’ s is n - N +1 (D) - N -1 (D). Therefore, f(D) is entirely determined by N +1 (D) and N -1 (D) by anonymity.

Proof of May’s Theory Step 2: Suppose N +1 (D) = N -1 (D) and f(D) = r. Obviously N +1 (D) = N -1 (D) = N +1 (-D) #1 N -1 (D) = N +1 (D) = N -1 (-D). #2 And because f satisfies universal domain, so f is also defined at – D. Since f(-D) = -f(D) = -r by neutrality, and f(-D) = f(D) = r by #1 and #2. Combining above results, – r = r so r = 0. That is N +1 (D) = N -1 (D) implies f(D) = 0. Step 2: Suppose N +1 (D) = N -1 (D) and f(D) = r. Obviously N +1 (D) = N -1 (D) = N +1 (-D) #1 N -1 (D) = N +1 (D) = N -1 (-D). #2 And because f satisfies universal domain, so f is also defined at – D. Since f(-D) = -f(D) = -r by neutrality, and f(-D) = f(D) = r by #1 and #2. Combining above results, – r = r so r = 0. That is N +1 (D) = N -1 (D) implies f(D) = 0.

Proof of May’s Theory Step 3: Suppose N +1 (D) > N -1 (D) where there are n elements in D, so that N +1 (D) = N -1 (D) + m where 0 N -1 (D), then f(D) = +1 If N +1 (D) N -1 (D) where there are n elements in D, so that N +1 (D) = N -1 (D) + m where 0 N -1 (D), then f(D) = +1 If N +1 (D) < N -1 (D), then f(D) = -1

Proof of May’s Theory Summary of Proof: From step 1, 2, and 3: If N +1 (D)=N -1 (D), then f(D)=0. If N +1 (D)>N -1 (D), then f(D)=+1. If N +1 (D) N -1 (D), then f(D)=+1. If N +1 (D)<N -1 (D), then f(D)=-1. These results just satisfy the formal definition of simple majority voting. So May ’ s theory is proved.

Voting Paradox To be continued … To be continued …

References Kelly, Jerry S., 1988, Social Choice Theory An Introduction, Springer- Verlag, Berlin Heidelberg. Kelly, Jerry S., 1988, Social Choice Theory An Introduction, Springer- Verlag, Berlin Heidelberg.