Decision Theory CHOICE (Social Choice) Professor : Dr. Liang Student : Kenwa Chu

2 6.1 The Problem of Social Choice  what are the relationships between the demands of rationality and those of justice or fairness  characterized problem of social choice A group of individuals has two or more alternative group actions or policies A group of individuals has two or more alternative group actions or policies The members of the group (henceforth, called citizens) have their own preferences concerning the group choice The members of the group (henceforth, called citizens) have their own preferences concerning the group choice

3 O1-O8 (CH2)  O1: If xPy, then not yPx.  O2: If xPy, then not xIy.  O3: If xIy, then not xPy and not yPx.  O4: xPy or yPx or xIy, for any relevant outcomes x and y.  O5: If xPy and yPz, then xPz.  O6: If xPy and xIz, then zPy.  O7: If xPy and yIz, then xPz.  O8: If xIy and yIz, then xIz.

4collective  group preference profile A set of such individual orderings-one for each A set of such individual orderings-one for each generate an ordinal utility scale generate an ordinal utility scale Satisfies (conditions )  collective choice rule a method that operates on preference profiles and yields a social ranking of the alternatives a method that operates on preference profiles and yields a social ranking of the alternatives

5 group preference profile _EX.  same ordering  one preference profile

6 social welfare function  A collective choice rule a method that operates on preference profiles and yields a social ranking of the alternatives. a method that operates on preference profiles and yields a social ranking of the alternatives.  SWFs (social welfare functions) assume that both the number of alternatives and the number of citizens are finite and will concentrate on a a special type of collective choice rules assume that both the number of alternatives and the number of citizens are finite and will concentrate on a a special type of collective choice rules operate on all the preference profiles possible for a given set of alternatives and citizens operate on all the preference profiles possible for a given set of alternatives and citizens yield social orderings that satisfy conditions O1-O8. yield social orderings that satisfy conditions O1-O8.

7 voting paradox  Majority prefer a to b prefer a to b prefer b to c prefer b to c prefer c to a prefer c to a  cyclical social orderings violating the condition that requires not aPc if cPa violating the condition that requires not aPc if cPa  not a social welfare function

8 social orderings mechanisms  condition U unrestricted domain condition unrestricted domain condition requirement requirement it produces a social ordering for every preference profile  condition D SWF not be dictatorial SWF not be dictatorial  majority rule not satisfy condition U not satisfy condition U But it satisfies condition D But it satisfies condition D

9PROBLEMS  Does a collective choice rule, which merely selects one alternative from a set of alternatives and declares it to be the first choice, count as an SWF?  Suppose there are six citizens and three alternatives and the collective choice rule in use is the following: To decide how to rank a pair of alternatives in the social ordering, roll a die and take the ranking of the citizen whose number comes up as the social ranking of that pair. Will this method necessarily implement an SWF? Will this method necessarily implement an SWF? Could this method yield a dictatorial SWF? Could this method yield a dictatorial SWF?

Arrow's Theorem

11 6-2a. Arrow's Conditions  narrow the field of SWFs (Only U,D) do not agree with one and the same citizen on each profile do not agree with one and the same citizen on each profile SWFs impose the same social ordering no matter how the citizens happen to feel about the alternatives SWFs impose the same social ordering no matter how the citizens happen to feel about the alternatives  Other conditions CS CS PA PA P I  Arrow's theorem negative result: no SWF can satisfy five quite reasonable conditions-two of which happen to be conditions U and D negative result: no SWF can satisfy five quite reasonable conditions-two of which happen to be conditions U and D

12 Condition CS  citizen’s sovereignty condition  Requires each pair of distinct alternatives x and y there is at least one preference profile for which the SWF yields a social ordering that ranks x above y each pair of distinct alternatives x and y there is at least one preference profile for which the SWF yields a social ordering that ranks x above y  EX. at least one preference profile for which the SWF ranks macaroni above beef at least one preference profile for which the SWF ranks macaroni above beef  the social ordering could not be imposed on them in a way entirely independent of their own preferences

13 Condition PA  positive association between individual and social values  Requires if an SWF ranks an alternative x above an alternative y for a given profile, it must also rank x above y in any profile that is exactly like,the original one except that one or more citizens have moved x up in their own rankings if an SWF ranks an alternative x above an alternative y for a given profile, it must also rank x above y in any profile that is exactly like,the original one except that one or more citizens have moved x up in their own rankings  any SWF that satisfied PA and socially ordered chicken above beef for profile 1 would do the same for profile 2

14 Condition P  Pareto condition also be called the condition of unanimity rule also be called the condition of unanimity rule  Requires the SWF must rank x above y for a given profile if every citizen ranks x above y in that profile. It could the SWF must rank x above y for a given profile if every citizen ranks x above y in that profile. It could  Relationship (P-CS-PA)

15 Condition P-CS-PA  The Pareto condition implies the citizens' sovereignty condition  Condition P does not imply condition PA

16 Condition P-CS-PA _EX.  not meet condition PA  conditions PA and CS together imply condition P

17 Condition I  independence-of-irrelevant- alternatives  Requires social welfare functions to obtain social orderings by comparing alternatives two at a time taken in isolation from the other alternatives social welfare functions to obtain social orderings by comparing alternatives two at a time taken in isolation from the other alternatives

18 Condition I _EX.

19 condition I _extent  Arrow's theorem (and social choice theory in general) abstracts from the mechanisms used in deriving social orderings abstracts from the mechanisms used in deriving social orderings does not distinguish between SWFs that yield the same outputs for each possible input. does not distinguish between SWFs that yield the same outputs for each possible input.  Condition I If each citizen ranks the alternatives x and y in the same order in the preference profiles P1 and P2, x and y must be in the same order with respect to each other in the social orderings that the SWF yields for P1 and P2. If each citizen ranks the alternatives x and y in the same order in the preference profiles P1 and P2, x and y must be in the same order with respect to each other in the social orderings that the SWF yields for P1 and P2.

20 condition I _excludes rank-order methods  rank-order method each alternative receives six points each alternative receives six points ranked as socially indifferent ranked as socially indifferent

21 condition I _excludes rank-order methods  a and b stand in the same respective order in both profiles  Condition I requires the SWF to rank a and b alike in both social rankings the SWF to rank a and b alike in both social rankings  the rank-order method used to generate these two tables (and any other SWF giving rise to them) fails to satisfy condition I

22 THE PARETO LEMMA  Any SWF that satisfies conditions PA, CS, and I also satisfies condition P  Proof Assume Assume the SWF satisfies conditions PA, CS, and I. P1 is a profile in which every citizen ranks alternative x above another y We will suppose for P1 the SWF does not socially rank x above y and derive a contradiction We will suppose for P1 the SWF does not socially rank x above y and derive a contradiction there must be some profile P2 different from P1 for which the SWF socially ranks x above y, because CS is in force. Furthermore, P1 and P2 must differ in their placement of x and y. there must be some profile P2 different from P1 for which the SWF socially ranks x above y, because CS is in force. Furthermore, P1 and P2 must differ in their placement of x and y. some in P2 do not prefer x to y some in P2 do not prefer x to y the placement of x and y in P1 can be obtained from that of P2 by moving x up in the ranking of one or more citizens. Hence by PA, the social ordering for it must rank x above y. the placement of x and y in P1 can be obtained from that of P2 by moving x up in the ranking of one or more citizens. Hence by PA, the social ordering for it must rank x above y. But then by condition I, the social ordering for P1 must also rank x above y. But then by condition I, the social ordering for P1 must also rank x above y. And that contradicts the assumption that the SWF failed to socially order x above y for P1. And that contradicts the assumption that the SWF failed to socially order x above y for P1.

23PROBLEMS  Prove that every dictatorial SWF must satisfy condition P.  Explain why we cannot prove the Pareto lemma by applying condition PA directly to P2 and dispensing, thereby, with the use of condition I.  Suppose that in Heaven God keeps a book that lists for each person and for each possible alternative God's assessment of the value of that alternative to that person-on a scale of -1,000 to +1,000. Now consider an SWF that works as follows: To decide how to rank two alternatives x and y, we first use God's book to find their values for the citizens of the particular society at hand. Then we sum the values for x and y, rank x above y if its sum is greater, y above x if its sum is greater, and rank them as indifferent otherwise. Would this SWF necessarily violate condition I? Condition D? Condition CS?  Prove that the following condition implies condition I: If P1 and P2 are two profiles and S is any subset of the set of alternatives, then if the citizens' relative rankings of the members of S are the same for P1 and Pt, the SWF places the members of S in the same relative positions in both P1 and P2.  Prove that condition I implies the condition of exercise 4.

24 6-2b. Arrow's Theorem and Its Proof

25 Proof Arrow's Theorem from PA  THEOREM Where three or more alternatives and two or more citizens are involved, there is no SWF that meets all five conditions CS, D, I, PA, and U. Where three or more alternatives and two or more citizens are involved, there is no SWF that meets all five conditions CS, D, I, PA, and U. If no SWF meets conditions D, I, U, and P, neither can any SWF meet conditions CS, D, I, PA, and U.For if it satisfied the latter, it would automatically satisfy the former, given the Pareto lemma. Consequently, we can prove Arrow's theorem by proving its Pareto version, namely, that no SWF simultaneously meets D, I, U, and P: If an SWF satisfies conditions U, P, and I, it must be dictatorial.

26 several definitions  A set of citizens is decisive for x over y just in case x is socially preferred to y whenever each member of the set prefers x to y.  A citizen is a dictator for x over y just in case the set consisting of him alone is decisive for x over y.  A citizen is a dictator if and only if he is decisive for every pair of distinct alternatives.  An SWF is dictatorial just in case some citizen is a dictator under it.

27 LEMMA 1  For any set of citizens and any pair of distinct alternatives there is at least one decisive set.  PROOF The set of all citizens is decisive for every pair of alternatives. For if every citizen prefers an alternative x to another one y, then, by condition P, society prefers x to y. So there is at least one set that is decisive for x over y. The set of all citizens is decisive for every pair of alternatives. For if every citizen prefers an alternative x to another one y, then, by condition P, society prefers x to y. So there is at least one set that is decisive for x over y.

28 several definitions _con.  A set of citizens is almost decisive for x over y just in case the social ordering ranks x above y when (a) all members of the set do and (b) all members outside prefer y to x.  A citizen is almost decisive for x over y if and only if the set consisting of him alone is almost decisive for x over y.

29 LEMMA 2  There is a citizen who is almost decisive for some pair of alternatives  PROOF By lemma 1 there are decisive sets for each pair of alternatives. By lemma 1 there are decisive sets for each pair of alternatives. there must be sets that are almost decisive for each alternative as well. there must be sets that are almost decisive for each alternative as well. Since the number of citizens and alternatives is finite, there must be at least one nonempty set that is almost decisive for some pair of alternatives but that has no nonempty subsets that are almost decisive for any alternatives. Since the number of citizens and alternatives is finite, there must be at least one nonempty set that is almost decisive for some pair of alternatives but that has no nonempty subsets that are almost decisive for any alternatives. We can find such a set by starting with society as a whole, which we already know to be almost decisive for every pair, and proceed to check all sets obtained from it by deleting one member, and so on, until we find one with the desired property. We can find such a set by starting with society as a whole, which we already know to be almost decisive for every pair, and proceed to check all sets obtained from it by deleting one member, and so on, until we find one with the desired property. Let M be such a minimal almost decisive set and let it be almost decisive for x over y. Since M is nonempty, at least one citizen belongs to it. Let J be such a citizen. We will prove that only J belongs to M. That will show that J is almost decisive for x over y. Let M be such a minimal almost decisive set and let it be almost decisive for x over y. Since M is nonempty, at least one citizen belongs to it. Let J be such a citizen. We will prove that only J belongs to M. That will show that J is almost decisive for x over y.

30 LEMMA 2 _con.  assume more than one citizen belongs to M and derive a contradiction more than one citizen belongs to M and derive a contradiction  Alternatives: x, y, z  set of citizens: J, M-J  only alternative left is that of society preferring z to x, and that again leads to the contradictory conclusion that J is almost decisive for z over y. Thus we have derived a contradiction from our assumption that M did not consist of J alone.

31 LEMMA 3  Any citizen who is almost decisive for a single pair of alternatives is decisive for every pair of alternatives.  ROOF Assume that J is a citizen who is almost decisive for x over y. Assume that J is a citizen who is almost decisive for x over y. We will show that he is decisive for all pairs of distinct alternatives. We will show that he is decisive for all pairs of distinct alternatives. These pairs may be divided into seven cases: x over y, y over x, x over a, a over x, y over a, a over y, and a over by where a and b are alternatives distinct from each other and from x and y. These pairs may be divided into seven cases: x over y, y over x, x over a, a over x, y over a, a over y, and a over by where a and b are alternatives distinct from each other and from x and y.

32 Case: x over a  Assume : a is an alternative distinct from x and y  This is to indicate that no information is given about the relative ordering of x and a in the rankings of the remaining members of society.  each citizen but J ranks y above both x and a, but each might place the latter two in any order independently of his fellow citizens.  everyone- J included-ranks y over a and condition P holds, society ranks y over a  the ordering condition holds, society must also rank x over a  since condition I is in force. Thus whenever J prefers x to a, society' does

33 Case: a over y  Since every citizen ranks a over x, society does (by condition P)  Society must also rank x over y, because J is almost decisive for x over y  by the ordering condition it follows that society ranks a over y.

34 Case: y over a  By condition P society prefers y to x.  Since we have already established that J is decisive for x over a, we can conclude that society prefers x to a.  But then by the ordering condition, it must prefer y to a.  Condition I then lets us conclude that J is decisive for y over a.

35 Case: a over x  The a-over-y case permits us to infer ( 推論 )that society prefers a to y; condition P results in its preferring y to x.  The argument then proceeds as in the previous cases to conclude that J is for a over x.

36 Case: x over y  Let a be any alternative distinct from x and y and consider any profile in which J prefers x to a and a to y. By the x-over-a case, society prefers x to a. By the a-over-y case, society prefers a to y. We then proceed as usual to infer that J is decisive for x over y.

37 Problems (Case Study)  Case: y over x  Case: a over b