Someone ran across my CV and asked me if I could send anything I’d written on game theory. So I dug up my class notes (for an undergrad class at Hillsdale) on Kenneth Arrow’s famous “Impossibility Theorem” regarding social choice. I haven’t looked at these in 7 years, so I hope they’re right:
NOTES ON ARROW’S
History of Economic Thought II
Following the methodological revolution described in Hicks, by the 1940s most economists no longer believed in cardinal utility. At the very least, most economists considered it much safer to assume that people had merely ordinal preferences, rather than to take the stronger view that people actually received units of psychic happiness (“utils” or “wantabs”) from various goods and services.
Economists developed formal techniques to rigorously develop this line of thought. The first step is to define the set of all possible items to be valued. Depending on the context, this set consists of different types of things. At the most abstract level, it could be viewed as “the set of all possible universes.” In a much more specific example, it could merely refer to “the set of all possible pizza orders the class could phone in to Hungry Howie’s.”
Once we have defined the appropriate set, we then can talk about how each individual ranks each element in the set. Since we are not going to assume cardinal utility, we can only discuss an individual’s ordinal rankings; that is, we can only take two elements at a time, and ask the individual, “Which of these do you prefer, or are you indifferent between them?” We can never ask—indeed, it doesn’t make sense to ask—the individual, “How much more do you like this element over this other element?”
Formally, we can summarize an individual’s answers to these questions by use of a preference relation. Normally we indicate this by a symbol that looks like a curvy greater-than-or-equal-to sign, but here I’ll just use the symbol @. If we take two elements, let’s call them x and y, from the set of all possible things to be valued, then the statement x@y means that the individual thinks that x is at least as good as y. If it were not true (at the same time) that y@x, then we would conclude that x is better than y (not merely just as good), because we know x is at least as good as y, but y is not at least as good as x. And if we knew that x@y and y@x at the same time, then we would conclude that this individual is indifferent between x and y.
NOTE: In order to construct a coherent ranking (from best to worst) of an individual’s preferences, it is necessary that his or her @ be complete and transitive. If @ is complete, that means it can be applied to any two elements from the set. I.e., for any elements (call them a and b), the individual could report either a@b, b@a, or both. If @ were incomplete, then the individual might say, “I really don’t know how I feel about those two elements; I can’t tell you which is at least as good as the other.”
If @ is transitive, then whenever x@y and y@z, it must also be the case that x@z.
Economists often want to use their science in order to make policy recommendations, or at least to make “objective” statements about various social arrangements. By analogy with an individual preference ranking, we can ask how “society” does (or should) value the different possible elements in the set of all valued things.
Because society is ultimately composed of individuals, most economists think that “social” preferences ought to be constructed from the preferences of the individuals in society. But at this point, a problem emerges: If people do not agree on how to rank, say, x with y—i.e. some people feel that x@y while other people do not—then how can we say how “society” should rank these two possible outcomes?
Economists thus began a search for plausible social welfare orderings. These are functions that take the @ for each individual—and we could keep them distinct by putting a superscript on them, so that @1 is the preference relation of person #1, etc.—and then use this information to generate a preference relation for society, which we will label @S.
So now the question is, what types of social welfare orderings are appealing, both on logical and moral grounds? In principle, there are billions of different rules we could invent, in order to generate a @S out of the individual @i of each member i in the society.
Kenneth Arrow intended to weed out the “silly” or obviously distasteful social welfare orderings (henceforth SWO). So he came up with a quite reasonable list of criteria that any decent SWO would need to satisfy.
One basic requirement is that it should be complete and transitive. That is, whenever the individual @i of each person in society is complete and transitive, whatever our rule is that generates the @S, that list of social preferences should also end up being complete and transitive.
Another criterion is that the SWO should obey weak Pareto optimality. In the present context, this means that if x@iy for every single person i, then it should also be the case that x@Sy. In other words, if every single person in society thinks that x is at least as good as y, it would be ridiculous if our SWO then ended up saying that “society” should value y more than x.
A third criterion is the independence of irrelevant alternatives. This is the least intuitive of the criteria. What it requires is that the determination of the social ranking of x and y should depend only on how each individual ranks x and y.
The final criterion is no dictatorship. This means that there cannot be some individual j such that @S= @j no matter what every other person’s preferences are. Note that this is a very weak requirement. For any particular group of individual preferences @1, @2, @3, …, it’s perfectly acceptable if our SWO constructs a @S that happens to be identical to some individual @j; this alone would not christen individual j as a dictator. What would qualify him as a dictator is if @S= @j for any possible group of individual preferences @1, @2, @3, …
What Arrow proved is that there does not exist any SWO that satisfies all four of the above conditions (if we have at least a few people and a few different elements in the set of valued things). Specifically, Arrow proved that if we assume we are dealing with an SWO that meets the first three criteria, then that SWO necessarily must work by picking some individual j and then simply setting @S=@j.
Since Arrow’s Impossibility Theorem is a negative result, it’s best to illustrate it by showing SWOs that do not satisfy his criteria. For simplicity, we’ll assume there are only three people, Joe, Billy, and Martha, and only three possible states of the world, x, y, and z. We thus are looking for a set of rules to take @J, @B, and @M in order to construct a “social” ranking of the possible outcomes x, y, and z.
Suppose we have the very simple SWO that says, “No matter how Joe, Billy, and Martha rank the alternatives, @S should always be defined so that x@Sy, y@Sz, and x@Sz, and so that the reverse is not true, e.g. that it is not the case that y@Sx, etc.”
Which of Arrow’s criteria does this suggested SWO violate? Well, it’s complete and transitive, so it’s okay on those grounds. It doesn’t have a dictator, either (it’s always possible that any person will have preferences that differ from those indicated by @S). Although it’s not as easy to see, I’m pretty sure that this hypothetical SWO also obeys the independence criterion. (E.g. the social ranking of x and y will never be affected by changing the individuals’ rankings of, say, y and z.)
What this SWO does (obviously) violate is the weak Pareto condition. For example, if Joe, Billy, and Martha all strictly prefer z to y, then our suggested SWO will still say that “society” prefers y to z. Thus our suggested SWO does not meet Arrow’s criteria, and we must keep looking.
What about majority rule? That is, suppose we define @S such that x@Sy only if at least two people feel this way, etc.
Majority rule violates the criterion of transitivity. That is, there are possible preferences that Joe, Billy, and Martha could have, such that a @S constructed on the basis of majority rule would violate transitivity. (To see this, consider the case where Joe ranks the alternatives in the order x, y, z, Billy ranks them y, z, x, and Martha ranks them z, x, y.) Note that Arrow requires the SWO to be transitive for any possible list of individual preference relations; it’s not enough that the SWO might satisfy all four criteria for some particular list of individuals’ preferences.
Finally, let’s consider the SWO that proceeds like this: “We will say that x@Sy only if Joe, Billy, and Martha all agree that x is at least as good as y. If at least one of them disagrees, though, we will say that it is not the case that x@Sy. Etc.”
Which criterion does this rule violate? Well, it’s transitive (so long as the individual relations are); if everybody thinks x is better than y, and that y is better than z, then that means everybody thinks x is better than z, and thus so will “society.” There is also no dictator with this proposed SWO, and it is also true (I think) that there is no violation of independence. And of course this SWO obeys the weak Pareto condition.
But this proposed SWO is, unfortunately, incomplete. That is, the rule we defined will not always tell us how “society” should compare, say, x and z. For suppose that Joe thinks x is strictly better than z, but that Martha thinks that z is strictly better than x. Then according to our rule, it can neither be true that x@Sz nor z@Sx. And completeness requires that our preference relation be able to tell us that one (or both) of these items is at least as good the other. Hence this proposed SWO too fails to satisfy Arrow’s criteria.