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 proved by John Harsanyi in 1955 and has important implications for political, moral, and economic philosophy, as well as utilitarianism 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).
 * Individual rationality—individual preferences (i.e. measures of well-being) satisfy the von Neumann-Morgenstern axioms.
 * Pareto efficiency (unanimity)—if the entire population prefers A to B, the social choice function will prefer A to B. In other words, if an action would hurt someone but help no one, the social choice function will reject it.

The theorem implies that if society and individuals behave rationally and consistently under uncertainty, and society respects individual preferences, then society must adopt a utilitarian criterion for social decision making.

Mathematical statement
Let $$N = \{1, 2, ..., n\}$$ be a finite set of individuals, and let $$X$$ be a set of social states. A social welfare function is a function $$W: \Delta(X) \to \mathbb{R}$$, where $$\Delta(X)$$ is the set of all probability distributions over $$X$$. A utility function for individual $$i \in N$$ is a function $$u_i: \Delta(X) \to \mathbb{R}$$. 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 von Neumann-Morgenstern expected utility functions, and if $$W$$ satisfies the Pareto condition, then there exist constants $$, ..., c_n$$ such that

$$W(p) = \sum_{i=1}^n c_i u_i(p)$$

for all $$p \in \Delta(X)$$. Moreover, the constants $$c_i$$ are unique up to a positive affine transformation.

Open questions and criticisms
Harsanyi argued his theorems successfully demonstrated that any meaningful theory of morality must be utilitarian, a position criticized 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 refers to the theorem as only being successful in establishing that rational morality must be "a" utilitarianism, rather than "the" utilitarianism.

Choice of individual utility functions
Amartya Sen notes that the nature of the individual utility functions is not completely clear. While Harsanyi first connected the to the VNM utility function implied by revealed preferences, and this is a natural choice for motivating preference utilitarianism, the behavior of individual the Allais paradox

Interpersonal comparison
Even assuming a single measure of objective "individual utility" were to exist, Harsanyi's theorem leaves the choice of weights $$c_i$$ undefined, unless paired with additional assumptions—in other words, Harsanyi's theorem fails to establish a way to perform.

Generalizations
Harsanyi's theorem can be generalized to cases of utility functions where.

(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.

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. 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.

Source: Conversation with Bing, 1/27/2024 (1) (2) Harsanyi's 'Utilitarian Theorem' and Utilitarianism - JSTOR. https://www.jstor.org/stable/3506225. (3) Harsanyi’s Utilitarian Theorem: A Simpler Proof and Some Ethical .... https://bing.com/search?q=Harsanyi%27s+utilitarian+theorem. (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.