<p>Universal domain and dictatorship together imply that, whether or not the individual who is a dictator is representative, society will only care about the dictator's preferences. For example, if everyone in the society prefers alternative 1 to alternative 2, except for the dictator, who prefers alternative 2 to alternative 1, then the dictator is clearly not representative. However, society still follows the dictator and prefers alternative 2 to alternative 1. This is what is implied by Arrow's axioms of Universal Domain, Weak Pareto, and IIA: that there is necessarily a dictator in society. In other words, it is impossible to have all four—Universal Domain, Weak Pareto, IIA, and no dictatorship—simultaneously. I don't see why this wouldn't be considered 'bad' for society.</p>

CC BY-SA 4.0
2024-09-20T01:40:57.880Z
Depends on what you mean by "bad". Arrow's dictator is not a dictator in the ordinary sense of the word. S/he could very well be the benevolent dictator that economists have long dreamed for.
Herr K.
CC BY-SA 4.0
2024-09-20T02:51:22.207Z
I agree. I just meant that the existence of an individual (dictator) whose preference becomes the social preference, regardless of what everyone else prefers, does not 'sound good' and may not be desirable. In reality, he may be benevolent and may not exercise his power to choose outcomes in a 'bad' way, as he might take others' preferences into account when determining his own preference over the alternatives and hence can act representative.
Amit
CC BY-SA 4.0
2024-09-19T20:52:40.633Z
2024-09-19T21:00:59.513Z
2024-09-19T21:00:59.513Z
Amit
<blockquote>
<p>What if there exists a person i
who is not the decision maker, but they just get lucky and it happens that the 'optimal' social welfare function happens to yield whatever decision i
prefers?</p>
</blockquote>
<p>I am not really sure where you see the possibility of this in math. The voter becomes dictator precisely because in the ranking voting setup there is an very analogous result median voter theorem that shows that the choice selected by voting ranked voting system will depend critically on that one person, the dictator (e.g. if the 'dictator' prefers A, A will be selected, when 'dictator' prefers C and changes the vote that will be the selection for whole society). Similarly, in first past post system model, assuming rationality, only the median voter's vote matters all other votes are irrelevant for the result. This is what makes the agent dictator. This is what you get from the Arrows seminal 1950 work.</p>
<p>I am not sure how someone can get lucky in the model. It is deterministic model, the only luck there is to be born with given preferences.  Moreover, there is no 'optimal' social welfare function. Idea of optimal social welfare function implies that you can somehow objectively say Utilitarian social welfare function is more optimal than Rawlsian.</p>
<p>Regarding your practical example. The preferences of those voters might happen to be around the median preferences. But that does not mean those voters do not impose their preferences on rest of the society if they vote according to their preferences and if their vote is tie breaker. If there is a committee of 13 judges who need to decide  whether whole society will be Rawlsian or Utilitarian, 6 are more Utilitarian aligned and 6 more Rawlsian, and finally in the middle there is one judge who is kind of undecided but leads toward Utilitarian preferences, then it is that last judge that decides which preferences will be imposed on the whole society.</p>
<p>Just because some outcome is arrived at through democratic means it does not mean it is not imposed, or even not imposed by the treat of force. For example, in democratic nations tax levels are decided by democratically elected politicians. Yet that does not make the taxes voluntary compared to non-democratic countries. They are still being imposed, under threat of physical violence.</p>
<p>Regarding the Arrow point, I think the direct interpretation is that Arrow simply points out that obviously there is a qualitative difference between dictator that rules by military might, and dictator that imposes their will based on some mechanism to which many members of society consented. However, using the definitions, if outcome depends on one single person then the person is still dictator whatever other qualitative differences there might be.</p>
<p>Deciding by elections is not the same as voluntarism. In fact, in most models when you allow agents to decide on voluntary provisions of public goods or other government actions you will get typically completely different results when you let the same agents vote and take collective action that then has to be imposed back on everyone (e.g. see examples in Mueller Public Choice III).</p>
<p>Hence those people in that example are imposing their preferences, as these words are traditionally understood in the public economics, public choice and political economy literature.</p>
<blockquote>
<p>Suppose there exists an individual whose preference is defined as 'I want what society as a whole wants'. Maybe they're very well informed about voter preferences, and are selfless.</p>
</blockquote>
<p>This is kind of moot point. The whole point of Arrow paper is to show that you simply cannot discover what society as whole wants through ranked voting. There might be ways how to discover preferences of society, but ranked voting system are not able to do that. This is the whole point of the conditions. Either the ranked voting system gives you cycling results, or tells you just about preferences of one individual and not preference of everyone and not ignoring completely preferences of voters at the same time.</p>
<p>Moreover, again consider example of median voter who is tie breaker between option A and B. Again suppose there are 6 voters voting for A and 6 for B. What should this selfless and informed voter that wants to just pursue preferences of society choose? The voter know that 6 people want A and 6 people want B. How does the voter selflessly decide? That voter is in position to become Buridan's ass.</p>
<p>We are dealing here with discrete choices. This 'selfless' voter will either impose A on all voters or B on all waters. Or if you end up in tie you get a median voter equivalent to the Condorcet cycle (this is why Arrow simply assumes this option away, as in that case you can satisfy other properties but no choice can be shown to be strictly preferred and chosen).</p>
<blockquote>
<p>Why is it bad if a 'fair' or 'optimal' social welfare function happens to yield decisions that coincidentally always align to one particular individual's preferences?</p>
</blockquote>
<ol>
<li><p>I do not know where you read that this is considered to be bad. I will admit that some time already passed since I read Arrow, but I can't recall any passage where he would claim that this is bad.</p>
<p>I searched the original <a href="https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf" rel="nofollow noreferrer">Arrow 1950</a> article and I couldn't find any instance of Arrow claiming that this result is somehow bad or even undesirable. It is just property of ranked voting system.</p>
<p>In fact its not even property of every voting system, it only applies to ordinal voting systems. Rated voting system, for example, does not have this particular result.</p>
</li>
<li><p>I do not believe there is any consensus on what would be 'fair' or 'optimal' social welfare function. If such consensus would exist then the whole idea of democracy does not make sense anyway. If there is fair or optimal social welfare function then dictatorship of benevolent social planner would be desirable.</p>
<p>If you believe some social welfare function is fair or optimal then indeed as long as some voting system, like ranked choice voting, delivers this outcome it does not matter if the system is de facto dictatorship because the outcome is just due to single individual.</p>
<p>The point of Arrows impossibility theorem is to show that if you want to use democracy as a democratic social preference aggregator, ranked choice voting system is not a system that can do that for you.</p>
</li>
</ol>

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2024-09-20T01:51:07.167Z
Arrow does say that non-dictatorship (cf. Condition 5 of OP's linked book) "is certainly a reasonable one to impose on the form of the social welfare function." So he does seem to view non-dictatorship as a desirable property of a SWF.
Herr K.
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2024-09-20T11:08:30.107Z
@HerrK. but I believe that is in positive context of making SWF that is democratic, I do not believe he is making normative judgement, i.e. this is better than benevolent social planner imposing SWF from above
1muflon1
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2024-09-19T17:31:42.447Z
2024-09-19T17:37:14.753Z
2024-09-19T17:37:14.753Z
1muflon1
<p>I'm trying to understand the motivation for the 'no dictator' axiom in Arrow's Impossibility theorem.</p>
<p>Applying the standard colloquial meaning of &quot;dictator&quot; in this context means that the question of &quot;what should society do?&quot; is always answered with &quot;Whatever dictator Dave decides we should do&quot;. The <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem#Basic_assumptions" rel="nofollow noreferrer">Wikipedia</a> definition is:</p>
<blockquote>
<p>the system does not <em>depend</em> on only one voter's ballot</p>
</blockquote>
<p>Which seems to align to the idea of the &quot;dictator&quot; being the decision maker.</p>
<p>The text of <a href="https://www.google.fr/books/edition/Social_Choice_and_Individual_Values/y_rkX6QWOYMC?hl=fr&amp;gbpv=1&amp;dq=Social+Choice+and+Individual+Values+kenneth+arrow&amp;printsec=frontcover&amp;pli=1" rel="nofollow noreferrer">Arrow's book</a> (first paragraph of section 5) seems to say the same thing:</p>
<blockquote>
<p>social choices are the be <strong>based solely</strong> on the preferences of one man</p>
</blockquote>
<p>But then the math says something else:</p>
<blockquote>
<p>A social welfare function is said to be dictatorial if there exists an individual <span class="math-container">$i$</span> such that , for all <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, <span class="math-container">$x P_i y$</span> implies <span class="math-container">$x P y$</span> regardless of the orderings <span class="math-container">$R_1, ... R_n$</span> of all individuals other than <span class="math-container">$i$</span>, where <span class="math-container">$P$</span> is the social preference relation ....</p>
</blockquote>
<p>What if there exists a person <span class="math-container">$i$</span> who is not the decision maker, but they just get lucky and it happens that the 'optimal' social welfare function happens to yield whatever decision <span class="math-container">$i$</span> prefers?</p>
<p>e.g. consider a bellweather, like the electorate of <a href="https://en.wikipedia.org/wiki/Bellwether#Australia" rel="nofollow noreferrer">Eden-Monaro</a>. Excluding recent elections, it just so happens that the electorate's favorite party each election wins that election overall nationally. This is not because the federal election is decided by them in particular. They are not <em>imposing</em> their preferences on the whole society. They just happen to be representative. This seems to align to the mathematical definition of dictator. Why is this situation so bad that we need to prevent it with an axiom? This seems like a situation that is good for Eden-Monaro, and neutral for everyone else.</p>
<p>Arrow's next paragraph seems to consider this:</p>
<blockquote>
<p>it cannot be claimed that Definition 6 is a true model of actual dictatorship. There is normally an element of consent by the community ...</p>
</blockquote>
<p>Suppose there exists an individual whose preference is defined as 'I want what society as a whole wants'. Maybe they're very well informed about voter preferences, and are selfless. This seems like a well defined preference. Allowing all well defined preferences is one of the other axioms, right? From wikipedia again:</p>
<blockquote>
<p>Universal domain — the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.</p>
</blockquote>
<p>Why is it bad if a 'fair' or 'optimal' social welfare function happens to yield decisions that coincidentally always align to one particular individual's preferences?</p>

CC BY-SA 4.0
2024-09-20T01:31:30.757Z
Loaded terms like 'fair' and 'optimal' are easily misconstrued. I'd rephrase your last paragraph as:  *Why is it undesirable that a social welfare function happens to yield decisions that coincidentally always align to one particular individual's preference?* or *Why is being non-dictatorial a desirable property for a social welfare function?*
Herr K.
CC BY-SA 4.0
2024-09-20T01:36:37.217Z
It is important to note that Arrow's "dictator" *is* part of the voting population. S/he participates in the voting process in the same way as everybody else. S/he is not, as you seem to suggest, a "decision maker" who exists apart from the electorate and approves/vetoes or implements/rejects the voting results by (the rest of) the electorate.
Herr K.
<social-welfare><social-choice>
Why does Arrow's impossibility theorem have the 'no dictator' axiom?
falsePockets
# Why does Arrow's impossibility theorem have the 'no dictator' axiom?

(Asked by: username_0 on 2024-09-19)

I'm trying to understand the motivation for the 'no dictator' axiom in Arrow's Impossibility theorem.

Applying the standard colloquial meaning of "dictator" in this context means that the question of "what should society do?" is always answered with "Whatever dictator Dave decides we should do". The [Wikipedia](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem#Basic_assumptions) definition is:

> the system does not *depend* on only one voter's ballot

Which seems to align to the idea of the "dictator" being the decision maker.

The text of [Arrow's book](https://www.google.fr/books/edition/Social_Choice_and_Individual_Values/y_rkX6QWOYMC?hl=fr&gbpv=1&dq=Social+Choice+and+Individual+Values+kenneth+arrow&printsec=frontcover&pli=1) (first paragraph of section 5) seems to say the same thing:

> social choices are the be **based solely** on the preferences of one man

But then the math says something else:

> A social welfare function is said to be dictatorial if there exists an individual $i$ such that , for all $x$ and $y$, $x P\_i y$ implies $x P y$ regardless of the orderings $R\_1, ... R\_n$ of all individuals other than $i$, where $P$ is the social preference relation ....

What if there exists a person $i$ who is not the decision maker, but they just get lucky and it happens that the 'optimal' social welfare function happens to yield whatever decision $i$ prefers?

e.g. consider a bellweather, like the electorate of [Eden-Monaro](https://en.wikipedia.org/wiki/Bellwether#Australia). Excluding recent elections, it just so happens that the electorate's favorite party each election wins that election overall nationally. This is not because the federal election is decided by them in particular. They are not *imposing* their preferences on the whole society. They just happen to be representative. This seems to align to the mathematical definition of dictator. Why is this situation so bad that we need to prevent it with an axiom? This seems like a situation that is good for Eden-Monaro, and neutral for everyone else.

Arrow's next paragraph seems to consider this:

> it cannot be claimed that Definition 6 is a true model of actual dictatorship. There is normally an element of consent by the community ...

Suppose there exists an individual whose preference is defined as 'I want what society as a whole wants'. Maybe they're very well informed about voter preferences, and are selfless. This seems like a well defined preference. Allowing all well defined preferences is one of the other axioms, right? From wikipedia again:

> Universal domain — the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.

Why is it bad if a 'fair' or 'optimal' social welfare function happens to yield decisions that coincidentally always align to one particular individual's preferences?

**Tags:** social-welfare, social-choice

> **Comments:**
> **username_1**: Loaded terms like 'fair' and 'optimal' are easily misconstrued. I'd rephrase your last paragraph as:  *Why is it undesirable that a social welfare function happens to yield decisions that coincidentally always align to one particular individual's preference?* or *Why is being non-dictatorial a desirable property for a social welfare function?*  
> **username_1**: It is important to note that Arrow's "dictator" *is* part of the voting population. S/he participates in the voting process in the same way as everybody else. S/he is not, as you seem to suggest, a "decision maker" who exists apart from the electorate and approves/vetoes or implements/rejects the voting results by (the rest of) the electorate.  

---

## 2 answers

### Answer by username_2 on 2024-09-19. Score: 2

Universal domain and dictatorship together imply that, whether or not the individual who is a dictator is representative, society will only care about the dictator's preferences. For example, if everyone in the society prefers alternative 1 to alternative 2, except for the dictator, who prefers alternative 2 to alternative 1, then the dictator is clearly not representative. However, society still follows the dictator and prefers alternative 2 to alternative 1. This is what is implied by Arrow's axioms of Universal Domain, Weak Pareto, and IIA: that there is necessarily a dictator in society. In other words, it is impossible to have all four—Universal Domain, Weak Pareto, IIA, and no dictatorship—simultaneously. I don't see why this wouldn't be considered 'bad' for society.

> **Comments:**
> **username_1**: Depends on what you mean by "bad". Arrow's dictator is not a dictator in the ordinary sense of the word. S/he could very well be the benevolent dictator that economists have long dreamed for.  
> **username_2**: I agree. I just meant that the existence of an individual (dictator) whose preference becomes the social preference, regardless of what everyone else prefers, does not 'sound good' and may not be desirable. In reality, he may be benevolent and may not exercise his power to choose outcomes in a 'bad' way, as he might take others' preferences into account when determining his own preference over the alternatives and hence can act representative.  

---

### Answer by username_3 on 2024-09-19. Score: 0

> What if there exists a person i
> who is not the decision maker, but they just get lucky and it happens that the 'optimal' social welfare function happens to yield whatever decision i
> prefers?

I am not really sure where you see the possibility of this in math. The voter becomes dictator precisely because in the ranking voting setup there is an very analogous result median voter theorem that shows that the choice selected by voting ranked voting system will depend critically on that one person, the dictator (e.g. if the 'dictator' prefers A, A will be selected, when 'dictator' prefers C and changes the vote that will be the selection for whole society). Similarly, in first past post system model, assuming rationality, only the median voter's vote matters all other votes are irrelevant for the result. This is what makes the agent dictator. This is what you get from the Arrows seminal 1950 work.

I am not sure how someone can get lucky in the model. It is deterministic model, the only luck there is to be born with given preferences. Moreover, there is no 'optimal' social welfare function. Idea of optimal social welfare function implies that you can somehow objectively say Utilitarian social welfare function is more optimal than Rawlsian.

Regarding your practical example. The preferences of those voters might happen to be around the median preferences. But that does not mean those voters do not impose their preferences on rest of the society if they vote according to their preferences and if their vote is tie breaker. If there is a committee of 13 judges who need to decide whether whole society will be Rawlsian or Utilitarian, 6 are more Utilitarian aligned and 6 more Rawlsian, and finally in the middle there is one judge who is kind of undecided but leads toward Utilitarian preferences, then it is that last judge that decides which preferences will be imposed on the whole society.

Just because some outcome is arrived at through democratic means it does not mean it is not imposed, or even not imposed by the treat of force. For example, in democratic nations tax levels are decided by democratically elected politicians. Yet that does not make the taxes voluntary compared to non-democratic countries. They are still being imposed, under threat of physical violence.

Regarding the Arrow point, I think the direct interpretation is that Arrow simply points out that obviously there is a qualitative difference between dictator that rules by military might, and dictator that imposes their will based on some mechanism to which many members of society consented. However, using the definitions, if outcome depends on one single person then the person is still dictator whatever other qualitative differences there might be.

Deciding by elections is not the same as voluntarism. In fact, in most models when you allow agents to decide on voluntary provisions of public goods or other government actions you will get typically completely different results when you let the same agents vote and take collective action that then has to be imposed back on everyone (e.g. see examples in Mueller Public Choice III).

Hence those people in that example are imposing their preferences, as these words are traditionally understood in the public economics, public choice and political economy literature.

> Suppose there exists an individual whose preference is defined as 'I want what society as a whole wants'. Maybe they're very well informed about voter preferences, and are selfless.

This is kind of moot point. The whole point of Arrow paper is to show that you simply cannot discover what society as whole wants through ranked voting. There might be ways how to discover preferences of society, but ranked voting system are not able to do that. This is the whole point of the conditions. Either the ranked voting system gives you cycling results, or tells you just about preferences of one individual and not preference of everyone and not ignoring completely preferences of voters at the same time.

Moreover, again consider example of median voter who is tie breaker between option A and B. Again suppose there are 6 voters voting for A and 6 for B. What should this selfless and informed voter that wants to just pursue preferences of society choose? The voter know that 6 people want A and 6 people want B. How does the voter selflessly decide? That voter is in position to become Buridan's ass.

We are dealing here with discrete choices. This 'selfless' voter will either impose A on all voters or B on all waters. Or if you end up in tie you get a median voter equivalent to the Condorcet cycle (this is why Arrow simply assumes this option away, as in that case you can satisfy other properties but no choice can be shown to be strictly preferred and chosen).

> Why is it bad if a 'fair' or 'optimal' social welfare function happens to yield decisions that coincidentally always align to one particular individual's preferences?

1. I do not know where you read that this is considered to be bad. I will admit that some time already passed since I read Arrow, but I can't recall any passage where he would claim that this is bad.

   I searched the original [Arrow 1950](https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf) article and I couldn't find any instance of Arrow claiming that this result is somehow bad or even undesirable. It is just property of ranked voting system.

   In fact its not even property of every voting system, it only applies to ordinal voting systems. Rated voting system, for example, does not have this particular result.
2. I do not believe there is any consensus on what would be 'fair' or 'optimal' social welfare function. If such consensus would exist then the whole idea of democracy does not make sense anyway. If there is fair or optimal social welfare function then dictatorship of benevolent social planner would be desirable.

   If you believe some social welfare function is fair or optimal then indeed as long as some voting system, like ranked choice voting, delivers this outcome it does not matter if the system is de facto dictatorship because the outcome is just due to single individual.

   The point of Arrows impossibility theorem is to show that if you want to use democracy as a democratic social preference aggregator, ranked choice voting system is not a system that can do that for you.

> **Comments:**
> **username_1**: Arrow does say that non-dictatorship (cf. Condition 5 of OP's linked book) "is certainly a reasonable one to impose on the form of the social welfare function." So he does seem to view non-dictatorship as a desirable property of a SWF.  
> **username_3**: @HerrK. but I believe that is in positive context of making SWF that is democratic, I do not believe he is making normative judgement, i.e. this is better than benevolent social planner imposing SWF from above  

---

