A Note on Dhillon (1998)∗ Tilman B¨orgers† and Yan-Min Choo‡ Department of Economics University of Michigan, Ann Arbor August 2015
Abstract We provide a counterexample to Theorem 1 (A) in Dhillon [3].
1
Dhillon’s Multi-Profile Version of Harsanyi’s Theorem on Utilitarianism
In an important paper, Amrita Dhillon [3] provided a multi-profile version of Harsanyi’s [5] single-profile theorem on utilitarianism. Like Harsanyi did, she assumed that the social alternatives are lotteries. Individuals’ preferences satisfy the von Neumann-Morgenstern (vN-M) axioms, and have therefore an expected utility representation. A social welfare function maps profiles of individuals’ preferences into society’s preference. Society’s preference also satisfies the vN-M axioms. Harsanyi [5] showed that for every preference profile for which the social welfare function satisfies the Pareto axiom society’s preferences can be represented by a weighted sum of individuals’ utility functions. Harsanyi’s result did not, however, provide any connection between the weights applied at different profiles of individuals’ preferences. The Pareto axiom is a single-profile axiom, and does not imply any constraints across different preference profiles. Any preference aggregation rule that operates across all possible profiles of individuals’ preferences over lotteries must violate (appropriately adapted versions of) Arrow’s [1] axioms for preference aggregation if there are at least three alternatives, even if we only consider the restricted domain of vN-M preferences. This was shown by Kalai and Schmeidler [7] and Hylland [6]. Dhillon [3] considered a system of axioms that is different from ∗
We are grateful to Amrita Dhillon for conversations about this note.
[email protected] ‡
[email protected] †
1
Arrow’s, and that, she asserts, is satisfied if and only if social welfare can be represented by the equally weighted sum of individuals’ utility functions where these are normalized so that the lowest utility is 0, and the highest utility is 1. In the literature, this particular social welfare function is called the “relative utilitarian” welfare function. Dhillon’s result is only one of two results that we are aware of that axiomatize social welfare functions, and that can be viewed as multi-profile versions of Harsanyi’s [5] theorem. The other result is in Dhillon and Mertens [4] and also axiomatizes relative utilitarianism.1 This latter result is in our view, however, a little less interesting because it is based on a somewhat opaque continuity axiom. Dhillon’s [3] result therefore occupies in our view a central place in modern welfare theory. This note shows by means of a counterexample that a major intermediate result in Dhillon’s [3] is not correct. We have not been able to establish whether her main theorem is correct nonetheless. The intermediate result that we prove to be incorrect is interesting in its own right. In a related paper (B¨orgers and Choo [2]) we prove a new version of Dhillon’s main theorem that is in spirit closely related to her original theorem. We also simplify her framework, re-interpret her axioms as “revealed preference” axioms where the preferences in question are those of the social planner, and we provide a much shortened proof of the result.
2
Dhillon’s Theorem 1 (A)
The major step in Dhillon’s [3] argument that we prove in this note to be incorrect is her Theorem 1 (A). This result is central to the proof of her main result, her Theorem 2. Theorem 1 (A) characterizes social welfare functions that satisfy the so-called “Extended Pareto Axiom.” We begin by re-producing the “Extended Pareto Axiom” and then Theorem 1 (A) from Dhillon [3]. We argue that both, Dhillon’s formulation of the “Extended Pareto Axiom,” and her formulation of Theorem 1 (A) are not entirely clear, and we provide the precise interpretations of the axiom and the theorem with which we shall work in this note. We freely use the terminology and notation defined in Dhillon [3] and assume that the reader is familiar with these. Here is Dhillon’s axiom, quoted from her paper: Extended Pareto Axiom (EP) For any profile of preferences R N ∈ L N and for any 2 element partition {G1 , G2 } of N, ∃ψG1 , ψG2 such that: for any pair of lotteries p and q pRGi q ⇒ pRq
i = 1, 2
And if further, pPG1 q, then 1
There are, of course, several axiomatizations of utilitarianism that assume, unlike Harsanyi [5] did, that the argument of the social welfare function consists not only of individuals’ preferences over lotteries, but also of certain interpersonal comparisons of utility. An example is Maskin [8]. We discuss the relation between these results and the results in Dhillon [3] and Dhillon and Mertens [4] in B¨ orgers and Choo [2].
2
pPq. As stated, this axiom does, in fact, not say what Dhillon has in mind. This is because, as stated, in the axiom the group aggregation functions ψG1 and ψG2 are allowed to be different for different preference profiles R N . But then it is clear that the axiom cannot have any implications for the way in which the weights of an individual’s utility function in social welfare depend on the preference profile. So, Theorem 1 (A), stated below, cannot possibly follow from EP. In this note, we assume that this is a simple oversight, and that Dhillon, in fact, meant to write:2 Extended Pareto Axiom (EP) For every G ⊂ N there exists a group aggregation rule ψG such that for any profile of preferences R N ∈ L N , any 2 element partition {G1 , G2 } of N , and any pair of lotteries p and q pRGi q i = 1, 2 ⇒ pRq And if further, pPG1 q, then pPq. We next reproduce Theorem 1 (A) from Dhillon [3]. Theorem 1(A) If A ≥ 4 and N ≥ 4, a SWF satisfies EP iff it can be represented by: X U= u0n (Rn ), whenever d(~u) > 2, n∈N
where U is a vN-M utility representation of social preferences, and each u0n is a (unique, up to the function Fn ) representation of individual preferences, such that u0n (a) = (h(un )(a))/Fn ((h(un )(·))) where h(un ) = un − mina∈A un (a), is a utility function in RA , and Fn : RA → R+ is positively homogeneous of degree 1 (if un is not constant) and translation invariant.3 If un is constant define Fn (un ) = 1. The result thus claims necessary and sufficient conditions for a social welfare function to satisfy EP. However, Theorem 1 (A) imposes no restrictions on social preferences when the rank of the collection of real vectors corresponding to the individuals’ vN-Mutility functions, i.e. the rank of the collection of vectors (ui (a))a∈A for every i ∈ N , is less than 3. The conditions provided therefore cannot be sufficient. A sufficient condition must also impose restrictions on profiles of utility functions with a rank below 3. If, for example, all utility functions are identical, thus the rank of the collection is 1, but society’s 2
We are grateful to Amrita Dhillon for confirming that what follows is the intended meaning of the EP axiom. 3 Note that Fn ((hn (un )(·)) + Fn ((un )(·)) by translation invariance. (Dhillon’s footnote.)
3
preference is represented by a vN-M utility function that corresponds to preferences that are different from all individuals’ preferences, then the extended Pareto axiom does not hold for this social welfare function, even if it satisfies the condition of Dhillon’s Theorem 1 (A). We assume that this is a simple oversight, and that Dhillon intended to define SWFs to only apply to preference profiles that can be represented by profiles of utility functions satisfying the condition d(~u) > 2.4 Below, we re-formulate Theorem 1 (A) adding the domain restriction. But before we do so, we shall make the assertion of the theorem (hopefully) a little more transparent. The transformation h(un ) in Theorem 1(A) renormalizes the vNM utility function un by subtracting a constant, so that the utility of the lowest ranked alternative is zero. To obtain u0n , all utilities are then divided by a constant Fn (h(un )). The assumption that Fn is homogeneous of degree 1 ensures that for all un that represent the same preferences, u0n is the same. Thus, social preferences are independent of which representation of individual preferences we chose. To simplify the presentation, one can choose a fixed representation for every individual preference. We shall focus on the representation that assigns the lowest alternative utility 0 and the best alternative utility 1. If the decision maker is indifferent between all alternatives, we assign all alternatives utility zero. Denote this representation of a given utility function by u(R). Let us denote 1/F (u(R)) by λn (R). Then we can reformulate Dhillon’s Theorem 1(A) as follows: Theorem 1(A) Suppose A ≥ 4 and N ≥ 4. Suppose we restrict the domain of the SWF φ to all preference profiles that have at least one utility representation such that d(~u) > 2. Then φ satisfies EP if and only if it can be represented by: X U= λn (Rn )un (Rn ). n∈N
where for every n ∈ N the function λn is of the form λn : L → R++ . What is important in Theorem 1(A) is that the numbers λn (Rn ) are allowed to depend on individual n’s identity, and on his or her preferences, but they do not depend on other agents’ preferences. Thus, the weight of an individual in social welfare may not depend on how this individual’s preferences compare to other agents’ preferences. This is the asserted main implication of EP. Dhillon’s condition is obviously sufficient for EP. We shall show in the next section that it is not necessary. 4
We are grateful to Amrita Dhillon for confirming that this is the correct interpretation of her Theorem 1 (A).
4
3
A Counterexample
We now construct an example of a SWF φ that satisfies EP, but not the condition in Dhillon’s theorem. Let N = {1, 2, 3, 4} and A = {a1 , a2 , a3 , a4 }. For all preference profiles R N = (R1 , R2 , R3 , R4 ) in the domain of φ the social preference φ(R N ) can be represented by: 6u(R1 ) + 9u(R2 ) + 5u(R3 ) + u(R4 ), (1) except for the preference profile RbN which we define by: 1 0 0 1 0 1 0 1 5 u(Rb1 ), u(Rb2 ), u(Rb3 ), u(Rb4 ) = 0 , 0 , 1 , 0 . 0 0 1 0
(2)
Note that RbN satisfies the dimension condition d(~u) > 2. The social preference φ(RbN ) is the preference represented by the utility vector (1, 2, 0, 0). This completes the construction of the counterexample. Note that, ignoring for the moment the special case of the preference profile RbN , the weights λn (R) in this counterexample are, in fact, for each n ∈ N , independent of n’s preference. Therefore Dhillon’s condition seems to be satisfied. The problem arises, of course, from the special case of RbN . In this case we have not given a weighted average representation of φ(RbN ). But we could have. In fact, many such representations are ˆ n (for n ∈ N ), the weights have to satisfy: possible. Denoting the weights by λ 1 1 0 0 1 2 1 0 1 0 ˆ4 = c + d ˆ3 + λ ˆ2 + λ ˆ1 + λ (3) λ 0 0 1 0 0 0 0 1 0 0 for some c > 0 and some d. Eliminating c and d from this system of equations and inequalities, we obtain: ˆ2 > λ ˆ 1 and λ ˆ2 + λ ˆ 3 = 2λ ˆ1 + λ ˆ4. λ (4) There are many solutions to (4), but the weights that we used to define φ for preference profiles other than RbN , that is, 6, 9, 5, and 1, do not satisfy (4). Thus, if we use the weights 6, 9, 5, and 1 for all preference profiles other than RbN , we do not obtain a representation of φ that satisfies the conditions in Dhillon’s theorem 1(A), because for the particular preference profile RbN the weights assigned to the preferences Rbn differ from the weights that the same preferences receive when different preference profiles are aggregated. 5
Here and in the following we identify utility functions u : A → R and the vector of utility values (u(a))a∈A .
5
For a proof that φ as we have defined it does not satisfy Dhillon’s condition, which is what we want to prove here, we have to show, however, slightly more. It might be the case that on the domain of all preference profiles other than RbN there is an alternative representation of φ that, for the preferences Rbn , involves weights that do satisfy (4). To show that this is not the case, we need to do a slightly tedious calculation that we provide in the next four paragraphs. Consider the preference profile R N defined by: 1 0 0 1 0 1 0 1 (u(R1 ), u(R2 ), u(R3 ), u(R4 )) = 0 , 0 , 1 , 1 . 0 0 1 0
(5)
All agents’ except agent 4’s preferences are the same as their preferences in RbN . Note that R N satisfies the dimension condition d(~u) > 2. Thus, it is in the domain of φ. The social preference φ(R N ) is the preference represented by the utility vector (7, 10, 6, 5). Thus, whatever weights we use to represent φ, for this particular profile they have to satisfy:6 7 1 0 0 1 10 0 1 0 1 (6) λ1 0 + λ2 0 + λ3 1 + λ4 1 = c 6 + d 5 0 0 1 0 for some c > 0 and some d. Using only the first three of the above equations, and eliminating c, d and λ4 , we obtain: ˆ1 − λ ˆ 2 − 3λ ˆ 3 = 0, 4λ (7) ˆ n for n = 1, 2, 3 because these three agents have the same where we have replaced λn by λ N N b preferences in R as in R , and thus their weights have to be the same as in RbN . Next, consider the preference profile R N defined by: 1 0 1 1 0 1 1 1 (u(R1 ), u(R2 ), u(R3 ), u(R4 )) = 0 , 0 , 1 , 0 . 0 0 0 0
(8)
All agents’ except agent 3’s preferences are the same as their preferences in RbN . Note that R N satisfies the dimension condition d(~u) > 2. Thus, it is in the domain of φ. The social 6 Here, and in the following, we abuse the symbol λi to denote not only the function λi : L → R++ but also to denote numbers λi > 0. Hopefully, the context ensures that no misunderstandings can arise.
6
preference φ(R N ) is the preference represented by the utility vector (12, 15, 5, 0). Thus, whatever weights we use to represent φ, for this particular profile they have to satisfy: 1 0 1 1 12 0 1 1 1 15 λ1 (9) 0 + λ2 0 + λ3 1 + λ4 0 = c 5 + d 0 0 0 0 0 Eliminating c, d and λ3 , we obtain: ˆ 1 − 7λ ˆ 2 + 3λ ˆ 4 = 0, 10λ
(10)
ˆ n for n = 1, 2, 4 because these three agents have the same where we have replaced λn by λ N N b preferences in R as in R , and thus their weights have to be the same as in RbN . Finally, consider the preference profile R N defined by: 1 0 1 1 0 1 0 1 (u(R1 ), u(R2 ), u(R3 ), u(R4 )) = 0 , 1 , 1 , 0 . 0 1 0 0
(11)
All agents’ except agent 2’s preferences are the same as their preferences in RbN . Note that R N satisfies the dimension condition d(~u) > 2. Thus, it is in the domain of φ. The social preference φ(R N ) is the preference represented by the utility vector (16, 10, 14, 5). Thus, whatever weights we use to represent φ, for this particular profile they have to satisfy: 16 1 1 0 1 10 0 1 0 1 (12) λ1 0 + λ2 1 + λ3 1 + λ4 0 = c 14 + d 5 0 0 1 0 Eliminating c, d and λ2 , we obtain: ˆ 1 − 3λ ˆ 3 + 3λ ˆ 4 = 0, 2λ
(13)
ˆ n for n = 1, 3, 4 because these three agents have the same where we have replaced λn by λ N N b preferences in R as in R , and thus their weights have to be the same as in RbN . ˆ1, λ ˆ2, λ ˆ3, λ ˆ 4 — conditions (7), (10), We have obtained three conditions for the weights λ and (13). Solving these, we obtain: ˆ1 λ 2 = , ˆ2 3 λ
ˆ2 λ 9 = , ˆ3 5 λ
7
ˆ3 λ 5 = , ˆ4 1 λ
(14)
and thus, to represent φ on its domain except RbN , we need to use weights for the preferences included in this profile that are proportional to 6, 9, 5, 1, and which thus violate (4). Thus, we cannot use the same weights to represent φ on its domain without RbN , and to represent φ for RbN , and thus φ violates Dhillon’s condition. Our goal is to prove that φ is a counterexample to Dhillon’s Theorem 1 (A), and what remains to do is to show that φ does satisfy EP. For this purpose, we first construct group aggregation rules ψij : L 2 → L (where i, j ∈ N and i < j) and ψijk : L 3 → L (where i, j, k ∈ N and i < j < k). We define coefficients ai for i ∈ N as follows: a1 = 6, a2 = 9, a3 = 5, a4 = 1. For i, j ∈ N with i < j we let ψi,j (Ri , Rj ) be the preference represented by the utility function: ai u(Ri ) + aj u(Rj ). For i, j, k ∈ N with i < j < k we let ψi,j,k (Ri , Rj , Rk ) be the preference represented by the utility function: ai u(Ri ) + aj u(Rj ) + ak u(Rk ). We now verify EP using these group aggregation rules. We do this by showing that whenever we partition N into two subsets, society’s preference can be represented by a utility function that is a weighted sum of utility functions that represent the two groups preferences, where all weights are strictly positive, and where the groups’ preferences are determined by the group aggregation rules defined above. This clearly implies EP. For preference profiles R N 6= RbN this is trivial, and we omit the proof. We from now on restrict attention to the preference profile RbN . Consider the partitioning of N into the two subsets: G = {1, 2} and G0 = {3, 4}. If we add the group preferences for the two groups with weights w > 0 and w0 > 0, then the sum is a preference with weights wa1 , wa2 , w0 a3 , w0 a4 which equals 6w, 9w, 5w0 , 4w0 . We need to choose w and w0 so that condition (4) is satisfied, i.e. so that: 12w + w0 = 9w + 5w0 ⇔
w 4 = 0 w 3
(15)
Recalling that for the profile RbN the utility function (1, 2, 0, 0) represents the social preference, we obtain: (1, 2, 0, 0)
∼
4u12 (Rb1 , Rb2 ) + 3u34 (Rb3 , Rb4 )
(16)
where the symbol “∼” means that the utility function to the left of the symbol and the utility function to the right of the symbol represent the same preferences. We have thus verified EP for this partition. Similarly, one can show: (1, 2, 0, 0) ∼ ∼ ∼
8u13 (Rb1 , Rb3 ) + 7u24 (Rb2 , Rb4 ) ∼ u123 (Rb1 , Rb2 , Rb3 ) + 2u(Rb4 ) ∼ u134 (Rb1 , Rb3 , Rb4 ) + 8u(Rb2 ) ∼
14u14 (Rb1 , Rb4 ) + 13u23 (Rb2 , Rb3 ) u124 (Rb1 , Rb2 , Rb4 ) + 4u(Rb3 ) 2u234 (Rb2 , Rb3 , Rb4 ) + 13u(Rb1 ) (17)
and this completes the proof that φ satisfies EP, and thus is a counterexample to Dhillon’s Theorem 1 (A). 8
4
Comments
The key to our counterexample is that a large variety of weights can be used to construct representations of the social preference φ(RbN ) as a weighted sum of the 0-1 normalized vNM utility functions representing the individual preferences Rbn . Indeed the set of all such weights is the 3-dimensional convex cone described by (4). This flexibility makes it possible to satisfy EP in the example even though the relative weights attached to individuals in subgroups and the relative weight attached to the same individuals in the society consisting of all individuals differ. No preference profile for which any representation of society’s preferences as the weighted sum of individuals’ vN-M utility functions involves the same relative weights of individuals’ 0-1 normalized vN-M utility functions could have served as a counterexample. It seems that at the heart of the complex proof in [3] is a mistaken belief about the relation between the rank of the collection of individuals’ utility functions and the uniqueness of relative weights in any representation of society’s preferences (assuming that society’s preferences are represented by at least one weighted sum of individuals’ utility functions). In our counterexample, the rank of the collection of individuals’ utility functions is 3, that is, one less than the number of individuals. This makes condition (4) particularly easy to satisfy, and therefore it makes calculations in the counterexample comparatively simple. Suppose we had instead focused on the case that the collection of individuals’ utility functions has dimension equal to the number of agents. It is clear in [3] that Dhillon believes that this case is straightforward. Indeed, she believes that, if the vectors of normalized utility functions are linearly independent, and if the social preference can represented by a weighted sum of individuals’ utility functions, then the weights in this sum are the same, regardless of which representation of individuals’ and society’s utility function is chosen.7 But even this is not the case. To see this, consider the example in which individuals’ preferences are given by: 1 0 0 1 1 0 (u(R1 ), u(R2 ), u(R3 )) = (18) 0 , 0 , 1 0 0 1 and suppose that society’s preference is described by the utility vector (1,2,1,1). Then the weights attached to individuals’ preferences must satisfy: 0 1 1 0 1 0 2 1 λ1 (19) 0 + λ2 0 + λ3 1 = c 1 + d 0 0 1 1 7
See the sentence preceding equation (10) on page 529 in [3].
9
This reduces to: λ1 = λ3 .
(20)
Evidently, the ratio of the weights attached to individuals 1 and 2, i.e. λ1 /λ2 , is not uniquely determined by (20).8 In fact, a condition under which the relative weights in any representation of society’s preferences as the sum of individuals’ utility functions are uniquely determined is easy to see: it is that the rank of the collection consisting of the individuals’ utility functions and the vector consisting of 1s only is equal to one plus the number of agents. This ensures that systems of equations such as (3) have single-dimensional solution sets. This condition has been called in the literature “Independent Prospects” (see Weymark [9]). In B¨orgers and Choo [2] we assume that all preference profiles in the domain of the social welfare function satisfy the Independent Prospects condition, and obtain a drastically simplified proof of Dhillon’s result. We also re-formulate her axioms, and provide a “revealed preference” interpretation of the axioms.
References [1] Kenneth J. Arrow (1951), Social Choice and Individual Values, New York: Wiley. [2] Tilman B¨ orgers and Yan-Min Choo (2015), Revealed Relative Utilitarianism, University of Michigan. [3] Amrita Dhillon (1998), Extended Pareto Rules and Relative Utilitarianism, Social Choice and Welfare 15, 521-542. [4] Amrita Dhillon and Jean-Fran¸cois Mertens (1999), Relative Utilitarianism, Econometrica 67, 471-498. [5] John Harsanyi (1955), Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility, Journal of Political Economy 63, 309-321. 8
The mistaken belief that we have describe here leads to the mistaken claim made in equation (10) on page 529 in [3]. This equation claims that, if the dimension of the collection of utility vectors equals the number of agents, which in this particular passage of the paper is 3, EP implies that the ratio of the weights attached to individuals 1 and 2 is the same regardless of which representation of the aggregated preferences for the group {1, 2} and for the group {1, 2, 3} one considers. To see that this is erroneous, consider the example described in the text. Suppose the preferences of {1, 2} are represented by (1,2,0,0), which is just the sum of the two individuals’ normalized utility functions. Then the ratio of 1’s and 2’s weights is 1. Suppose also the preferences of the group {1, 2, 3} are obtained simply by adding up the utility functions, so that the group preferences are the ones described in the text. Condition (20) shows that there are representations of the group preference in which the ratio of the weights of 1 and 2 is not equal to 1, contrary to what is claimed in equation (10) in [3].
10
[6] Aanund Hylland (1980), Aggregation Procedure for Cardinal Preferences: A Comment, Econometrica 48, 539-542. [7] Ehud Kalai and David Schmeidler (1977), Aggregation Procedure for Cardinal Preferences: A Formulation and Proof of Samuelson’s Impossibility Conjecture, Econometrica 45, 1431-1438. [8] Eric Maskin (1978), A Theorem on Utilitarianism, Review of Economic Studies 45, 93-96. [9] John A. Weymark (1994), Harsanyi’s Social Aggregation Theorem With Alternative Pareto Principles, in: Wolfgang Eichhorn (editor), Models and Measurement of Welfare and Inequality, Heidelberg: Springer Verlag, 869-887.
11