Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility1955

Social welfare as the sum of cardinal individual utility (measurable hedenometer style) was criticized by the Pareto school. According to the HET website the Paretians were Neoclassical general equilibrium theorists of the 1910-1950s. Here we are concerned with the Fundamental Welfare Theorems (first proved by Lange (1942) which I have to read before the Arrow-Debreu improvements) and the social optimum best, Pareto-optimal allocation. One approach was the Bergson-Samuelson SWF: an arbitrary function of socioeconomic variables of a form chosen by one's ethical value judgements, with everyone having different ones. Given prevalence of individual value judgements, it is agreed that the SWF should be increasing function of individual utilities: if \(X \succsim Y\) and no \(Y \succsim X \Rightarrow X\) is socially preferable. Alright. Fleming showed that with one additional postulate the SWF is restricted to a narrow class of functions to express it as weighted sum of individual utilities. Harsanyi shows Fleming's postulate implies an individualistic value judgement, and argues is perfectly reasonable. With VNMM axioms on choice, a much weaker postulate suffices.

Fleming's postulate is expressed in two frameworks: G.E. Moore ideal utilitarianism and preference terminology. Postulate A: If \(X \succ Y \Rightarrow Y \not\succ X\). Postulate B: If \(X \succ Y\) and \(Y \succ Z \Rightarrow X \succ Z\). Classic transitivity, if you prefer Pessi to Pendu and Pendu to Peymar, you prefer Pessi to Peymar. Postulate C: If \(X \sim Y\) and \(Y \sim Z \Rightarrow X \sim Z\). All of these are from a social standpoint. Using \(\succsim\) instead of \(\succ\) removes Postulate C. Then, Postulate D: If \(i\) has \(X \succ Y\) and \(\forall \: j \neq i\) we have \(Y \not\succ X \Rightarrow X \succ Y\) from a social standpoint. Postulate E: Suppose \(i\) has \(X \sim X^{\prime}\) and \(Y \sim Y^{\prime}\) but \(X \succsim Y\) and \(X^{\prime} \succsim Y^{\prime}\). Suppose \(j\) has \(X \sim X^{\prime}\) and \(Y \sim Y^{\prime}\) but \(Y \succsim X\) and \(Y^{\prime} \succsim X^{\prime}\). Suppose \(k \neq i,j\) has \(X \sim Y\) and \(X^{\prime} \sim Y^{\prime} \Rightarrow\) social preferences should be as follows: if \(X \sim Y \Rightarrow X^{\prime} \sim Y^{\prime}\), if \(X \succsim Y \Rightarrow X^{\prime} \succsim Y^{\prime}\), and if \(Y \succsim X \Rightarrow Y^{\prime} \succsim X^{\prime}\), which means that social choice must depend only on \(i\) and \(j\)'s interests even if they are in conflict. So social choice depends only on individual interests directly affected and leaves no room for the interests of interpersonal cultural values or super-individual states. He concedes on the ideals of equity embedded in the postulates, and tries to argue that community context enters through external economies and diseconomies.

Postulate E's separability assumption is not acceptable to me, insofar as we must recognize communities as morally prior to the individuals they are comprised of. The normative weight of a conflict between \(i\) and \(j\) is not intrinsic to \(i\) and \(j\) but is derived from the relational social structure they exist in. Harsanyi's rebuttal reduces the community to a utility shifter that is consulted instrumentally and then discarded once individual utility is given, whereas the deontological claim is that relational obligations between a worker and his class, or a serf and his lord, generate independent moral facts about what each party is owed that are not exhausted by any moral algebra. Postulate E eliminates relational moral structure by way of either a utility effect or an irrational sentiment, and thus is ideological ... not neutral at all.

Harsanyi recognizes this, and then moves to try to make away with such postulate. He wants to also move from certain outcomes to ones that involve probabilities, which he claims is more representative of the world. He dedicates the third section to prove a few primitives. I will re-write the setting to what I believe is better notation. We start with a finite set of social states \(\mathcal{X}\), and we call a prospect a probability distribution over \(\mathcal{X}\). We call \(\Delta(\mathcal{X})\) the set of such prospects. Prospects are also called lotteries in economics. There are \(n\) individuals that are indexed by \(i=1,\cdots,n\), each with preference relation \(\succsim_i\) over \(\Delta(\mathcal{X})\), while society has a preference relation \(\succsim_S\) over \(\Delta(\mathcal{X})\). For \(P,Q \in \Delta(\mathcal{X})\) and \(\lambda \in [0,1] \Rightarrow \lambda P \oplus (1-\lambda)Q\) is the prospect assigning probability \(\lambda p(x) + (1-\lambda)q(x)\) to each \(x \in \mathcal{X}\). Harsanyi requires us to know the von Neumann-Morgenstern axioms or Marschak's postulates, which are equivalent. M1: \(\forall \: P,Q \in \Delta(\mathcal{X})\) either \(P \succsim Q\) or \(Q \succsim P\) or both, \(P \succsim P\), and having \(P\succsim Q\) with \(Q \succsim R \Rightarrow P \succsim R\). So complete, reflexive, and transitive binary relation. M2: \(\forall \: P,Q,R \in \Delta(\mathcal{X})\) with \(P \succ Q \succ R\: \exists \: \lambda \in (0,1)\) s.t. \(\lambda P \oplus (1-\lambda)R \sim Q\) M3: \(\exists \: P_1,P_2,P_3,P_4 \in \Delta(\mathcal{X})\) s.t. \(\forall \: k \neq l \Rightarrow P_k \not\sim P_l\) M4: \(\forall \: P,Q,Q^{\prime},R \in \Delta(\mathcal{X})\) and \(\lambda \in [0,1]\) having \(Q \sim Q^{\prime} \Rightarrow \lambda P \oplus (1-\lambda)Q \sim \lambda P \oplus (1-\lambda)Q^{\prime}\).

Using these, Harsanyi uses three postulates. H1: \(\succsim_S\) satisfies M1, M2, M3, M4. Here we encounter the more egregious ideological move imo. H2: Each \(\succsim_i\) also satisfies M1, M2, M3, M4. H3: \(\forall \: P,Q \in \Delta(\mathcal{X})\) if \(\forall \: i \in \{ 1,\cdots,n\}\) they have \(P \sim_i Q \Rightarrow P \sim_S Q\).

Then we move to theorems. T1: \(\exists \: W : \Delta(\mathcal{X})\to \mathbb{R}\) s.t. \(\forall \: P,Q \in \Delta(\mathcal{X}) \Rightarrow P \succsim_SQ \iff W(P) \geqslant W(Q)\) and \(W\) satisfies the expected utility property \(W(\lambda P \oplus (1-\lambda)Q)=\lambda W(P)+(1-\lambda)W(Q)\). Also, if \(W^{\prime}\) also represents \(\succsim_S\) with that property \(\Rightarrow W^{\prime}=\alpha W + \beta\) for some \(\alpha \in \mathbb{R}^{+}\) and \(\beta \in \mathbb{R}\).

T2: \(\forall \: i \: \exists \: U_i : \Delta(\mathcal{X})\to \mathbb{R}\) s.t. \(\forall \: P,Q \in \Delta(\mathcal{X}) \Rightarrow P \succsim_i Q \iff U_i(P) \geqslant U_i(Q)\) and \(U\) satisfies the expected utility property \(U_i(\lambda P \oplus (1-\lambda)Q)=\lambda U_i(P)+(1-\lambda)U_i(Q)\). Also, if \(U_i^{\prime}\) also represents \(\succsim_S\) with that property \(\Rightarrow U_i^{\prime}=\alpha U_i + \beta\) for some \(\alpha \in \mathbb{R}^{+}\) and \(\beta \in \mathbb{R}\).

So both are unique up to positive linear transformations. Both of these proofs need us to apply the VNM representation theorem. T3: \(\exists \: f : \mathbb{R}^n \to \mathbb{R}\) s.t. \(W(P) = f(U_1(P),\cdots,U_n(P))\). So \(W\) is a single-valued function of all the \(U_i\). T4: \(W\) is homogeneous of degree one in \((U_1,\cdots,U_n)\). So \(\forall \: P \) with \(U_i(P)=u_i\) and \(W(P)=w\) and \(k \in \mathbb{R} \Rightarrow W(P^{\prime})=kw\) whenever \(\forall \: i \Rightarrow U_i(P^{\prime})=ku_i\). Both of these proofs look incredibly boring so I skipped them.

T5: \(\exists \: a_1,\cdots,a_n \in \mathbb{R}\) s.t. \[W=\sum_{i=1}^{n}a_iU_i\] This is the key theorem, and we will sketch a proof. \(\forall \: i\) we define \(S_i\) as a prospect that satisfies \(U_i(S_i)=1\) and \(\forall \: j\neq i \Rightarrow U_j(S_i)=0\). By T3 we have \(W(S_i)=f(0,\cdots,1,\cdots,0).\) Let \(a_i\equiv W(S_i)\). By T4 applied to \(S_i\) with scalar \(U_i\) we have that at utility vector \((0,\cdots,U_i,\cdots,0) \Rightarrow W= a_iU_i\). Take any \(P\) with utility vector \((u_1,\cdots,u_n)\). \(\forall \: i\) let \(\tilde{S}_i\) be the prospect with \(U_i(\tilde{S}_i)=u_i\) and \(U_j(\tilde{S}_i)=0\) for \(j\neq i\). By the previous application of T4 we have \(W(\tilde{S}_i)=a_iu_i\). Now, we deine a mixed prospect \[ T = \frac{1}{n} \bigoplus_{i=1}^{n} \tilde{S}_i \] As the paper says \(T\) is the mixed prospect of obtaining either \(S_1\) or \(S_2\) or \(\cdots S_n\), each with probability \(\tfrac{1}{n}\). By the expected utility property of each \(U_j\) we have \[U_j(T) = \frac{1}{n} \sum_{i=1}^{n} U_j (\tilde{S}_i)=\frac{1}{n} u_j\] since \(U_j(\tilde{S}_i)=0\) for \(i \neq j\) and \(U_j(\tilde{S}_j)=u_j\). Analogously, by the expected utility property of \(W\) we have \[W(T) = \frac{1}{n} \sum_{i=1}^{n} W (\tilde{S}_i)=\frac{1}{n} \sum_{i=1}^{n} a_i u_i\] By T3 \(W(T) = f(U_1(T),\cdots,U_n(T)) = f(\frac{u_1}{n},\cdots,\frac{u_n}{n})\). By T4 with \(k=n \Rightarrow\) \[f(u_1,\cdots,u_n)=n \cdot f\left(\frac{u_1}{n},\cdots,\frac{u_n}{n}\right) = n \cdot W(T) = n \cdot \frac{1}{n} \sum_{i=1}^{n} a_iu_i = \sum_{i=1}^{n}a_iu_i\] This proves that \(\forall \: P \in \Delta(\mathcal{X}) \Rightarrow\) \[ W(P) = \sum_{i=1}^{n} a_i U_i(P) \] However, this does not guarantee that there will not be negative utilities, but this can be easily done with another postulate. We then proceed to the fourth section, where Harsanyi tries to convince us that he has suceeded in separating people's preferences for themselves with their preferences for society. Using a sort of Rawlsian argument, he tells us that under the veil of ignorance a rational actor would maximize average utility given that they would judge society by the average utility that people have.

Finally, on the last section that includes a little bit of trans erasure and some unc sexism, Harsanyi posits (and tries to rescue it from the Paretian critique) that interpersonal utility comparisons are not value judgements but something empirical, measurable, and factual. The more perfect information (psychological) the more these individual SWF converge on the unweighted mean of utilities, which would make the SWF objective. While the math is all good and correct, Harsanyi never provides justification as to why the premises are acceptable. Look at H2 which forces \(\succsim_S\) to become social evaluation, thus subsuming it t the expected utility model of rational choice, and excluding deontological and non-consequentialist moral ontology from the start. These ethics violate M4 and are thus never refuted, but rather hidden away. A child who elbows their sibling accidentally and one who does so deliberately will generate identical utility vectors. By treating class, domination, and institutional position as explanatory variables affecting utility, Harsanyi assumes rather than argues that their moral significance is exhausted by their contribution to individual welfare. Any independent moral status of exploitation, domination, or unequal social relations is excluded from the outset because only utilities appear as arguments of the social welfare function.