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This question is a quantitative analogy to this question on philosophy.SE. Within the framework of economic decision-theory using utility functions, how do you characterise the moral principles of utilitarianism and altruism, and how do they relate? (Please ignore uncertainty in the outcome of actions, since I don't want to complicate the question unnecessarily.)

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Descriptive Framework: Consider a society composed of $n$ individuals, and consider a person in this society who is faced with a decision problem where he has to choose an action $a \in \mathscr{A}$. For simplicity, we will presume that the marginal utility from this action for each individual is known to the decision-maker, and is given by the function $\Delta U_i : \mathscr{A} \rightarrow \mathbb{R}$ for person $i$. A positive value of a marginal utility function means that there is a gain from the chosen action to that individual, and a negative value means there is a loss. Note that utility is interpreted here in the cardinal sense, so it is not merely an ordinal measure (i.e., it can be aggregated by summation).

The vector function of all the marginal utility functions is $\boldsymbol{\Delta U} = (\Delta U_1, ..., \Delta U_n): \mathscr{A} \rightarrow \mathbb{R}^n$, and this gives the total change in utility from the action, for all individuals in society. We can represent an ethical system dependent on these utilities as an ordinal ranking $\succeq$ over $\mathscr{A}$ that depends on $\boldsymbol{\Delta U}$. This ranking describes which actions are more or less good under the ethical systems, based on the vector of marginal utilities in the society.


How to describe utilitarianism and altruism: Both of these ethical systems involve judging the goodness of an action by the benefit to oneself and others, which we will take to be a judgement in relation to some aggregation of the marginal utilities of the action. We characterise these two ethical systems with the following rankings.

  • Utilitarianism: This ethical system holds that a person should act to maximise the aggregate benefit of their action to the whole of society. This is represented by the ranking $\succeq_{\text{U}}$ given by:

    $$a \succeq_{\text{U}} a' \quad \quad \iff \quad \quad \sum_{i} \Delta U_i (a) \geqslant \sum_{i} \Delta U_i (a').$$

  • Altruism: This ethical system holds that a person should act to sacrifice his own welfare to others, to the greatest degree. For each person $k$ in the society, this is represented by the ranking $\succeq_{\text{A},k}$ (each individual has a different ethical ranking) given by:

    $$a \succeq_{\text{A},k} a' \quad \quad \iff \quad \quad \sum_{i \neq k} \Delta U_i (a) - \Delta U_k (a) \geqslant \sum_{i \neq k} \Delta U_i (a') - \Delta U_k (a').$$

Both of these rankings involve an inequality on sums of marginal utilities, but they differ in terms of the signs applied to the parts of the aggregation. Both can be written in the general form:

$$a \succeq_{\mathbf{w}} a' \quad \quad \iff \quad \quad \sum_{i} w_i \Delta U_i (a) \geqslant \sum_{i} w_i \Delta U_i (a'),$$

for some specified vector of weights $\mathbf{w} = (w_1, ..., w_n)$. In the case of utilitarianism we have weights $w_i = 1$, so that each individual contributes their utility equally to the aggregation. In the case of altruism we have $w_i = \mathbb{I}(i \neq k) - \mathbb{I}(i=k)$ so that the decision-maker contributes his marginal utility negatively to the aggregation (i.e., self-sacrifice is considered good).

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