I have thought a little bit about the question why Non-Dictatorship might be a reasonable axiom for Arrow's Theorem or in other words what might be bad with Dictatorship defined as in Arrow's Axiom. Here are my thoughts:
I take it that the focus of Arrow's theorem is finding a plausible voting-rule for social decisions. I know that this is not what Arrow intended, but Sen's article has convinced me that this is the most plausible interpretation or in other words the only interpretation in which the results actually hold.
The axioms of Arrow's theorem are the following (According to Heap et al. The Theory of Choice, Blackwell 1992, p. 210):
Collective rationality: The collective choice should be represented by an ordering of all alternatives. An ordering is complete: every pair of alternatives can be ranked against each other, even if the ranking is one of indifference. The ordering should also be transitive, so that for any alternative, x, y and z, if x is preferred or indifferent to y and y is preferred or indifferent to z, then x is preferred to z.
Universal Domain: The function mapping a statement of individual preference into a statement of collective choice should be capable of taking as its domain of operation all logically possible orderings by individuals.
Pareto inclusiveness: If all individuals prefer x to y, then society should prefer x over y.
Independence of irrelevant alternatives: The social preference between two alternatives is restricted to information solely on how individuals rank these alternatives.
Non-Dictatorship: No named individual should be able to determine the social choice in all circumstances in the sense that the social ordering coincides with the ordering of that individual whatever others might think.
The task is finding a rule to determine a social preference function from the individual preference functions which satisfies these axioms. The famous conclusion is that there is no such rule.
We all know that the majority-rule can't fulfill the Arrow's axiom's because it leads to intransitivities in the social prederence (which violates the collective rationality axiom) in cases where the preferences of the individuals have a structure like the following (where set of individuals is {A, B, C} and the set of alternatives is {x, y, z}:
A: x > y > z
B: y > z > x
C: z > x > y
The Social Preference would be according to the majority rule:
x > y (since for A and C x > y) y > z (since for A and C y > z) and z > x (since for B and C z > x)
So, we can rule out the majoritiy rule as a rational rule for collective decision-making (as long as we want to uphold all the axioms). If we now turn to other rules. we can stipulate that these rules will hace at least one feature: They will violate the majority rule at least on one occasion that is they will determine that at least in one case something is preferend to another thing although the majority prefers them the other way round. We can stipulate this of course, because otherwise the rule would just be the majority rule! Now, Arrow's Theorem shows that if we have at least one violation of the majority rule then, by the pareto condition and collective rationality if follows that there will be a dictator defined in the following way:
Dictatorship: "The social ordering coincides with the ordering of that individual whatever others may think."
Now, on first sight there seems to be nothing problematic with a Dictator defined in such a way: He could be just a lucky person for whom it just happens to be the case that all his wishes are fullfilled without him having any causal power over this. So, why should we uphold the Non-Dictatorship axiom? Why not exchange it for a much weaker axiom?
Here is a reason why we might want to uphold Non-Dictatorship: First there has to be some feature of the rule which determines why someone is the dictator. This feature is either something people can influence or it is one which people can not influence. But, if it is a feature people can influence then the rule which allows Dictatorship in the weak sense, also would allow Dictatorship in the causal sense. On the other hand, if it is a feature which people cannot influence, it might seem unfair that the rule favours people with features like that on which they have no causal influence. If the rule favours the biggest person, it might seem unfair that this person is favoured for a property which he just happens to have. If the feature becomes something arbitrary (like winning a flip of a coin), then it might seem irrational to base a social decision on something arbitrary like this. Remember that we are talking about voting rules.
The second horn of this "dilemma" is obviously the weaker one. There might be at least some features on which people have no causal influence which might be a good base for social decision making. The rule could - for example - include that the least favoured group has their hearing sometimes even if the majority is against it. But then again: If this allows that the social ordering is sometimes always determined by the preferences of the least favoured group, this might seem like an unfair or arbitrary rule.
I take it that the focus of Arrow's theorem is finding a plausible voting-rule for social decisions. I know that this is not what Arrow intended, but Sen's article has convinced me that this is the most plausible interpretation or in other words the only interpretation in which the results actually hold.
The axioms of Arrow's theorem are the following (According to Heap et al. The Theory of Choice, Blackwell 1992, p. 210):
Collective rationality: The collective choice should be represented by an ordering of all alternatives. An ordering is complete: every pair of alternatives can be ranked against each other, even if the ranking is one of indifference. The ordering should also be transitive, so that for any alternative, x, y and z, if x is preferred or indifferent to y and y is preferred or indifferent to z, then x is preferred to z.
Universal Domain: The function mapping a statement of individual preference into a statement of collective choice should be capable of taking as its domain of operation all logically possible orderings by individuals.
Pareto inclusiveness: If all individuals prefer x to y, then society should prefer x over y.
Independence of irrelevant alternatives: The social preference between two alternatives is restricted to information solely on how individuals rank these alternatives.
Non-Dictatorship: No named individual should be able to determine the social choice in all circumstances in the sense that the social ordering coincides with the ordering of that individual whatever others might think.
The task is finding a rule to determine a social preference function from the individual preference functions which satisfies these axioms. The famous conclusion is that there is no such rule.
We all know that the majority-rule can't fulfill the Arrow's axiom's because it leads to intransitivities in the social prederence (which violates the collective rationality axiom) in cases where the preferences of the individuals have a structure like the following (where set of individuals is {A, B, C} and the set of alternatives is {x, y, z}:
A: x > y > z
B: y > z > x
C: z > x > y
The Social Preference would be according to the majority rule:
x > y (since for A and C x > y) y > z (since for A and C y > z) and z > x (since for B and C z > x)
So, we can rule out the majoritiy rule as a rational rule for collective decision-making (as long as we want to uphold all the axioms). If we now turn to other rules. we can stipulate that these rules will hace at least one feature: They will violate the majority rule at least on one occasion that is they will determine that at least in one case something is preferend to another thing although the majority prefers them the other way round. We can stipulate this of course, because otherwise the rule would just be the majority rule! Now, Arrow's Theorem shows that if we have at least one violation of the majority rule then, by the pareto condition and collective rationality if follows that there will be a dictator defined in the following way:
Dictatorship: "The social ordering coincides with the ordering of that individual whatever others may think."
Now, on first sight there seems to be nothing problematic with a Dictator defined in such a way: He could be just a lucky person for whom it just happens to be the case that all his wishes are fullfilled without him having any causal power over this. So, why should we uphold the Non-Dictatorship axiom? Why not exchange it for a much weaker axiom?
Here is a reason why we might want to uphold Non-Dictatorship: First there has to be some feature of the rule which determines why someone is the dictator. This feature is either something people can influence or it is one which people can not influence. But, if it is a feature people can influence then the rule which allows Dictatorship in the weak sense, also would allow Dictatorship in the causal sense. On the other hand, if it is a feature which people cannot influence, it might seem unfair that the rule favours people with features like that on which they have no causal influence. If the rule favours the biggest person, it might seem unfair that this person is favoured for a property which he just happens to have. If the feature becomes something arbitrary (like winning a flip of a coin), then it might seem irrational to base a social decision on something arbitrary like this. Remember that we are talking about voting rules.
The second horn of this "dilemma" is obviously the weaker one. There might be at least some features on which people have no causal influence which might be a good base for social decision making. The rule could - for example - include that the least favoured group has their hearing sometimes even if the majority is against it. But then again: If this allows that the social ordering is sometimes always determined by the preferences of the least favoured group, this might seem like an unfair or arbitrary rule.
4 comments:
Interesting! A very nice description of Arrows (im)possibility theorem. However, I will try to challenge one of your points.
You wrote: "there has to be some feature of the rule which determines why someone is the dictator" (emph. added). But why does this have to be the case? Why must the rule determine who the dictator is, can we not think of any other fair/good/reasonable method of assigning the dictator for cases where the majority rule runs into circles? Like we do when we assign a chairmen of a committee for example. The chairman often sits in the role of a dictator because she can determine a verdict if it is a tie for example, or she can formulate the proposition order in pair-wise voting. But the chairman is not assigned uniquely for each question. Furthermore, who is chairman often depends on factors outside the question at hand. Of course, one can claim that the method of assigning chairman in the first place also runs into Arrow's problem. But it seems at least possible to assign a chairman on other grounds than preference-based voting.
Hmmm. Maybe I was not clear enough on this. By saying "there has to be some feature of the rule which determines why someone is the dictator" I just meant that there must be something in the rule - a part of the rule - by which it is determined who plays the role of the dictator (that is who is the person whoms preference ordering can in at least one instance overrule the majority). Even if we determine this person by election or through a lottery, this method will surely be part of the rule or specified by it. I can't see any other way for rational decision making: We surely do not want to come up with a new method every time we get problems with the majority rule wouldn't we?
I don't rule out that there might be a fair method to determine this, but we have to keep in mind that whatever method this will be it will enable (in some instances) the person to force (in some sense) her preference ordering on the social ordering. Only if we find a method to determine which person this should be for which we can accept that the person will be a dictator (in the arrow sense), will we have good method.
Ok, maybe that wasn't more clear than before... The feature I was refering to does not have to be a explicit part of the rule. I could just be a consequence of how the rule is been phrased or something. The rule for example could say: Majority rule, but if intransitivity is possible as a result of the last vote, then the social preference in the last vote shall be that which will ensure transitivity. This feature doesn't determine a dictator explicitly, but as a consequence of the rule a dictator might be picked. Hope my point is clear now...
Hi, me again. I think I expressed my point a bit too sloppy before. Assigning the dictator does indeed have to be a part of the rule. My point was only that who is assigned does not have to be a result of each specific group vote on. For example: Many different questions might have to be decided over the course of a year, but the if-intransitivity-you-make-the-call-person for all the questions that year can be assigned in a process quite independent of each issue. In that case who is dictator will be exogenously given with regard to each specific social decision. Who the dictator is must still be part of each decision rule, and the voters will "have no causal influence" over who it is on each occation, but it may still be a fair and reasonable way of appointing the dictator (seen over the longer run).
Am I making any sense?
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