aises_6_8

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<h1 id="social-welfare-functions">6.8 Social Welfare Functions</h1>
<p><strong>Should we have AIs maximize total wellbeing?</strong> We have
explored different ways to increase social wellbeing, such as material
wealth, preferences, and happiness. This section explores social welfare
functions as a way of moving from the wellbeing of individuals in a
society to a measure of the welfare of the society as a whole. They are
drawn from the disciplines of social choice theory, welfare economics,
and wellbeing science.<p>
We begin by defining social welfare functions and their role in
measuring the overall wellbeing of a society. We then discuss how these
functions can be used to compare different outcomes and guide
decision-making by powerful agents such as governments or beneficial
AIs. We consider two types of social welfare functions - utilitarian and
prioritarian - and consider some of the advantages and drawbacks of each.
By understanding these key concepts, we can see how we might design AIs
to use social welfare functions to maximize the good they do in
society.</p>
<p><strong>Social welfare functions are a way to measure the overall
wellbeing of a society.</strong> They aggregate a collection of
individual levels of wellbeing into a single value that represents
societal welfare. These functions help us to understand how to balance
individuals’ various needs and interests within a society. A social
welfare function tackles the challenge of resource and benefit
distribution within a community. It assists in determining how to value
one person’s happiness against another’s and how to weigh the needs of a
majority against those of a minority. This helps solve the <em>problem
of aggregation</em>: the challenge of integrating varied individual
wellbeing into a single collective measure that represents the entire
society. By expressing societal welfare in a systematic, numeric manner,
social welfare functions contribute to decisions designed to optimize
societal welfare overall.</p>
<p><strong>Different social welfare functions can recommend taking
different actions.</strong> Consider an AI-powered decision support
system used in a city planning committee. The system suggests three key
project proposals for the betterment of the community: (1) developing a
local health clinic, (2) initiating an after-school education program,
and (3) constructing a community green park. Additionally, it estimates
what the wellbeing of each of the three individuals in this community,
Ana, Ben, and Cara, would be if these proposals were implemented,
summarized by their wellbeing values in the table below.<p>

<br>
<table class="tableLayout">
<caption>A city planning committee chooses between three projects with
different wellbeing impacts</caption>
<thead>
<tr class="header">
<th style="text-align: left;">Individuals</th>
<th style="text-align: center;">Health Clinic</th>
<th style="text-align: center;">Education Program</th>
<th style="text-align: center;">Green Park</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">Ana</td>
<td style="text-align: center;">6</td>
<td style="text-align: center;">8</td>
<td style="text-align: center;">3</td>
</tr>
<tr class="even">
<td style="text-align: left;">Ben</td>
<td style="text-align: center;">8</td>
<td style="text-align: center;">6</td>
<td style="text-align: center;">7</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Cara</td>
<td style="text-align: center;">4</td>
<td style="text-align: center;">5</td>
<td style="text-align: center;">10</td>
</tr>
</tbody>
</table>
<br>
<p>None of these options stands out. Each person has a different top
ranking, and none of them would be harmed too much by the planning
committee choosing any one of these. However, a decision must be made.
More generally, we want a systematic rule to move from these individual
data points to a collective ranking. This is where <em>social welfare
functions</em> come into play. We can briefly consider two common
approaches that we expand upon later in this section:<p>
</p>
<ol>
<li><p>The <em>utilitarian</em> approach ranks alternatives by the total
wellbeing they bring to all members of society. Using this rule in the
example above, the system would rank the proposals as follows:</p>
<ol>
<li><p>Green Park, where the total wellbeing is <span
class="math inline">3 + 7 + 10 = 20</span>.</p></li>
<li><p>Education Program, where the total wellbeing is <span
class="math inline">8 + 6 + 5 = 19</span>.</p></li>
<li><p>Health Clinic, where the total wellbeing is <span
class="math inline">6 + 8 + 4 = 18</span>.</p></li>
</ol></li>
<li><p>On the other hand, the <em>Rawlsian maximin</em> rule prioritizes
the least fortunate person’s well-being. It would rank the proposals
according to how the person who benefits the least fares in each
scenario. Using the maximin rule, the system would rank the proposals in
this order:</p>
<ol>
<li><p>Education Program, where Cara is worst off with a wellbeing of
5.</p></li>
<li><p>Health Clinic, where Cara is worst off with a wellbeing of
4.</p></li>
<li><p>Green Park, where Ana is worst off with a wellbeing of
3.</p></li>
</ol></li>
</ol>
<p><strong>Deciding on a social welfare function is important for clear
decision-making.</strong> The choice of social welfare function provides
a structured quantitative approach to decisions that can impact
everyone. It sets a benchmark for decision-makers, like our hypothetical
AI in city planning, to optimize collective welfare in a way that is
transparent and justifiable. When we have a framework to quantify
society’s wellbeing, we can use it to inform decisions about allocating
resources, planning for the future, or managing risks, among other
things.</p>
<p><strong>Social welfare functions can help us guide AIs.</strong> By
measuring social wellbeing, we can determine which actions are better or
worse for society. Suppose we have a good social welfare function and
AIs with the ability to accurately estimate social welfare. Then it
might be easier to train these AIs to increase social wellbeing, such as
by giving them the social welfare function as their objective function.
Social welfare functions can also help us judge an AI’s actions against
a transparent metric; for instance, we can evaluate an AI’s
recommendations for our city-planning example by how well its choices
align with our social welfare function.<p>
However, there are technical challenges to overcome before this is
feasible, such as the ability to reliably estimate individual wellbeing
and the several problems explored in the Single Agent Safety
chapter. Additionally, several normative choices need to be made. What
theory of wellbeing—preference, hedonistic, objective goods—should be
the basis of the social welfare function? Should aggregation be
utilitarian or prioritarian? What else, if anything, is morally valuable
besides the aggregate of individual welfare? The idea of using social
welfare functions to guide AIs is promising in theory but requires more
exploration.</p>
<h2 id="measuring-social-welfare">6.8.1 Measuring Social Welfare</h2>
<p><strong>Overview.</strong> In this section, we will consider how
social welfare functions work. We’ll use our earlier city planning
scenario as a reference to understand the fundamental properties of
social welfare functions. We will discuss how social welfare functions
help us compare different outcomes and the limitations of such
comparisons. Lastly, we’ll touch on ordinal social welfare functions,
why they might be insufficient for our purposes, and how using
additional information can give us a more holistic approach to
determining social welfare.</p>
<p><strong>Social welfare functions take the total welfare distribution
as an input and give us a value of that distribution as an
output.</strong> We can use the city planning example above to consider
the basic properties of social welfare functions. The input to a social
welfare function is a vector of individuals’ wellbeing values: for
instance, after the construction of a health clinic, the wellbeing
vector of the society composed of Ana, Ben, and Cara would be<p>
<span
class="math display"><em>W</em><sub><em>H</em></sub> = (6, 8, 4).</span></p>
<p>which tells us that three individuals have wellbeing levels equal to
seven, eight, and six. The social welfare function is a rule of what to
do with this input vector to give us one measure of how well off this
society is.</p>
<p><strong>The function applies a certain rule to the input vector to
generate its output.</strong> We can apply many possible rules to a
vector of numbers quantifying wellbeing that can generate one measure of
overall wellbeing. To illustrate, we can consider the utilitarian social
welfare function, which implements a straightforward rule: adding up all
the individual wellbeing values. In the case of our three-person
community, we saw that the social welfare function would add 6, 8, and
4, giving an overall social welfare of 18. However, social welfare
functions can be defined in numerous ways, offering different
perspectives on aggregating individual wellbeing. We will later examine
continuous prioritarian functions, which emphasize improving lower
values of wellbeing. Other social welfare functions might emphasize
equality by penalizing high disparities in wellbeing. These different
functions reflect different approaches to understanding and quantifying
societal wellbeing.</p>
<p><strong>Social welfare functions help us compare outcomes, but only
within one function.</strong> The basic feature of social welfare
functions is that a higher output value signifies more societal welfare.
In our example, a total welfare score of 20 would indicate a society
that is better off than one with a score of 18. However, it’s important
to remember that the values provided by different social welfare
functions are not directly comparable. A score of 20 from a utilitarian
function, for instance, does not correspond to the same level of
societal wellbeing as a 20 from a Rawlsian minimax social welfare
function, since they apply different rules to the wellbeing vector. Each
function carries its own definition of societal wellbeing, and the
choice of social welfare function plays a crucial role in shaping what
we perceive as a better or worse society. By understanding these
aspects, we can more effectively utilize social welfare functions as
guideposts for AI behavior and decision-making, aligning AI’s actions
with our societal values.</p>
<p><strong>Some social welfare functions might just need a list ranking
the different choices.</strong> Sometimes, we might not need exact
numerical values for each person’s wellbeing to make the best decision.
Think of a simple social welfare function that only requires individuals
to rank their preferred options. For example, three people could rank
their favorite fruits in the order “apple, banana, cherry”. From this,
we learn that everyone prefers apples over cherries, but we don’t know
by how much they prefer apples. This level of detail might be enough for
some decisions: clearly, we should give them apples instead of cherries!
Such a social welfare function won’t provide exact measures of societal
wellbeing, but it will give us a ranked list of societal states based on
everyone’s preferences.</p>
<p><strong>Voting is a typical example of an ordinal social welfare
function.</strong> Instead of trying to estimate each person’s potential
wellbeing for each option, the city planning committee might ask
everyone to vote for their top choice. We assume that each person votes
for the option they believe will enhance their wellbeing most. This
relates to the <em>Preference View of Wellbeing</em>, which says that
individuals know what’s best for them, and their choices reflect their
wellbeing. Through this voting process, we can create a list ranking the
options by the number of votes each one received, even if we don’t
assign specific numerical values to each option—this is ordinal
information, where all we know are the rankings.<p>
As an illustration, if most people vote for the park, that indicates the
park might contribute the most to overall wellbeing, according to this
social welfare function. The function would look at the number of votes
for each option, and the one with the most votes would be deemed the
best choice. This is a simpler approach than calculating everyone’s
wellbeing for each option, so it’s less likely to lead to errors.
However, this type of social welfare function does have its own
challenges, such as the problem highlighted by Arrow’s Impossibility
Theorem.</p>
<p><strong>Arrow’s impossibility theorem is one reason we might want
more than ordinal information.</strong> In our city planning scenario,
Ana would vote for the Education Program, Ben would vote for the Health
Clinic, and Cara would vote for the Green Park. If we tried to aggregate
these rankings without any additional information, such as the numerical
scores assigned to each option, we would not have any reason to choose
one option over the other. This is an example of a general problem:
Arrow’s Impossibility Theorem. In essence, it suggests it’s impossible
to create a perfect voting system that fulfills a set of reasonable
criteria while only using ordinal information. These criteria involve
metrics of fairness like “non-dictatorships” and coherency like “if
everyone prefers apples to cherries, then the social welfare function
prefers apples to cherries”. Adding information about the strength of
preferences is one solution to Arrow’s Impossibility Theorem.
Additionally, if we have access to such information, using it will mean
that we arrive at better answers. Next, we will consider two classes of
social welfare functions that use such information: utilitarian and
prioritarian social welfare functions.</p>
<h3 id="utilitarian-social-welfare-functions">Utilitarian Social Welfare
Functions</h3>
<p><strong>Overview.</strong> This section is focused on utilitarian
social welfare functions-—functions that sum up individual wellbeing to
calculate social welfare, in line with the utilitarian theory of
morality. We’ll start by exploring cost-benefit analysis, a common
social welfare function that draws inspiration from utilitarian
reasoning, through the example of a decision to build a health clinic.
While cost-benefit analysis provides a convenient and quantitative
approach to decision-making, we’ll discuss its limitations in capturing
the full ideas of wellbeing. Further, we’ll consider how utilitarian
social welfare functions can foster equity, especially when diminishing
returns of resources are considered and how AI systems, optimized using
these functions, could potentially improve inequality. Lastly, we’ll
discuss how the utilitarian social welfare function, under certain
assumptions, is the only method that satisfies key decision-making
criteria.</p>
<p><strong>Cost-benefit analysis is a popular but crude approximation of
utilitarian social welfare.</strong> Governments and other
decision-makers often use cost-benefit analysis to ground their
decision-making quantitatively. Simply, this involves adding up the
expected benefits of an action or a decision and comparing it to the
anticipated costs. We can apply this to whether to build a health clinic
that’ll operate for 10 years. If the benefits outweigh the costs over
the considered time span, the committee may decide to proceed with
building the clinic.<p>


<div>
    <table class="tableLayout">
        <caption>Table 6.3: A cost-benefit analysis assigns monetary values to all factors and compares the total
                 benefits with the overall costs. </caption>
        <thead>
            <tr class="header">
                <td style="text-align: left;"> <b>Item</b> </td>
                <td style="text-align: center;"> <b>Value per Person</b> </td>
                <td style="text-align: center;"> <b>People Affected</b> </td>
                <td style="text-align: center;"> <b>Frequency <br> of Item</b> </td>
                <td style="text-align: center;"> <b>Total Value</b> </td>
            </tr>
        </thead>

        <tbody>
            <tr class="odd">
                <td style="text-align: left;"> <b>Costs</b> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> <b>$22,500,000</b> </td>
            </tr>

            <tr class="even">
                <td style="text-align: left;"> Construction Expenses </td>
                <td style="text-align: center;"> $15,000,000 </td>
                <td style="text-align: center;"> - </td>
                <td style="text-align: center;"> One-time </td>
                <td style="text-align: center;"> $15,000,000 </td>
            </tr>
            <tr class="odd">
                <td style="text-align: left;"> Staffing and Maintenance </td>
                <td style="text-align: center;"> $750,000 </td>
                <td style="text-align: center;"> - </td>
                <td style="text-align: center;"> Every year for <br> 10 years</td>
                <td style="text-align: center;"> $7,500,000 </td>
            </tr>

            <tr class="even">
                <td style="text-align: left;"> <b>Benefits</b> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> </td>
                <td style="text-align: center;"> <b>$40,000,000</b> </td>
            </tr>

            <tr class="odd">
                <td style="text-align: left;"> Fewer Hospital Visits </td>
                <td style="text-align: center;"> $1,000 </td>
                <td style="text-align: center;"> 1,000 </td>
                <td style="text-align: center;"> Every year for <br> 10 years</td>
                <td style="text-align: center;"> $10,000,000 </td>
            </tr>

            <tr class="even">
                <td style="text-align: left;"> Increased Employment: Doctors</td>
                <td style="text-align: center;"> $100,000</td>
                <td style="text-align: center;"> 5</td>
                <td style="text-align: center;"> Every year for <br> 10 years</td>
                <td style="text-align: center;"> $5,000,000</td>
            </tr>

            <tr class="odd">
                <td style="text-align: left;"> Increased Employment: Staff</td>
                <td style="text-align: center;"> $25,000</td>
                <td style="text-align: center;"> 20</td>
                <td style="text-align: center;"> Every year for <br> 10 years</td>
                <td style="text-align: center;"> $5,000,000</td>
            </tr>

            <tr class="even">
                <td style="text-align: left;"> Increased Life Expectancy by 1 year</td>
                <td style="text-align: center;"> $20,000</td>
                <td style="text-align: center;"> 1,000</td>
                <td style="text-align: center;"> One-time</td>
                <td style="text-align: center;"> $20,000,000</td>
            </tr>

            <tr class="odd">
                <td style="text-align:left;"> <b>Net Benefit</b></td>
                <td style="text-align:left;"> </td>
                <td style="text-align:left;"> </td>
                <td style="text-align:left;"> </td>
                <td style="text-align:center;"> <b>+$17,500,000</b></td>
            </tr>
        </tbody>
    </table>
</div>
</p>

This method allows the government to assess multiple options and choose
the one with the highest net benefit. By doing so, it approximates
utilitarian social welfare. For the health clinic, the committee assigns
monetary values to each improvement and then multiplies this by the
number of people affected by the improvement. In essence, cost-benefit
analysis assumes that wellbeing can be approximated by financial losses
and gains and considers the sum of monetary benefits instead of the sum
of wellbeing values. Using monetary units simplifies comparison and
limits the range of factors it can consider.</p>
<p><strong>Cost-benefit analysis is not a perfect representation of
utilitarian social welfare.</strong> Money is not a complete measure of
wellbeing. While money is easy to quantify, it doesn’t capture all
aspects of wellbeing. For instance, it might not fully account for the
psychological comfort a local health clinic provides to a community.
Additionally, providing income to five doctors who are already high
earners might be less important than employing 20 support staff, even
though both benefits sum to $5,000,000 over the ten years. Cost-benefit
analysis lacks this fine-grained consideration of wellbeing. AI systems
could, in theory, maximize social welfare functions, considering a
broader set of factors that contribute to wellbeing. However, we largely
rely on cost-benefit analysis today, focusing on financial measures, to
guide our decisions. This brings us to the challenge of improving this
method or finding alternatives to better approximate utilitarian social
welfare in real-world decision-making, including those involving AI
systems. Utilitarian social welfare functions would promote some level
of equity. Usually, additional resources are less valuable when we
already have a lot of them. This is diminishing marginal returns: the
benefit from additional food, for instance, is very high when we have no
food but very low when we already have more than we can eat. Extending
this provides an argument for a more equitable distribution of resources
under utilitarian reasoning. Consider taxation: if one individual has a
surplus of wealth, say a billion dollars, and another has only one
dollar, redistributing a few dollars from the wealthy individual to the
less fortunate one may elevate overall societal wellbeing. This is
because the added value of a dollar to the less fortunate individual is
likely very high, allowing them to purchase necessities like food and
shelter, whereas it is likely very low for the wealthy individual.</p>
<p><strong>AI systems optimizing utilitarian social welfare functions
might therefore improve inequality.</strong> Utilitarian AI systems
might recognize the diminishing returns of resource accumulation,
leading them to suggest policies that lead to a more equal distribution
of resources. It is important to remember that the utilitarian has no
objection to inequality, except that those with less can better use
resources than those who are already happy. This counters a frequent
critique that utilitarianism neglects inequality. Current methods like
cost-benefit analysis may not fully capture this aspect. Therefore, AI
guided by a utilitarian social welfare function might propose approaches
for a more equitable distribution of resources that conventional methods
could overlook.</p>
<p><strong>A utilitarian social welfare function is the only way to
satisfy some basic requirements.</strong> Let us reconsider the city
planning committee deciding what to build for Ana, Ben, and Cara. If
they all have the same level of wellbeing whether a new Education
Program or a Green Park is built, then it seems right that the city’s
planning committee shouldn’t favor one over the other. Suppose we
changed the scenario a bit. Suppose Ana benefits more from a Health
Clinic than an Education Program, and Ben and Cara don’t have a strong
preference either way. It now seems appropriate that the committee
should favor building the Health Clinic.</p>
<p><strong>Harsanyi’s Social Aggregation Theorem.</strong> Harsanyi
showed that—assuming the individuals are rational, in the sense of
maximizing expected utility—if we want our social welfare function to
make choices like this consistently, we need to use a model where we add
up everyone’s wellbeing <span class="citation"
data-cites="weymark1993harsanyi">[1]</span>. This is the foundation of a
utilitarian social welfare function. Harsanyi’s aggregation theorem
proved that it is the only kind of social welfare function that always
is indifferent between options if everyone is equally happy with them
and favors the option that makes someone better off, as long as it
doesn’t make anyone worse off. This has been seen as a compelling reason
to pick utilitarian social welfare functions over other ones.</p>
<h3 id="prioritarian-social-welfare-functions">Prioritarian Social
Welfare Functions</h3>
<p><strong>Overview.</strong> In this section, we will consider
prioritarian social welfare functions and how they exhibit differing
degrees of concern for the wellbeing of various individuals. We will
start by describing prioritarian social welfare functions, which give
extra weight to the wellbeing of worse-off people. This discussion will
include some common misconceptions about prioritarianism and how it
contrasts with utilitarianism in resource allocation. We will focus on
two types of prioritarian functions: the Rawlsian minimax social welfare
function and continuous prioritarian functions.</p>
<p><strong>There are two common misunderstandings about prioritarian
social welfare functions.</strong> Firstly, prioritarians are not
focused on reducing inequality itself, unlike egalitarians. Their main
goal is to increase the wellbeing of those who need it most. They think
giving an extra unit of wellbeing to someone with less is more valuable
than giving it to someone already well-off. The level of inequality in a
society doesn’t affect their measure of social welfare, and reducing
inequality isn’t their goal unless it improves individual wellbeing.
Secondly, prioritarians are not driven by the belief that it’s easier to
improve the wellbeing of the worst off. They would rather see benefits
go to the worse off even if it costs the same as improving the lives of
those who are better off. This also sets them apart from utilitarians,
who might typically help the worst off because they see more value due
to diminishing marginal utility. The prioritarian approach isn’t about
the efficiency of resources but about focusing resources on those who
need them most. Thus, while utilitarian and prioritarian social welfare
functions aim for a better society, they have distinct ways of achieving
this goal.</p>
<p><strong>A special case of prioritarian social welfare functions is
the Rawlsian “maximin” function.</strong> The Rawlsian social welfare
function takes the idea of protecting the worse off to the extreme:
social welfare is simply the lowest welfare of anyone in society. It is
called the “<em>maximin</em>” function because it seeks to maximize the
minimum wellbeing, or in other words, to ensure that the worst-off
individual is as well off as possible. When we applied the maximin
function to the city planning committee, it decided that the worst
option was the Green Park, despite the utilitarian social welfare
function determining that was the best one. Using the maximin principle
here, we want to go with the choice to ensure the worst-off person is as
happy as possible. When we look at the least happy person for each
option, we see that for the Health Clinic it’s 6 (Cara), for the
Education Program it’s 5 (Cara), and for the Green Park it’s 4 (Ana).
So, if we follow the maximin rule, the committee would choose the Health
Clinic because it would bring up the wellbeing of the person doing the
worst. This way, we ensure we’re looking out for the people who need it
most.<p>
However, this maximin approach has its own problems, like the “grouch”
issue. This is when someone always rates their wellbeing low, no matter
what. If we’re always looking out for the worst-off person, we might end
up always catering to the grouch, even though their low score might not
be due to real hardship. It’s important to remember this when
considering how to use the Rawlsian maximin approach.</p>
<p><strong>Continuous prioritarian social welfare functions offer a
middle ground.</strong> We might want our social welfare function to
account for both the positive effects of increasing anyone’s welfare and
the extra positive effects of increasing the welfare of someone who
isn’t doing well. This is the tradeoff between efficiency and equity.
Many social welfare functions embrace prioritarian principles without
going to the extreme of the maximin function’s concern for equity. This
can be achieved using a social welfare function that shows diminishing
returns relative to individual welfare: the boost it gives to social
welfare when any individual’s wellbeing improves is larger if that
person initially had less wellbeing. For example, we could use the
<em>logarithmic social welfare function</em><p>
<span
class="math display"><em>W</em>(<em>w</em><sub>1</sub>,<em>w</em><sub>2</sub>,...,<em>w</em><sub><em>n</em></sub>) = log <em>w</em><sub>1</sub> + log <em>w</em><sub>2</sub> + ... + log <em>w</em><sub><em>n</em></sub>,</span>
where <span class="math inline"><em>W</em></span> is the social welfare
and each <span class="math inline"><em>w</em><sub>1</sub></span>,<span
class="math inline"><em>w</em><sub>2</sub></span>,...,<span
class="math inline"><em>w</em><sub><em>n</em></sub></span> are the
individual wellbeing values of everyone in society. The logarithmic
social welfare function is (ordinally) equivalent to the Bernoulli-Nash
social welfare function, which is the product of wellbeing values.<p>
Suppose there are three individuals with wellbeing 2, 4, and 16 and we
are using log base 2. Social welfare is<p>
<span
class="math display"><em>W</em>(2, 4, 16) = log<sub>2</sub>(2) + log<sub>2</sub>(4) + log<sub>2</sub>(16) = 1 + 2 + 4 = 7.</span></p>
<p>Compare the following two changes: increasing someone from 2 to 4 or
increasing someone from 4 to 8. The first change results in<p>
<span
class="math display"><em>W</em>(4, 4, 16) = log<sub>2</sub>(4) + log<sub>2</sub>(4) + log<sub>2</sub>(16) = 2 + 2 + 4 = 8.</span>
The second change results in<p>
<span
class="math display"><em>W</em>(2, 8, 16) = log<sub>2</sub>(2) + log<sub>2</sub>(8) + log<sub>2</sub>(16) = 1 + 3 + 4 = 8.</span></p>
<p>Even though the second change is larger, social welfare is increased
by the same amount as when improving the wellbeing of the individual who
was worse off by a smaller amount. This highlights the principle that
improving anyone’s welfare is beneficial, and it’s easier to increase
social welfare by improving the welfare of those who are worse off.
Additionally, even though the second change doesn’t affect the worst
off, the social welfare function shows society is better off. This
approach allows us to consider both the overall level of wellbeing and
its distribution.</p>
<p><strong>We can specify exactly how prioritarian we are.</strong> Let
the parameter <span class="math inline"><em>γ</em></span> represent the
degree of priority we give worse-off individuals. We can link the
Rawlsian maximin, logarithmic, and utilitarian social welfare functions
by using the isoelastic social welfare function<p>
<span class="math display">$$W(w_1, w_2, ..., w_n) =
\frac{1}{1-\gamma}\left(w_1^{1-\gamma} + w_2^{1 - \gamma} + ... +
w_n^{1-\gamma}\right).$$</span></p>
<p>If we think we should give no priority to the worse-off, then we can
set <span class="math inline"><em>γ</em> = 0</span>: the entire equation
is then just a sum of welfare levels, which is the utilitarian social
welfare function. By contrast, if we were maximally prioritarian, taking
the limit of <span class="math inline"><em>γ</em></span> as it gets
infinitely large, then we recover Rawls’ maximin function. Similarly, if
we took the limit of <span class="math inline"><em>γ</em></span> as it
approached 1, we would recover the logarithmic social welfare function.
This function is widespread in economic literature due to this
versatility; among other things, it allows researchers to determine from
observed preferences what a typical person’s parameter <span
class="math inline"><em>γ</em></span> is. By adopting this most general
form, we can pick exactly how we want to prioritize individuals in
society. This enables a principled way of trading off efficiency and
equity.</p>
<h3 id="conclusions-about-social-welfare-functions">Conclusions About
Social Welfare Functions</h3>
<p><strong>Using AI to estimate social welfare.</strong> Artificial
intelligence might be used to estimate the inputs and calculate the
outputs of social welfare functions to help us decide between different
actions and implement our ideal decision processes. By processing large
amounts of data, AI systems might be able to predict individual welfare
under various scenarios, paving the way for more informed
decision-making. It has been demonstrated, for instance, that
context-aware machine learning models like large language models can be
used for predicting disease incidence faster than conventional
epidemiological methods, which can be used to evaluate different public
health proposals from a social welfare perspective. The application of
AI can lead to more refined estimates of social welfare.</p>
<p><strong>Using social welfare functions to train AI.</strong> A second
connection between social welfare functions and AI lies in creating AI
systems. We can shape AI systems to generally promote social welfare,
ensuring they are aligned with human interests even when optimizing for
distinct goals, such as corporate profit. Particularly in reinforcement
learning, where decision-making is based on rewards or penalties, these
can be tethered to societal wellbeing. Even for other agents, AI
behavior can be guided towards promoting social welfare, aligning AI
performance with societal wellbeing and making AI a compelling tool for
social good.</p>
<p><strong>Using AI to maximize social welfare.</strong> Beyond creating
AI systems in line with social welfare, we can steer advanced AI to
actively optimize social wellbeing by using social welfare functions as
their objective functions. This harnesses AI’s power to promote societal
welfare effectively. Given AI’s data processing and predictive
capabilities, it can evaluate various strategies to identify what would
best increase social welfare. By aligning AI objectives with social
welfare functions, we can develop beneficial AI systems that not only
recognize and understand social wellbeing but also actively work towards
enhancing it. This proactive use of AI bolsters our ability to build
societies where individual wellbeing is optimized and evenly
distributed, reflecting our social preferences.</p>
<p><strong>Challenges.</strong> Social welfare functions require
choosing a theory of wellbeing. AI systems that maximize social welfare
may also inherit various problems of current models such as
vulnerability to adversarial attacks and proxy gaming and difficulties
in interpreting the internal logic that led them to make certain
decisions (both discussed in the Single Agent Safety chapter),</p>
<p><strong>Summary.</strong> In this section, we discussed social
welfare functions: mathematical functions that tell us how to aggregate
information about individual welfare into one society-wide measure. We
considered why social welfare functions might be relevant in the context
of decision making by beneficial AI systems and explored a few
properties that all social welfare functions possess. We analyzed how
governments today use cost benefit analysis as an approximation to
social welfare functions and why that falls short of what we desire. The
main point of this section was to document and understand some of the
most common social welfare functions—-the utilitarian function, the
Rawlsian maximin function, and continuous prioritarian functions-—and
how they differ in their levels of concern for the well-being of
different individuals, and can be represented within the framework of a
general isoelastic social welfare function.<p>
We think that the use of AI can help estimate social welfare inputs and
calculate outputs, and AI systems can be shaped to generally promote
social welfare and actively optimize it using social welfare functions
as objective functions. Ultimately, the proactive use of AI bolsters our
ability to build societies where individual well-being is both optimized
and evenly distributed, reflecting our social preferences.<p>
</p>
<br>
<br>
<h3>References</h3>
<div id="refs" class="references csl-bib-body" data-entry-spacing="0"
role="list">
<div id="ref-weymark1993harsanyi" class="csl-entry" role="listitem">
<div class="csl-left-margin">[1] J.
A. Weymark, <span>“Harsanyi’s social aggregation theorem and the weak
pareto principle,”</span> <em>Social Choice and Welfare</em>, 1993, doi:
<a
href="https://doi.org/10.1007/BF00182506">https://doi.org/10.1007/BF00182506</a>.</div>
</div>
</div>