Draft:Harsanyi's utilitarian theorem

In social choice theory and ethics, Harsanyi's utilitarian theorems are a set of closely-related results showing that under certain conditions, the only coherent social choice function is given by the utilitarian rule. The theorem was first proven by John Harsanyi in 1955 and has important implications for political, moral, and economic philosophy, as well as electoral systems and welfare economics.

The theorem states that any social choice function is given by a weighted sum of individual utility functions, so long as the social choice function satisfies three conditions:

• Social rationality—the social choice function satisfies von Neumann-Morgenstern rationality (or equivalently, philosophical coherence).
• Rationality of well-being—individual preferences (i.e. measures of well-being) satisfy the von Neumann-Morgenstern axioms.
• Pareto efficiency (unanimity)—if an action would hurt someone but help no one, the social choice function will reject it.

The theorem thus says that if a society wishes to behave rationally and consistently under uncertainty, and society respects individual preferences, society must adopt a utilitarian rule for social decision making.

Mathematical statement

Let $X$ be the set of all social states, with $\Delta (X)$ being the set of all probability distributions over $X$. A utility function for individual $i\in N$ is a function $u_{i}:\Delta (X)\to \mathbb {R}$, satisfying the following properties (the von Neumann–Morgenstern axioms):

The four axioms of VNM-rationality are completeness, transitivity, continuity, and independence. These axioms, apart from continuity, are often justified using the Dutch book theorems (whereas continuity is used to set aside lexicographic or infinitesimal utilities).

Completeness assumes an individual has well defined preferences:

Axiom 1 (Completeness) For any lotteries $L$ and $M$, either $\,L\succeq M$ or $\,M\succeq L$.

(the individual must express some preference or indifference).

Transitivity assumes that preferences are consistent across any three options:

Axiom 2 (Transitivity) If $\,L\succeq M$ and $\,M\succeq N$, then $\,L\succeq N$.

Axiom 1 and Axiom 2 together can be restated as the statement that the individual's preference is a total preorder.

Continuity assumes that there is a "tipping point" between being better than and worse than a given middle option, where the individual is indifferent between the two states.

Axiom 3 (Continuity): If $\,L\preceq M\preceq N$, there is some probability $\,p\in [0,1]$ such that $\,pL+(1-p)N\,\sim \,M$.

where the notation on the left side refers to a situation in which L is received with probability p and N is received with probability (1–p).

Independence assumes that a preference holds independently of the probability of another outcome.

Axiom 4 (Independence): For any $M$ and $p\in [0,1)$ (with the "irrelevant" part of the lottery underlined):

$L\preceq N\quad {\text{if and only if}}\quad \,(1-p)\,L+{\underline {pM}}\preceq (1-p)\,N+{\underline {pM}}$

In other words, the probabilities involving $M$ cancel out and don't affect our decision, because the probability of $M$ is the same in both lotteries.

A social welfare function is a function $W:\Delta (X)\to \mathbb {R}$, where is the We say that $W$ satisfies the Pareto criterion if for any $p,q\in \Delta (X)$, $W(p)\geq W(q)$ whenever $u_{i}(p)\geq u_{i}(q)$ for all $i\in N$.

Harsanyi's utilitarian theorem states that if $W$ and $u_{i}$ for all $i\in N$ are VNM-rational utility functions, and if $W$ satisfies the Pareto and VNM-rationality conditions, there exist constants $c_{i}$ such that

$W(p)=\sum _{i=1}^{n}c_{i}u_{i}(p)$

for all $p\in \Delta (X)$.

Open questions and criticisms

Harsanyi argued his theorems successfully demonstrated that any meaningful theory of morality must be utilitarian, a position disputed by other philosophers and economists. While Harsanyi's theorem is successful in establishing that under reasonable assumptions, the social utility function must be a linear combination of individual utilities, it is much more difficult to say whether or not it establishes utilitarianism as philosophers commonly understand the phrase, assuming this is even a well-defined notion. Hilary Greaves argues the theorem only successfully establishes that rational morality must be "a" utilitarianism, rather than "the" utilitarianism of classical philosophy. Other philosophers reject Pareto efficiency, typically appealing to some external non-humanist or non-naturalist origin of morality such as divine command.

Choice of individual utility functions

Harsanyi attempted to connect the utility functions in his theorem to the VNM utility function implied by revealed preferences, a natural choice for motivating preference utilitarianism; such an approach can be grounded in a respect for autonomy, allowing individuals to choose what they consider meaningful or important for themselves.

However, human behavior does not generally follow the axioms of rational choice, as shown by the Allais paradox and other major results of behavioral economics. Amartya Sen notes that the nature of the individual utility functions is therefore not completely clear. Revealed preferences are therefore not sufficient on their own to establish an individual's "true" utility function, as there are situations where an individual's revealed preferences will contradict themselves.

Interpersonal comparison

Even assuming a single measure of objective "individual utility" exists, Harsanyi's theorem leaves the choice of weights $c_{i}$ undefined, unless paired with additional assumptions—in other words, Harsanyi's theorem does not establish a way to perform interpersonal comparisons of utility.

https://philpapers.org/rec/NEBAWI

Generalizations

Harsanyi's theorem can be generalized to social utility functions that do not satisfy the axioms of rational choice; under weaker constraints than full rationality, the social choice function can be shown to be isoelastic.

refs

(3) Harsanyi's 'Utilitarian Theorem' and Utilitarianism - JSTOR. https://www.jstor.org/stable/3506225. (4) Harsanyi’s Utilitarian Theorem: A Simpler Proof and Some Ethical .... https://link.springer.com/chapter/10.1007/978-3-662-09664-2_17. (5) Harsanyi’s utilitarianism via linear programming. http://personal.rhul.ac.uk/uhte/035/Harsanyi%20utilitarianism%20Economics%20Letters%20revised%20proofs.pdf.

https://philpapers.org/rec/NEBAWI

References:

¹: Harsanyi, J. C. (1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy, 63(4), 309-321. ²: Hammond, P. J. (1991). Harsanyi's utilitarian theorem: A simpler proof and some ethical connotations. In M. J. Holler (Ed.), Power, voting, and voting power (pp. 247-266). Physica-Verlag. https://link.springer.com/chapter/10.1007/978-3-662-09664-2_17. ³: Sen, A. K. (1977). Social choice theory: A re-examination. Econometrica, 45(1), 53-88. ⁴: Risse, M. (2002). Harsanyi's 'utilitarian theorem' and utilitarianism. Noûs, 36(4), 550-577. https://www.jstor.org/stable/3506225

Harsanyi’s utilitarianism via linear programming. http://personal.rhul.ac.uk/uhte/035/Harsanyi%20utilitarianism%20Economics%20Letters%20revised%20proofs.pdf.