Sumiyo Nishiguchi Department of Liberal Arts, Faculty of Science Division I, Tokyo University of Science 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601 Japan Email: nishiguchi@rs.tus.ac.jp
Abstract
Disambiguation of pronoun reference has been an important issue for both theoret- ical and computational linguists.  While linguistic theories on binding conditions eliminate impossible readings to a cer- tain extent, many inter-sentential anaphora remain ambiguous. Nishiguchi  (2011, 2012a,b, 2014, 2016a,b) consider pronoun resolution as a social choice among dis- course participants which obeys Arrow's Impossibility Theorem (Arrow 1963). This paper further discusses discourse up- date of Social Welfare Function which provides updated variable assignment. In (1), she has multiple candidates for its antecedent— Emma, Lisa and Lisa's mom. Prox- imity and saliency of antecedents have been con- sidered to be key factors to decide (Leass 1991). In (1), the most proximate antecedent her (Lisa)'s mom is identified to be the antecedent for she. (1) Frances: …Not while Emma’s not here.  You know Emma Billy: Mm. Frances: she's, she was walking with Lisa and I weren't there and her Mum sh– jus– , like she muc– , she mucks about a lot and she told Leigh that if he don't serve her he's gonna die, she’s gonna punch him right! However, proximity does not always resolve referential ambiguity of pronouns. Him in (2a) un- ambiguously means someone other than the clos- est John —some discourse-salient entity.  In (2b), the pronoun is ambiguous. (2)  a.  John ε likes him{*i/j≠i}· b.  John i said he{i/j} likes himself{i/j}. In linguistic binding theory (Chomsky 1981, Reinhart 1983), antecedents are called binders, which bind bindees that are anaphoric pronouns, e.g., him or himself.  Condition B is that pro- nouns must be free in their local domain, mean- ing that they are not bound by the antecedent by means of coindexing and c-commanding relation. C-command is roughly equivalent to precedence, with some restrictions. However, (3) is ambiguous in four ways and can have either one of the following interpretations: i) John broke John’s leg, ii) John broke Bill’s leg, iii) Bill broke Bill's leg, or iv) Bill broke John's leg. He and his can be bound by either John or Bill. The binding theories have no way of disambiguat- ing these pronouns since there is no way of know- ing speaker intention. Proximity does not predict the different readings in (3) either. (3) Anna: Bill 7 is a good goalkeeper. Kim:  John չ  said he i / j  broke his i / j  leg  re- cently.
Social Choice Theory
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Although Social Choice Theory (Arrow 1963, Moulin 1988, Taylor 2005, Gaertner 2009) has only been briefly mentioned in van Rooij (2011) in relation with interadjective comparison, Arrow's Impossibility Theorem is obeyed in a social choice of pronominal reference.  Typically, social choice theory explains collective decision making in case of voting and has solved the problems with ma- jority decision. Preferences are ordering between alternatives and should satisfy the following ax- ioms. When R stands for a knowledge of all pairs and x, y and z for alternatives, Axiom 1.  For all x and y, either xRy or yRx. Axiom 2.  For all x, y, and z, xRy and yRz imply xRz. Axiom 1 states that the relation R is connected—every candidate is related to each other.  Relations that satisfy Axiom 2 are transi- tive. In (4), N, a finite set of individuals or voters, consists of five individuals and χ , a nonempty set of alternatives or candidates, has three members. Let L(χ) denote the set of all linear orders on A profile R is a vector of linear orders, or X・ preferences.  R is a vector of preferences of an individual i. N x >y denotes the set of individuals that prefer the candidate x to y.  Supposing R the profile given in this model, N o >c is a set of people who prefers Obama to Clinton, that are, Anna, Heather and George (cf. Endriss 2016).
A social welfare function (SWF) F is a function which takes individual's preferences and returns collective preference.  Arrow demonstrated that any SWF for three or more alternatives the follow- ing conditions must be a dictatorship. Condition 2 states that the relative ranking of two candidates remains unchanged regardless of other candidates. Theorem 1 (General Possibility Theorem (Impos- sibility Theorem)). If there are at least three alter- natives which the members of the society are free to order in any way, then every social welfare func- tion satisfying Conditions 1 and 2 and yielding a social ordering satisfying Axioms 1 and 2 must be either imposed or dictatorial. Condition 1 (Pareto condition).  A SWF F satisfies the Pareto condition if, whenever all individuals rank x above y, then so does society: N x > y = N implies xF(R)y Condition 2 (Independence of irrelevant alterna- tives (IIA)). A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings: Nx>y = Nx>y implies xF(R)y ⇔ xF(R')y Condition 3 (Nondictatorship). There is no indi- vidual i such that for every element in the domain of rule f, ∀x, y ∈ X: xP i y → xPy (Sen 1979)
Application to Pronoun Resolution
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SWF for pronoun resolution satisfies Arrow's Im- possibility Theorem, or General Possibility The- orem, by satisfying Axioms 1, 2, Pareto Condi- tion and IIA but demonstrating dictatorship. Pro- noun resolution is compared with voting by multi- ple voters, discourse participants.  The candidates or choices would be different interpretation of the sentence.  In (5), the referent of he is ambigu- ous.  Chris meant he to be Bob , while Naomi in- terpreted him to be John. As the disagreement on pronominal reference is consolidated in the dis- course, pronoun resolution is certainly a social choice and Social Choice Function (SCF) decides the antecedent. (5) Chris: John said he broke his leg. Naomi: Did he? John looked fine when I saw him this morning. Chris: It is Bob who broke his leg. Naomi:  I thought you were talking about John. When individuals I = {c, n}, candidates χ = {j, b}, Chris and Naomi's ordering is jRcb ∧ bRnj, denote the set of linear orders on χ by L(χ). Pref- erences (or ballots) are taken to be elements of L(χ).  A profile R ∈ L(χ) 1 is a vector of prefer- ences.  SCF or voting rule is a function F : L(χ) 1 → 2 × \ 0 mapping a given profile to a nonempty set of winners; e.g., a singleton set {b} for (5). SWF is a function F : L(χ) 1 → L(χ) mapping any given profile to a (single) collective preference or- der.  Although the preferences between the candi- dates vary between the individuals, SWF returns a single preference order and ambiguities are re- solved during the conversation. There are three possible antecedents for she in (1)—Emma (e), Lisa (l) and Lisa's mother (m). Let us say that Billy (b) prefers e to l, and also l to m to be the antecedent.  On the other hand, the speaker Francis (f) prefers m to l, and l to e ac- cording to the proximity.  All three candidates are ordered in accordance with Axiom 1, i.e., eR b l ∧ lR b m and mR f l ∧ lR f e.  Transitivity also holds for pronoun antecedent preferences. Each of them implies eR b lR b m and mR f lR f e. SWF for pronoun resolution also meets Pareto condition.  When the interpretation of the addressees agrees with the one of the speaker, the decision of the society fol- lows. It is unlikely that pronouns refer to someone else other than speaker's intention and hearer's in- terpretation. A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings.  Let us say that the preference relations are denoted by R and R'. Assume that IIA does not hold and consider a dialogue in (7) where the relative rankings be- tween Bob and John is affected by irrelevant can- didate Victor's ranking. The social decision differs from the relative ranking between John and Bob of speaker and hearer, which does not happen, in (8). (6) Chris: Bob is a good skier. But John said he broke his leg. Naomi: Did he? Poor Bob! (7) Chris: Victor is a good skier and so is Bob. But John said he broke his leg. Naomi: Did he? Poor Bob! (8) bR’cvR’cj ∧ bR’njR’nv̸jF(R)b Then, N b > j = N b > j implies bF(R)j ⇔ bF(R')j The speaker's decision on pronominal reference dictates the social preference.  Even when there is disagreement or misunderstanding, the speaker corrects unifies interpretation in general, as in (9). Pronoun resolution is dominated, or dictated, by the speaker's meaning. (9) Chris: Bob is a good skier. But John said he broke his leg. Naomi: Did he? Poor Bob! Chris: No. I mean John broke his leg. (10) xPcy → xPy Proof. Suppose: xP c y → ～xPy, that is, xP c y → yRx, where R is weak preference.  However, the dialogue normally proceeds jP c b → jPb as in (10). Contradiction. ● Lemma 1. The social welfare function for pro- noun resolution is IIA and Pareto but is dictato- rial.
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Dynamic Update of SCF
In linguistic literature, a variable assignment func- tion g has been assumed to assign the referent to indices indexed to pronouns. For example, g may assign John to the variable x: g(x) = John. Now, g can be considered to be SCF which selects a ref- erent for a pronoun socially.  Let us define g and the space as in (11). The assignment function g is updated throughout the discourse as in (12). b. Information state σ consists of Social Welfare Function F, Social Choice Func- tion g for variable assignment, individ- ual's preferences R, individuals in the dis- course X, a set of indices such as i, a set of discourse participants V, and relation be- tween decisions B. Σ = < F, G, R, X, I, V, B> (12) σ1 There were ooh's and aah's when he x 1 finished, and some unbridled laugh- ter.  Aileen a  was looking dubiously at her y 1 husband h but he x 2 was in no mood to disap- prove. σ2 He x 3 winked at the Duke d and called across to him x4 , 'What a grand thing, your Honour, to have a wedding without a min- ister!’ The Duke d did his x 5 stately bow at that and then Donald m was calling for an- other song. σ3 Some of the veterans were on the point of giving tongue but young Donald McCulloch m  was on his x 6  feet and moving into the middle of the ring, he x 7 was full of himself x 8 , sparkling with mischief but with an undertow of ardour. σ4 ‘Duncan Ban MacIntyre ь wrote a song for his x 9 wife Mary r . σ5 I do not know if Alex 1 used it to court his10 Maryr – he x 11 must have used some- thing —‘The joke was unconscious but crow- ing laughter came from the young men n be- side the whisky jar.  (BNC A0N1311-1315, King Cameron) (13) a. g1= {<y1,a>,<x2,h>} I = {a, r} (a: author, r: reader) S = {a, h} b.  g2 = { <x3, h>, <x4, d>} S = {a, h, d, m} c.  g3 = { <x6, m>, <x7, m>, <x8, m>} S = {a, h, d, v, m} d. g4 = { <x9, b>} S = {a, h, d, v, m, b, r}
G is regarded as SCF. Also, the set of best ele- ments S’ can be called its choice set of the whole set of alternaties, and is denoted g(S', R) (cf. Sen 1979) R is a sequence of individual's preferences where R x is a preference ordering of x. (14)  g1(S, R) = {a, h}
g3(S, R) = {m} g4(S, R) = {b} g5(S, R) = {l} As the author's dynamic preferences change in the discourse as in (15a), g is updated throughout the discourse by means of a relation B. (15) a. σ2: hR a d for he x 3 ∧ dR a h for he x 4 ∧ dR a hI a m for he x 5 (aI x b: x is indifferent between a and b, ∧:  dynamic conjunc- tion) b.  Social Decision: hRd ∧ dRh ∧ dRhIm c. B(gn,gn+1) (16) Dynamic Social Welfare Function: F n BF n+1 BF n+2,…
Comparison with Other Studies
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Dynamic Predicate Logic (Groenendijk and Stokhof 1991) consider update semantics where two states differ with respect to variable assign- ment.  When h[x]g, the state g is updated with respect to the assignment to x.  The current pa- per consider an abstract function B between two Parkes and Procaccia (2013) model dy- SCFs. namic decision making under constantly changing preferences using Markov decision processes, in which the states coincide with preference profiles and a policy corresponds to a social choice func- tion.
Detection of Speaker Intention
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In order to implement Dynamic Social Choice for pronoun disambiguation, speaker's intention needs to be detected from the text.  The phrases such as “I mean” are used to resolve ambiguity of pronominal reference in the discourse. as in (17). (17)  ‘…And Sarah Morgan likes the idea of An- gela marrying someone in the government.' McLeish considered this cold and rational as- sessment.  ‘When did you last see her? Miss Angela Morgan, I mean.’ (BNC AB9) Out of 18 instances of “I mean PNP” (PNP stands for proper name" ) found with the query “I mean N” in BNC, 7 instances had a preceding pronoun, the caraphor.