6 CHOOSING ON INFLUENCEwell-defined preferences for all those alternatives that an individual’s preferencesare incomplete, the choice outcome will not be unique.Now assume that we observe Ian and Jane’s choice behaviour as summarizedin the table above, without any information regarding their underlyingpreferences. Could we infer that they are choosing as if they are getting influencedby each other whenever they cannot decide on their own? Moreover,can we actually recover the hidden preferences and identify the influence theycreate on each other consistent with this choice behaviour? We answer bothof these questions affirmatively relying on our characterization theorem. In afirst result, we show that three behavioural properties of the choice data oftwoindividualsarenecessaryandsufficienttolinktheseindividuals’behaviouraccording to Choice on Mutual Influence. The first property, Expansion is anindividual rationality property, known to be satisfied by two stage maximizationmodels. Unlike Expansion, the remaining two properties are novel to ourmodel since they are defined over a pair of choice behaviours. Nullipotencystates that all the influence that could be created among these individuals isalready inherent in their behaviour; no further interaction could cause furtherrefinement of their final choices. The key property of our model is the lastproperty, Consistency of Influence, which ensures that any violation of consistency,in a standard sense, observed in an individual’s choice data indicatesthe influence acquired. In a way, Consistency of Influence allows us to link individualsto each other by tracing the inconsistencies in their choice behaviour.For two individuals to be interacting according to Choice on Mutual Influence,they have to account for each other’s inconsistencies.Given the choice behaviour of Ian and Jane, the representation theoremensures that the underlying preferences are uniquely identified, and hence theinfluence they acquire from each other. Nonetheless, in general the identificationdoes not always sustain uniqueness. In other words, in certain situations,

CHOOSING ON INFLUENCE 7there are more than one pair of preferences that would yield the same choicedata according to Choice on Mutual Influence. In a subsection we explore theextent of this identification problem and find out a regularity condition on thepreferences that would guarantee uniqueness.Although Choice on Mutual Influence is based on only two individuals, thecharacterization exercise gives us clues about how to uncover the links betweenindividuals in more general settings. By exploiting these clues, it is immediateto extend the analysis to multi-individual settings, where interaction structurestake more complicated forms. The second section presents two differentextensions for this purpose. In the first one, we focus on a group of interactingindividuals that use the unanimity rule in the second stage of decision-makingto resolve indecisiveness. The second extension, on the other hand, considersindividuals that are influenced by particular individuals on specific problems.To the best of our knowledge, Choosing on Influence is the first paper todevelop a choice-theoretic approach to interaction. The novelty of the paperlies in this approach: We study multiple choice behaviours together and aimto figure out the unobservable elements of interaction, such as the relativeidentities of the influencer and influencee or the problems for which influenceis acquired, out of the observed choice behaviour.As a two stage maximization process, Choice on Mutual Influence mechanismis clearly related to the other two stage maximization processes studiedin the boundedly rational choice literature. 6 The baseline model of twostage maximization would be Rational Shortlist Methods (RSM) proposed byManzini and Mariotti (2007). RSM essentially refers to a single-valued choice6 In addition to the papers cited, see: Bajraj and Ulku (2015); Garca-Sanz and R. Alcantud(2015); Manzini and Mariotti (2012); Yildiz (2015). For a detailed account of two stagemechanisms, see Horan (2014).

8 CHOOSING ON INFLUENCEmechanism where a two-stage maximization process yields the chosen alternativeuniquely. Two rationality properties, Expansion and a weakening ofWARP, Weak WARP are shown to be necessary and sufficient for the choicedata to reveal the two binary relations of RSM. A natural and interesting subclassof RSM models, not only for our purposes but also in general, would beRSM with transitive binary relations. Au and Kawai (2011) show that an additionalaxiom that ensures acyclicity of the revealed preference relation doesalso ensure transitivity. Horan (2014) proposes a behavioural axiom that doesnot impose acyclicity directly but does guarantee the existence of transitiverationales. Our model is inherently different from the existing two stage maximizationprocedures: we consider multiple individuals. The second criteriathat the individual uses to choose is not simply another criteria in her mind,but another individual that is also equipped with a choice structure. Hencethe behavioural properties we search for are to reveal the mutual relationshipbetween these two individuals, and to identify the specific choice problemsthat influence is acquired.The outline of the paper is as follows: The first section is devoted to ChoiceonMutualInfluence. Wedescribethemodel,presentthecharacterizingaxiomsand the theorem. We also tackle the identification problem in this section.The second section extends the model to multi-individual settings. The thirdsection presents several other uses of our model. The final section concludes.All proofs are left to an appendix.1. Choice on Mutual Influence1.1. The Model. Let X be a nonempty finite set of alternatives and Ω X bethe set of all nonempty subsets of X. Let 1 and 2 denote two individuals.For any i ∈ {1,2}, we define the decision outcomes of i on Ω X as a choicecorrespondence C i : Ω X ⇒ X with ∅ ≠ C i (S) ⊆ S for every S ∈ Ω X .

12 CHOOSING ON INFLUENCETheorem 1. Let C i : Ω X ⇒ X for i ∈ {1,2}. C 1 and C 2 satisfy EXP, NULLand CoI if and only if (C 1 ,C 2 ) is a Choice on Mutual Influence mechanism.The key point of our characterization exercise lies in the following observation:i’s choice behaviour not only reveals information about i’s underlyingpreferences but also j’s underlying preferences. Any inconsistency in i’s behaviourcorresponds to a pair of alternatives that is uncompared according toi’s preferences but ranked by j’s, as assured by CoI. Any multi-valued choicebehaviour, on the other hand, corresponds to alternatives that are neitherranked by i’s, nor by j’s preferences, as assured by NULL. All properties areindependent as we show in the appendix.1.3. Identification. The characterization theorem ensures that given a particularpairofchoicebehaviour(C1 ,C 2 )satisfyingEXP,NULLandCoI,wecanrecover ‘a’ pair of revealed preferences (≻ 1 ,≻ 2 ) that would represent (C 1 ,C 2 )according to Choice on Mutual Influence. But not necessarily this pair is ‘the’pair of underlying preferences. In some situations one can actually find otherpairs of revealed preferences that would explain the same choice behaviour. 11But then the following questions arise immediately:- How accurately can we actually identify the underlying preferences (andhence the influence)?- Under which conditions the underlying preferences (and hence the influence)can be uniquely identified?To answer the first question we investigate the part of the preferences thatis uniquely identified; i.e., the set of binary pairs possessed by all preferencepairs explaining the same choice data. We show that we can indeed recovera major part of underlying preferences. As an answer to the second question,11 We say that (≻ 1 ,≻ 2 ) explains (rationalizes) (C 1 ,C 2 ) if C 1 (S) = Max(Max(S,≻ 1 ),≻ 2 )and C 2 (S) = Max(Max(S,≻ 2 ),≻ 1 ).

CHOOSING ON INFLUENCE 13we come up with a regularity condition on the preferences that would allow torestore uniqueness.According to standard revealed preference argument, choices from binarymenus reveal the underlying preferences. Here we cannot apply the standardargument since choices from binary menus may also reflect the influence acquired.On that account we first aim to distinguish the choices from binarymenus where an agent influences the other, from the ones that there is no interactionfor sure. The latter is indeed simple: for those binary menus wherethe individuals choose a different alternative uniquely we can be sure that thechoices reflect the underlying preferences. In order to detect the former, welook for choice inconsistencies in larger menus. First notice that choices frombinary menus define the following mutually exclusive sets of binary comparisons:Disagreements, influences acquired, influences formed, and agreements,where;• Disagreements of i from j: P i = {xy ∈ X ×X : x = C i (xy) ≠ C j (xy)}• Influence of i over j: Q i = {xy ∈ X ×X : x = C i (xy) = C j (xy) andthere exists S ∈ Ω X with x ∈ S such that C j (S) ≠ C j (S \y)}• Agreements: R = {xy ∈ X × X : x = C i (xy) = C j (xy) and C i (S) =C i (S \y) and C j (S) = C j (S \y) for all S ∈ Ω X with x ∈ S}.In the proof of Theorem 1, we show that (≻ 1 ,≻ 2 ) such that ≻ i = (P i ∪Q i ∪R) = {xy ∈ X × X : C i (S) = C i (S \ y) for all S ∈ Ω X with x ∈ S} fori ∈ {1,2} explains a given pair of choice behaviours (C 1 ,C 2 ). Indeed this pairof (≻ 1 ,≻ 2 ) is the largest pair that would explain (C 1 ,C 2 ). However, one canfind subsets of ≻ 1 and ≻ 2 that would explain the same choices. The severity ofthis overidentification problem is understood by investigating the intersectionof any pair of preferences explaining the same choice data.

14 CHOOSING ON INFLUENCEA first observation is that any pair of preferences explaining a given (C 1 ,C 2 )has to possess the binary pairs that two individuals disagree on, P 1 and P 2 .These refer to the pairs of alternatives about which the individuals have reversetastes. Hence, (P 1 ,P 2 ) will be common to any preference pair (≻ 1 ,≻ 2 )explaining a given (C 1 ,C 2 ).Moreover, we are able to detect the pairs of alternatives such that an influenceis acquired for sure, Q 1 and Q 2 . Consider a binary problem such thatboth individuals have chosen x over y. If one of the individuals shows inconsistentbehaviour in a larger problem, then the choice of x over y from the binaryproblem can only be the result of being influenced and any preference pairresulting in this behaviour will recognize that. Q i identifies the pairs x,y suchthat individual j has been influenced by i to choose x over y. Thus, (Q 1 ,Q 2 )will be common to any preference (≻ 1 ,≻ 2 ) for a given (C 1 ,C 2 ) as well.Finally, since preferences are defined to be transitive, the ordered pairs thatarenotnecessarilyinP i orQ i , butimpliedbytransitivityof≻ i willbecommonto any (≻ 1 ,≻ 2 ) for the given (C 1 ,C 2 ). Let us denote the transitive closure ofP i ∪Q i astr(P i ∪Q i ). 12 Thefollowingtheoremstatesthattheintersectionofallpairsofpreferencesexplainingagivenchoicedatais(tr(P 1 ∪Q 1 ),tr(P 2 ∪Q 2 )): 13Theorem 2. Let C i : Ω X ⇒ X for i ∈ {1,2} such that (C 1 ,C 2 ) is a Choiceon Mutual Influence mechanism. Then the intersection of all preference pairs(≻ 1 ,≻ 2 ) explaining (C 1 ,C 2 ) is (tr(P 1 ∪Q 1 ),tr(P 2 ∪Q 2 )).Theorem 2 ensures that we can recover a major part of underlying preferencesand hence the influence individuals form on each other. The overidentificationproblem relates to those ordered pairs in R, the pairs of alternatives12 Thetransitiveclosureofabinaryrelationisthesmallesttransitiverelationthatcontainsit.13 Notice that this means not only tr(P i ∪ Q i ) is a common part of all preference pairsexplaining a given choice data, but also it is the largest common part.

CHOOSING ON INFLUENCE 15that both individuals choose the same out of the binary problem, and none ofthem has shown any inconsistency in a larger menu that would indicate the influenceacquired. Hence the model does not help us to associate these orderedpairs uniquely to one of the individuals unless they are a part of the transitiveclosure of (P i ∪ Q i ). Consider the extreme case where two individuals showexactly the same choice behaviour all over. Then since P i = Q i = ∅ ≠ R,we cannot conclude whether two individuals have actually the same sincerepreferences or one of them has null preferences and is getting fully influencedby the other. On the other extreme, if we observe a pair of choice behaviourswith a null R, we can completely identify the underlying preferences and theinfluence. However an empty R is not a necessary condition for full identification.Theidentificationissuearisesduetothefactthatalthoughanindividual,say i, is actually indecisive between two alternatives, x and y, she does notshow any choice inconsistency regarding those, although she is influenced byj to choose, say x over y. But this only happens if x and y are too ‘similar’in terms of their relative comparison to the other alternatives for both of theagents: whatever is better than x for i is also not worse than y and whateveris worse than y for i is also not better than x at least for one of the agents.Moreover no alternative that is better than x is better than any alternativethat is worse than y, again, at least for one of the agents. 14 This observationtranslates as a regularity condition on the pair of preferences that allows torecover underlying preferences uniquely.Definition 2. (≻ 1 ,≻ 2 ) is regular if xy,yx /∈≻ i but xy ∈≻ j imply that at leastone of the following holds:14 Noticethatifweallowforindifferenceinadditiontoindecisiveness,(sincexisindifferentto y implies that whatever is better (worse) than x is also better (worse) than y) we willnever be able to uniquely identify the indifference part of underlying preferences apart fromthose alternatives that both i and j are indifferent on.

16 CHOOSING ON INFLUENCE(i) there exists a ∈ X with ax ∈≻ i but ay /∈ (≻ 1 ∪ ≻ 2 )(ii) there exists b ∈ X with yb ∈≻ i but xb /∈ (≻ 1 ∪ ≻ 2 )(iii) there exist a,b ∈ X with ax,yb ∈≻ i but ab /∈ (≻ 1 ∪ ≻ 2 ).With regularity assumption we sustain uniqueness in our characterizationresult.Theorem 3. Let C i : Ω X ⇒ X for i ∈ {1,2}. C 1 and C 2 satisfy EXP, NULLand CoI if and only if there exists a unique regular pair of asymmetric andtransitive preferences (≻ 1 ,≻ 2 ) that explain (C 1 ,C 2 ).2. Choice on Social InfluenceChoice on Mutual Influence is a simple decision making mechanism for interactingindividuals. Despite its simplicity, it is powerful enough to easilyextend to more complicated structures and explain more convoluted formsof social interactions. In this section, we present two different extensions tomulti-individual settings to demonstrate this.Through out this section, we consider a group N of n individuals, N ={1,2,...,n}, where ≻ i and C i denote the preference relation and choice behaviourof i ∈ N, respectively. Facing a decision problem, individuals refer totheir own preferences in a first stage, as before. If the first stage maximizationresults in a unique alternative, there is no room for social influence. Ifotherwise, they conduct a second stage where social influence is acquired.In the first model we present, Choice on Unanimous Influence, the secondstage decision making of an individual involves a consideration of the opinionsof all other individuals. If all others agree, then the individual behaves accordingly.In other words, the second stage rationale is the Pareto aggregation ofall other individuals’ preferences.

CHOOSING ON INFLUENCE 17The second model, Choice on Expert Influence, allows individuals to beinfluenced by different individuals over different choice problems. We considerindividuals with expertise on certain sets of alternatives. If i is an expert on aset of alternatives, any question regarding those alternatives is only asked toi by all other individuals of the society. 152.1. Choice on Unanimous Influence. Let i ∈ N. Facing a decision problemS, i first maximizes ≻ i over S. Then, in the second stage, i maximizes thePareto binary relation for the rest of the group, ≻ N\i , which is the intersectionof all ≻ j , for all j ∈ N \{i}:≻ N\i = {xy ∈ X ×X : xy ∈≻ j for all j ∈ N \{i}}.Definition 3. Wesay that (C 1 ,C 2 ,...,C n ) isaChoice on UnanimousInfluencemechanism, if there exists n asymmetric and transitive binary relations (≻ 1,≻ 2 ,...,≻ n ) such that:C i (S) = Max(Max(S,≻ i ),≻ N\i )for all S ∈ Ω X and for all i ∈ N.The behavioural characterization of Choice on Unanimous Influence closelyfollows that of Choice on Mutual Influence. EXP stays the same, but we modifyNULL and CoI properties to reflect the multi-individual requirements of15 Notice that both of the models we present only accommodate direct influence betweenmembers of a group, since influence is acquired from preferences. Another interesting settingwould be the one where individuals are influenced by another individual’s choices, insteadof preferences. In those environments, there is also room for indirect influence betweenindividuals. One generalization of our model that would be consistent with this approachwould be the following: Individual j influences i if C i (S) = C j (Max(S,≻ i )) for all S.Accordingtothismodel,ifirstreferstoherownpreferencesandwheneverindecisive,directlycopies j’s actions. The behaviour generated by this model is actually only a refinement ofthe behaviour generated by our model and does not satisfy well-known rationality propertiesincluding EXP.

18 CHOOSING ON INFLUENCEthe setting.Expansion (EXP). For any x ∈ X, S,T ∈ Ω X and i ∈ N, if x ∈ C i (S) ∩C i (T), then x ∈ C i (S ∪T).Nullipotency’ (NULL’). For any S ∈ Ω X and i ∈ N, ⋃ j∈N\i C j(C i (S)) =C i (S).Consistency of Influence’ (CoI’). For any x,y ∈ X and i ∈ N, ifx = C i (xy) and there exists S ∈ Ω X with x ∈ S such that C i (S) ≠ C i (S \y),then C j (T) = C j (T \y) for all T ∈ Ω X with x ∈ T and for all j ∈ N \{i}.These three properties are necessary and sufficient to detect a group ofindividualsthatarechoosingconsistentlywithChoiceonUnanimousInfluencemodel.Theorem 4. Let C i : Ω X ⇒ X for i ∈ N. C 1 ,C 2 ,...,C n satisfy EXP, NULL’and CoI’ if and only if (C 1 ,C 2 ,...,C n ) is a Choice on Unanimous Influencemechanism.2.2. Choice on Expert Influence. Let E i ⊂ X denote expertise of individuali, i.e., the set of alternatives for which all the members of the grouprefer to i’s opinion whenever they are indecisive. For simplicity purposes, weassume that areas of expertise are disjoint; E i ∩E j = ∅ for i ≠ j. Notice thatif an individual j does not possess any expertise, then E j = ∅. Moreover, notnecessarily all alternatives belong to an area of expertise, i.e., there may existsome alternatives for which no one is an expert.Choice on Expert Influence works as follows: Facing a decision problem S,i maximizes her own preference ≻ i . If this maximization does not give outa single alternative, then in a second stage, i seeks expert influence. If thereis j ∈ N with expertise on any alternative for which i is indecisive about, i

CHOOSING ON INFLUENCE 19acquires influence from j. Notice that i can acquire influence from more thanone individual in the same choice problem, but that will never be for the sameset of alternatives.Let (≻ j |E j ) denote the part of the preference of j over the alternatives inher expertise, i.e., (≻ j |E j ) = {xy ∈≻ j : x,y ∈ E j }. Then, the expert rationaleof this society will be: ≻ E = ⋃ j∈N (≻ j |E j ).Definition 4. We say that (C 1 ,C 2 ,...,C n ) is a Choice on Expert Influencemechanism, if there exists n asymmetric and transitive binary relations (≻ 1,≻ 2 ,...,≻ n ) and n areas of expertise E 1 ,E 2 ,...,E n such thatC i (S) = Max(Max(S,≻ i ),≻ E )for all S ∈ Ω X and for all i ∈ N.Now, consider (C 1 ,C 2 ,...,C n ). What kind of properties on the choice behavioursin this group indicate a Choice on Expert Influence mechanism?Moreover, can we recover the underlying preferences and areas of expertise?The characterizing properties of Choice on Expert Influence are also basedon the properties of Choice on Mutual Influence. Apart from individual rationalityproperties, NULL and CoI will be modified to this more sophisticatedform of interaction.Let I = {xy ∈ X × X : There exists i ∈ N and S ∈ Ω X with x ∈ S suchthat x = C i (xy) and C i (S) ≠ C i (S\y)}. Hence, I is the set of all binary pairsthat are associated with at least one inconsistency in the group. Followingthe logic of CoI, these are the binary pairs for which an influence has beenacquired. Consistency of Expert Influence brings some structure on this set:Consistency of Expert Influence (CoEI). For any xy ∈ I, there existj ∈ N such that C j (T) = C j (T \y) for all T ∈ Ω X with x ∈ T. Moreover, for

20 CHOOSING ON INFLUENCEanyz ∈ X withyz ∈ I,wehaveC j (T ′ ) = C j (T ′ \z)forallT ′ ∈ Ω X withy ∈ T ′ .CoEI makes sure that for any influence acquired, there is an expert j behindit. Moreover, if j is the expert on a pair of alternatives, than any related influencealso has to come from j; it is not possible to be influenced by individualj to choose, say, x over y, but then to be influenced by another individual k tochoose, say, y over z. Notice that this indeed the case since areas of expertiseare disjoint.CoEI accounts for the influence acquired. We now introduce the NULLcounterpart of this setting, a property that accounts for the multi-valued decisionoutcomes. Binding Influence states that if there is an expert on a certainset of issues, it is not possible to be indecisive, yet not to be influenced by thisexpert:Binding Influence (BI). For any binary chain x 1 x 2 ,x 2 x 3 ,...,x t−1 x t ∈ I, wehave x l x k ≠ C i (x l x k ) for any l,k ∈ {1,2,..,t}, for all i ∈ N.The existence of a binary chain in I indicates the existence of an expertfor the alternatives that constitute a part of this chain. BI ensures that theexpert is influential. In other words, if an individual is not able to choose asingle alternative from a binary set, then there cannot exist an expert with astrict preference over these alternatives.CoEI and BI are the properties that build the interaction links between individuals.Apart from these, we need two individual rationality properties: EXPand a weakening of WWARP for correspondences, another common property

CHOOSING ON INFLUENCE 21of two stage maximization procedures (Manzini and Mariotti, 2007). 16Weak WARP* (WWARP*). For any x,y ∈ X and i ∈ N, if x = C i (xy)and x ∈ C i (S) for some S ∈ Ω X with x,y ∈ S, then y /∈ C i (S).WWARP* prohibits the choice of an alternative y from a set where anotheralternative x, which is uniquely chosen over y from the binary problem, hasbeen chosen. 17Theorem 5. Let C i : Ω X ⇒ X for i ∈ N. C 1 ,C 2 ,...,C n satisfy EXP,WWARP*, CoEI and BI if and only if (C 1 ,C 2 ,...,C n ) is a Choice on ExpertInfluence mechanism.Hence, given a group of individuals whose choice behaviours satisfy theabove properties, we can detect the experts, the areas of expertise and theinfluence that individuals acquire from the experts simply by following choiceinconsistencies.3. Further CommentsThe previous section aimed to show that Choice on Mutual Influence caneasily be extended to more sophisticated interaction environments and therevealed preference approach can be used to detect the particular manner ofinteraction, underlying preferences and influence acquired in those environments.In this section, we return back to our baseline model and demonstrate16 WWARP for choice functions states that if x is chosen over y from a binary menu, andx is chosen from a set S, where y ∈ S, then y cannot be chosen from any set T with T ⊂ S.17 It is also possible to characterize Choice on Mutual Influence with EXP, WWARP*,CoI and a weakening of NULL to binary problems. Since weakening NULL comes with thecost of additional WWARP* axiom, we prefer the previous characterization.

22 CHOOSING ON INFLUENCEother uses of Choice on Mutual Influence. Let us list several settings whereour model and analysis become particularly helpful:(i) Influence vs. Homophily: Homophily refers to the tendency to createsocial ties with people that are similar to one’s self. 18 Both homophily andsocial influence result in behavioral resemblances between connected people.However it is not trivial to distinguish these two forces from one another: Dosocially connected individuals behave similarly because one of them influencesthe other or do they behave similarly because they indeed share similar tastesthat has led them to behave similarly and these similar tastes are the reasonthey have got socially connected to begin with? This phenomenon is known asthe identification problem of homophily and social influence and it is mainlychallenged by economists from an econometrical perspective. Many studiesdocument that both effects prevail simultaneously and distinguishing one fromthe other requires strong parametrical assumptions (Aral et al., 2009; La Fondand Neville, 2010; Manski, 1993; Noel and Nyhan, 2011; Shalizi and Thomas,2010). 19Our model provides a novel approach to this identification problem. Behaviorconsistent with our testable axioms confirms the existence of social tiesbetweenindividualsandrevealedpreferenceargumentallowstorecovertheunderlyingpreferences and the influence created. Then, identifying homophily18 For an overview of research on homophily in general see McPherson et al. (2001), incouples see Blackwell and Lichter (2004), on economic networks see Currarini et al. (2009).19 A major part of the works concentrates around adolescent behavior such as schoolachievement, use of drugs and recreational activities among high school children (Bramoulléet al., 2009; Calvo-Armengol et al., 2009; Manski, 1993) and innovation diffusion (Aral,2011; Iyengar et al., 2011). Recently online networks have drawn particular attention sincethey provide a powerful data source where the network structure is easily observable, hencethis structure itself may provide additonal information to solve this identification problem(Anagnostopoulos et al., 2008; Aral et al., 2009; Aral and Walker, 2012; Lewis et al., 2012).

CHOOSING ON INFLUENCE 23becomes an issue of comparing individual preferences. Let us illustrate thispoint by the example of Ian and Jane. Consider once again their choice behaviorfrom the menus including the alternatives A,B and C:Menus Ian JaneABC BC CAB A ABC BC BCAC C CWeobserveiandj choosingexactlythesameoutofthemenus{A,B},{B,C}and {A,C} but showing slightly different behaviours in the larger menu,{A,B,C}. How similar i and j are to each other? Do we have any evidenceof peer influence in these choices?As we have discussed in the introduction, choice behaviours of i and j doindeed satisfy Expansion, Nullipotency and Consistency of Influence axioms,indicating a Choice on Mutual Influence mechanism. Our identification strategyuniquely reveals the underlying preferences as C ≻ i A and A ≻ j B. 20Hence the influence i creates on j is to choose A over B, and the influence iacquires from j is in her choice of C over A both in the binary menu and thelarger menu. What about the level of homophily in this pair? Homophily isabout the similarity of individuals’ tastes, that can be assessed by comparingtheir preferences. The closer the preferences are to each other, the more similarare individuals’ tastes. ≻ i and ≻ j do not have any binary comparison incommon. However they both fail to compare B and C, suggesting rather a20 Since both of them fails to choose uniquely from {B,C}, neither ≻ i nor ≻ j ranks Band C. C i (ABC) ≠ C i (AC), although A = C i (AB) implies that ≻ i is indecisive over A andB, whereas A ≻ j B. Similarly, C j (ABC) ≠ C j (BC), although C = C j (AC) implies that≻ j is indecisive over A and C, whereas C ≻ i A.

24 CHOOSING ON INFLUENCElow level of similarity then no similarity. A proper measure of homophily canbe constructed by making use of a similarity function between preferences. 21(ii) Group Influence vs. Group Identification: Social identity refers to aperson’s sense of self, based on perceived memberships in social groups. 22 Animportant insight of social identity theory is about identification; the attainmentof the prescribed behaviour of the corresponding social group. Oncean individual identifies herself with a group, she begins adopting behavioursconsistent with the norms of that group. According to the interpersonalintergroupcontinuum argument (Tajfel and Turner, 1986), this behaviour willbe somewhere in between two extreme forms of social behaviour. At one extreme,there is the interpersonal behaviour, which refers to the interactionbetween a group of individuals and is purely determined by individual characteristics.The other extreme, intergroup behaviour, on the other hand, iscompletely determined by group membership characteristics. We claim thatour model accommodates interpersonal-intergroup continuum argument andallows to identify the degree to which group identification has taken place.21 One such measure would be Kendall’s correlation coefficient τ, which measures thecorrelation between two preferences based on the distance between them: τ(≻ 1 ,≻ 2 ) = 1−2 d(≻1,≻2)max d(≻ 1,≻ 2) , where d(≻ 1,≻ 2 ) is the generalization of Kemeny-Snell distance to incompletepreferences and max d(≻ 1 ,≻ 2 ) denotes the maximum possible distance between two binaryrelations defined over X (Bogart, 1973). This distance function simply counts the number ofbinary pairs on which these preferences do not agree, i.e., d(≻ i ,≻ j ) = |≻ i \ ≻ j |+|≻ j \ ≻ i |.Notice that τ is equal to 1 when two preferences are identical, and −1 when they arecompletely reverse. As noted in Bogart (1973), τ is the only linear function of d that takesa value 1 when two preferences are identical, and −1 when they are completely reverse.22 Social identity theory is developed by Tajfel and Turner (1986) and introduced to theanalysis of economic behaviour by Akerlof and Kranton (2000), leading to a wide empirical,theoretical and experimental literature. For an extensive review of identity economics, seeAklerlof and Kranton (2010).

CHOOSING ON INFLUENCE 25To see this, consider a group N of individuals and a binary relation thatrepresents the norms of the group (prescriptions in Akerlof-Kranton terminology):≻ N . Assume that ≻ N is common knowledge and individuals adopt itin their two-stage decision making procedures. If an individual uses ≻ N asthe first stage relation and her own preference, ≻ i as the second stage relation,only to refine her choices further, she is said to be identifying herselfcompletely with her group. Hence she will be appealing to the intergroupextreme of the continuum. If, instead, an individual uses ≻ i in the first stageand refers to ≻ N in the second stage, then she is only being influenced by thegroup norms whenever she cannot decide on her own. In this case we can talkabout group influence, but to a much lesser degree than the former case. If,on the other hand, an individual uses ≻ i in the first stage, and gets influencedby some other member(s) of the group at the second stage, then she will be atthe other extreme of the continuum, demonstrating interpersonal behaviour.Given the individual decision outcomes, as ≻ N is known, our identificationstrategy allows to distinguish these behaviours from each other and detect thedegree of group identification.4. Concluding RemarksInteraction is an essential element of daily life and economic decisions andyetmicrofoundationsofinteractionhavenotbeenexplored. Thisstudypresentsa first attempt in providing a choice-theoretic approach to interaction.We model interaction as a means to deal with an inherent individual necessity,inability to choose due to incomplete preferences. Our baseline model,Choice on Mutual Influence, despite its simplicity, is flexible enough to graspat more convoluted forms of interactions. The characterization of Choice onMutual Influence lays out two important properties of the choice behaviour

26 CHOOSING ON INFLUENCEconsistent with the model: First, choice inconsistencies in one’s behaviour correspondtotheinfluencesacquired,andhencearetracedbacktothepreferencesof the influential individual. Second, all the influence that can be acquired isalready inherent in the choice behaviour. These properties correspond to ourCoI and NULL properties respectively. A key observation, which we also explorein the second section of this paper, is that modifications of these twoproperties indeed aid to characterize various interaction environments.As the studies of the last decades have shown, expanding the realm of choicedata has provided new explanations to the behaviours that were once classifiedas ‘irrational’. The novelty of this paper lies in its focus on multiple choicebehaviours instead of one. We believe there is still a lot to explore once we gobeyond more than one individual’s behaviour.ReferencesAkerlof, G. A. and R. E. Kranton (2000). Economics and identity. The QuarterlyJournal of Economics 115(3), 715–753.Aklerlof, G. and R. Kranton (2010). Identity economics: How our identitiesshape our work, wages, and well-being.Anagnostopoulos, A., R. Kumar, and M. Mahdian (2008). Influence and correlationin social networks. In Proceedings of the 14th ACM SIGKDD internationalconference on knowledge discovery and data mining, KDD ’08,New York, NY, USA, pp. 7–15. ACM.Apesteguia, J. and M. A. Ballester (2013). Choice by sequential procedures.Games and Economic Behavior 77(1), 90–99.Apesteguia, J. and M. A. Ballester (2014). A measure of rationality andwelfare. Working Papers (Universitat Pompeu Fabra. Departamento deEconomía y Empresa) (1220), 1.

CHOOSING ON INFLUENCE 27Aral, S. (2011, 03-04). Commentary–Identifying social influence: A commenton opinion leadership and social contagion in new product diffusion. MarketingScience 30(2), 217–223.Aral, S., L. Muchnik, and A. Sundararajan (2009). Distinguishing influencebasedcontagion from homophily-driven diffusion in dynamic networks. ProcNatl Acad Sci U S A 106(51), 21544–9.Aral, S. and D. Walker (2012). Identifying influential and susceptible membersof social networks. Science 337(6092), 337–41.Au, P. H. and K. Kawai (2011). Sequentially rationalizable choice with transitiverationales. Games and Economic Behavior 73(2), 608–614.Bajraj, G. and L. Ulku (2015). Choosing two finalist and the winner. SocialChoice and Welfare.Bernheim, B. D. and A. Rangel (2007). Toward choice-theoretic foundationsfor behavioral welfare economics. The American Economic Review 97(2),pp. 464–470.Blackwell, D. L. and D. T. Lichter (2004). Homogamy among dating, cohabiting,and married couples. Sociological Quarterly 45(4), 719–737.Blume, L. E., W. A. Brock, S. N. Durlauf, and Y. M. Ioannides (2010, October).Identification of Social Interactions. Economics Series 260, Institutefor Advanced Studies.Bogart, K. (1973). Preference structures i: Distances between transitive preferencerelations. Journal of Mathematical Sociology 3, 49–67.Bramoullé, Y., H. Djebbari, and B. Fortin (2009, May). Identification of peereffects through social networks. Journal of Econometrics 150(1), 41–55.Calvo-Armengol, A., E.Patacchini, andY.Zenou(2009, October). Peereffectsand social networks in education. Review of Economic Studies 76(4), 1239–1267.

28 CHOOSING ON INFLUENCECherepanov, V., T. Feddersen, and A. Sandroni (2013). Rationalization. TheoreticalEconomics 8(3), 775–800.Currarini, S., M. O. Jackson, and P. Pin (2009, 07). An economic model offriendship: Homophily, minorities, and segregation. Econometrica 77(4),1003–1045.Evans, W. N., W. E. Oates, and R. M. Schwab (1992). Measuring peer groupeffects: A study of teenage behavior. Journal of Political Economy 100(5),966–91.Garca-Sanz, M. D. and J. C. R. Alcantud (2015). Sequential rationalizationof multivalued choice. Mathematical Social Sciences 74(0), 29 – 33.Glaeser, E. L., B. Sacerdote, and J. A. Scheinkman (1996, May). Crime andsocial interactions. The Quarterly Journal of Economics 111(2), 507–48.Horan, S. (2014). A simple model of two-stage choice. Technical report.Iyengar, R., C. V. den Bulte, and T. W. Valente (2011, 03-04). Rejoinder–Further reflections on studying social influence in new product diffusion.Marketing Science 30(2), 230–232.La Fond, T. and J. Neville (2010). Randomization tests for distinguishingsocial influence and homophily effects. In Proceedings of the 19th internationalconference on World Wide Web, WWW ’10, New York, NY, USA,pp. 601–610. ACM.Lewis, K., M. Gonzalez, and J. Kaufman (2012, January). Social selectionand peer influence in an online social network. Proceedings of the NationalAcademy of Sciences 109(1), 68–72.Manski, C. F. (1993, July). Identification of endogenous social effects: Thereflection problem. Review of Economic Studies 60(3), 531–42.Manzini, P. and M. Mariotti (2007, December). Sequentially rationalizablechoice. American Economic Review 97(5), 1824–1839.

CHOOSING ON INFLUENCE 29Manzini, P. and M. Mariotti (2012). Categorize then choose: Boundedlyrational choice and welfare. Journal of the European Economic Association10(5), 1141–1165.Manzini, P. and M. Mariotti (2014). Welfare economics and bounded rationality:the case for model-based approaches. Journal of Economic Methodology21(4), 343–360.Mas, A.andE.Moretti(2009, September). Peersatwork. American EconomicReview 99(1), 112–45.Masatlioglu, Y., D. Nakajima, and E. Y. Ozbay (2012). Revealed attention.American Economic Review 102(5), 2183–2205.Masatlioglu, Y. and E. A. Ok (2005). Rational choice with status quo bias.Journal of Economic Theory 121(1), 1 – 29.McPherson, M., L. Smith-Lovin, and J. M. Cook (2001). Birds of a feather:Homophily in social networks. Annual Review of Sociology 27(1), 415–444.Noel, H. and B. Nyhan (2011, July). The “unfriending” problem: The consequencesof homophily in friendship retention for causal estimates of socialinfluence. Social Networks 33(3), 211–218.Rubinstein, A. and Y. Salant (2007, July). (A,f) Choice with Frames. Levine’sBibliography 843644000000000029, UCLA Department of Economics.Rubinstein, A. and Y. Salant (2011). Eliciting welfare preferences from behaviouraldata sets. The Review of Economic Studies.Shalizi, C. R. and A. C. Thomas (2010). Homophily and contagion are genericallyconfounded in observational social network studies. Sociological Methodsand Research 40, 211–39.Tajfel, H. and J. C. Turner (1986). The social identity theory of intergroupbehavior. In Psychology of Intergroup Relations [ed. by S. Worchel; W. G.Austin], Chicago, USA, pp. 7–24. IL: Nelson-Hall.Yildiz, K. (2015). List rationalizable choice. Forthcoming.

30 CHOOSING ON INFLUENCEZimmerman, D. J. (2003, February). Peer effects in academic outcomes: Evidencefrom a natural experiment. The Review of Economics and Statistics85(1), 9–23.5. AppendixProof of Theorem 1. Necessity is fairly straightforward, thus omitted. Weprove the sufficiency part.For i ∈ {1,2}, define ≻ i ⊆ X ×X as follows:xy ∈≻ i iff C i (S) = C i (S \y) for all S with x,y ∈ S.Notice that ≻ i is asymmetric by definition since xy,yx ∈≻ i implies that ∅ =C i (xy). To see transitivity of ≻ i take any x,y,z ∈ X with xy,yz ∈≻ i . Takeany S ∈ Ω X with x,z ∈ S. If y ∈ S, then by yz ∈≻ i , C i (S) = C i (S \ z).Let y /∈ S and consider S ∪ y. By xy ∈≻ i , C i (S ∪ y) = C i (S). By yz ∈≻ i ,C i (S ∪y) = C i (S ∪y \z). By xy ∈≻ i , C i (S \z ∪y) = C i (S \z). But then,C i (S) = C i (S \z).Take any S ∈ Ω X . We now show that Max(Max(S,≻ i ),≻ j ) ⊆ C i (S).Take any x ∈ Max(Max(S,≻ i ),≻ j ). Obviously x ∈ Max(S,≻ i ). Take anyz ∈ Max(S,≻ i ). Assume for a contradiction that z = C i (xz). Since x ∈Max(S,≻ i ), there exists T ∈ Ω X with x,z ∈ T such that C i (T) ≠ C i (T \x).But then, CoI implies that zx ∈≻ j , contradicting with x ∈ Max(Max(S,≻ i),≻ j ), as expected. Hence x ∈ C i (xz) for all z ∈ Max(S,≻ i ). But then, EXPimplies that x ∈ C i (Max(S,≻ i )).For any y ∈ S\Max(S,≻ i ), there exists y ′ ∈ Max(S,≻ i ) such that y ′ y ∈≻ isince ≻ i is transitive. But then C i (Max(S,≻ i )) = C i (Max(S,≻ i )∪y). Iterativeapplication of the same argument yields that C i (Max(S,≻ i )) = C i (S),concluding that x ∈ C i (S).

CHOOSING ON INFLUENCE 31We finally show that C i (S) ⊆ Max(Max(S,≻ i ),≻ j ). Take any x ∈ C i (S).By definition of ≻ i , x ∈ Max(S,≻ i ). Assume for a contradiction that thereexists y ∈ Max(S,≻ i ) such that yx ∈≻ j . Notice that this implies x /∈ C j (T)for any T ∈ Ω X with x,y ∈ T.- Case 1: Let y ∈ C i (S). By NULL, C j (C i (S)) = C i (S). Since bothx,y ∈ C i (S), we contradict with yx ∈≻ j .-Case2: Lety /∈ C i (S): Then,aswehaveshownearlier,y /∈ Max(Max(S,≻ i),≻ j ). Since y ∈ Max(S,≻ i ), there exists z ∈ Max(Max(S,≻ i ),≻ j ) withzy ∈≻ j by transitivity of ≻ j . But then z ∈ C i (S). By transitivity zx ∈≻ j ,implying that x /∈ C j (T) for any T ∈ Ω X with x,z ∈ T. But then, by NULL,C j (C i (S)) = C i (S), creates the desired contradiction since both x,z ∈ C i (S).□Independence of the properties. Consider X = {x,y,z}, x = C 1 (xy),y =C 1 (yz),z = C 1 (xz) and x = C 1 (xyz). If x = C 2 (xy),y = C 2 (yz),x =C 2 (xz),y = C 2 (xyz), CoI and NULL are satisfied but EXP is not. If instead,y = C 2 (xy),z = C 2 (yz),x = C 2 (xz),x = C 2 (xyz), CoI is not satisfied butthe others are. Finally, if xy = C 2 (xy),z = C 2 (yz),z = C 2 (xz),z = C 2 (xyz),NULL is the only one that is not satisfied.Proof of Theorem 2. Let (C 1 ,C 2 ) be a Choice on Mutual Influence mechanism.Fix i,j ∈ {1,2} with i ≠ j, without loss of generality.Necessity. Consider any (≻ 1 ,≻ 2 ) explaining (C 1 ,C 2 ). We will show thattr(P i ∪Q i ) ⊂≻ i , hence a subset of the intersection as well. Take any xy ∈ P i .Since y = C j (xy), we have xy /∈≻ j . But then x = C i (xy) implies that xy ∈≻ i .Now take any xy ∈ Q i . As there exists S ∈ Ω X with x ∈ S and C j (S) ≠C j (S \y), we have xy /∈≻ j . But then, x = C i (xy) implies that xy ∈≻ i . Thusfor xy ∈ (P i ∪Q i ) we also have xy ∈≻ i . Transitivity of ≻ i proves the claim.

32 CHOOSING ON INFLUENCESufficiency. Wenowshowthatfor≻ ∗ i= tr(P i ∪Q i ), thereexists≻ ∗ j suchthat(≻ ∗ 1,≻ ∗ 2) also explains (C 1 ,C 2 ). In the proof of Theorem 1 we have shown that(≻ 1 ,≻ 2 ) explains (C 1 ,C 2 ) for ≻ 1 = {xy ∈ X ×X : C 1 (S) = C 1 (S \y) for allS ∈ Ω X with x ∈ S} and ≻ 2 = {xy ∈ X×X : C 2 (S) = C 2 (S\y) for all S ∈ Ω Xwith x ∈ S}. Now let ≻ ∗ j=≻ j . We will show that Max(Max(S,≻ ∗ i),≻ j ) =Max(Max(S,≻ i ),≻ j ) and Max(Max(S,≻ j ),≻ ∗ i) = Max(Max(S,≻ j ),≻ i )for all S ∈ Ω X .First notice that ≻ i = P i ∪ Q i ∪ R and ≻ j = P j ∪ Q j ∪ R. Now, take anyS ∈ Ω X and x ∈ Max(Max(S,≻ ∗ i),≻ j ). Assume for a contradiction thatx /∈ Max(Max(S,≻ i ),≻ j ). There are two possible cases, where both of themresult in the desired contradiction as we show below:-Case 1: x is eliminated in the first stage: Since ≻ i is transitive, thereexists y ∈ Max(S,≻ i ) with yx ∈ (≻ i \ ≻ ∗ i). But then, yx ∈ R, which meansyx ∈≻ j . Since ≻ ∗ i⊂≻ i , we have y ∈ Max(S,≻ i ) ⊂ Max(S,≻ ∗ i). But then,yx ∈≻ j creates a contradiction with x ∈ Max(Max(S,≻ ∗ i),≻ j ).-Case2: xiseliminatedinthesecondstage: Thenthereexistsy ∈ Max(S,≻ i) with yx ∈≻ j . But since y ∈ Max(S,≻ i ) ⊂ Max(S,≻ ∗ i), this creates a contradictionwith x ∈ Max(Max(S,≻ ∗ i),≻ j ).Now take any x ∈ Max(Max(S,≻ i ),≻ j ) and assume for a contradictionthat x /∈ Max(Max(S,≻ ∗ i),≻ j ). Similarly, we have the following cases, whichend with contradictions as desired:-Case 1: x is eliminated in the first stage: Since Max(S,≻ i ) ⊂ Max(S,≻ ∗ i),we contradict with x ∈ Max(Max(S,≻ i ),≻ j ).-Case 2: x is eliminated in the second stage: Then, there exists y ∈Max(S,≻ ∗ i) with yx ∈≻ j . Since y /∈ Max(S,≻ i ), by transitivity of ≻ i , thereexists z ∈ Max(S,≻ i ) with zy ∈ (≻ i \ ≻ ∗ i). But then zy ∈ R, which impliesthat zy ∈≻ j . But then, by transitivity zx ∈≻ j , creating the desiredcontradiction with x ∈ Max(Max(S,≻ i ),≻ j ).

CHOOSING ON INFLUENCE 33Finally, we show that Max(Max(S,≻ j ),≻ ∗ i) = Max(Max(S,≻ j ),≻ i ) forall S ∈ Ω X . Take any S ∈ Ω X and x ∈ Max(Max(S,≻ j ),≻ ∗ i) and assumefor a contradiction that x /∈ Max(Max(S,≻ j ),≻ i ). Then, there exists y ∈Max(S,≻ j ) with yx ∈ (≻ i \ ≻ ∗ i). But then, yx ∈ R and hence, yx ∈≻ j , contradictingwith x ∈ Max(S,≻ j ). Now, take any x ∈ Max(Max(S,≻ j ),≻ i ).Since ≻ ∗ i⊂≻ i , we directly have x ∈ Max(Max(S,≻ j ),≻ ∗ i), establishing theproof. □Proof of Theorem 3. We only prove sufficiency. Let (C 1 ,C 2 ) as defined.Consider ≻ i = {xy ∈ X ×X : C i (S) = C i (S \y) for all S ∈ Ω X with x ∈ S}for i ∈ {1,2}. By the proof of Theorem 1, we know that (≻ 1 ,≻ 2 ) explains(C 1 ,C 2 ). We only need to show regularity and uniqueness.To see regularity, take any x,y ∈ X with xy,yx /∈≻ i , but xy ∈≻ j , fori,j ∈ {1,2} and i ≠ j. Assume for a contradiction that (≻ 1 ,≻ 2 ) is notregular, which means:(i) for all a ∈ X with ax ∈≻ i , ay ∈ (≻ 1 ∪ ≻ 2 )(ii) for all b ∈ X with yb ∈≻ i , xb ∈ (≻ 1 ∪ ≻ 2 )(iii) for all a,b ∈ X with ax,yb ∈≻ i , ab ∈ (≻ 1 ∪ ≻ 2 ).By definition of ≻ i , there exists S ∈ Ω X with x ∈ S such that C i (S) ≠C i (S \ y). There are two possible cases, where both of them result in thedesired contradiction as we show below:-Case 1: There exists z ∈ C i (S), but z /∈ C i (S\y): First, notice that z ≠ y,since xy ∈≻ j and transitivity of ≻ i means that there exists a ∈ Max(S,≻ i )such that ax ∈≻ i , for y to survive the second stage elimination. But, by (i),ay ∈ (≻ 1 ∪ ≻ 2 ), hence y /∈ C i (S). Now, let z ≠ y. Then, there exists b ∈ Ssuch that bz ∈≻ j , yb ∈≻ i . Moreover there does not exist any a ∈ S withab ∈≻ i , in particular xb /∈≻ i (*). But then, by (ii), xb ∈≻ j , which impliesx /∈ Max(S,≻ i ) for z to survive the second stage elimination. By transitivity

34 CHOOSING ON INFLUENCEof ≻ i , there exists a ∈ Max(S,≻ i ) with ax ∈≻ i . By (iii), ab ∈ (≻ 1 ∪ ≻ 2 ). By(*), ab ∈≻ j , but then ab,bz ∈≻ j implies that z /∈ C i (S), giving the desiredcontradiction.-Case 2: There exists z ∈ C i (S \ y), but z /∈ C i (S): First assume thatz survives the first stage elimination in S but there exists a ∈ (Max(S,≻ i)\Max(S \y,≻ i )) with az ∈≻ j . Since Max(S,≻ i ) ⊂ Max(S \y,≻ i )∪{y},we have a = y. But then, by (ii), xz ∈≻ 2 , which implies x /∈ Max(S \y,≻ i )for z to survive the second stage elimination. By transitivity of ≻ i , there existsa ∈ Max(S \ y,≻ i ) with ax ∈≻ i . By (iii), az ∈≻ j , but then z /∈ C i (S \ y).Hencez cannotsurvivethefirststageeliminationinS, althoughitsurvivesthefirst stage elimination in (S \y), which implies yz ∈≻ i and for all a ∈ S witha ≠ y, we have az /∈≻ i , in particular xz /∈≻ i (*). But then, by (ii), xz ∈≻ j ,which implies x /∈ Max(S\y,≻ i ) for z to survive the second stage elimination.By transitivity of ≻ i , there exists a ∈ Max(S \y,≻ i ) with ax ∈≻ i . By (iii),az ∈ (≻ 1 ∪ ≻ 2 ). By (*), az ∈≻ j , but then z /∈ C i (S \y), giving the desiredcontradiction in order to establish that (≻ i ,≻ j ) is regular.Now we only need to show uniqueness. Assume for a contradiction thatthere exists another regular pair (≻ ′ 1,≻ ′ 2) ≠ (≻ 1 ,≻ 2 ) that explains (C 1 ,C 2 ).Take any ≻ ′ i≠≻ i for i ∈ {1,2}. There are two possible cases, where both ofthem result in the desired contradiction as we show below:-Case 1: There exists x,y ∈ X with xy ∈ (≻ ′ i \ ≻ i ): Then, there existS ∈ Ω X with x,y ∈ S such that C i (S) ≠ C i (S \y). But since ≻ ′ i is transitiveMax(S,≻ ′ i) = Max(S \y,≻ ′ i and hence C i (S) = C i (S \y), absurd.-Case 2: There exists x,y ∈ X with xy ∈ (≻ i \ ≻ ′ i): Since for all S ∈ Ω Xwith x,y ∈ S, C i (S) = C i (S \ y), yx /∈≻ ′ i and xy ∈≻ ′ j. Now notice that ifC i (S) = C i (S \y) for all S with x ∈ S, then the following holds:

CHOOSING ON INFLUENCE 35(i)yb ∈≻ i impliesxb ∈ (≻ 1 ∪ ≻ 1 )foranybinX: Assumeforacontradictionthat there exists b ∈ X with yb ∈≻ i but xb /∈ (≻ 1 ∪ ≻ 2 ). Then, C i (bxy) ≠xb = C i (xb), absurd.(ii) ax,yb ∈≻ i implies ay,ab ∈ (≻ 1 ∪ ≻ 2 ) for any a,b ∈ X: Assume for acontradiction that there exists a,b ∈ X with ax,yb ∈≻ i but ab /∈ (≻ 1 ∪ ≻ 2 ).But then C i (abxy) ≠ ab = C i (abx), absurd. Now assume for a contradictionthat there exists a,b ∈ X with ax,yb ∈≻ i but ay /∈ (≻ 1 ∪ ≻ 2 ). But thenC i (abxy) = ay ≠ C i (abx), which is absurd.But then, since xy,yx /∈≻ ′ i and xy ∈≻ ′ j, (i) and (ii) contradicts with regularity,establishing the proof. □Proof of Theorem 4. For i ∈ N, define ≻ i ⊆ X ×X as follows:xy ∈≻ i iff C i (S) = C i (S \y) for all S with x,y ∈ S.The proof replicates the proof of Theorem 1 by using ≻ N\i instead of ≻ j andusing the quantifier ‘for all j’, instead of ‘j’, wherever necessary. The propertiesare used exactly the same way with their two individual counterparts. □Proof of Theorem 5. We only prove sufficiency. For i ∈ N, define ≻ i ⊆X ×X as follows:xy ∈≻ i iff C i (S) = C i (S \y) for all S with x,y ∈ S.As shown in the proof of Theorem 2, ≻ i is asymmetric, transitive and C i (S) =C i (Max(S,≻ i ) for any S ∈ Ω X . Now, for j ∈ N, define E j as follows:x,y ∈ E j iff xy ∈ tr(≻ j |E j)′where,x,y ∈ E j ′ iff xy ∈ (I∩ ≻ j ) and for all yz,tx ∈ I, we have yz,tx ∈≻ j .Notice that by CoEI, for any xy ∈ I, there exists j ∈ N such that x,y ∈ E j ′ ⊆E j . If E j ∩ E k ≠ ∅ for some j,k ∈ N, then discard E j ∩ E k either from E j

36 CHOOSING ON INFLUENCEor E k , randomly. Notice that this does not break transitivity of (≻ j |E j ) or(≻ k |E k ) thanks to CoEI.Finally, let ≻ E = ⋃ j∈N (≻ j |E j ). ≻ E is asymmetric and transitive, since each(≻ i |E i ) is asymmetric and transitive and areas of expertise are disjoint.Now take i ∈ N and S ∈ Ω X . First, take x ∈ Max(Max(S,≻ i ),≻ E ). Forany z ∈ Max(S,≻ i ), we have x ∈ C i (xz). This holds since z = C i (xz) andx ∈ Max(S,succ i ) imply that zx ∈ I. But then zx ∈≻ E , contradicting withx ∈ Max(Max(S,≻ i ),≻ E ). Hence, x ∈ C i (xz) for all z ∈ Max(S,≻ i ). Butthen, by EXP, x ∈ C i (Max(S,≻ i ) = C i (S,≻ i ).Now take x ∈ C i (S). By definition of ≻ i , x ∈ Max(S,≻ i ). Assume for acontradiction that x /∈ Max(Max(S,≻ i ),≻ E ). But then, since ≻ E is transitive,there exists y ∈ Max(Max(S,≻ i ),≻ E ) such that yx ∈≻ E . By theprevious part, y ∈ C i (S). If x = Ci(xy), since x ∈ C i (S), by WWARP*, wehave y /∈ C i (S), contradiction. If y = C i (xy), since y ∈ C i (S), by WWARP*,we have x /∈ C i (S), contradiction. Finally, if xy = C i (xy), then by BI, theredoes not exist a binary chain yz 1 ,z 1 z 2 ,...,z t x ∈ I, which contradicts withyx ∈≻ E , establishing the proof.

More magazines by this user
Similar magazines