Added Computational Social Choice notes.

Computational Social Choice/0 - Index and Summary.md

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+![[course-summary.png]]
+
+### Social Choice Functions (and related types of functions)
+- Pareto SCF
+- majority SCF
+- silly SCF $f_{silly}$
+- approval voting
+- median voting
+- Borda's rule
+- Black's rule
+- Baldwin's rule
+- Kemeny's Rule
+- Top Cycle $TC$
+- Uncovered Set $UC$
+- Banks set $BA$
+- Tournament equilibrium set $TEQ$
+- Bipartisan Set $BP$
+- Random Dictatorship
+- Maximal Lotteries
+
+### Algorithms
+- Single-Peakedness Algorithm (Bartholdi; Escoffier)
+- Top Cycle Algorithm
+- Uncovered Set Algorithm (= Matrix Multiplication)
+- Finding some ("random") alternative in the Banks set (in linear time)
+
+
+### Terms
+- rationalizable choice function
+- base relation
+- Contraction consistency $\alpha$
+- Expansion consistency $\gamma$
+- Strong expansion consistency $\gamma^+$
+- transitive rationalizability
+- weak axiom of revealed preference (WARP)
+- anonymity
+- neutrality
+- Pareto optimality
+- resoluteness
+- manipulability by a voter, strategyproofness
+- monotonicity
+- positive responsiveness
+- most decisive SCF
+- Condorcet winner
+- Independence of infeasible alternatives (IIA)
+- dictatorship
+- social welfare function (SWF)
+- decisive group of voters for $a$ against $b$, decisive and  semidecisive group of voters
+- weak dictatorship
+- oligarchy
+- collegium
+- strategic abstention and participation
+- dichotomous preferences
+- declaration of oddity ($R(U)$ = the linear orders)
+- single-peaked preferences $D^>_{SP}$
+- median voting
+- value-restricted domains
+- scoring rule
+- Condorcet extension
+- Fishburn classification: C1, C2, C3 SCFs
+- reinforcement
+- composed scoring rule
+- cancellation (technical axiom)
+- social preference function (SPF)
+- majority graph (with majority margin weights)
+- anonymity, neutrality, Pareto optimality for SPFs
+- local independence of irrelevant alternatives (LIIA)
+- Feedback Arc Set (FAS)
+- Kemeny Ranking decision problem
+- dominant set
+- binary SCF
+- majoritarian SCF
+- Dominion $D(x)$, Dominator $\overline{D}(x)$, $D^k(x)$, $\overline{D}^k(x)$, $D^\ast(x)$ and $\overline{D}^\ast(x)$
+- covering relation $C$
+- transitive subset of a tournament
+- strong retentiveness $\rho^+$
+- retentiveness $\rho$
+- $S$-retentiveness
+- $\mathring{\phantom{I}}: S \mapsto \mathring{S}$ operator
+- set rationalizability
+- set consistency conditions $\hat{\alpha}$ and $\hat{\gamma}$
+- stability ($\hat{\alpha}$ + $\hat{\gamma}$)
+- quasi-transitive rationalizability ($\alpha$ + $\hat{\alpha}$ + $\hat{\gamma}$)
+- optimal probability distribution
+- more discriminating SCF (selects fewer alternatives on average)
+- strong monotonicity (= invariance under weakening of unchosen alternatives)
+- lower contour set
+- non-imposing SCF
+- Kelly's preference extension $R^K$
+- Kelly strategyproofness/participation
+- Fishburn's preference extension $R^F$
+- set-non-imposing SCF
+- social decision scheme (SDS)
+- manipulability of an SDS
+- non-imposing SDS
+- (uniform) random dictatorship
+- majority margin matrix $M_{x,y} = n_{xy} - n_{yx}$
+- maximality of a lottery
+
+
+### Theorems
+- Sen's Theorem
+- Schwartz' characterization of $\alpha$ and $\gamma$
+- (Arrow, 1959): transitively rationalizable <=> $\alpha$ + $\beta^+$
+- anonymity + neutrality => not (resolute for an arbitrary number of voters and alternatives)
+- (Moulin 1983): $\exists$ anonymous, neutral and PO <=> n cannot be divided by any $2 \leq q \leq m$.
+- resoluteness + monotonicity + 2 alternatives => strategyproof
+- May's Theorem
+- Condorcet(-May) paradox
+- Arrow's Impossibility Theorem
+- Field Expansion Lemma, Group Contraction Lemma
+- Variants of Arrovian Impossibility Results
+- Arrow's Theorem holds for strict preferences
+- transitive-in-domain $R_M$ --> $\max$-SCF satisfies strategy-proofness and participation in that domain
+- (Inada, 1964) $R_M$ transitive on dichotomous preferences
+- (Black, Arrow) $R_M$ transitive on single-peaked preferences
+- (Sen & Pattanaik) $R_M$ transitive exactly on the value-restricted domains
+- scoring rules invariant under positive affine transformations
+- scoring rule monotonic <==> score vector monotonic (given enough voters)
+- Scoring rules are not Condorcet extensions
+- (Smith, 1973) Condorcet winners are never Borda losers; Condorcet losers are never Borda winners
+- (Gehrlein, 1978) Borda's rule among all scoring rules maximizes the probability of choosing a Condorcet winner, if it exists
+- (Smith; Young) Neutral + anon. SCF is a composed scoring rule iff it satisfies reinforcment
+- (Young, 1974) Borda's rule is the unique SCF satisfying neutrality, Pareto-optimality, reinforcement and cancellation
+- (Young & Levenglick 1978) No Condorcet extension satisfies reinforcement when $m \geq 3$
+- Condorcet Jury theorem: for two alternatives, majority rule is the maximum likelihood SCF
+- (Young 1988) For $p$ close to 0.5, Borda's rule is the maximum likelihood SCF
+- (Young 1988) Kemeny's rule is the maximum-likelihood SPF for any $p$
+- (Young & Levenglick 1978) Kemeny's rule is the only neutral SPF that satisfies Condorcet consistency and reinforcement
+- (Young 1988) Kemeny's rule is the only SPF that satisfies anonymity, neutrality, Pareto optimality, reinforcement and LIIA
+- (McGarvey, 1953) Every majority graph (with 1 on every edge) can be induced by a preference profile with an odd number of voters
+- Kemeny Ranking hardness results
+	- FAS is NP complete (even if restricted to tournaments)
+	- (Bartholdi 1989) Kemeny Ranking decision problem is NP-complete
+	- (Dwork 2001) Finding a Kemeny ranking is NP-hard for $n \in \{4, 6, 7, 8, \dots\}$
+	- deciding for a ranking if it is a Kemeny ranking is coNP-complete
+	- deciding whether a given alternative is a Kemeny winner is $\Theta_2^P$-complete
+	- FAS is APX-hard (cannot be approximated efficiently); However, there is a polynomial-time approximation scheme for weighted tournament FAS.
+- Strong Condorcet-May Impossibility
+- Strong Mas-Colell/Sonenschein impossibility
+- TC is a Condorcet extension
+- (Bordes 1976) TC is the finest SCF satisfying anonymity, neutrality, pos-resp. and $\beta^+$.
+- majoritarian SCF selects everything from a 3-cycle (or generally, a "completely symmetric" majority graph)
+- (Deb 1977) Maximal elements of $P_M^\ast$ = top cycle
+- covering relation $C$ is a transitive subrelation of $P_M$
+- (Moulin 1986) UC is the finest majoritarian SCF satisfying $\gamma$
+- $UC \subseteq TC$, i.e. $UC$ is a refinement of $TC$
+- (Brandt & Geist 2014) $UC$ is the largest majoritarian SCF that satisfies Pareto optimality
+- (Shepsle & Weingast 1984) UC = the alternatives that can reach any other alternative in at most two steps
+- $BA \subseteq UC$, i.e. $BA$ is a refinement of $UC$
+- majoritarian SCF that satisfies $\gamma$ satisfies $\rho^+$
+- $BA$ is the finest majoritarian SCF that satisfies $\rho^+$
+- computing $BA$ is NP-complete
+- $BA, UC, TC$ all select a single alternative if and only if it is a Condorcet winner
+- Computing $TEQ$ is NP-hard
+- Schwartz' conjecture is false
+- set-rationalizability <=> $\hat{\alpha}$
+- many SCFs violate $\hat{\alpha}$:
+	- non-trivial monotonic scoring rules violate $\hat{\alpha}$
+	- almost all other rules we studied: instant-runoff, plurality with runoff, Baldwin's rule, Black's rule, Kemeny's rule, maximin, Young's rule, Copeland's rule, uncovered set, Banks set
+- $TC$ satisfies both $\hat{\alpha}$ and $\hat{\gamma}$
+- every tournament has a unique optimal probability distribution
+- $BP$ and optimal probability distribution properties
+	- $p(x) > 0$ iff $u_p(x) = 0$
+	- $|BP(A, P_M)|$ is odd
+	- $p(x)$ is the quotient of odd numbers or 0
+- $BP \subseteq UC$
+- $BP$ is a Condorcet extension
+- $BP$ is a (non-unique) most discriminating stable majoritarian SCF
+- $BP$ can be computed in polynomial time ("solving a symmetric zero-sum game")
+- $BP$ is $P$-complete
+- any monotonic scoring rule satisfies participation
+- a resolute SCF an two alternatives is strategyproof iff it is monotonic
+- in any domain with $R_M$ acyclic, $Max(R_M, A)$ satisfies strategproofness and participation
+- strong monotonicity implies monotonicity
+- a resolute SCF is strongly monotonic iff any single voter can move around the alternatives below the winning alternative without changing the result
+- (Muller & Satterthwaite 1977) A resolute SCF is strategyproof iff it is strongly monotonic
+- no resolute Condorcet extension satisfies strong monotonicity if $m, n \geq 3$
+- Gibbard-Satterthwaite Impossibility: Any non-imposing, strategyproof, resolute SCF on at least 3 alternatives is dictatorial
+- (Brandt et al. 2016) No resolute Condorcet extension satisfies participation when $n \geq 12$, $m \geq 4$
+- The second-order Copeland, Instant Runoff, and Baldwin's rules are NP-hard to manipulate
+- Every strongly monotonic SCF is $R^K$-strategyproof
+- An SCF that satisfies monotonicity, $\hat{\alpha}$ and IIA satisfies strong monotonicity
+- $BP$, $TC$ and $UC$ are $R^K$-strategyproof
+- Every majoritarian $R^K$-strategyproof SCF satisfies $R^K$-participation
+- (Barberà 1977; Kelly 1977) Every non-imposing $R^K$-strategyproof, quasi-transitively rat. SCF is weakly dictatorial for $m \geq 3$
+- (Barberà 1977) Every Pareto-optimal, $R^K$-strategyproof, positive responsive SCF is dictatorial for $m \geq 3$
+- (Brandt & Lederer, 2021) $TC$ is the only majoritarian SCF that satisfies $R^F$-strategyproofness and set non-imposition
+- (Brandt & Lederer, 2021) $TC$ is the finest majoritarian $R^F$-strategyproof SCF
+- (Brandl et al. 2015) No Pareto-optimal majoritarian SCF satisfies $R^F$-participation for $m \geq 5$
+- (Brandt et al. 2018) For weak preferences and $m \geq 5$, there is no Pareto-optimal, anonymous and $R^F$-strategyproof SCF
+- every SDS that puts probability 1 on Condorcet winners can be manipulated if $m, n \geq 3$
+- (Gibbard 1977) The only non-imposing, non-manipulable SDS for $m \geq 3$ are random dictatorships
+- (Barberà 1979) There are probabilistic variants of Borda's rule and Copeland's rule that are non-manipulable
+- (Kreweras 1965; Fishburn 1984) Maximal lotteries "essentially pick a randomized Condorcet winner"
+- A unique maximal lottery always exists
+- Maximal lotteries are Condorcet extensions, Pareto optimal, monotonic, satisfy reinforcement, participation, etc. etc... (all the good properties!)

Computational Social Choice/0 - everything.md

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+ ![[Week 1 - Introduction]]
+![[Week 2 - Choice Theory, Rationalizability, Consistency]]
+![[Week 3 - Formalizing Social Choice, May's Theorem, Condorcet Paradox]]
+![[Week 4 - Arrow's Impossibility Theorem]]
+![[Week 5 - Escape Route 1, Domain Restrictions]]
+![[Week 6 - Escape Route 2, Variable-Electorate Condition]]
+![[Week 7 - Maximum Likelihood SCFs and Kemeny's Rule]]
+![[Week 8 - Computing Kemeny, Escape Route 3, Top Cycle]]
+![[Week 9 - Computing Top Cycle, Uncovered Set]]
+![[Week 10 - Banks Set, Tournament Equilibrium Set]]
+![[Week 11 - Escape Route 4, Weaken Both Consistency Conditions; Bipartisan Set]]
+![[Week 12 - Strategyproofness, Gibbard-Satterthwaite]]
+![[Week 13 - Hardness of Manipulation, Kelly and Fishburn Extensions, Randomized Social Choice]]

Computational Social Choice/README.md

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+# About these notes
+
+These notes are my lecture notes on the Computational Social Choice course held in the winter term 2021/2022 by Prof. Felix Brandt. The notes are based on the course slides [^1]. Images are taken from the course slides.
+
+- [[0 - everything.pdf]] contains the notes of all chapters exported as a single PDF
+- [[cheatsheet.pdf]] contains the cheatsheet I used as a template for the handwriten sheet allowed in the exam 
+
+The notes are written in [Obsidian markdown](https://obsidian.md/) and are best viewed in Obsidian.
+
+[^1]: Felix Brandt -- Computational Social Choice, lecture slides

Computational Social Choice/Week 1 - Introduction.md

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+# Computational Social Choice
+
+### Common Voting rules
+#### Plurality
+Whoever is ranked first by more voters than any other candidate wins.
+
+*Usage:* most democratic elections
+
+#### Borda's Rule
+The $i$-th ranked alternative of each voter gets $m-i$ points (the bottom alternative gets zero points). The points from each voter are summed together, and the alternative with the greatest total score wins.
+
+*Usage:* in Slovenia for elections, in academic institutions, in the Eurovision Song Contest.
+
+#### Sequential majority comparisons (SMC)
+There is a fixed sequence of comparisons of the alternatives, e.g. (((a vs. b) vs. c) vs. d). Majority comparisons are performed between the first two alternatives, then between the winner and the next alternatives, etc. The winner of a majority comparison between two alternatives is the alternative that is preferred by the majority of voters.
+
+*Usage:* e.g. for the US congress *amendment procedure*.
+
+#### Plurality with runoff
+The two alternatives with the highest plurality scores (i.e. which are first-ranked by the most voters) face off in a final round with a majority comparison.
+
+(Runoff = "Stichwahl")
+
+*Usage:* In France, Brazil and Russia for elections.
+
+#### Instant Runoff
+Successively delete alternatives that are ranked first by the lowest number of voters (by deleting an alternative on top of some voter's profile, the next undeleted alternative below becomes the top alternative of this voter). Repeat until all alternatives that remain are first-ranked by the same number of voters: these are the winners.
+
+*Usage:* Canada and UK (for elections?), Oscar nominations.
+
+
+
+##### Example: "A Curious Preference Profile"
+Consider the preference profile
+
+| 5   | 4   | 3   | 2   |
+| --- | --- | --- | --- |
+| a   | e   | d   | b   |
+| c   | b   | c   | d   |
+| b   | c   | b   | e   |
+| d   | d   | e   | c   |
+| e   | a   | a   | a   |
+
+Then
+- by plurality, a wins
+- by Borda's rule, b wins
+- by SMC, c wins
+- by instant-runoff, d wins
+- by plurality with runoff, e wins
+
+
+### Desirable Properties (Axioms)
+Here are informal definitions of some axioms that will be defined more formally.
+- anonymity - the voting rule treats voters equally
+- neutrality - the voting rule treats alternatives equally
+- monotonicity - a chosen alternative will still be chosen if it is ranked higher in some individual rankings (and nothing else changes)
+- Pareto-optimality - no alternative is chosen if there is another alternative which all voters prefer to it
+
+![[common-voting-rules-axioms.png]]
+
+SMC fails neutrality (obviously), and Pareto-optimality (say, a Pareto-dominates c, but a loses against b and b loses against c).
+
+Runoff rules fail monotonicity: Say, voters reinforce b, and because of that b faces off against c (against which it loses) instead of a (against which it would have won).
+
+### Strategic Manipulation / Strategic Abstention
+Manipulation: mis-represent preferences to obtain a better outcome. By *Gibbard-Satterthwaite impossibility theorem*: every reasonable single-winner voting rule is prone to manipulation!
+
+Abstention: don't participate in the election to obtain a better outcome. Plurality and Borda's rule are resistant to strategic abstention.