The Logic of Collective Choice: Formal Conditions and Theorems

Vocabulary practice flashcards covering formal conditions, transitivity properties, choice definitions, power structures, and tradeoff theorems in social choice theory and collective choice.

Weak Pareto (P)

A condition requiring that if every individual strictly prefers alternative $x$ to alternative $y$, then society must strictly prefer $x$ to $y$: $\forall x, y \in X, [\forall i: x P_i y] \rightarrow x P y$.

Independence of Irrelevant Alternatives (I)

A condition requiring that social preference between any pair of alternatives $x$ and $y$ depends exclusively on individual preferences between $x$ and $y$, remaining invariant to rankings of other alternatives: $\forall x, y \in X: [(\forall i: x R_i y \leftrightarrow x R'_i y) \wedge (\forall i: y R_i x \leftrightarrow y R'_i x)] \rightarrow [(x R y \leftrightarrow x R' y) \wedge (y R x \leftrightarrow y R' x)]$.

Non-Dictatorship (D)

A condition stating that there is no single individual whose strict preference always determines the strict social preference for every pair of alternatives: $\nexists i \text{ such that } \forall x, y \in X: x P_i y \rightarrow x P y$.

Strong Pareto (P+)

A condition stating that if every individual weakly prefers $x$ to $y$ and at least one individual strictly prefers $x$ to $y$, then society must strictly prefer $x$ to $y$: $\forall x, y \in X, [\forall i: x R_i y \wedge \exists j: x P_j y] \rightarrow x P y$.

Anonymity (A)

A condition requiring all individuals to be treated symmetrically, such that any permutation of the individual preference profiles yields identical social preferences.

Neutrality (N)

A condition requiring all alternatives to be treated symmetrically, such that permuting alternative labels across individual preferences produces the identical permutation in social preferences.

Monotonicity / Positive Responsiveness (M)

A condition requiring that if an alternative $x$ improves or maintains its position relative to $y$ in all individual preferences and strictly rises for at least one individual, an existing social indifference between $x$ and $y$ is broken strictly in favor of $x$ ($x I y \rightarrow x P' y$).

Non-Negative Responsiveness (M1 or R)

A monotonicity condition requiring that if alternative $x$ rises or stays the same relative to $y$ in everyone's preference orderings and rises strictly for at least one individual, the social standing of $x$ relative to $y$ cannot worsen ($x P y \rightarrow x P' y$ and $x I y \rightarrow x R' y$).

Knife-Edge Indifference (M2 or K)

A condition stating that if alternative $x$ improves relative to $y$ in at least one individual's preference while never falling for any individual, an existing social indifference must become a strict social preference: $x I y \rightarrow x P' y$.

Full Transitivity (T or RR)

The property that for all alternatives $x, y, z \in X$, if society weakly prefers $x$ to $y$ and $y$ to $z$, then society must weakly prefer $x$ to $z$: $\forall x, y, z \in X: (x R y \wedge y R z) \rightarrow x R z$.

Quasi-Transitivity (QT or PP)

A relaxation of transitivity requiring transitivity only over strict preferences: $\forall x, y, z \in X: (x P y \wedge y P z) \rightarrow x P z$.

Acyclicity (AC)

A rationality condition requiring that there are no strict preference cycles of any length, formalized as: $(x_1 P x_2 \wedge x_2 P x_3 \wedge \dots \wedge x_{k-1} P x_k) \rightarrow x_1 R x_k$.

Triple Acyclicity (TAC)

Acyclicity restricted to any subset of three alternatives: $\forall x, y, z \in X: (x P y \wedge y P z) \rightarrow x R z$.

Maximal Set

The set of elements $x \in M(S)$ in a subset $S$ such that no alternative in $S$ is strictly preferred to $x$ ($\nexists y \in S \text{ such that } y P x$).

Choice Set

The set of elements $x \in C(S)$ in a subset $S$ that are weakly preferred to all alternatives in $S$ ($\forall y \in S: x R y$).

Decisive Group

A group of individuals $V$ is decisive for alternative $x$ against $y$ if whenever every individual in $V$ strictly prefers $x$ to $y$, society strictly prefers $x$ to $y$: $\forall i \in V: x P_i y \rightarrow x P y$.

Semidecisive Group

A group of individuals $V$ is semidecisive for alternative $x$ against $y$ if whenever every individual in $V$ strictly prefers $x$ to $y$, society weakly prefers $x$ to $y$: $\forall i \in V: x P_i y \rightarrow x R y$.

Veto

The power of an individual who is semidecisive for all pairs of alternatives, allowing that individual to block any strict social preference against their strictly preferred alternative.

Dictator

An individual who is decisive for all pairs of alternatives (global strict-preference dictator).

Local Dictator

An individual who is decisive for a specific ordered pair of alternatives $x, y$.

Weak-Preference Local Dictator

An individual $j$ for a specific pair $x, y$ such that whenever $j$ weakly prefers $x$ to $y$, society weakly prefers $x$ to $y$ ($x R_j y \rightarrow x R y$).

Oligarchy

A unique group $G$ of individuals that is decisive for all pairs of alternatives, where every individual member of $G$ possesses a veto.

Collegium

The set of individuals who belong to every decisive group, which includes any individual possessing a veto.

Value Restriction (VR)

Sen's condition satisfied for alternative $x$ in triple $\{x, y, z\}$ iff every individual agrees that $x$ is never worst, never best, or never medium: $[\forall i: x P_i y \vee x P_i z] \vee [\forall i: y P_i x \vee z P_i x] \vee [\forall i: (x P_i y \wedge x P_i z) \vee (y P_i x \wedge z P_i x)]$.

Limited Agreement (LA)

Sen's condition satisfied for an ordered pair $\{x, y\}$ in a triple $\{x, y, z\}$ iff all individuals weakly agree that $x$ is at least as good as $y$: $[\forall i: x R_i y]$.

Extremal Restriction (ER)

Sen's condition satisfied for an ordered triple $\{x, y, z\}$ iff: $[\exists i: x P_i y \wedge y P_i z] \rightarrow [\forall j: z P_j x \rightarrow (z P_j y \wedge y P_j x)]$.

Arrow's Theorem

An impossibility theorem establishing that no Social Welfare Function (SWF) can simultaneously satisfy Universal Domain (U), Weak Pareto (P), Independence of Irrelevant Alternatives (I), and Non-Dictatorship (D).

May's Theorem

A characterization theorem showing that the Method of Majority Rule (MMR) is uniquely characterized by Universal Domain (U), Anonymity (A), Neutrality (N), and Monotonicity / Positive Responsiveness (M).

Gibbard's 1969 Theorem

A theorem proving that any Quasi-Transitive Social Decision Function (QTSDF) satisfying Universal Domain (U), Weak Pareto (P), and Independence of Irrelevant Alternatives (I) must result in an oligarchy.

Pareto Extension Rule (PER)

A social aggregation rule specifying that $x P y$ if and only if all individuals strictly prefer $x$ to $y$ ($\forall i \in N: x P_i y$), and $x I y$ otherwise; it constitutes a QTSDF satisfying UPID.

Mas-Colell & Sonnenschein Theorem (1972a)

A theorem proving that any Quasi-Transitive Social Decision Function (QTSDF) with at least 3 alternatives satisfying Universal Domain (U), Independence of Irrelevant Alternatives (I), Weak Pareto (P), and Monotonicity (M) implies the existence of a dictator.

Mas-Colell & Sonnenschein Theorem (1972b)

A theorem proving that any Social Decision Function (SDF) with at least 4 alternatives satisfying Universal Domain (U), Independence of Irrelevant Alternatives (I), Weak Pareto (P), and Monotonicity (M) guarantees the existence of a vetoer (even if only Triple Acyclicity is required).

Muller-Satterthwaite Theorem

A theorem showing that any single-winner Social Decision Function (SDF) satisfying Universal Domain (U), Weak Pareto (P), and Independence of Irrelevant Alternatives (I) must be dictatorial.

Brown's Theorem

A theorem stating that any Social Decision Function (SDF) satisfying Universal Domain (U), Weak Pareto (P), Independence of Irrelevant Alternatives (I), and Non-Dictatorship (D) must have a non-empty collegium, provided the number of alternatives is greater than or equal to the number of individuals.