Social Choice Theory
Social Preference Aggregation
 different people may have different preferences over social outcomes
 in the US, polarized issues include gun control, immigration, civil rights, etc
 big question: how do we aggregate individual preferences into social preferences?
 which group's opinion is "correct"?
 when (and when not to) implement social change?
Social Choice Function
 input:
 society of \(N\) people
 choice set \(X\)
 example: \(X=\{ \text{Biden}, \text{Harris}, \text{Sanders} \}\)
 each person has her own preference relation over \(X\)
 output:
Simple Majority: Condorcet Paradox

voter 1 
voter 2 
voter 3 
most preferred 
Biden 
Harris 
Sanders 
middle 
Harris 
Sanders 
Biden 
least preferred 
Sanders 
Biden 
Harris 
 how many voters think Biden \(\succ\) Harris?
 how many people think Harris \(\succ\) Sanders?
 how many people think Sanders \(\succ\) Biden?
Simple Majority: Agenda Manipulation

voter 1 
voter 2 
voter 3 
most preferred 
Biden 
Harris 
Sanders 
middle 
Harris 
Sanders 
Biden 
least preferred 
Sanders 
Biden 
Harris 
 problem: agenda manipulation
 suppose agenda setter wants Biden to win
 step 1: \(\text{Sanders vs Harris}\)
 step 2: \(\text{winner of step 1 vs Biden}\)
Simple Majority: Strategic Voting

voter 1 
voter 2 
voter 3 
most preferred 
Biden 
Harris 
Sanders 
middle 
Harris 
Sanders 
Biden 
least preferred 
Sanders 
Biden 
Harris 
 another problem: strategic voting
 agenda is \(\text{Sanders vs Harris}\), then \(\text{winner vs Biden}\)
 focus on voter 2 and her vote at step 1
 vote sincerely: Harris wins step 1, Biden wins step 2
 vote strategically: Sanders wins step 1, Sanders wins step 2 \(\leftarrow\) voter 1 prefers this so she will vote for the "wrong" candidate at step 1
Desirable Properties of SCF: Completeness
 full domain / complete ranking (C):
 returns a social preference for any profile \(\{ \succsim_1, \dots, \succsim_N \}\) of individual preferences
Desirable Properties of SCF: Transitivity
 transitivity of social ranking / no cycles (T):
 if \(x \succsim_S y\) and \(y \succsim_S z\), then \(x \succsim_S z\)
 note: all individual preferences satisfy C and T; we don't want to miss these properties of rationality during aggregation
Desirable Properties of SCF: Pareto
 Pareto (P):
 if every person agrees that \(x \succsim_i y\) AND for at least one person \(x \succ_i y\), then \(x \succ_S y\)
Dictatorial Preferences
 so far we want social ranking to be complete, transitive, and Paretian
 "easy" way to satisfy all three is pick a dictator
 person \(i\) is a dictator if \(x \succ_i y \Longrightarrow x \succ_S y\)
 regardless of others' preferences, social ranking is same as dictator's
Desirable Properties of SCF: No Dictator
 no dictator (ND):
 minimum requirement for "democracy"
 necessary, but not sufficient for democracy
(Logic Reference): Necessity
 let \(A\) and \(B\) be two (logical) statements
 \(A\) is necessary for \(B\) if \(\text{not } A \Longrightarrow \text{not } B\) (same as \(B \Longrightarrow A\))
 example:
 \(A\) is "there is no dictator" and \(B\) is "society is a democracy"
 \(\text{not } A \Longrightarrow \text{not } B\): "if not no dictator then not democracy"
 same as \(B \Longrightarrow A\): "if democracy then no dictator"
(Logic Reference): Sufficiency
 \(A\) is sufficient for \(B\) if \(A \Longrightarrow B\)
 example:
 \(A\) is "there is no dictator" and \(B\) is "society is a democracy"
 "if no dictator then democracy"
(Back to) Desirable Properties of SCF: IIA
 independence of irrelevant alternatives (IIA):
 social preferences between \(x\) and \(y\) depend only on the individual preferences between \(x\) and \(y\), and not other alternatives
 let \(\{\succsim_1, \dots, \succsim_N\}\) and \(\{\succsim_1', \dots, \succsim_N'\}\) be two profiles of individual preferences
 ranking of \(x\) and \(y\) is the same in both: \(x \succsim_i y \iff x \succsim_i' y\) for every \(i = 1\dots N\)
 IIA: social preference must be the same as well: \(x \succ_S y \iff x \succ_S' y\)
Independence of Irrelevant Alternatives
 consider preference profiles \(\{ \succ_1,\succ_2,\succ_3 \}\) and \(\{ \succ_1',\succ_2',\succ_3' \}\):

voter 1 
voter 2 
voter 3 

voter 1 
voter 2 
voter 3 
most preferred 
Biden 
Harris 
Sanders 

Biden 
Harris 
Biden 
middle 
Harris 
Sanders 
Biden 

Sanders 
Sanders 
Sanders 
least preferred 
Sanders 
Biden 
Harris 

Harris 
Biden 
Harris 
 same individual rankings between \(\text{Biden}\) and \(\text{Harris}\)
 IIA: social ranking between Biden and Harris should be the same
Why IIA?
 individual's intensity of preferences should not determine the social ranking
 intensity of relative political preference is private info and can easily be misrepresented
 a good SCF should be stable with respect to adding and removing of alternatives
 ranking of \(\text{Biden}\) vs \(\text{Harris}\) should not depend on whether \(\text{Sanders}\) is running
Desirable Properties of SCF: Summary
 we would like our social choice function to satisfy
 completeness (C)
 transitivity (T)
 Pareto (P)
 no dictator (ND)
 independence of irrelevant alternatives (IIA)
Example of SCF: Unanimity Rule Without Status Quo
\[
x \succsim_S y \iff x \succsim_i y \text{ for every person } i = 1\dots N
\]
 if everyone prefers \(x\) to \(y\), then \(x\) is socially preferred to \(y\)
 if everyone prefers \(y\) to \(x\), then \(y\) is socially referred to \(x\)
 if people do not unanimously agree that \(x\) is better than \(y\) or vice versa, SCF does not return a social preference
Borda "Scoring" Rule
 every person provides a ranking of all alternatives, worst \(\to\) best
 worst choice receives \(1\) point; second worst  \(2\) points, and so on
 social score of an alternative is the sum of individual scores
 winner is the alternative with highest score
Borda Rule: Example

voter 1 
voter 2 
voter 3 
most preferred 
Biden 
Harris 
Biden 
middle 
Sanders 
Sanders 
Sanders 
least preferred 
Harris 
Biden 
Harris 
 Biden: \(3+1+3=7\) \(\leftarrow\) Borda winner
 Sanders: \(2+2+2 = 6\)
 Harris: \(1+3+1 = 5\)
Borda Rule: Properties
 C: we can calculate score for any alternatives and we can compare any pair of alternatives by comparing scores
 T: if \(x \succsim_S y\) and \(y \succsim_S z\), then \(\text{score}(x) \geq \text{score}(y) \geq \text{score}(z)\), meaning that \(x \succsim_S z\)
 P: if every voter prefers \(x\) to \(y\), then \(\text{score}(x) > \text{score} (y)\) and \(x \succsim_S y\)
 ND: social scores depend on all people's preferences so no individual is a dictator
Borda Rule: IIA Fails
 consider preference profiles \(\{ \succ_1,\succ_2,\succ_3 \}\) and \(\{ \succ_1',\succ_2',\succ_3' \}\):
voter 1 
voter 2 
voter 3 

voter 1 
voter 2 
voter 3 
\(x\) 
\(y\) 
\(z\) 

\(x\) 
\(y\) 
\(z\) 
\(y\) 
\(z\) 
\(x\) 

\(y\) 
\(x\) 
\(x\) 
\(z\) 
\(w\) 
\(y\) 

\(z\) 
\(z\) 
\(y\) 
\(w\) 
\(x\) 
\(w\) 

\(w\) 
\(w\) 
\(w\) 
 do all voters rank \(x\) and \(y\) the same way in \(\succ\) and \(\succ'\)? \(\text{Yes}\)
 is social ranking of \(x\) and \(y\) the same for \(\succ\) and \(\succ'\)? \(\text{No}\)
Borda Rule: IIA Fails, Another Example
60 voters 
40 voters 



60 voters 
40 voters 
Navalny 
Ivanov 



Navalny 
Ivanov 
Medvedev 
Medvedev 



Medvedev 
Medvedev 
Ivanov 
Zuganov 



Ivanov 
Navalny 
Zuganov 
Navalny 





 social score is
 Navalny: \(4\cdot 60+1\cdot 40=280\) \(\longrightarrow\) \(3\cdot 60 + 1\cdot 40 = 220^*\)
 Medvedev: \(3\cdot 60 + 3\cdot 40 = 300^*\) \(\longrightarrow\) \(2\cdot 60 + 2\cdot 40 = 200\)
 Ivanov: \(2\cdot 60 + 4\cdot 40 = 280\) \(\longrightarrow\) \(1\cdot 60 + 3\cdot 40 = 180\)
 Zuganov: \(1\cdot 60 + 2\cdot 40 = 140\) \(\longrightarrow\) not participating
 IIA fails: Medvedev wins but when Zuganov drops out, Navalny wins
Borda Rule: Why Failure of IIA is Bad
60 voters 
40 voters 



60 voters 
40 voters 
Navalny 
Ivanov 



Navalny 
Medvedev 
Medvedev 
Medvedev 



Medvedev 
Ivanov 
Ivanov 
Navalny 



Ivanov 
Navalny 
Big Question of Social Choice Theory
 so far, all our examples of social choice functions failed at least of desirable property
 is there any social choice function that satisfies C, T, P, ND, IIA?
Arrow's Impossibility Theorem
 for any social choice problem with at least three alternatives, there does not exist a social choice function that satisfies C, T, P, ND, IIA
 interpretation: groups of people do not have welldefined preferences, even if each person is rational
Arrow's Theorem: Alternative Statement
 "for any social choice problem with at least three alternatives, if a SCF satisfies C, T, P, and IIA, then it must be dictatorial
 resist the urge to interpret as "we need to have a dictator for society to be rational"
 AIT merely states that nondictatorial SCFs have some problems
AIT: Interpretation
 no nondictatorial SCF can satisfy all 5 axioms of social choice but many can satisfy \(4/5\)
 unanimity rule without status quo: all but C
 simple majority: all but T
 Borda scoring rule: all but IIA
 "constant rule" (constant social preferences, e.g. alphabetical ranking): all but P
Are Axioms Too Demanding?
 IIA has weakest moral basis; Borda rule satisfies all but IIA but has problems:
 how to measure intensity of preferences?
 how to avoid strategic manipulation of social ranking by addition/removal of candidates?
 "my rule is intended for honest men only": Borda, probably
Other Ways Around AIT
 AIT: "or any social choice problem with at least three alternatives, there does not exist a social choice function that satisfies C, T, P, ND, IIA"
 does not apply if choice is between two alternatives
 does not apply if we restrict preferences