<p><strong>Descriptive Framework:</strong> Consider a society composed of <span class="math-container">$n$</span> individuals, and consider a person in this society who is faced with a decision problem where he has to choose an action <span class="math-container">$a \in \mathscr{A}$</span>.  For simplicity, we will presume that the <strong>marginal utility</strong> from this action for each individual is known to the decision-maker, and is given by the function <span class="math-container">$\Delta U_i : \mathscr{A} \rightarrow \mathbb{R}$</span> for person <span class="math-container">$i$</span>.  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 <em>cardinal</em> sense, so it is not merely an ordinal measure (i.e., it can be aggregated by summation).</p>
<p>The vector function of all the marginal utility functions is <span class="math-container">$\boldsymbol{\Delta U} = (\Delta U_1, ..., \Delta U_n): \mathscr{A} \rightarrow \mathbb{R}^n$</span>, and this gives the total change in utility from the action, for all individuals in society.  We can represent an <strong>ethical system</strong> dependent on these utilities as an ordinal ranking <span class="math-container">$\succeq$</span> over <span class="math-container">$\mathscr{A}$</span> that depends on <span class="math-container">$\boldsymbol{\Delta U}$</span>.  This ranking describes which actions are more or less good under the ethical systems, based on the vector of marginal utilities in the society.</p>
<hr />
<p><strong>How to describe utilitarianism and altruism:</strong> 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.</p>
<ul>
<li><p><strong>Utilitarianism:</strong> 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 <span class="math-container">$\succeq_{\text{U}}$</span> given by:</p>
<p><span class="math-container">$$a \succeq_{\text{U}} a' \quad \quad \iff \quad \quad \sum_{i} \Delta U_i (a) \geqslant \sum_{i} \Delta U_i (a').$$</span></p>
</li>
<li><p><strong>Altruism:</strong> This ethical system holds that a person should act to sacrifice his own welfare to others, to the greatest degree.  For each person <span class="math-container">$k$</span> in the society, this is represented by the ranking <span class="math-container">$\succeq_{\text{A},k}$</span> (each individual has a different ethical ranking) given by:</p>
<p><span class="math-container">$$a \succeq_{\text{A},k} a' \quad \quad \iff \quad \quad \sum_{i \neq k} \Delta U_i (a) - \Delta U_k (a) \geqslant \sum_{i \neq k} \Delta U_i (a') - \Delta U_k (a').$$</span></p>
</li>
</ul>
<p>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:</p>
<p><span class="math-container">$$a \succeq_{\mathbf{w}} a' \quad \quad \iff \quad \quad \sum_{i} w_i \Delta U_i (a) \geqslant \sum_{i} w_i \Delta U_i (a'),$$</span></p>
<p>for some specified vector of weights <span class="math-container">$\mathbf{w} = (w_1, ..., w_n)$</span>.  In the case of utilitarianism we have weights <span class="math-container">$w_i = 1$</span>, so that each individual contributes their utility equally to the aggregation.  In the case of altruism we have <span class="math-container">$w_i = \mathbb{I}(i \neq k) - \mathbb{I}(i=k)$</span> so that the decision-maker contributes his marginal utility negatively to the aggregation (i.e., self-sacrifice is considered good).</p>

CC BY-SA 4.0
2021-09-17T08:37:42.973Z
Appreciate your explanation! Could you provide some literature regarding the utility and altruism?
Royun
CC BY-SA 4.0
2021-09-17T09:41:27.480Z
@Royun: Sorry, it's been too long since I studied this stuff, so I don't recall the literature.
Ben
CC BY-SA 4.0
2018-05-24T01:19:35.377Z
2018-05-24T01:19:35.377Z
2020-06-17T09:40:54.643Z
Ben
<p>This question is a quantitative analogy to this <a href="https://philosophy.stackexchange.com/questions/32331/is-it-self-evident-that-utilitarianism-requires-altruism">question on philosophy.SE</a>.  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.)</p>

<utility>
Characterising utilitarianism and altruism in economics
Ben
# Characterising utilitarianism and altruism in economics

(Asked by: username_0 on 2018-05-24)

This question is a quantitative analogy to this [question on philosophy.SE](https://philosophy.stackexchange.com/questions/32331/is-it-self-evident-that-utilitarianism-requires-altruism). 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.)

**Tags:** utility

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## 1 answers

### Answer by username_0 on 2018-05-24. Score: 3

**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.

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**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).

> **Comments:**
> **username_1**: Appreciate your explanation! Could you provide some literature regarding the utility and altruism?  
> **username_0**: @Royun: Sorry, it's been too long since I studied this stuff, so I don't recall the literature.  

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