Logic and Social Choice Theory. A Survey. ILLC, University of Amsterdam. November 16, staff.science.uva.

Logic and Social Choice Theory A Survey Eric Pacuit ILLC, University of Amsterdam staff.science.uva.nl/ epacuit epacuit@science.uva.nl November 16, 2007

Setting the Stage: Logic and Games Game Logics Logics for social interactive situations (ATL, DEL/ETL, STIT, etc.) When are two games the same? Epistemic program in game theory (Formally) Verifying that a social procedure is correct (Social Software) Develop ( well-behaved ) logical languages that can express game theoretic concepts, such as the Nash equilibrium

Setting the Stage: Logic and Games Game theory is full of deep puzzles, and there is often disagreement about proposed solutions to them. The puzzlement and disagreement are neither empirical nor mathematical but, rather, concern the meanings of fundamental concepts ( solution, rational, complete information ) and the soundness of certain arguments...logic appears to be an appropriate tool for game theory both because these conceptual obscurities involve notions such as reasoning, knowledge and counter-factuality which are part of the stock-in-trade of logic, and because it is a prime function of logic to establish the validity or invalidity of disputed arguments. M.O.L. Bacharach. Logic and the Epistemic Foundations of Game Theory..

Setting the Stage: Logic and Games M. Pauly and W. van der Hoek. Modal Logic form Games and Information. Handbook of Modal Logic (2006). G. Bonanno. Modal logic and game theory: Two alternative approaches. Risk Decision and Policy 7 (2002). J. van Benthem. Extensive games as process models. Journal of Logic, Language and Information 11 (2002). J. Halpern. A computer scientist looks at game theory. Games and Economic Behavior 45:1 (2003). R. Parikh. Social Software. Synthese 132: 3 (2002).

What about logic and social choice theory?

Plan Quick Review Some Classic Results of Social Choice Theory Formalizing Social Choice Theory A Logician Looks at Aggregation Results A General Framework Concluding Remarks Goal: Illustrate where logic naturally shows up in social choice theory and point to some relevant literature.

Aggregating Preferences: Some Notation Suppose that there are n individuals and two alternatives x and y Let xpiy denote that i prefers x to y and xiiy denote that i is indifferent between x and y

Aggregating Preferences: Some Notation For each i there is a variable Di { 1, 0, 1} where 1 if ypix D = 0 if xiiy 1 if xpiy f : { 1, 0, 1} n { 1, 0, 1} is the group decision function

Simple Majority Procedure For k { 1, 0, 1}, let Nk(D1,..., Dn) = {i Di = k} Let D = D1,..., Dn f is a simple majority decision method iff 1 if N1( D) N 1( D) < 0 f( D) = 0 if N1( D) N 1( D) = 0 1 if N1( D) N 1( D) > 0

Properties of group decision funcitons A group decision function f is Decisive if it is a total function Symmetric if f(d1,..., Dn) = f(d j(1),..., D j(n) ) for all permutations j. I.e., f is symmetric in all of its arguments. Neutral if f( D1,..., Dn) = f(d1,..., Dn) Positively Responsive if D = f(d1,..., Dn) = 0 or 1, and D i = D i for all i i0, and D > D i0 i0, then D = f(d 1,..., D n) = 1

May s Theorem Theorem (May, 1952) A group decision function is the method of simple majority decision if and only if it is always decisive, symmetric, neutral and positively responsive

Generalizing May s Theorem In May s Theorem, the agents are making a single binary choice between two alternatives. What about more general situations?

Generalizing May s Theorem In May s Theorem, the agents are making a single binary choice between two alternatives. What about more general situations? Agents choose between between more than two alternatives. There are multiple interconnected propositions on which simultaneous decisions are to be made.

Generalizing May s Theorem In May s Theorem, the agents are making a single binary choice between two alternatives. What about more general situations? Agents choose between between more than two alternatives. (Condorcet, Arrow) There are multiple interconnected propositions on which simultaneous decisions are to be made. (Judgement Aggregation)

Arrow s Theorem: Some Notation Let R be the set of all reflexive, transitive and connected relations on a set of candidates X. A social welfare function F is a function F : R n R Suppose that R = F (R1,..., Rn)

Arrow s Theorem: Conditions Universal Domain: F is a total function Weak Pareto Principle: For any two candidates x, y if xriy for each agent i then xf ( R)y Independence of Irrelevant Alternatives: Suppose that R and R are two preference profiles and x and y are two candidates such that for all individuals i, if xriy iff xr i y then xf ( R)y iff xf ( R )y. Dictatorship: i is a dictator if for all profiles R R n, if xriy then xf ( R)y.

Arrow s Theorem Theorem (Arrow 1951/1963) A social welfare function satisfies Universal Domain, Weak Pareto Principle, Independence of Irrelevant Alternatives iff it is a Dictatorship.

Judgement Aggregation (Kornhauser and Sager 1993) P : a valid contract was in place Q: the defendant s behaviour was such as to breach a contract of that kind R: the court is required to find the defendant liable. P Q (P Q) R R 1 yes yes yes yes 2 yes no yes no 3 no yes yes no

Should we accept R? P Q (P Q) R R 1 yes yes yes yes 2 yes no yes no 3 no yes yes no

Should we accept R? No, a simple majority votes no. P Q (P Q) R R 1 yes yes yes yes 2 yes no yes no 3 no yes yes no

Should we accept R? Yes, a majority votes yes for P and Q and (P Q) R is a legal doctrine. P Q (P Q) R R 1 yes yes yes yes 2 yes no yes no 3 no yes yes no

List and Pettit Impossibility Result Suppose there are n agetns and let L be a propositional language. Personal judgement sets: a consistent, complete and deductively closed set of formulas a maximally consistent set. A collective judgement aggregation function: Let M = {Γ Γ is a maximally consistent set} then a collective aggregation function is defined as follows: F : M n M

Some Conditions Universal Domain F is a total function Anonymity For all Γ M n, F (Γ1,..., Γn) = F (Γ π(1),..., Γ π(n) ) for all permutations π Systematicity There exists a function f : {0, 1} n {0, 1} such that for any Γ M n, F (Γ1,..., Γn) = {ϕ X f(δ1(ϕ),..., δn(ϕ)) = 1}, where, for each agent i and each ϕ X, δi(ϕ) = 1 if ϕ Γi and δi(ϕ) = 0 if ϕ Γi

Theorem (List and Pettit, 2001) There exists no judgement aggregation function generating complete, consistent and deductively closed collective sets of judgements which satisfies Universal Domain, Anonymity and Systematicity. Theorem (Pauly and van Hees, 2003) Given that the agenda is atomically closed, an aggregation function is responsive and independent of irrelevant alternatives iff it is a dictatorship.

Brief Survey of the Literature See personal.lse.ac.uk/list/doctrinalparadox.htm for a detailed overview of the current state of affairs. Some highlights: Other impossibility results: Pauly and van Hees (2003), van Hees (2004), Gärdenfors (2004), and others List and Pettit (2005) compare their impossibility result with Arrow s Theorem

Plan Quick review of some classic results of Social Choice Theory Formalizing Social Choice Theory A Logician Looks at Aggregation Results A General Framework Concluding Remarks Goal: Illustrate where logic naturally shows up in social choice theory and point to some relevant literature.

Judgement Aggregation Logic T. Agotnes, W. van der Hoek, M. Wooldridge. Reasoning about Judgement and Preference Aggregation. AAMAS, 2007. Some Notation: N = {1,..., n} a set of agents A is the agenda (set of formulas of some logic L on the table satisfying certain fullness conditions ) Let J(A, L) is the set of judgements (eg. maximally consistent subsets of A) γ J(A, L) n is a judgement profile with γi agent i s judgement set

Judgement Aggregation Logic: Semantics Tables F, γ, p

Example: Judgement Aggregation Logic: Semantics Tables F, γ, p P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)}

Judgement Aggregation Logic: Semantics Tables F, γ, p Example: P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)} F is an aggregations function F : J(A, L) n J(A, L)

Judgement Aggregation Logic: Semantics Tables F, γ, p Example: P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)} γ J(A, L) n (assuming consistency and completeness)

Judgement Aggregation Logic: Language Atomic Formulas: At = {i, σ, hp p A, i N} Formulas: ϕ ::= α ϕ ϕ ϕ ϕ ϕ

Judgement Aggregation Logic: Language Atomic Formulas: At = {i, σ, hp p A, i N} Formulas: ϕ ::= α ϕ ϕ ϕ ϕ ϕ F, γ, p = ϕ

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p F, γ, p = i iff p γi F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p The current proposition on the table is q F, γ, p = i iff p γi F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p The current proposition on the table is q F, γ, p = i iff p γi Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p The current proposition on the table is q F, γ, p = i iff p γi Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p The current proposition on the table is q F, γ, p = i iff p γi Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ Quantification over the set of judgement profiles F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual

Judgement Aggregation Logic: Truth F, γ, p = hq iff q = p The current proposition on the table is q F, γ, p = i iff p γi Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ Quantification over the set of judgement profiles F, γ, p = ϕ iff p A, F, γ, p = ϕ Quantification over the agenda Boolean connectives as usual

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)}

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)} Fmaj, γ, P = 1 2 3

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False Fmaj True True False A = {P, Q, P Q, P, Q, (P Q)} Fmaj, γ, P = σ

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 True True True Fmaj True True True A = {P, Q, P Q, P, Q, (P Q)} Fmaj, γ, P = ((1 2) (2 3) (1 3))

Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True True True Individual 3 True True True Fmaj True True True A = {P, Q, P Q, P, Q, (P Q)} Fmaj, γ, P = ((1 2) (2 3) (1 3))

Judgement Aggregation Logic: Results

Judgement Aggregation Logic: Results Sound and complete axiomatization

Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult

Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P )

What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. 2005.

What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. 2005. Given a semantic domain D and a target class T D

What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. 2005. Given a semantic domain D and a target class T D Fix a language L and a satisfaction relation = D L L be a set of axioms absolutely axiomatizes T iff for all M D, M T iff M = (i.e., defines T )

What do the (Im)possibility results say? May s Theorem: is the set of aggregation functions w.r.t. 2 candidates, T is majority rule, L is the language of set theory, is the properties of May s theorem, then absolutely axiomatizes T. Arrow s Theorem: is the set of aggregation functions w.r.t. 3 or more candidates, T is a dictatorship, L is the language of set theory, is the properties of May s theorem, then absolutely axiomatizes T.

A Minimal Language M. Pauly. Axiomatizing Collective Judgement Sets in a Minimal Logical Language. 2006. Let ΦI be the set of individual formulas (standard propositional language) VI the set of individual valuations ΦC the set of collective formulas: α ϕ ψ ϕ α: The group collectively accepts α. VC the set of collective valuations: v : ΦC {0, 1}

A Minimal Language Let CON n = {v VC v( α) = 1 iff i n, vi(α) = 1} E. ϕ ψ provided ϕ ψ is a tautology M. (ϕ ψ) ( ϕ ψ) C. ( ϕ ψ) ( ϕ ψ) N. D. Theorem [Pauly, 2005] VC(KD) = CON n, provided n 2 Φ 0. (D = VC, T = CON n = EMCND, then absolutely axiomatizes T.)

A Minimal Language Let MAJ n = {v VC v([>]α) = 1 iff {i vi(α) = 1} > n 2 } STEM contains all instances of the following schemes S. [>]ϕ [>] ϕ T. ([ ]ϕ1 [ ]ϕk [ ]ψ1 [ ]ψk) 1 i k ([=]ϕ i [=]ψi) where v VI : {i v ( ϕi) = 1} = {i v(ψi) = 1} E. [>]ϕ [>]ψ provided ϕ ψ is a tautology M. [>](ϕ ψ) ([>]ϕ [>]ψ) Theorem [Pauly, 2005] VC(STEM) = MAJ. (D = VC, T = MAJ n = ST EM, then absolutely axiomatizes T.)

Related Work T. Daniëls. Social Choice and Logic via Simple Games. ILLC, Masters Thesis, 2007. General discussion of preference aggregation and judgement aggregations Interprets formulas directly on aggregations rules F, γ = α iff α F (γ) Studies the relationship with neighborhood models for modal logic Goldblatt-Thomason Result about expressivity of the modal language

A General Framework T. Daniëls, EP. A General Framework for Aggregation Results. forthcoming, 2007. Let N be a set of agents and Z a lattice (ordered set). A choice function is a map π : N Z. If π(i) = z then we will say that agent i accepts z.

A General Framework Let Π := {π π : N Z} be the set of all choice functions. An aggregation function is a map F : Π Z. Given π, F (π) is the socially accepted element of Z.

A General Framework: Consistency Assumption For all i N, π(i) < and for all π Π, F (π) <

A General Framework: Judgement Aggregation Let Φ be an agenda, a set of sentences of some logical language. Let Z = {z z Φ and z is consistent } z z = if z z is inconsistent z z otherwise z z iff z z. is the inconsistent set. =

A General Framework: Preference Aggregation Let X be a set of candidates Let Z = {z z X X z satisfies appropriate order axioms } z z = if z z is inconsistent with the order axioms z z otherwise z z iff z z. is the universal relation. =

Relevant Sets of Agents Fix π Π and z Z. Acceptance set of z under π: [z ] π = {i N z π(i)} Blocking set of z under π: [z\] π = {i N z π(i) = }. Set with no opinion about z under π: N ([z ] π [z\] π ).

Decisive Sets Let F be an aggregation function and suppose z Z, M N. M is decisive for F with respect to z iff for all π Π, Whenever z π(i) for all i M and z π(j) = for all j N M, then z F (π).

Families of Decisive Sets Fix F : Π n Z. Define ΩF : Z 2 2N where Ω(z) = {X X N is decisive for z}

Families of Decisive Sets Fix F : Π n Z. Define ΩF : Z 2 2N where Ω(z) = {X X N is decisive for z} What we want: ΩF = {X X N is decisive for F }

Families of Decisive Sets Fix F : Π n Z. Define ΩF : Z 2 2N where Ω(z) = {X X N is decisive for z} What we want: ΩF = {X X N is decisive for F } General Project: Properties of the Lattice and properties of F correspond to proeprties of ΩF

Concluding Remarks Logic naturally appears in Social Choice Theory: Judgement Aggregation Logic for Social Choice Preference Representation (cf. the work of U. Endriss, J. Lang, F. Rossi, etc.)

Concluding Remarks Logic naturally appears in Social Choice Theory: Judgement Aggregation Logic for Social Choice Preference Representation (cf. the work of U. Endriss, J. Lang, F. Rossi, etc.) Logic provides another dimension to compare various axiomatization results from social choice (expressivity of the underlying logical language). Meta-Social Choice Theory : decidability results, non-axiomatizability, etc.

Thank you.