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Belief-Averaged Relative Utilitarianism
Florian BrandlPrinceton University
We study preference aggregation under uncertainty when individual and collec-tive preferences are based on subjective expected utility. A natural procedure fordetermining the collective preferences of a group then is to average its members’beliefs and add up their (0 , -normalized utility functions. This procedure extendsthe well-known relative utilitarianism to decision making under uncertainty. Weshow that it is the only aggregation function that gives tie-breaking rights to agentswho join a group and satisfies an independence condition in the spirit of Arrow’sindependence of irrelevant alternatives as well as four undiscriminating axioms.A group of policymakers seeks to decide between action plans whose consequences depend onunknown facts about the world. Their preferences over the alternatives hinge on their beliefsabout the world, such as the probability that an economic intervention is efficacious, and theirtastes for consequences like enhanced labor rights or higher GDP. How should they arrive at agroup preference?To put this question on formal grounds, we assume that each agent is rational in the sense thather preferences meet Savage’s (1954) criteria and we shall hold groups to the same standard.The beauty of Savage’s assumptions is that they allow us to express preferences in an intuitiveway: agents assign probabilities to possible states of the world—expressing their belief—andutilities to consequences—capturing tastes, so that more preferred acts have higher expectedutility. To tackle the aggregation problem, we consider aggregation functions, which map eachpossible configuration of preferences for the members of a group to a preference of the group.Our approach is thus multi-profile in that it contemplates hypothetical configurations that couldarise rather than a single observed one.A seemingly sensible way to arrive at the preferences of a group—or, equivalently, its beliefand utility function—is by averaging its members’ beliefs and adding up their utility functions,each normalized so that the minimal and maximal values agree across agents. We call thisaggregation method belief-averaged relative utilitarianism .For decision making under risk, that is, when all agents hold the same objective belief, relativeutilitarianism is well-known and has been characterized several times (see Karni, 1998; Dhillon,1998; Dhillon and Mertens, 1999; Segal, 2000; Börgers and Choo, 2017b). For decision makingunder uncertainty, Gilboa et al. (2004) introduced a restricted Pareto condition that is necessary1 raft – May 11, 2020 and sufficient for the collective belief and utility function to be linear combinations of the agents’beliefs and utility functions, but allows the weights to vary across profiles. We combine boththemes, which leads to a characterization of belief-averaged relative utilitarianism. Our axiomsare inspired by those of Dhillon and Mertens and Gilboa et al. and discussed in detail below.It is instructive to contrast our ex post relative utilitarianism with Sprumont’s (2019) ex ante relative utilitarianism: the former derives a collective belief and compares acts based on thesum of the agents’ expected utilities under the collective belief; the latter first calculates theexpected utility of an act for each agent based on their own belief and then compares acts by thesum of expected utilities. The upshot is that ex post aggregation gives rise to a collective beliefand utility function (and thus Savage-type collective preferences), whereas ex ante aggregationmeets the Pareto principle. Only dictatorships can achieve both as was shown by Mongin(1995). In subsequent work, Mongin argues that preferences based on different beliefs can leadto “spurious unanimities” to which the Pareto condition need not apply. This motivates therestricted Pareto axiom introduced by Gilboa et al. (2004), which ex post relative utilitarianismdoes satisfy and is the most we can hope for given our assumptions about collective preferences.We discuss related literature more extensively in Section 5.Our two distinguishing axioms are restricted monotonicity (or monotonicity for short) and independence of redundant acts .To understand monotonicity, consider the following situation: a group has determined acollective preference (represented by a belief and a utility function). Say they are indifferentbetween two acts f and g . Now an additional agent who prefers f to g joins the group.Moreover, f induces the same probability distribution over consequences for her belief and thegroup belief. That is, for every consequence, she assigns the same probability to f resulting inthis consequence as does the group. Likewise, g induces the same distribution for both beliefs.Monotonicity demands that the augmented group with the additional agent prefers f to g . Inother words, agents get to break ties when joining a group if their belief does not differ fromthe group’s belief in a way that is relevant for the acts in question. To see why the restrictionto conforming beliefs is reasonable, perhaps even desirable, consider two examples: Example 1.
Alice and Bob are planning a weekend activity. They are considering hiking andplaying board games at home. There is uncertainty whether it will rain or not, which creates twodifferent states of the world: one in which it will rain and one in which it will not rain. Afterdiscussing their beliefs about the weather and their tastes for both activities under differentconditions, they decide that they, as a group, are indifferent. Later they receive a call fromtheir friend Charlie, who will join them for the activity. Charlie prefers hiking to playing boardgames, and so they decide to go hiking.Since the uncertainty is about the weather and every one of them is unlikely to have accessto more relevant information than the others, it is safe to assume that Charlie shares Alice andBob’s belief. (Indeed, it may well be that all three agents have the same belief.) Hence, Charlieshould get to break the tie, so that the group of all three agents prefers hiking to playing boardgames. 2 raft – May 11, 2020 rain/win rain/lose no rain/win no rain/lose { Alice, Bob } belief 10% 40% 10% 40%utility from f f { Alice, Bob, Charlie } belief 30% 20% 30% 20%utility from f f ) for the corresponding group of agents in each state. Weassume playing board games ( g ) yields utility 0 for all groups in all states. Then Aliceand Bob have expected utility 0 for both f and g and are thus indifferent; Charlie’sexpected utility for f is − . , so that he prefers g to f . However, the group of all threeagents prefers f to g , since they assign utility . to f . Example 2.
Later Alice sprains her foot, which makes hiking impossible. Alice and Bob bringup going to a football match of their favorite team ( f ) as an alternative to playing board games( g ). Now there is not only uncertainty about the weather but also about whether their team willwin or lose the match. As before, they are indifferent between both options. They call Charlie,who prefers playing board games over the match. Nevertheless, the three of them decide to goto the football match.Now there are four states of the world resulting from two possibilities for the weather andtwo for the outcome of the match. We can view playing board games as a constant act, onewhich yields the same consequence in all states since it depends neither on the weather nor onthe outcome of the match; but going to the match depends on both factors. In contrast to thebelief about the weather, Charlie may well disagree with Alice and Bob’s belief about the game,and so monotonicity does not apply. Say Alice and Bob assign a probability of 50% to rain andof 20% to a win (and both events are independent); Charlie agrees with the probability of rainbut assigns 80% to a win. If Charlie is a football expert, the group may assign a large weightto his belief and rate a win at 60% (with again 50% for rain). On the other hand, they assignthe same weight to everyone’s utilities. For suitable numerical values (see Table 1), the groupdecision in Example 2 can occur.Our definition of monotonicity requires that the set of agents can vary. We will thus consideraggregation functions that take as input the preferences of an arbitrary finite set of agents thatmay vary in size. When requiring that the preferences of every single-agent group are thoseof its sole member (which we call faithfulness), restricted monotonicity implies the restrictedPareto condition of Gilboa et al. (2004). It prescribes that whenever all agents are indifferent3 raft – May 11, 2020 between two acts f and g and both induce the same distribution over consequences for everyagent’s belief, then the group is indifferent between f and g . It is necessary and sufficient forthe collective belief and utility function to be linear combinations of the agents’ beliefs andutility functions. The weights in both these linear combinations can, however, vary arbitrarilyacross profiles. Monotonicity implies that the weight of an agent in either linear combinationcan only depend on her own preferences as we will see. To also eliminate this dependency, weneed an axiom that connects profiles with different preferences for the same agent.Typical candidates are independence axioms, which state that the collective preferences overa set of acts can only depend on the individual preferences over those acts. The most well-known candidate from this family, Arrow’s independence of irrelevant alternatives (or acts inour case), demands the above assertion for arbitrary sets of acts. It will lead to an impossibilityresult even with only mild additional axioms (see, for example, Kalai and Schmeidler (1977)for decision making under risk). Dhillon and Mertens (1999) proposed a weaker version calledindependence of redundant alternatives. It is the blueprint for our independence of redundantacts , which requires the above independence for sufficiently large sets of acts. More precisely,say we have two preference profiles (on the same set of agents) and a set of acts G so that everyagent has the same preferences over acts in G in both profiles and every act is unanimouslyindifferent to some act in G in both profiles. Then the collective preferences over acts in G should be the same in both profiles. The second assumption on G is equivalent to saying thatfor both profiles, G has the same image in utility space as the set of all acts. That is, the vectorof expected utilities of every act is equal to the expected utilities of an act in G .On top of restricted monotonicity and independence of redundant acts, we make four addi-tional assumptions about the aggregation function: the preferences of any single-agent groupare those of its sole member ( faithfulness ), no agent can impose her belief on a group ( no beliefimposition ), the preferences of any group depend continuously on the preferences of its members( continuity ), and relabeling agents does not change the collective preferences ( anonymity ). Weshow that these six conditions characterize belief-averaged relative utilitarianism.The proof is modular and yields two intermediary results that are interesting in their ownright. First, dropping anonymity, we characterize the class of aggregation functions that assigntwo positive (and possibly different) weights to every agent, one for her belief and one forher utility function, and then derive the collective preferences of any group from the weightedlinear combinations of the beliefs and utility functions of its members. Second, additionallydropping independence of redundant acts allows the weights of an agent to depend on her ownpreferences. More precisely, the weights of every agent can now be arbitrary continuous andpositive functions of her preferences. However, the weights of an agent cannot depend on theother agents’ preferences. We give details about the necessity of the axioms for all three resultsin Section 4. 4 raft – May 11, 2020
1. Preferences and Aggregation Functions
Let Ω be a set of states of the world and Σ be a sigma-algebra over Ω . We refer to elements of Σ as events. A probability measure π on (Ω , Σ) is non-atomic if for every A ∈ Σ with π ( A ) > ,there is B ⊂ A with < π ( B ) < π ( A ) . We denote by Π the set of all non-atomic and countablyadditive probability measures. Let X be a set of consequences endowed with a sigma-algebra.An act is a measurable function f : Ω → X that maps states to consequences.We assume that preferences over acts are the result of maximizing expected utility accordingto a belief π ∈ Π and a utility function u : X → R that is measurable and bounded. We saythat π and u represent the preference relation < ⊂ F × F and conversely that π and u are thebelief and the utility function associated with < if f < g if and only if Z Ω ( u ◦ f ) dπ ≥ Z Ω ( u ◦ g ) dπ, ( ⋆ )for all acts f, g . The strict and symmetric part of < are ≻ and ∼ , respectively. We denote by ¯ R the set of all preference relations that can be represented by expected utility maximization; R consists of ¯ R minus complete indifference, which corresponds to a constant utility functionand an arbitrary belief.All preference relations in R give rise to a unique belief. Utility functions are only unique upto positive affine transformations. To establish a one-to-one correspondence between preferencerelations and utility functions, let U be the set of all non-constant utility functions normalizedto the unit interval, that is, inf { u ( x ) : x ∈ X } = 0 and sup { u ( x ) : x ∈ X } = 1 ; ¯ U consists of U plus the utility function that is constant at . When π ∈ Π and u ∈ ¯ U represent < , we write E < ( f ) = R Ω ( u ◦ f ) dπ for the expected utility of f under π and u .We postulate an infinite set of potential agents N . A group I consists of a non-empty andfinite subset of agents; the collection of all groups is I . Symbols in bold face refer to tuplesindexed by a set of agents. Every agent has a preference relation < i ∈ R . (Notice that noagent may be completely indifferent.) A preference profile <<< ∈ R I for agents in I specifies thepreferences of each agent in I . For i ∈ N − I and < i ∈ R , we obtain a preference profile <<< + i foragents in I ∪ { i } by adding < i to <<< . Similarly, when | I | > and i ∈ I , <<< − i is the profile where < i is deleted. We seek to aggregate the preferences of all members of a group into a collectivepreference. To this end, we consider an aggregation function Φ that maps every preferenceprofile for every group to an element of ¯ R .
2. Conditions for Belief-Averaged Relative Utilitarianism
Our axioms for characterizing belief-averaged relative utilitarianism are in part generalizationsof Dhillon and Mertens’s axioms for relative utilitarianism in the context of risk. The first two,restricted monotonicity and independence of redundant acts, carry the most power in the sensethat they rule out other aggregation functions one might come up with.The restricted monotonicity axiom applies if a group I is indifferent between two acts f and g and is joined by an agent i so that f induces the same distribution over consequences according5 raft – May 11, 2020 to the group’s belief and agent i ’s belief and so does g . In that case, the augmented group I ∪ { i } should prefer f to g if and only if i does. Formally, for all I ∈ I , i ∈ N − I , <<< ∈ R I ,and < i ∈ R with < = Φ( <<< ) and < + i = Φ( <<< + i ) , f ∼ g and f < i g implies f < + i g if π ◦ f − = π i ◦ f − and π ◦ g − = π i ◦ g − (restricted monotonicity)Moreover, a strict preference between f and g for agent i implies a strict collective preference.In the terminology of Gilboa et al. (2004), f and g are lotteries.The reasoning for independence of redundant acts is that two acts so that all agents areindifferent between them are perfect substitutes for each other, thus making each redundant inthe presence of the other. It then prescribes that for two profiles that agree on a set of actsthat makes every other act redundant, the collective preferences over this set should also agree.For all I ∈ I , <<< , <<< ′ ∈ R I , and G ⊂ F , <<< | G = <<< ′ | G implies Φ( <<< ) | G = Φ( <<< ′ ) | G if for all f ∈ F there are g, g ′ ∈ G such that f ∼ i g and f ∼ ′ i g ′ for all i ∈ I (independence of redundant acts)By the conditions on G , its image in utility space is the same as that of F for both profiles <<< and <<< ′ . That is, { ( E < i ( g )) i ∈ I : g ∈ G} = { ( E < i ( f )) i ∈ I : f ∈ F } and similarly for <<< ′ .The remaining four axioms are mostly standard. Monotonicity and independence of redun-dant acts relate different profiles to each other, but do not anchor the aggregation function. Forexample, there could be a phantom agent with fixed preferences and the collective preferencesof every group are derived from belief-averaged relative utilitarianism of the preferences of thegroup’s members and the phantom agent. We can rule out phantom agents by requiring thepreferences of a single-agent group to be those of its sole member. For all i ∈ N and < i ∈ R , Φ( < i ) = < i (faithfulness)Part of our definition of monotonicity is that an additional agent can break ties betweencertain acts in her favor if she has a strict preference. It thus rules out that Φ ignores the utilityfunction of some agent altogether. But so far nothing prevents us from ignoring the beliefs ofan agent. To counter this, it suffices that no agent can impose her belief on a group. That is,the belief of a group is not identical to that of one of its members unless the rest of the groupwould arrive at that belief anyway. Formally, for all I ∈ I with | I | > , i ∈ I , and preferenceprofiles <<< ∈ R I where Φ( <<< ) is not complete indifference and π and π ′ are the beliefs associatedwith Φ( <<< ) and Φ( <<< − i ) , π ′ = π i implies π = π i (no belief imposition)Continuity requires that a small changes in the agents’ preferences can only leads to smallchanges in the collective preferences. To make this precise, we need to equip R and ¯ R withtopologies. The uniform metric sup {| E < ( f ) − E < ′ ( f ) | : f ∈ F } induces a topology on R . Theset of profiles R I gets the product topology of R . The topology on ¯ R is that of R plus the entire6 raft – May 11, 2020 set ¯ R (which is thus the only neighborhood of the relation expressing complete indifference).Thus, ¯ R is the closure of R in ¯ R . Φ is continuous (continuity)Lastly, anonymity prescribes that relabeling the agents within a group does not change thecollective preferences. For all I ∈ I and <<< ∈ R I , Φ( <<< ) = Φ( <<< ◦ η ) for all permutations η on I (anonymity)Notice that anonymity as defined here is in general weaker than allowing η to be a bijectionbetween two groups J and I of the same size.Our results remain true if we require all axioms except restricted monotonicity and faithfulnessto hold only for groups of size 2. Independence of redundant acts, continuity, and anonymityare used only for profiles of two agents. The assumption that Φ does not allow belief impositionis used for larger profiles, but it is not hard to check that this could be avoided.
3. Three Characterizations of Aggregation Functions
Conceptually, our most interesting result is a characterization of belief-averaged relative utili-tarianism. We will obtain it as a corollary of Theorem 1, which uses the first five axioms (thusexcluding anonymity) to characterize affine aggregation functions. These functions assign twopositive weights to every agent, one for their belief and one for their utility function, and de-termine the preferences of every group by adding up the weighted beliefs and utility functionsof its members. Importantly, the weights are constant across all profiles, that is, they cannotdepend on an agent’s own preferences, the preferences of any other agent, or who is a memberof the group.
Theorem 1.
Let | X | ≥ and Φ be an aggregation function. Then the following are equivalent.(i) Φ satisfies restricted monotonicity, independence of redundant acts, faithfulness, no beliefimposition, and continuity(ii) There are λ , µ ∈ R N ++ such that for all I ∈ I and <<< ∈ R I , Φ( <<< ) is represented by P i ∈ I λ i P i ∈ I λ i π i and P i ∈ I µ i u i When additionally requiring Φ to be anonymous, it follows at once that the weights of allagents have to be equal. To see this, consider the two-agent group I = { i, j } and any profile <<< ∈ R I so that the beliefs π i and π j and the utility functions u i and u j are distinct. Then λ i π i + λ j π j = λ i π j + λ j π i whenever λ i = λ j . Likewise, µ i u i + µ j u j = µ i u j + µ j u i if µ i = µ j .Thus, anonymity can only hold if λ i = λ j and µ i = µ j . Since multiplication of all weights bythe same positive constant does not change the collective preferences, we may assume that allweights are equal to 1. This gives a characterization of belief-averaged relative utilitarianism asthe only aggregation function that satisfies all our axioms. It derives the collective preferencesby averaging the beliefs and adding up the normalized utility functions of all agents.7 raft – May 11, 2020 Corollary 1.
Let | X | ≥ and Φ be an aggregation function. Then the following are equivalent.(i) Φ satisfies restricted monotonicity, independence of redundant acts, faithfulness, no beliefimposition, continuity, and anonymity(ii) For all I ∈ I and <<< ∈ R I , Φ( <<< ) is represented by | I | P i ∈ I π i and P i ∈ I u i The proof of Theorem 1 proceeds as follows. First, we only consider the implications ofrestricted monotonicity, faithfulness, and no belief imposition. The former two imply the re-stricted Pareto condition and thus that beliefs and utility functions are aggregated linearly (withpositive weights for utility functions by the strict part of monotonicity). The additional strengthof monotonicity lies in the fact that if an agent joins a group, the belief of the augmented groupis an affine combination of the belief of the original group and that of the agent. Assuming allbeliefs are affinely independent, it follows that the relative weights of the agents in the originalgroup cannot change. The analogous statement holds for utility functions.Now, given any profile, we can apply this conclusion to every agent and the subprofile exclud-ing this agent. It follows that the magnitude of the weight of the belief and the utility functionof an agent can only depend on her own preferences. The signs may however depend on thepreferences of the other agents. Since weights for utility functions have to be positive, any de-pendence on other agents’ preferences vanishes for the weights of utility functions. Surprisingly,continuity allows us to get the same conclusion for beliefs. If the weight for an agent’s belief everwere to change sign, by continuity, her weight would have to be zero in some (two-agent) profile.But then the second agent would get to impose her belief, which is ruled out. We conclude thatthe weight for an agent’s belief cannot change sign. If it were to be negative regardless of herpreferences, we could find a profile where the collective belief assigns a negative probability tosome event, which gives a contradiction.In summary, we derive the following intermediate result, which is interesting in its own right.
Proposition 1.
Let | X | ≥ and Φ be an aggregation function. Then the following are equiva-lent.(i) Φ satisfies restricted monotonicity, faithfulness, no belief imposition, and continuity(ii) There are continuous functions λ , µ : R → R N ++ such that for all I ∈ I and <<< ∈ R I , Φ( <<< ) is represented by P i ∈ I λ i ( < i ) P i ∈ I λ i ( < i ) π i and P i ∈ I µ i ( < i ) u i The last step is to show that additionally presuming independence of redundant acts givesthat the weights of an agent have to be constant, that is, that the functions λ i and µ i inProposition 1 are constant. By applying independence of redundant acts to a suitable two-agent profile, it follows relatively quickly that the weights of an agent cannot depend on herbelief. To conclude they are independent of her utility function as well, we consider a two-agent profile in which every act is unanimously indifferent to an act with a range of only threeconsequences { x , x , x ∗ } . If the focal agent changes her utility for other consequences in a waythat does not change the image of the profile in utility space, we can apply independence ofredundant acts and conclude that the collective preferences over acts with range { x , x , x ∗ } raft – May 11, 2020 do not change. This can only be if the weights of that agent remain the same. By repeatedlyapplying this assertion, we can construct a path between any pair of utility functions so thatneighboring utility functions result in the same weights.
4. Necessity of the Axioms
We discuss the necessity of the axioms for Corollary 1 first since the examples we give here willwork for Theorem 1 and Proposition 1 as well.Not much can be said about the aggregation function if it does not satisfy restricted mono-tonicity. For example, every faithful function that is constant on profiles with two or more agentssatisfies all axioms except monotonicity. Here is an example, adapted from Dhillon and Mertens(1999), that violates monotonicity but satisfies the restricted Pareto condition. The collectivebelief is the average of the agents’ beliefs. Consider the closure of the profile in utility spaceand let ( u i ) i ∈ I be the unique point that maximizes the product of utilities. Let the collectiveutility function be the linear combination of the agents’ utility functions where the weight ofagent i is Π j ∈ I \{ i } u j . Since the weights for an agent are the same in any two profiles with thesame image in utility space, this function satisfies independence of redundant acts.The class of functions which satisfy all axioms but independence of redundant acts andanonymity is characterized by Proposition 1. Anonymity holds if and only if the weight functions λ i and µ i are the same across agents.Without faithfulness, we could have a “phantom agent” whose belief and utility function arealways added on top of the beliefs and utility functions of real agents with a constant weight.If Φ is not continuous, agents with the same preferences could be handled specially. Forexample, let ( α n ) be a strictly increasing positive sequence. Then if Φ( <<< ) is represented by P < ∈R α n ( < ) P < ∈R α n ( < ) π i and P < ∈R α n ( < ) u i , where n ( < ) is the number of agents in the profile <<< with preferences < , then Φ satisfies all axioms but continuity.It is open whether no belief imposition is necessary for the conclusion of Corollary 1. Inthe absence of anonymity, that is, for Theorem 1, it is necessary, however. In that case, welose the decomposable form for the group belief if Φ allows belief imposition. For example, wecould have that the group belief is π whenever agent is part of the group and | I | P i ∈ I π i otherwise. Thus, whether the belief of an agent gets non-zero weight could depend on whethersome particular other agent is present.For Proposition 1, continuity is even more vital. Without it, the weight of an agent’s beliefcan even be negative. Consider Ω = [0 , equipped with the Borel sigma-algebra. Let ˜ π be theuniform distribution on Ω and, for a non-atomic measure π on Ω , let ρ ( π ) = sup { π ( E )˜ π ( E ) : E ∈ Σ } .(Since non-atomic measures have continuous density functions, ρ ( π ) is finite.) For i ≥ ,let λ i ( < i ) = i ρ ( π i ) , and λ as well as all µ i be constant at 1. If Φ( <<< ) is represented as inProposition 1 except that the collective belief is π − P i ∈ I −{ } λ i ( < i ) π i whenever ∈ I and π = ˜ π , it satisfies all axioms but continuity.Our proof requires | X | ≥ since it relies on profiles with three linearly independent utilityfunctions. We do not know if our results hold when | X | = 3 .9 raft – May 11, 2020
5. Relationship to the Literature on Preference Aggregation UnderUncertainty and Risk
Much of the early literature on group decision making under uncertainty has focused on thePareto condition, which requires that a unanimous preference among agents prevails in thecollective preferences. For decision making under risk, that is, when all agents have the samebelief and acts become lotteries, Harsanyi’s (1955) well-known theorem shows that under thePareto condition, the utility function of a group has to be a linear combination of the individ-ual utility functions. Hylland and Zeckhauser (1979) consider this condition in a multi-profileframework under uncertainty and show that it is incompatible with non-dictatorship and sep-arate aggregation of beliefs and utility functions. Mongin (1995) examines the implications ofdifferent degrees of the Pareto condition for single profiles when individual and group prefer-ences are of Savage-type as in the present paper. Roughly, he finds that it implies the existenceof a dictator for all but degenerate profiles. Mongin (1998) shows that this result persists inAnscombe and Aumann’s (1963) model of subjective expected utility. It disappears if one al-lows utility functions to be state-dependent, but reappears for intermediary preference models,which allow identification of beliefs.In contrast to these negative findings, Gilboa et al. (2004) show that the previously mentionedrestricted Pareto condition is equivalent to linear aggregation of beliefs and utility functions. Inparticular, the restricted Pareto condition does not apply to what Mongin (2016) calls “spuriousunanimities”. Gilboa et al. (2014) consider an intermediate between the full and the restrictedPareto condition, which they call no-betting-Pareto dominance. It states that one act dominatesanother if it Pareto dominates it in the usual sense and there is a belief such that if all agentsheld it, they would also unanimously prefer the first act to the second. They argue that no-betting-Pareto dominance characterizes situations in which agents can benefit from trade, butdo not seek to determine how it restricts preference aggregation.Among the few multi-profile approaches is that of Dietrich (2019), which adds consistencywith Bayesian updating and continuity of the aggregation function to the restricted Pareto con-dition. These conditions imply that utilities are aggregated linearly and beliefs are aggregated geometrically , that is, the group belief is a geometric mean of the individual beliefs. More-over, the weight of an agent in either of these combinations can depend on the profile of utilityfunctions, but not on beliefs.Several authors dispense with the assumption that collective preferences are based on subjec-tive expected utility maximization. Most relevantly, Sprumont (2019) characterizes a differentform of relative utilitarianism where acts are compared by their cumulative expected utility P i ∈ I E < i ( f ) = P i ∈ I R Ω ( u i ◦ f ) dπ i . By contrast, belief-averaged relative utilitarianism yieldsthe order induced by R Ω (cid:0)P i ∈ I u i ◦ f (cid:1) dπ , where π = | I | P i ∈ I π i . Thus, the difference is whetherthe expectation is taken before or after summing over agents. For his characterization, Sprumontassumes the full Pareto axiom, independence of inessential expansions (a strengthening of inde-pendence of redundant acts), belief irrelevance (the ranking of constant acts is independent of More precisely, independence of inessential expansions requires that if two profiles agree on a set of acts G so raft – May 11, 2020 beliefs), and that collective preferences are continuous and satisfy Savage’s sure-thing principle.As a high-level summary, one could say that the assumptions about the aggregation functionare stronger whereas those about collective preferences are weaker.Alon and Gayer (2016) assume that groups have Gilboa and Schmeidler’s (1989) max-minexpected utility preferences, where acts are compared based on their minimal expected utilitywithin a set of beliefs. In addition to the restricted Pareto condition, they assume that if allagents believe that one event is more likely than another, then so does the group. These twoaxioms imply that utility functions are aggregated linearly and the set of group beliefs consistsof convex combinations of individual beliefs.Nascimento (2012) studies the aggregation of the preferences of experts that agree on theranking of risky prospects, but are otherwise very general; in particular, they need not resultfrom subjective expected utility maximization. He gives a set of assumptions about the experts’and the aggregated preferences under which the latter are the result of a compromise betweenutilitarian aggregation and the Rawlsian criterion.Some results from the literature on preference aggregation under risk are particularly rel-evant. First, Dhillon and Mertens’s (1999) characterization of relative utilitarianism, whichderives the collective preference by adding up the normalized utility functions of the agents.As Harsanyi, they assume that individual and collective preferences are given by linear utilityfunctions on lotteries. They require the aggregation function to satisfy continuity, anonymity,a monotonicity condition, and independence of redundant alternatives. Monotonicity and in-dependence of redundant alternatives are analogs of restricted monotonicity and independenceof redundant acts. In the same vein, Dhillon (1998) characterizes relative utilitarianism with aPareto condition for groups of agents instead of monotonicity. It requires that if two disjointgroups agree on the preference between two lotteries, then the union of both groups enter-tains the same preference. Like our monotonicity condition, it acts on variable sets of agents.Börgers and Choo (2017a) discovered a flaw in Dhillon’s proof, which presumably stems fromthe fact that she allows agents to have equal or opposite utility functions without assumingcontinuity. Börgers and Choo (2017b) provide a more accessible characterization of relativeutilitarianism along the same lines as Dhillon and Mertens.
6. Conclusions and Open Problems
We have shown that restricted monotonicity and independence of redundant acts in conjunctionwith three axioms which require the aggregation function to be well-behaved, that is, faithful,continuous, and free from belief imposition, necessitate affine aggregation of beliefs and utilityfunctions with constant weights. If anonymity holds in addition, all weights have to be equal andbelief-averaged relative utilitarianism remains as the only possibility. These results are basedon a characterization of aggregation functions that satisfy monotonicity but not necessarily that G contains a most-preferred and a least-preferred act for every agent, then the collective preferences overacts in G are the same for both profiles. Dhillon and Mertens (1999) use a weaker monotonicity condition, but one which also allows them to prove theequivalent of our Lemma 5. raft – May 11, 2020 independence of redundant acts. We have thus drawn a clear picture of monotonic aggregationfunctions.We leave open if there is a succinct representation of the class of functions that satisfyindependence of redundant acts (plus perhaps some standard axioms). In particular, it isunclear how much independence of redundant acts restricts the aggregation of beliefs. Finally,it would be desirable to determine whether ruling out belief imposition is necessary for theconclusion of Corollary 1. Acknowledgments
This material is based on work supported by the Deutsche Forschungsgemeinschaft under grantBR 5969/1-1. The author thanks Dominik Peters and Omer Tamuz for helpful comments.
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APPENDIX: Proofs
An aggregation function Φ gives rise to two functions φ and ψ , which take a profile <<< ∈ R I foran arbitrary group I to a collective belief φ ( <<< ) ∈ Π and a collective utility function ψ ( <<< ) ∈ ¯ U .Whenever the collective utility function is trivial, that is, ψ ( <<< ) = 0 , φ ( <<< ) is not uniquelydetermined. In fact, it can be arbitrary.Preferences are invariant under multiplication of the belief by a positive constant and positiveaffine transformations of utility functions. For measures π, π ′ on Ω and utility functions u, u ′ on X , we write π ≡ π ′ if π = απ ′ for some α > and u ≡ u ′ if u = αu ′ + β for α > and β ∈ R .Theorem 1 requires that we find λ , µ ∈ R N ++ such that φ ( <<< ) ≡ P i ∈ I λ i π i and ψ ( <<< ) ≡ P i ∈ I µ i u i for all I ∈ I and <<< ∈ R I . The proof proceeds in three main steps. First, weexamine the implications of restricted monotonicity in conjunction with faithfulness and nobelief imposition. These axioms imply that in almost all profiles, the weights assigned toan agent’s belief and utility function can only depend on her own preferences. The weightsfor beliefs may be negative however. Second, we add continuity, which allows us to rule out13 raft – May 11, 2020 negative weights and to extend the obtained representation to all profiles (Proposition 1). Lastly,independence of redundant acts implies that the weights of an agent cannot depend on her ownpreferences either and thus have to be constant across all profiles. A. Implications of Restricted Monotonicity
The proofs that the weight of the belief and the weight of the utility function of an agent canonly depend on her own preferences in Sections A.1 and A.2 proceed along the same lines.For the most part, the proof for beliefs requires more work, since we cannot rule out negativeweights. Thus, we advise readers interested in the proofs to take a look at Appendix A.2 first.
A.1. Aggregation of Beliefs
The first lemma is the basis from which we will derive all further conclusions about beliefaggregation. It states that if an agent joins a group, the new group belief is an affine combinationof the previous group belief and the belief of the agent. The restricted Pareto condition ofGilboa et al. (2004) already implies that the collective belief is an affine combination of itsmembers’ beliefs. The additional strength of this conclusion lies in the fact that no matterwhich belief the new agent holds, it is always combined with the same belief of the originalgroup. If we assume that an agent cannot impose her belief on the group, her weight in theaffine combination cannot be 1.
Lemma 1.
Assume that Φ satisfies restricted monotonicity and rules out belief imposition. Let I ∈ I , i ∈ I , and <<< ∈ R I with ψ ( <<< ) = 0 . Then, φ ( <<< ) = (1 − α ) φ ( <<< − i ) + απ i for some α ∈ R − { } .Proof. Let < = Φ( <<< ) and < − i = Φ( <<< − i ) . Monotonicity implies that f ∼ g whenever f ∼ − i g , f ∼ i g and φ ( <<< − i ) ◦ f − = π i ◦ f − and φ ( <<< − i ) ◦ g − = π i ◦ g − . Thus, (the two-agent case of)Theorem 1 of Gilboa et al. (2004) implies that φ ( <<< ) = (1 − α ) φ ( <<< − i ) + απ i for some α ∈ R .If π i = φ ( <<< − i ) , we can choose α arbitrarily. Otherwise, π i = φ ( <<< ) , since Φ rules out beliefimposition, and so α = 1 . Lemma 2.
Assume that Φ satisfies restricted monotonicity and faithfulness and rules out beliefimposition. Let I ∈ I and <<< ∈ R I with ψ ( <<< ) = 0 . Then φ ( <<< ) = P i ∈ I λ i π i for some λ ∈ R I with P i ∈ I λ i = 1 . Moreover, if ( π i ) i ∈ I are affinely independent, then λ ∈ ( R − { } ) I and λ isunique.Proof. Since Φ is faithful, we have that φ ( < ) = π . Now let one agent after another join. Weapply Lemma 1 at each step and get φ ( <<< ) = P i ∈ I λ i π i for some λ ∈ R I with P i ∈ I λ i = 1 .If ( π i ) i ∈ I are affinely independent, λ is unique. We prove by induction over | I | that λ i = 0 for all i . If | I | = 1 , then λ = 1 is forced. Now suppose that | I | > and let i, j ∈ I . By theinduction hypothesis, we have φ ( <<< − i ) = P k ∈ I −{ i } λ ′ k π k and φ ( <<< − j ) = P k ∈ I −{ j } λ ′′ k π k for some λ ′ ∈ ( R − { } ) I −{ i } and λ ′′ ∈ ( R − { } ) I −{ j } . Lemma 1 implies that φ ( <<< ) = (1 − α ) φ ( <<< − i ) + απ i = (1 − β ) φ ( <<< − j ) + βπ j raft – May 11, 2020 for some α, β ∈ R − { } . We set λ = ((1 − α ) λ ′ , α ) = ((1 − β ) λ ′′ , β ) , where α and β appear inposition i and j respectively. Since α = 1 and λ ′ k = 0 , it follows that λ k = 0 for all k ∈ I − { i } .Similarly, β = 1 implies that λ k = 0 for all k ∈ I − { j } .We define the dimension of a vector of beliefs π ∈ Π I as the maximal number of affinelyindependent probability distributions in { π i : i ∈ I } . Equivalently, the dimension of π is thedimension of the subset { ( π i ( E )) i ∈ I : E ∈ Σ } of R I . For later use, we prove a fact for π withdimension at least 3. Lemma 3.
Assume that Φ satisfies restricted monotonicity and rules out belief imposition. Let I ∈ I and <<< ∈ R I with ψ ( <<< ) = 0 . If π has dimension at least 3, there are distinct i, j ∈ I suchthat φ ( <<< − i,j ) , π i , and π j are affinely independent.Proof. Suppose { , , } ⊂ I . Since π has dimension at least 3, we may assume that π , π ,and π are affinely independent. So φ ( <<< ) cannot be in the affine hull of all three pairs from { π , π , π } , for if say φ ( <<< ) is in the affine hull of { π , π } and { π , π } , then φ ( <<< ) = π and sois not in the affine hull of { π , π } . Assume that φ ( <<< ) is not in the affine hull of { π , π } . ThenLemma 1 implies that φ ( <<< − , ) is not in the affine hull of { π , π } . Since π = π , φ ( <<< − , ) , π ,and π are affinely independent.Lemma 2 ensures that the group belief is always an affine combination of the agents’ beliefs.To show that φ has the form claimed in Theorem 1, we have to prove that the relative weightof an agent in this affine combination depends only on her own belief and utility function. Fornow, we have to be contempt with a weaker conclusion, which allows negative weights. For therest of this section, we will assume that beliefs and utility functions are pairwise distinct in allprofiles . Lemma 4.
Assume that Φ satisfies restricted monotonicity and faithfulness and rules out beliefimposition. Then there are λ : R → ( R − { } ) N and for all I ∈ I , σ I : R I → { − , } I suchthat for all I ∈ I and <<< ∈ R I , φ ( <<< ) ≡ P i ∈ I σ Ii ( <<< ) λ i ( < i ) π i . Moreover, σ Ii ( <<< ) σ Ij ( <<< ) = σ { i,j } i ( < i , < j ) σ { i,j } j ( < i , < j ) forall I , i, j ∈ I , and <<< ∈ R I .Proof. For l ∈ R − { , } , let I l = { , , l } and R l ⊂ R I l be the set of all profiles for agents I l such that π , π , and π l are affinely independent. Let l ∈ N − { , } be arbitrary andfix some ˜ <<< ∈ R l . By Lemma 2, there is a unique function κ : R l → ( R − { } ) I l such that φ ( <<< ) = P i ∈ I l κ i ( <<< ) π i for all <<< ∈ R l . For i, j ∈ I l and <<< ∈ R l , let λ i,j ( <<< ) = κ j ( <<< − j , ˜ < j ) κ i ( <<< ) κ j ( ˜ <<< ) κ i ( <<< − j , ˜ < j ) .The fact that κ maps to ( R − { } ) I l ensures that λ i,j is well-defined. Then, let λ i ( < i ) = | κ i ( <<< ) || λ i,j ( <<< ) | and σ I l i ( <<< ) = sign( κ i ( <<< )) .We show that λ i is independent of j and <<< − i and thus well-defined. We proceed in threesteps. Before, note that the projection of R l to R that returns the preferences of i is onto, andso λ i is a function on all of R . 15 raft – May 11, 2020 Step . Let k ∈ I l − { i, j } . We show that κ i ( <<< ) κ j ( <<< ) is independent of < k . To this end, let <<< ′ ∈ R l such that <<< − k ′ = <<< − k . By Lemma 1, we have that φ ( <<< ) = (1 − α ) φ ( <<< − k ) + απ k = κ i ( <<< ) π i + κ j ( <<< ) π j + κ k ( <<< ) π k , and φ ( <<< ′ ) = (1 − β ) φ ( <<< − k ′ ) + βπ ′ k = κ i ( <<< ′ ) π i + κ j ( <<< ′ ) π j + κ k ( <<< ′ ) π ′ k , for some α, β ∈ R − { } . Affine independence of π , π , π l and π , π , π ′ l implies that κ i ( <<< ) π i + κ j ( <<< ) π j ≡ φ ( <<< − k ) = φ ( <<< − k ′ ) ≡ κ i ( <<< ′ ) π i + κ j ( <<< ′ ) π j . In particular, κ i ( <<< ) κ j ( <<< ) = κ i ( <<< ′ ) κ j ( <<< ′ ) . Repeatedapplication yields the desired independence. Step . We show that λ i,j ( <<< ) is independent of i and j . This is tedious, but only uses Step 1.Let k ∈ I l − { i, j } . First we show independence of j . λ i,j ( <<< ) = κ j ( <<< − j , ˜ < j ) κ i ( <<< ) κ j ( ˜ <<< ) κ i ( <<< − j , ˜ < j )= κ j ( <<< − j,k , ˜ < k , ˜ < j ) κ i ( <<< ) κ j ( ˜ π ) κ i ( <<< − j,k , ˜ < k , ˜ < j )= κ j ( <<< − j,k , ˜ < k , ˜ < j ) κ i ( <<< ) κ k ( <<< − j,k , ˜ < k , ˜ < j ) κ j ( ˜ <<< ) κ i ( <<< − j,k , ˜ < k , ˜ < j ) κ k ( <<< − j,k , ˜ < k , ˜ < j )= κ j ( ˜ <<< ) κ k ( <<< − k , ˜ < k ) κ i ( <<< ) κ j ( ˜ <<< ) κ i ( <<< − k , ˜ < k ) κ k ( ˜ <<< ) = λ i,k ( <<< ) Verifying independence of i is very similar. λ i,j ( <<< ) = κ j ( <<< − j , ˜ < j ) κ i ( <<< ) κ j ( ˜ <<< ) κ i ( <<< − j , ˜ < j )= κ j ( ˜ <<< − k,i , < k , < i ) κ i ( <<< ) κ j ( ˜ <<< ) κ i ( ˜ <<< − k,i , < k , < i )= κ j ( ˜ <<< − k,i , < k , < i ) κ i ( <<< ) κ k ( ˜ <<< − k,i , < k , < i ) κ j ( ˜ <<< ) κ i ( ˜ <<< − k,i , < k , < i ) κ k ( ˜ <<< − k,i , < k , < i )= κ j ( <<< − j , ˜ < j ) κ i ( <<< ) κ k ( <<< ) κ j ( ˜ <<< ) κ i ( <<< ) κ k ( <<< − j , ˜ < j ) = λ k,j ( <<< ) Step . We show that λ i ( <<< ) is independent of <<< − i and j . λ i ( < i ) = | κ i ( <<< ) || λ i,j ( <<< ) | = | κ j ( ˜ <<< ) κ i ( <<< − j , ˜ < j ) || κ j ( <<< − j , ˜ < j ) | = | κ j ( ˜ <<< ) κ i ( ˜ <<< − i , < i ) || κ j ( ˜ <<< − i , < i ) | , where we use Step 1 for the last equality. The resulting term is independent of <<< − i and, byStep 2, of j .Now it is easy to see that φ ( <<< ) = X i ∈ I l κ i ( <<< ) π i ≡ X i ∈ I l κ i ( <<< ) | λ i,j i ( <<< ) | π i = X i ∈ I l σ I l i ( <<< ) λ i ( < i ) π i , where j i ∈ I l − { i } for all i . For the second equality, we used the fact that λ i,j is independentof i and j . 16 raft – May 11, 2020 Since l was arbitrary, we have now defined λ i for each i ∈ N . However, we have defined λ and λ multiple times, once for each l ∈ N − { , } . So we have to check that these definitionsare not conflicting. It follows from Lemma 1 that the ratio between λ and λ is the same foreach triple { , , l } . Thus, we can define λ and λ as obtained for, say, l = 3 and scale thetriples ( λ , λ , λ l ) obtained for the remaining l appropriately. Step . Now we define σ for the remaining profiles. Our strategy will be to first define it fortwo-agent profiles, then inductively for all profiles such that π has dimension at least 3, andthen for the remaining profiles. At each point, we maintain that σ Ii ( <<< ) σ Ij ( <<< ) = σ { i,j } i ( % i , % j ) σ { i,j } j ( % i , % j ) , which wewill refer to as the ratio condition on σ . We omit the superscript in expressions like σ Ii ( <<< ) fromnow on, since it is clear from the profile.Let I ∈ I and <<< ∈ R I . If | I | = 2 , say I = { , } , then φ ( <<< ) = α π + α π for a unique α ∈ ( R − { } ) I . We define σ i ( <<< ) = sign( α i ) for i ∈ I .Now assume that | I | ≥ and π has dimension at least 3; assume further that we have defined σ for all profiles of dimension three on fewer than | I | agents such that the ratio condition holds.Let i ∈ I such that π i = φ ( <<< − i ) , which exists by Lemma 3. We show that there is s ∈ {− , } such that sσ j ( <<< − i ) = σ i ( % i , % j ) σ j ( % i , % j ) for all j ∈ I − { i } . If not, there are j, k which require s = 1 and s = − respectively. It is not hard to see that then there must be j, k with this property suchthat ( π i , π j , π k ) has dimension 3. Then we have σ j ( <<< − i ) σ k ( <<< − i ) = σ j ( % j , % k ) σ k ( % j , % k ) = σ j ( % i , % j ) σ i ( % i , % j ) σ i ( % i , % k ) σ k ( % i , % k ) = σ j ( <<< i,j,k ) σ i ( <<< i,j,k ) σ i ( <<< i,j,k ) σ k ( <<< i,j,k ) = σ j ( <<< i,j,k ) σ k ( <<< i,j,k ) where we use the fact that σ I −{ i } satisfies the ratio condition for the first equality. This is acontradiction, since σ { i,j,k } also satisfies the ratio condition. Thus we can find s as required.Lemma 1 implies that φ ( <<< ) = (1 − α ) φ ( <<< − i ) + απ i for some unique α ∈ R − { } . If α < ,we set σ I = ( σ I −{ i } , s ) ; if α > , set σ I = − ( σ I −{ i } , s ) .Lastly, if π has dimension , let i ∈ N − I and consider a profile <<< ′ for agents in I ∪{ i } such that <<< − i ′ = <<< and π ′ has dimension 3. By Lemma 2, π ′ i = φ ( <<< ) and so φ ( <<< ′ ) = (1 − α ) φ ( <<< ) + απ ′ i for some α = 1 . If α < , we set σ j ( <<< ) = σ j ( <<< ′ ) for all j ∈ I ; if α > , set σ j ( <<< ) = − σ j ( <<< ′ ) .We still have to make sure that these definitions of λ and σ are consistent with φ . For I ∈ I and <<< ∈ R I , let ¯ φ ( <<< ) ≡ P i ∈ I σ Ii ( <<< ) λ i ( < i ) π i . We show by induction over | I | that φ and ¯ φ agreeon all profiles <<< . First we assume that π has dimension at least 3. Later we will take care ofthe remaining profiles.We start with two observations. Step . Let <<< be a profile for agents in I such that π has dimension 3 and π − i has dimension2. If φ and ¯ φ agree on <<< , then they also agree on <<< − i . By Lemma 1 and the assumption, wehave φ ( <<< ) = (1 − α ) φ ( <<< − i ) + απ i ≡ X j ∈ I −{ i } σ j ( <<< ) λ j ( < j ) π j + σ i ( <<< ) λ i ( < i ) π i for some α ∈ R − { } . Since π i is not in the affine hull of ( π j ) j ∈ I −{ i } , we have to have (1 − α ) φ ( <<< − i ) ≡ P j ∈ I −{ i } σ j ( <<< ) λ j ( < j ) π j . If α < , then by definition, σ j ( <<< − i ) = σ j ( <<< ) raft – May 11, 2020 for all j ∈ I − { i } , and so φ ( <<< − i ) ≡ P j ∈ I −{ i } σ j ( <<< − i ) λ j ( < j ) π j ≡ ¯ φ ( <<< − i ) . If α > , then σ j ( <<< − i ) = − σ j ( <<< ) for all j , and again φ ( <<< − i ) = ¯ φ ( <<< − i ) follows. Step . Let <<< be a profile for agents in I and i, j ∈ I . If φ and ¯ φ agree on <<< − i and <<< − j and ¯ φ ( <<< − i,j ) , π i , and π j are affinely independent, then they also agree on <<< . Lemma 1 implies that φ ( <<< ) = (1 − α ) φ ( <<< − j ) + απ j = (1 − β ) φ ( <<< − i ) + βπ i for some α, β ∈ R − { } . To make notation less cumbersome, we write σ k ( <<< − J ) = σ Jk and λ k ( < k ) = λ k for k ∈ I and J ⊂ I − { i } for the rest of this step. Four cases arise, depending onwhether α and β are greater of smaller than 1. Case . Assume α, β < . By definition of σ , we have that σ k = σ jk for all k ∈ I − { j } and σ k = σ ik for all k ∈ I − { i } . In particular, σ ik = σ jk for k ∈ I − { i, j } . Moreover, either σ ijk = σ ik for all k ∈ I − { i, j } or σ ijk = − σ ik for all k . Let s = 1 in the former case and s = − otherwise.Then, φ ( <<< ) ≡ s X k ∈ I −{ i,j } σ ijk λ k π k | {z } φ ( <<< − i,j ) + σ ji λ i π i + α ′ π j = s X k ∈ I −{ i,j } σ ijk λ k π k + β ′ π i + σ ij λ j π j for some α ′ , β ′ ∈ R . Affine independence implies that α ′ = σ ij λ j = σ j λ j . Moreover, σ ji = σ i and sσ ijk = σ ik = σ k . So φ ( <<< ) = P k ∈ I σ k λ k π k , which concludes this case. Case . Assume α > and β < . By definition of σ , we have that σ k = − σ jk for all k ∈ I − { j } and σ k = σ ik for all k ∈ I − { i } . In particular, σ ik = − σ jk for k ∈ I − { i, j } . Moreover, either σ ijk = σ jk for all k ∈ I − { i, j } or σ ijk = − σ jk for all k . Let s = 1 in the former case and s = − otherwise. Then, φ ( <<< ) ≡ − φ ( <<< − j ) + α ′ π j ≡ − s X k ∈ I −{ i,j } σ ijk λ k π k − σ ji λ i π i + α ′ π j , and ≡ φ ( <<< − i ) + β ′ π i ≡ − s X k ∈ I −{ i,j } σ ijk λ k π k + β ′ π i + σ ij λ j π j for some α ′ , β ′ ∈ R . The second equality in the second line follows from − sσ ijk = − σ jk = σ ik for k ∈ I − { i, j } . Affine independence implies that α ′ = σ ij λ j = σ j λ j . Moreover, − σ ji = σ i and − sσ ijk = − σ jk = σ k . So φ ( <<< ) = P k ∈ I σ k λ k π k .The remaining two cases are analogous to the two we have examined and therefore omitted. Step . We have shown that φ and ¯ φ agree for groups I = { , , l } for all l , which we use forthe base case | I | = 3 . Let I = { , i, j } for distinct i, j ∈ N − { } and <<< ∈ R I such that π , π i ,and π j are affinely independent. Observe that φ and ¯ φ agree on the subprofiles <<< − i and <<< − j of <<< by Step 5. Since ¯ φ ( <<< − i,j ) = π , π i , and π j are affinely independent, Step 6 implies that φ and ¯ φ agree on <<< . With a second application of the same argument, we get that φ and ¯ φ agreeon all profiles of three agents with affinely independent beliefs.In the rest of the proof, we deal with the case | I | ≥ . Moreover, we assume that π hasdimension at least for now. 18 raft – May 11, 2020 Case . Suppose π − k has dimension 2 for some k ∈ I . Thus, all beliefs in π − k are linearcombinations of π i and π j for distinct but otherwise arbitrary i, j ∈ I − { k } . Since π hasdimension 3, π k is not in the affine hull of the beliefs in π − k . So any subprofile of π with atleast three agents one of which is k has dimension 3. By the induction hypothesis, φ and ¯ φ agree on such profiles except for possibly π itself. In particular, they agree on <<< − i and <<< − j .Moreover, ¯ φ ( <<< − i,j ) = απ i + βπ j + γπ k for α, β, γ ∈ R with γ = 0 by Lemma 2. So ¯ φ ( <<< − i,j ) , π i ,and π j are affinely independent. By Step 6, we get that φ and ¯ φ agree on <<< . Case . The remaining case is that π − k has dimension 3 for all k ∈ I . The induction hypothesisimplies that φ and ¯ φ agree on <<< − j and <<< − i . The argument in the proof of Lemma 3 also appliesto ¯ φ , and so we can choose distinct i, j ∈ I such that ¯ φ ( <<< − i,j ) , π j , and π i are affinely independent.We again conclude from Step 6 that φ and ¯ φ agree on <<< .Now let <<< be an arbitrary profile for agents in I . We have covered the case when π hasdimension at least 3. If π has dimension 2, let i ∈ N − I and ( ˜ <<< ) be a profile for agents in I ∪ { i } such that ˜ <<< − i = <<< and ˜ π has dimension 3. Then φ ( ˜ <<< ) = ¯ φ ( ˜ <<< ) and so Step 5 applies,which gives φ ( <<< ) = ¯ φ ( <<< ) . Single-agent profiles are covered by Lemma 2. A.2. Aggregation of Utility Functions
To begin with, it is useful to clarify the linear algebra on ¯ U . Elements of ¯ U are normalizedrepresentatives of a class of utility functions, consisting of all its positive affine transformations.Thus, we say that ( u i ) i ∈ I are linearly independent if their span does not include any utilityfunction that is equivalent to the element of ¯ U , that is, any constant utility function.We show that the collective utility function of a group containing agent i is a linear combina-tion of the utility function of i and that of the group without i . If the latter two utility functionsare not equal to completely opposed, then i has positive weight in this linear combination. InLemma 6, we leverage this fact to prove that the collective utility function of any group is a positive linear combination of the utility functions of its members. Lemma 5.
Assume that Φ satisfies monotonicity. Let I ∈ I , i ∈ I , and <<< ∈ R I . Then ψ ( <<< ) = αψ ( <<< − i ) + βu i for some α, β ∈ R . Moreover, if u i = ± ψ ( <<< − i ) , then β > and β isunique.Proof. Let < = Φ( <<< ) , < − i = Φ( <<< − i ) , and π − i = φ ( <<< − i ) . We start in the same way as forLemma 1. Monotonicity implies that f ∼ g whenever f ∼ − i g , f ∼ i g , π − i ◦ f − = π i ◦ f − ,and π − i ◦ g − = π i ◦ g − . Thus, it follows from Theorem 1 of Gilboa et al. (2004) that ψ ( <<< ) = αψ ( <<< − i ) + βu i for some α, β ∈ R .If u i = ± ψ ( <<< − i ) , then β is unique. Moreover, we can find probability distributions p and q on X with finite support such that ψ ( <<< − i )( p ) = ψ ( <<< − i )( q ) and u i ( p ) > u i ( q ) . Liapounoff’s (1940)theorem allows us to construct acts f and g with the following properties: they induce thedistributions p and q under π − i and π i , that is, p = π − i ◦ f − = π i ◦ f − and q = π − i ◦ g − = π i ◦ g − ; Thus, f ∼ − i g and f ≻ i g . Since Φ is monotonic, we get that f ≻ g . From Lemma 1, weknow that φ ( <<< ) is an affine combination of π − i and π i , and so φ ( <<< ) ◦ f − = p and φ ( <<< ) ◦ g − = q .It follows that f ≻ g if and only if ψ ( <<< )( p ) > ψ ( <<< )( q ) . Thus, β > .19 raft – May 11, 2020 Lemma 6.
Assume that Φ satisfies monotonicity and faithfulness. Let I ∈ I and <<< ∈ R I .Then ψ ( <<< ) = P i ∈ I µ i u i for some µ ∈ R I . If ( u i ) i ∈ I are linearly independent, µ ∈ R I ++ and µ is unique.Proof. The first part is a straightforward corollary of Lemma 5. For the second part, assume that ( u i ) i ∈ I are linearly independent. Let µ ∈ R I such that ψ ( <<< ) = P i ∈ I µ i u i . Linear independenceimplies that µ is unique and u i = ± ψ ( <<< − i ) for all i ∈ I . Thus, Lemma 5 implies that ψ ( <<< ) = αψ ( <<< − i ) + βu i for α ∈ R and β > . Since ψ ( <<< − i ) is a linear combination of ( u j ) j ∈ I −{ i } and µ is unique, it follows that µ i = β > .The dimension of an vector of utility functions u ∈ U I is the maximal number of linearlyindependent utility functions in { u i : i ∈ I } . The next lemma is the analogue of Lemma 3. Itsproof is similar and therefore omitted. Lemma 7.
Assume that Φ satisfies restricted monotonicity. Let I ∈ I and <<< ∈ R I with ψ ( <<< ) = 0 . If u has dimension at least 3, there are distinct i, j ∈ I such that ψ ( <<< − i,j ) , u i , and u j are linearly independent. In general, µ may depend on <<< . The content of the next lemma is that µ i must not dependon <<< − i . For the rest of this section, we assume that beliefs and utility functions are pairwisedistinct in any profile . Lemma 8.
Assume that | X | ≥ and Φ satisfies monotonicity and faithfulness. Then there is µ : R → R N ++ such that ψ ( <<< ) ≡ P i ∈ I µ i ( < i ) u i for all I ∈ I and <<< ∈ R I .Proof. The first part is very similar to the construction of λ in the proof of Lemma 4. For l ∈ N − { , } , let I l = { , , l } and R l be the set of all <<< ∈ R I l such that u , u , and u l arelinearly independent. Let l ∈ N − { , } be arbitrary and fix some ˜ <<< ∈ R l . By Lemma 6,there is a unique function ν : R l → R I l ++ such that ψ ( <<< ) = P i ∈ I l ν i ( <<< ) u i for all <<< ∈ R l . For i, j ∈ I l and <<< ∈ R l , let λ i,j ( <<< ) = ν j ( <<< − j , ˜ < j ) ν i ( <<< ) ν j ( ˜ <<< ) ν i ( <<< − j , ˜ < j ) . The fact that ν maps to R I l ++ ensures that λ i,j is well-defined and positive. Then, let µ i ( < i ) = ν i ( <<< ) λ i,j ( <<< ) . Note that the projection of R l to R that returns the preferences of i is onto, and so µ i is a function on all of R . Here we use theassumption that | X | ≥ , as otherwise R l is empty.With the same arguments as in the proof of Lemma 4, we can show that ν i ( <<< ) ν j ( <<< ) is independentof < k for k ∈ I l − { i, j } , that λ i,j is independent of i and j , and that µ i is well-defined. Thenwe have ψ ( <<< ) = X i ∈ I l ν i ( <<< ) u i ≡ X i ∈ I ν i ( <<< ) λ i,j i ( <<< ) u i = X i ∈ I µ i ( < i ) u i , where j i ∈ I − { i } for all i ∈ I .Since l was arbitrary, we have now defined µ i for each i ∈ N . However, we have defined µ and µ multiple times, once for each l ∈ N − { , } . So we have to check that these definitionsare not conflicting. It follows from Lemma 1 that the ratio between µ and µ is the same foreach triple { , , l } . Thus, we can define µ and µ as obtained for, say, l = 3 and scale thetriples ( µ , µ , µ l ) obtained for the remaining l appropriately.20 raft – May 11, 2020 µ defines a function that returns a collective utility function for every profile. For I ∈ I and <<< ∈ R I , let ¯ ψ ( <<< ) ≡ P i ∈ I µ i ( < i ) u i . The following two observations will carry us a long way inthe rest of the proof. Step . Let <<< be a profile for agents in I such that u has dimension 3 and u − i has dimension2. If ψ and ¯ ψ agree on <<< , then they also agree on <<< − i . By Lemma 5 and the assumption, wehave ψ ( <<< ) = αψ ( <<< − i ) + βu i ≡ X j ∈ I −{ i } µ j ( < j ) u j + µ i ( < i ) u i for some α, β ∈ R . Since u i is not in the span of ( u j ) j ∈ I −{ i } , we get that ψ ( <<< − i ) ≡ P j ∈ I −{ i } µ j ( < j ) u j ≡ ¯ ψ ( <<< − i ) . Step . Let <<< be a profile for agents in I and i, j ∈ I . If ¯ ψ ( <<< − i,j ) , u i , and u j are linearlyindependent and ψ and ¯ ψ agree on <<< − i and <<< − j , then they also agree on <<< .Lemma 5 and Lemma 6 imply that ψ ( <<< ) ≡ ψ ( <<< − j ) + αu j ≡ ≡ ¯ ψ ( <<< − j ) z }| {X k ∈ I −{ i,j } µ k ( < k ) u k + µ i ( < i ) u i + α ′ u j , and ψ ( <<< ) ≡ ψ ( <<< − i ) + βu i ≡ X k ∈ I −{ i,j } µ k ( < k ) u k | {z } ≡ ¯ ψ ( <<< − i,j ) + β ′ u i + µ j ( < j ) u j for some α, α ′ β, β ′ ∈ R ++ . Linear independence of ¯ ψ ( <<< − i,j ) , u i , and u j implies that α ′ = µ j ( < j ) ,and so ψ and ¯ ψ agree on <<< .With all this in place, we can finish the proof. First we show by induction over | I | that ψ and ¯ ψ agree on all profiles where u has dimension at least 3. Later we take care of the remainingprofiles later.The base case is | I | = 3 . Let <<< ∈ R I . First assume that I = { , i, j } for distinct i, j ∈ N −{ } .We have shown that ψ and ¯ ψ agree for groups of the form { , , l } for any l . Thus Step 1 impliesthat they agree on <<< − i and <<< − j . Moreover, ¯ ψ ( <<< − i,j ) = u , u i , and u j are linearly independent.So Step 2 implies that ψ and ¯ ψ agree on <<< . A second iteration of the same argument impliesthat they agree on profiles for three arbitrary agents.Now we deal with the case | I | ≥ and again assume that u has dimension at least . Case . Suppose ψ ( <<< ) = 0 or ¯ ψ ( <<< ) = 0 . We show that ψ ( <<< ) = 0 if and only if ¯ ψ ( <<< ) = 0 .Assume for contradiction that ψ ( <<< ) = 0 and ¯ ψ ( <<< ) = 0 . By Lemma 7, we can find distinct i, j such that ψ ( <<< − i,j ) , u i , and u j are linearly independent. Since | I | ≥ , u − i,j has dimension atleast 2. If it has dimension exactly 2, then either u − i or u − j has dimension 3, as otherwise u would have dimension 2. Suppose u − j has dimension at least 3. By the induction hypothesis, ψ ( <<< − j ) = ¯ ψ ( <<< − j ) . Since ψ ( <<< − i,j ) , u i , and u j are linearly independent, we can conclude that u j = ± ψ ( <<< − j ) . But ¯ ψ ( <<< ) ≡ ¯ ψ ( <<< − j ) + αu j for some α > , and so since ± u j = ψ ( <<< − j ) =¯ ψ ( <<< − j ) , we get ¯ ψ ( <<< ) = 0 . 21 raft – May 11, 2020 The proof is similar if ψ ( <<< ) = 0 and ¯ ψ ( <<< ) = 0 . Note however, that we find i, j suchthat ¯ ψ ( <<< − i,j ) , u i , and u j are linearly independent not directly by Lemma 7, but by the sameargument as in its proof. Case . Suppose u − k has dimension 2 for some k ∈ I . Thus, all beliefs in u − k are linearcombinations of u i and u j for distinct but otherwise arbitrary i, j ∈ I − { k } . Since u hasdimension 3, u k is not in the span of the utility functions in u − k . So any subprofile of u with atleast 3 agents one of which is k has dimension 3. By the induction hypothesis, ψ and ¯ ψ agree onsuch profiles except for possibly <<< itself. In particular, they agree on <<< − i and <<< − j . Moreover, ¯ ψ ( <<< − i,j ) = αu i + βu j + γu k for α, β, γ ∈ R with γ > by definition of ¯ ψ . So ¯ ψ ( <<< − i,j ) , u i , and u j are linearly independent. Step 2 implies that ψ ( <<< ) = ¯ ψ ( <<< ) . Case . The remaining case is that u − k has dimension at least 3 for all k ∈ I . The inductionhypothesis implies that ψ and ¯ ψ agree on <<< − i and <<< − j . By Lemma 7 we can choose distinct i, j ∈ I such that ψ ( <<< − i,j ) , u i , and u j are linearly independent. If u − i,j has dimension 3, theinduction hypothesis implies that ψ and ¯ ψ agree on <<< − i,j . If u − i,j has dimension 2, then we usethe fact that u − i has dimension 3 to apply Step 1 and conclude that ψ and ¯ ψ agree on <<< − i,j .In either case, ¯ ψ ( <<< − i,j ) , u i , and u j are linearly independent. So Step 2 implies that ψ and ¯ ψ agree on <<< .Now let <<< be an arbitrary profile for agents in I . We have covered the case when u hasdimension at least 3. If u has dimension 2, let i ∈ N − I and ˜ <<< be a profile for agents in I ∪ { i } such that ˜ <<< − i = <<< and ˜ u has dimension 3. Then ψ ( ˜ <<< ) = ¯ ψ ( ˜ <<< ) , and so Step 1 implies ψ ( <<< ) = ¯ ψ ( <<< ) . Single-agent profiles are covered by Lemma 6. B. Implications of Continuity
Let us first define topologies on Π and U ∗ , which we do in the same way as for ¯ R . For π, π ′ ∈ Π ,the uniform metric sup {| π ( E ) − π ′ ( E ) | : E ∈ Σ } gives a topology on π . For u, u ′ ∈ U (note theabsence of constant utility function 0), we also use the uniform metric sup {| u ( x ) − u ′ ( x ) | : x ∈ X } . The topology on ¯ U is that of U plus the entire set ¯ U . So the only neighborhood of 0 isthe set ¯ U itself. This is the topology ¯ U inherits from the space of all utility functions equippedwith the uniform metric when forming the quotient via normalization.The mappings from preference relations to beliefs and utility functions are now continuous.Likewise, the inverse operation mapping a pair of belief and utility function to a preferencerelation is continuous. Lemma 5 of Dietrich (2019) is the equivalent of this statement in hisframework. To ease notation in the proof of the next lemma, when E is an event and x, y areconsequences, we write xEy for the act which yields x for states in E and y for states in Ω − E . Lemma 9.
The correspondence π ( < ) and the function u ( < ) mapping < ∈ ¯ R to the beliefsand the utility function representing < are (upper-hemi) continuous. Moreover, the function < ( π, u ) mapping each pair of belief and utility function to the preference relation it induces iscontinuous. raft – May 11, 2020 Proof.
Let ( < n ) be a sequence that converges to < in ¯ R . For each n , let π n ∈ π ( < n ) and u n = u ( < n ) .First we show that ( u n ) converges to u = u ( < ) . Let x ∈ X and f x be the act that returns x in all states. We have sup {| u n ( x ) − u ( x ) | : x ∈ X } = sup {| E < n ( f x ) − E < ( f x ) | : x ∈ X } , and so ( u n ) converges uniformly to u .Second, assume that ( π n ) converges to π ′ ∈ Π . We have to show π ′ ∈ π ( < ) . If u = 0 , then π ( < ) = Π and there is nothing to show. Otherwise, π ( < ) = { π } for some π ∈ Π . We showthat ( π n ) converges to π , which implies π = π ′ . Since u = 0 , we can choose x, y ∈ X such that u ( x ) > u ( y ) .Then, for large enough n , sup {| π n ( E ) − π ( E ) | : E ∈ Σ } = sup {| E < n ( xEy ) − u n ( y ) u n ( x ) − u n ( y ) − E < ( xEy ) − u ( y ) u ( x ) − u ( y ) | : E ∈ Σ } ≤ u ( x ) − u ( y ) sup {| E < n ( xEy ) − E < ( xEy ) | : E ∈ Σ } , and so ( π n ) convergesuniformly to π .Conversely, assume that ( π n ) and ( u n ) converge to π and u , respectively, and let < n = < ( π n , u n ) and < = < ( π, u ) be the induced preference relations. For ǫ > , let n ∈ N such thatfor n ≥ n , sup {| π n ( E ) − π ( E ) | : E ∈ Σ } < ǫ and sup {| u n ( x ) − u ( x ) | : x ∈ X } < ǫ . Then, forall n ≥ n and f ∈ F , | E < n ( f ) − E < ( f ) | = | Z Ω ( u n ◦ f ) dπ n − Z Ω ( u ◦ f ) dπ |≤ | Z Ω ( u n ◦ f − u ◦ f ) dπ n | + | Z Ω ( u ◦ f ) dπ n − Z Ω ( u ◦ f ) dπ | < ǫ ǫ ǫ Thus E < n converges uniformly to E < .What still separates us from the desired statement of Proposition 1 after establishing Lemma 4and Lemma 8 is that Lemma 4 allows negative weights for beliefs and that both lemmas onlyapply if beliefs and utility functions are pairwise distinct. The latter is not hard to deal with,but showing that weights cannot be negative takes more work. Proposition 1.
Let | X | ≥ and Φ be an aggregation function. Then the following are equiva-lent.(i) Φ satisfies restricted monotonicity, faithfulness, no belief imposition, and continuity(ii) There are continuous functions λ , µ : R → R N ++ such that for all I ∈ I and <<< ∈ R I , Φ( <<< ) is represented by P i ∈ I λ i ( < i ) P i ∈ I λ i ( < i ) π i and P i ∈ I µ i ( < i ) u i Proof.
One can check easily that ( ii ) implies ( i ) . The rest of the proof will establish that ( i ) implies ( ii ) .Let λ , µ , and for all I ∈ I , σ I be the functions obtained from Lemma 4 and Lemma 8. Step . We show that µ is continuous. Let i, j ∈ N and <<< ∈ R { i,j } such that π i = π j and u i = ± u j ; let ( < in ) be a sequence in R converging to < i and <<< n = ( < in , < j ) . By assumption, Φ is continuous, and by Lemma 9, the mapping from preference relations to the correspondingutility functions is continuous. Thus, u n = ψ ( <<< n ) ≡ µ i ( u ni ) u ni + µ j ( u j ) u j converges to u = raft – May 11, 2020 ψ ( <<< ) ≡ µ i ( u i ) u i + µ j ( u j ) u j . First, µ i ( < in ) is bounded, as otherwise, a subsequence of ( u n ) would converge to u i . But this is impossible, since µ j ( u j ) = 0 and u i = ± u j . Now if α isan accumulation point of ( µ i ( u ni )) , then αu i + µ j ( u j ) u j ≡ u , since ( u n ) converges to u . But αu i + µ j ( u j ) u j ≡ βu i + µ j ( u j ) u j if and only if α = β . So ( µ i ( u ni )) is bounded and has a uniqueaccumulation point. Thus, it converges to µ i ( u i ) . Step . Let i, j ∈ N and <<< ∈ R { i,j } such that π i = π j , u i = ± u j , and Φ( <<< ) is not completeindifference. We show that σ { i,j } i λ i is continuous at <<< . (For convenience, we will omit thesuperscript { i, j } from now on.) Let ( <<< n ) be a sequence of profiles converging to <<< . Let α n = σ i ( <<< n ) λ i ( < in ) and β n = σ j ( <<< n ) λ j ( < jn ) , and α = σ i ( <<< ) λ i ( < i ) and β = σ j ( <<< ) λ j ( < j ) .First we prove convergence when j ’s preferences remain constant at < j . Let ¯ <<< n = ( < in , < j ) , ¯ α n = σ i ( ¯ <<< n ) λ i ( < in ) , and ¯ β n = σ j ( ¯ <<< n ) λ j ( < j ) . We need to show that ( ¯ α n ) converges to α . Notethat ¯ β n can only vary in sign but not in absolute value. Since Φ and the correspondence mappingpreference relations to the corresponding beliefs are continuous, we have that ¯ α n π ni + ¯ β n π j ≡ φ ( ¯ <<< n ) → φ ( <<< ) ≡ απ i + βπ j . With the same reasoning as in Step 1, we get that ( ¯ α n ) is boundedand has a unique accumulation point. Thus, it converges to α . Similarly, σ j ( < i , < j n ) λ j ( < jn ) converges to β .Now we show that ( α n ) converges α . We already know that the sequences of absolute valuesof ( α n ) and ( β n ) converge to α and β , respectively. So any subsequence ( α n k , β n k ) such thatall α n k and all β n k have the same sign converges. By the same reasoning as in the previousparagraph, we conclude that α n k π n k i + β n k π n k j ≡ φ ( <<< n k ) → φ ( <<< ) ≡ απ i + βπ j . Since π i = π j ,this implies that ( α n k , β n k ) converges to ( α, β ) . Thus ( α n , β n ) converges to ( α, β ) . Step . Now we deduce that σ i is always equal to 1. Assume for contradiction that there isa profile ˜ <<< ∈ R { i,j } such that ˜ π i = ˜ π j , ˜ u i = ± ˜ u j , Φ( ˜ <<< ) is not complete indifference, and σ i ( ˜ <<< ) = − . Since σ i λ i and σ j λ j are continuous at ˜ <<< by Step 2, we can find a neighborhoodof ˜ <<< such that σ i ( <<< ) = − for all profiles <<< contained in it. In particular, we can find ǫ > such that σ i ( <<< ) = − whenever < i = ˜ < i , u j = ˜ u j , and sup {| π j ( E ) − ˜ π j ( E ) | : E ∈ Σ } < ǫ . Let ˜ P be this set of profiles. By Liapounoff’s (1940) theorem, we can find an event E such that ˜ π i ( E ) = ˜ π j ( E ) = ǫ . But then ˜ P contains a profile <<< such that σ i ( <<< ) = − , π i ( E ) = ˜ π i ( E ) = ǫ and π j ( E ) = 0 . This is not possible, since σ i ( <<< ) λ i ( < i ) π i ( E ) + σ j ( <<< ) λ j ( < j ) π j ( E ) would benegative.Since j was arbitrary and σ I satisfies the restriction on the ratio of σ Ii and σ Ij stated inLemma 4, it follows that σ Ii is constant at 1 for all I . Since we have shown that σ i λ i iscontinuous, so is λ i .(Alternatively, one could show that the set of profiles where beliefs and utility functions arepairwise distinct and Φ is not complete indifference is connected. Then the fact that σ i λ i iscontinuous and never 0 implies that it cannot change sign.) Step . Let ¯Φ be the aggregation function where ¯Φ( <<< ) is represented by ¯ φ ( <<< ) ≡ P i ∈ I λ i ( < i ) π i and ¯ ψ ( <<< ) ≡ P i ∈ I µ i ( < i ) u i for every profile <<< with agents in I ∈ I . We know that Φ and ¯Φ agree on all profiles for which beliefs and utility functions are pairwise distinct. These profilesare dense in R I for all I . Our task is to show that they agree on an arbitrary profile <<< ∈ R I .24 raft – May 11, 2020 Case . Suppose neither Φ( <<< ) nor ¯Φ( <<< ) is complete indifference. By Lemma 9, Step 1, andStep 2, φ, ψ, ¯ φ , and ¯ ψ are continuous at <<< . Moreover, the pairs φ and ¯ φ and ψ and ¯ ψ agree ona set of profiles with <<< in its closure. Thus, φ ( <<< ) = ¯ φ ( <<< ) and ψ ( <<< ) = ¯ ψ ( <<< ) . It follows that Φ( <<< ) = ¯Φ( <<< ) . Case . Suppose that Φ( <<< ) is complete indifference. (The proof is analogous if ¯Φ( <<< ) is completeindifference.) Let i ∈ N − I with preferences < i such that u i = ± ¯ ψ ( <<< ) . Then for <<< + i =( <<< , < i ) , by Lemma 5, ψ ( <<< + i ) = u i and ¯ ψ ( <<< + i ) ≡ ¯ ψ ( <<< ) + αu i for some α > . In particular, ψ ( <<< + i ) , ¯ ψ ( <<< + i ) = 0 . Thus Case 1 implies that ψ ( <<< + i ) = ¯ ψ ( <<< + i ) . Since u i = ± ¯ ψ ( <<< ) , this canonly be if ¯ ψ ( <<< ) = 0 , and hence ¯Φ( <<< ) is complete indifference. C. Implications of Independence of Redundant Acts
Using independence of redundant acts, we derive a lemma which, together with Proposition 1,concludes the proof of Theorem 1. But first we need an auxiliary statement. Recall that afunction is simple if it has finite range.
Lemma 10.
Let I ∈ I and i ∈ I ; let <<< ∈ R I such that u j is simple for j ∈ I − { i } . Then forevery act f , there is a simple act g such that f ∼ j g for all j ∈ I .Proof. Put differently, we want to show that for every act f , there is a simple act g such that ( E < j ( f )) j ∈ I = ( E < j ( g )) j ∈ I .We first show that the sets X + = { x ∈ X : u i ( x ) ≥ E < i ( f ) } and X − = { x ∈ X : u i ( x ) ≤ E < i ( f ) } are non-empty. If X + is empty, then Ω = S k ∈ N { s ∈ Ω : u i ( f ( s )) ≤ E < i ( f ) − k } .Note that all sets in this union are measurable. Since π i is countably additive, there is k suchthat π i ( { s ∈ Ω : u i ( f ( s )) ≤ E < i ( f ) − k } ) = ǫ > . The fact that X + is empty then gives E < i ( f ) ≤ E < i ( f ) − ǫk , which is a contradiction. Similarly, one shows that X − is non-empty.Now let V = { u − i ( x ) : x ∈ X } ⊂ R I −{ i } be the range of u − i . Since all u j are simple, V isfinite. We partition the set of consequences X into the measurable sets X v = u − − i ( v ) and the setof states Ω into the measurable sets E v = f − ( X v ) with v ranging over V . To define g , considertwo cases. If π i ( E v ) = 0 , choose x v ∈ X v arbitrarily and let g ( s ) = x v for all s ∈ E v . If π i ( E v ) > , then π i ( E v ) π i | E v is a probability measure on E v , where π i | E v is π i restricted to events containedin E v . By the previous paragraph, the sets X + v = { x ∈ X v : u i ( x ) ≥ π i ( E v ) R E v ( u i ◦ f ) d ( π i | E v ) } and X − v = { x ∈ X v : u i ( x ) ≤ π i ( E v ) R E v ( u i ◦ f ) d ( π i | E v ) } are non-empty. Thus, there are x + ∈ X + v and x − ∈ X − v and α ∈ [0 , such that αu i ( x + ) + (1 − α ) u i ( x − ) = π i ( E v ) R E v ( u i ◦ f ) d ( π i | E v ) .Since π i is non-atomic, there is E + v ⊂ E v such that π i ( E + v ) = απ i ( E v ) . We define g ( s ) = x + for s ∈ E + v and g ( s ) = x − for s ∈ E v − E + v . This gives Z E v ( u i ◦ f ) d ( π i | E v ) = π i ( E v )( αu i ( x + ) + (1 − α ) u i ( x − )= π i ( E + v ) u i ( x + ) + π i ( E v − E + v ) u i ( x − ) = Z E v ( u i ◦ g ) d ( π i | E v ) Also, since u − i is constant on X v , R E v ( u j ◦ f ) d ( π j | E v ) = R E v ( u j ◦ g ) d ( π j | E v ) for all j ∈ I − { i } .In summary, we have ( E < j ( f )) j ∈ I = ( E < j ( g )) j ∈ I .25 raft – May 11, 2020 Lemma 11.
Let λ , µ : R → R N ++ be continuous functions; let Φ be an aggregation functionsuch that for every I ∈ I and <<< ∈ R I , Φ( <<< ) is represented by P i ∈ I λ i ( < i ) P i ∈ I λ i ( < i ) π i and P i ∈ I µ i ( < i ) u i . Then if Φ satisfies independence of redundant acts, λ and µ are constant.Proof. Let i, j ∈ N and < i , < i ′ ∈ R . We want to show that λ i ( < i ) = λ i ( < i ′ ) and µ i ( < i ) = µ i ( < i ′ ) . In the first step, we show that λ i and µ i are independent of π i . In the rest of the proof,we show that they are independent of u i , too. Step . Assume that u i = u ′ i . First we show λ i ( < i ) = λ i ( < i ′ ) . Let Λ = { E ∈ Σ : π i ( E ) = π ′ i ( E ) } .We construct a suitable belief for agent j . Let E ∈ Λ such that π i ( E ) = . If π i = π ′ i , thereis nothing to show. Otherwise, either π i | E = π ′ i | E or π i | Ω − E = π ′ i | Ω − E . Assume the former istrue. Then define π j so that π j ( F ) = 2 π i ( F ) for every F ⊂ E and π j (Ω − E ) = 0 . Moreover,choose u j ∈ U so that u j is simple and u j = ± u i and let < j be represented by π j and u j .The set of acts to which we will apply independence of redundant acts is G = { g ∈F : g is simple and g − ( x ) ∈ Λ for every x ∈ X } . To meet the antecedent of independenceof redundant acts, we have to show that for every f ∈ F , there is g ∈ G such that f ∼ i g and f ∼ j g . (The choice of G and u i = u ′ i ensure that also f ∼ ′ i g .)By Lemma 10, we may assume that f is simple. Define g as follows: let f (Ω) = { x , . . . , x k } be the range of f . For every x l , let α l = π i ( E ∩ f − ( x l )) and α cl = π i ((Ω − E ) ∩ f − ( x l )) .(Note that π j ( E ∩ f − ( x l )) = 2 α l .) Liapounoff’s theorem allows us to find events E l ⊂ E and E cl ⊂ Ω − E in Λ such that π i ( E l ) = α l and π i ( E cl ) = α cl . In fact, we can partition E and Ω − E into { E , . . . , E k } and { E c , . . . , E ck } , respectively. Then let g ( s ) = x l for s ∈ E l ∪ E cl . One cancheck that π j ( E l ∪ E cl ) = 2 π i ( E l ) = 2 α l . Thus, E < i ( f ) = E < i ( g ) and E < j ( f ) = E < j ( g ) and so f ∼ i g and f ∼ j g .Let < = f ( < i , < j ) and < ′ = f ( < i ′ , < j ) and π, u, π ′ , u ′ be the corresponding beliefs and utilityfunctions. Independence of redundant acts applied to the profiles ( < i , < j ) and ( < i ′ , < j ) gives g < g ′ if and only if g < ′ g ′ for all g, g ′ ∈ G .Assume that λ i ( < i ) = λ i ( < i ′ ) . First, since u j = ± u i and u ≡ λ i ( u i ) u i + λ j ( u j ) u j , < cannotbe complete indifference, and so we can find consequences x and y such that x ≻ y . Recall that π i ( E ) = π ′ i ( E ) = and π j ( E ) = 1 . It follows that π ( E ) = π ′ ( E ) and π ( E ) , π ′ ( E ) > . So thereis an event E ′ ⊂ E such that E ′ ∈ Λ , π ( E ′ ) = , and π ′ ( E ′ ) = . Thus, xE ′ y and yE ′ x are actsin G but xE ′ y yE ′ x and xE ′ y ∼ ′ yE ′ x . This contradicts independence of redundant acts andso λ i ( < i ) = λ i ( < i ′ ) .Second, assume that µ i ( < i ) = µ i ( < i ′ ) . Since u j = ± u i , it follows that u = u ′ and we can a findsimple lotteries p and q on X such that u ( p ) > u ( q ) but u ′ ( q ) ≥ u ′ ( p ) . Since λ i ( < i ) = λ i ( < i ′ ) by the previous paragraph, π ( F ) = π ′ ( F ) if and only if F ∈ Λ . So by Liapounoff’s theorem, wecan find acts g and g ′ in G with g ◦ π = p and g ′ ◦ π ′ = q . This gives g ≻ g ′ but g ′ < ′ g , whichcontradicts independence of redundant acts. We conclude that µ i ( < i ) = µ i ( < i ′ ) . Step . By Step 1, we can view λ i and µ i as functions λ i ( u i ) and µ i ( u i ) of u i . We show thatboth these functions are constant.Recall that U consists of utility functions which are normalized to the unit interval, thatis, inf x u ( x ) = 0 and sup x u ( x ) = 1 . Let U ′ = { u ∈ U : there exist x, y ∈ X with u ( x ) = raft – May 11, 2020 u i u j u ( x ) u ( x ) u ( x ∗ ) u ( y ) u ( y ) u ( y ) 0 1 u ′′ i u j u ′′ ( x ) u ′′ ( x ) u ′′ ( x ∗ ) = u ′′ ( y ) u ′′ ( y ) u ′′ ( y ) Figure 1: The images of u = ( u i , u j ) and u ′′ = ( u ′′ i , u j ) in utility space. For example, u ( x ) =( u i ( x ) , u j ( x )) = (0 , . The consequences y , y , and y are examples for the threecases in the definition of u ′′ i . and u ( y ) = 1 } be those utility functions for which the infimum and the supremum are attained.Observe that the closure of U ′ is U . Thus, since λ i and µ i are continuous, it suffices to showthat they are constant on U ′ . This we do now.Let u i ∈ U ′ ; let x , x ∈ X such that u i ( x ) = 0 and u i ( x ) = 1 and x ∗ ∈ X − { x , x } bearbitrary; let u ′ i be such that u ′ i ( x ) = u i ( x ) for x ∈ { x , x , x ∗ } . We show that λ i ( u i ) = λ i ( u ′ i ) and µ i ( u i ) = µ i ( u ′ i ) . Since | X | ≥ , repeated application of this statement gives the sameconclusion for all u ′ i ∈ U ′ .Let u ′′ i be such that u ′′ i ( x ) = u i ( x ∗ ) if u i ( x ) < u i ( x ∗ ) and u ′ i ( x ) > u i ( x ∗ ) u ′ i ( x ) if u i ( x ) < u i ( x ∗ ) and u ′ i ( x ) ≤ u i ( x ∗ ) u i ( x ) if u i ( x ) ≥ u i ( x ∗ ) Note that u ′′ i ( x ) = u ′ i ( x ) = u i ( x ) for x ∈ { x , x , x ∗ } . We want to apply independence ofredundant acts to profiles with utility functions ( u i , u j ) and ( u ′′ i , u j ) and the set of acts G = { f ∈ F : f (Ω) ⊂ { x , x , x ∗ }} . This requires choosing u j appropriately. Let u j be such that u j ( x ) = if u i ( x ) ≤ u i ( x ∗ ) u i ( x ) otherwiseFigure 1 depicts the images of ( u i , u j ) and ( u ′′ i , u j ) in utility space. From u i to u ′′ i , we adjustthe utility for consequences with u i ( x ) ≤ u i ( x ∗ ) toward u ′ i ( x ) without raising it above u i ( x ∗ ) .Setting u j as we did, we can now apply independence of redundant acts to the correspondingprofiles.Let π i , π j ∈ Π with π i = π j and < i , < i ′′ , and < j be represented by the pairs ( π i , u i ) , ( π i , u ′′ i ) ,and ( π j , u j ) , respectively. First, since u i ( x ) = u ′′ i ( x ) for x ∈ { x , x , x ∗ } , it is clear that <<< = ( < i , < j ) and <<< ′′ = ( < i ′′ , < j ) agree on the preferences over acts in G . Second, since u ( x ) raft – May 11, 2020 is in the convex hull of { u ( x ) , u ( x ) , u ( x ∗ ) } for all x ∈ X , we have that for every act f ∈ F ,there is an act g ∈ G such that f ∼ i g and f ∼ j g . The analogous assertion holds for < i ′′ and < j . It follows from independence of redundant acts that with < = Φ( <<< ) and < ′′ = Φ( <<< ′′ ) , wehave for all g, g ′ ∈ G , g < g ′ if and only if g < ′′ g ′ . Let ( π, u ) and ( π ′′ , u ′′ ) be the beliefs andutility functions associated with < and < ′′ , respectively. Note that u ( x ) = u ′′ ( x ) = 0 and u ( x ) = u ′′ ( x ) = 1 .If λ i ( u i ) = λ i ( u ′′ i ) , then π = π ′′ since π i = π j . So we can find an event E such that π ( E ) = = π ′′ ( E ) . It follows that x Ex ∼ x Ex but x Ex ′′ x Ex , which is a contradiction sinceboth acts are in G .If µ i ( u i ) = µ i ( u ′′ i ) , then u ( x ∗ ) = u ′′ ( x ∗ ) , since u i ( x ∗ ) = u ′′ i ( x ∗ ) = u j ( x ∗ ) . Let E be anevent such that π ( E ) = π ′′ ( E ) = u ( x ∗ ) . Then x ∗ ∼ x Ex but x ∗ x Ex , which is again acontradiction.We conclude that λ i ( u i ) = λ i ( u ′′ i ) and µ i ( u i ) = µ i ( u ′′ i ) . The function u ′′ i is closer to u ′ i than is u i , since we have constructed it by moving utilities toward those in u ′ i . Two more modificationsof agent 1’s utility function along the same lines will result in u ′ i . To this end, we apply thesame construction first to the profiles with utility functions ( u ′′ i , u ′ j ) and ( u ′′′ i , u ′ j ) and then toprofiles with utility functions ( u ′′′ i , u j ) and ( u ′ i , u j ) (and the same beliefs π i and π j ). u ′′′ i ( x ) = u ′′ i ( x ∗ ) if u ′′ i ( x ) ≥ u ′′ i ( x ∗ ) and u ′ i ( x ) < u ′′ i ( x ∗ ) u ′ i ( x ) if u ′′ i ( x ) ≥ u ′′ i ( x ∗ ) and u ′ i ( x ) ≥ u ′′ i ( x ∗ ) u ′′ i ( x ) if u ′′ i ( x ) < u ′′ i ( x ∗ ) u ′ j ( x ) = if u ′′ i ( x ) ≥ u ′′ i ( x ∗ ) u ′′ i ( x ) otherwiseIn summary, this gives λ i ( u i ) = λ i ( u ′ i ) and µ i ( u i ) = µ i ( u ′ i ))