1. Social Choice Theory By Shiyan Li

2. 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.

3. 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

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

5. 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

6. 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

7. 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 is to be chosen, simple majority voting assigns +1. When d 1 +d 2 +d 3 + … +d n-1 +d n < 0, y 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.

8. Formal Definition of Simple Majority Voting Use the sign function to formally define the simple majority voting: (d 1, d 2, …, d n ) sgn(d 1 +d 2 + … +d n ) Use the sign function to formally define the simple majority voting: (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 negative 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 negative

9. Formal Definition of Simple Majority Voting E.g. for absolute majority voting: 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)>N-1(D) -1 if N-1(D)>N+1(D) 0 otherwise if N+1(d1, d2, d3, …, dn) > n/2 if N-1(d1, d2, d3, …, dn) > n/2 Absolute Majority Voting

10. 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).

11. 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.

12. 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 distinct outcomes are 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 distinct outcomes are 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

13. 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 2 3 4 5 6 7 (+1, +1, +1, 0, 0, -1, -1) and voter: 1 2 3 4 5 6 7 (-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 2 3 4 5 6 7 (+1, +1, +1, 0, 0, -1, -1) and voter: 1 2 3 4 5 6 7 (-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.

14. 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.

15. 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.

16. 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

17. Property 4 of Rule f Purpose: Simple majority voting can not be strictly characterized by property 1~3 yet (unresponsiveness). Purpose: Simple majority voting can not be strictly characterized by property 1~3 yet (unresponsiveness). 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.

18. 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.

19. Property 4 of Rule f Positive responsiveness can be inferred by indirect 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 direct i-variants. Positive responsiveness can be inferred by indirect 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 direct i-variants.

20. 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 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 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

21. 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.

22. 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.

23. 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.

24. 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.

25. 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

26. 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.

27. General Social Choice Rules X: a nonempty set of alternatives. The elements of X must only be mutually incompatible. X: a nonempty set of alternatives. The elements of X must only be mutually incompatible. v: agenda, v ≠ Ø and v ⊆ X, a set of alternatives that are currently available. v: agenda, v ≠ Ø and v ⊆ X, a set of alternatives that are currently available. N: a set of individuals. N: a set of individuals.

28. General Social Choice Rules xR i y: i ∈ N; x, y ∈ X; individual i determines alternative x to be at least as good as alternative y; or i weakly prefers x to y. 1. R i is reflexive: xR i x for all x ∈ X. 2. R i is complete: xR i y or yR i x (or both) for all x, y ∈ X. 3. R i is transitive: For all x, y, z ∈ X, if both xR i y and yR i z then xR i z. xR i y: i ∈ N; x, y ∈ X; individual i determines alternative x to be at least as good as alternative y; or i weakly prefers x to y. 1. R i is reflexive: xR i x for all x ∈ X. 2. R i is complete: xR i y or yR i x (or both) for all x, y ∈ X. 3. R i is transitive: For all x, y, z ∈ X, if both xR i y and yR i z then xR i z.

29. General Social Choice Rules xP i y: xR i y and not yR i x; i strongly prefers x to y. xP i y: xR i y and not yR i x; i strongly prefers x to y. yP i x: yR i x and not xR i y; i strongly prefers y to x. yP i x: yR i x and not xR i y; i strongly prefers y to x. xI i y: xR i y and also yR i x; i is indifferent between x and y. xI i y: xR i y and also yR i x; i is indifferent between x and y.

30. General Social Choice Rules Profile: an assignment of one preference relation to each individual. Profile: an assignment of one preference relation to each individual. C(v): the elements chosen from agenda v by choice function C. (i) C(v) ⊂ v; (ii) C(v) ≠ Ø. C(v): the elements chosen from agenda v by choice function C. (i) C(v) ⊂ v; (ii) C(v) ≠ Ø.

31. General Social Choice Rules Social Choice Rule: A social choice rule assigns to each of a collection of profiles a corresponding choice function. Social Choice Rule: A social choice rule assigns to each of a collection of profiles a corresponding choice function. social choice rule, f profile of preferences, u choice function, C = f(u) agenda, v chosen set, C u (v)

32. Standard Domain Constraint Standard domain constraint includes: i) there are at least three alternatives in X; ii) there are at least three individuals in N; iii) the social choice rule has as domain all logically possible profiles of preference orderings on X; iv) each choice function that is an output of the rule has in its domain all finite nonempty agendas. Standard domain constraint includes: i) there are at least three alternatives in X; ii) there are at least three individuals in N; iii) the social choice rule has as domain all logically possible profiles of preference orderings on X; iv) each choice function that is an output of the rule has in its domain all finite nonempty agendas.

33. Pareto Condition Weak Pareto Condition: Let the social choice rule select choice function C u at profile u. Suppose at u everyone unanimously strictly prefers one alternative, say x, to another, say y; then if x is available (i.e., x ∈ v), y won ’ t be chosen (i.e., y ∉ C u (v)) Weak Pareto Condition: Let the social choice rule select choice function C u at profile u. Suppose at u everyone unanimously strictly prefers one alternative, say x, to another, say y; then if x is available (i.e., x ∈ v), y won ’ t be chosen (i.e., y ∉ C u (v))

34. Pareto Condition Strong Pareto Condition: Let the social choice rule select choice function C u at profile u. Suppose at u everyone unanimously find one alternative, x, to be at least as good as another, y, and at least one individual strictly prefers x to y. Then if x is available (i.e., x ∈ v), ywon ’ t be chosen (i.e., y ∉ C u (v)) Strong Pareto Condition: Let the social choice rule select choice function C u at profile u. Suppose at u everyone unanimously find one alternative, x, to be at least as good as another, y, and at least one individual strictly prefers x to y. Then if x is available (i.e., x ∈ v), ywon ’ t be chosen (i.e., y ∉ C u (v))

35. Pareto Condition Example: For agenda: 1: (x y 1 ) y 2 2: x y 1 y 2 3: x (y 1 y 2 ) In Weak Pareto Condition: y 2 ∉ C u (v) In Strong Pareto Condition: y 1, y 2 ∉ C u (v) Example: For agenda: 1: (x y 1 ) y 2 2: x y 1 y 2 3: x (y 1 y 2 ) In Weak Pareto Condition: y 2 ∉ C u (v) In Strong Pareto Condition: y 1, y 2 ∉ C u (v)

36. Pareto Condition X is Pareto-superior to y at profile u = (R 1, R 2, …, R n ) if: (i) xR i y for all individuals i in N; (ii) xP i y for at least one individual i in N. X is Pareto-superior to y at profile u = (R 1, R 2, …, R n ) if: (i) xR i y for all individuals i in N; (ii) xP i y for at least one individual i in N. Alternatives for which there are no available Pareto-superior alternatives are called Pareto optimal. Alternatives for which there are no available Pareto-superior alternatives are called Pareto optimal.

37. Dictator Weak Dictator Individual i is a weak dictator if for every pair of alternatives, x and y, every profile u = (R 1, R 2, …, R n ) and every agenda v, if xP i y then y ∈ C u (v) implies x ∈ C u (v). Weak Dictator Individual i is a weak dictator if for every pair of alternatives, x and y, every profile u = (R 1, R 2, …, R n ) and every agenda v, if xP i y then y ∈ C u (v) implies x ∈ C u (v).

38. Dictator Coalition A subset S of the set N of all individuals is called a coalition. Coalition A subset S of the set N of all individuals is called a coalition. Decisive Coalition For a social choice rule that maps u to C u, A coalition S is called decisive for alternative x against alternative y if: for ∀i: i ∈ S xR i y; ∃j: j ∈ S xP j y; then ∀v: v ⊂ X, x ∈ v y ∉ C u (v). If ∀x, y: x, y ∈ X S is decisive for alternative x against alternative y, then we simply say S is decisive. Decisive Coalition For a social choice rule that maps u to C u, A coalition S is called decisive for alternative x against alternative y if: for ∀i: i ∈ S xR i y; ∃j: j ∈ S xP j y; then ∀v: v ⊂ X, x ∈ v y ∉ C u (v). If ∀x, y: x, y ∈ X S is decisive for alternative x against alternative y, then we simply say S is decisive.

39. Dictator Dictator If a decisive coalition S = {i}, then i is a dictator. Dictator If a decisive coalition S = {i}, then i is a dictator.

40. Borda Rules Borda Count Assume that X is finite. Then associated with any preference ordering R i there is a ranking function r i that associates an integer with each alternative: r i (x) is the number of alternatives stictly preferred to x. Given a profile u = (R 1, R 2, …, R n ), there is a ranking function r given by r(x) = ∑ i r i (x). The value of r(x) is called Borda count of x. Borda Count Assume that X is finite. Then associated with any preference ordering R i there is a ranking function r i that associates an integer with each alternative: r i (x) is the number of alternatives stictly preferred to x. Given a profile u = (R 1, R 2, …, R n ), there is a ranking function r given by r(x) = ∑ i r i (x). The value of r(x) is called Borda count of x.

41. Borda Rules Global Borda Rule C u (v) = {x|r(x) ≤ r(y) for all y ∈ v}. This rule has us choose from v those alternatives with minimal Borda count. Global Borda Rule C u (v) = {x|r(x) ≤ r(y) for all y ∈ v}. This rule has us choose from v those alternatives with minimal Borda count.

42. Independence of Irrelevant Alternatives If two profiles u, u ’, restricted to an agenda v are identical, then the choices made from that agenda should be the same: C u (v) = C u’ (v). If two profiles u, u ’, restricted to an agenda v are identical, then the choices made from that agenda should be the same: C u (v) = C u’ (v).

43. Local Borda Rules Local Borda Count Given a profile u = (R 1, R 2, …, R n ), there is for each v: r v (x) = ∑ i r i v (x). Local Borda Count Given a profile u = (R 1, R 2, …, R n ), there is for each v: r v (x) = ∑ i r i v (x). Local Borda Rule C u (v)= {x|r v (x) ≤ r v (y) for all y ∈ v}. Local Borda Rule C u (v)= {x|r v (x) ≤ r v (y) for all y ∈ v}.

44. Transitive Explanation Explanation: A choice function C is explainable if there exists a relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}. Explanation: A choice function C is explainable if there exists a relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}. Transitive Explanation: A choice function C has transitive explainable if there is a reflexive, complete and transitive relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}. Transitive Explanation: A choice function C has transitive explainable if there is a reflexive, complete and transitive relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}. We say a social choice rule has transitive explainable if at every admissible profile u the associated C u has a transitive explainable. We say a social choice rule has transitive explainable if at every admissible profile u the associated C u has a transitive explainable.

45. Arrow’s Impossibility Theorem There does not exist any social choice rule satisfying all of: 1. the standard domain constraint; 2. the strong Pareto condition; 3. independence of irrelevant alternatives; 4. has transitive explanations; 5. absence of a dictator. There does not exist any social choice rule satisfying all of: 1. the standard domain constraint; 2. the strong Pareto condition; 3. independence of irrelevant alternatives; 4. has transitive explanations; 5. absence of a dictator.

46. Mechanism Design Implementing a social choice function f(u 1, …, u n ) using a game. Implementing a social choice function f(u 1, …, u n ) using a game. Center (auctioneer) does not know the agents’ preferences. Center (auctioneer) does not know the agents’ preferences. Agents may lie. Agents may lie. Goal is to design the rules of the game so that in equilibrium (s 1, …, s n ), the outcome of the game is f(u 1, …, u n ). Goal is to design the rules of the game so that in equilibrium (s 1, …, s n ), the outcome of the game is f(u 1, …, u n ).

47. Mechanism Design Mechanism designer specifies the strategy sets S i and how outcome is determined as a function of (s 1, …, s n )  (S 1, …, S n ). Mechanism designer specifies the strategy sets S i and how outcome is determined as a function of (s 1, …, s n )  (S 1, …, S n ). Variants Strongest: There exists exactly one equilibrium. Its outcome is f(u 1, …, u n ). Medium: In every equilibrium the outcome is f(u 1, …, u n ). Weakest: In at least one equilibrium the outcome is f(u 1, …, u n ). Variants Strongest: There exists exactly one equilibrium. Its outcome is f(u 1, …, u n ). Medium: In every equilibrium the outcome is f(u 1, …, u n ). Weakest: In at least one equilibrium the outcome is f(u 1, …, u n ).

48. 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.