# Review: “Social Choice and Individual Values”

In this book, Arrow ostensibly proves, using formidable mathematical tools, that social choice is impossible. Social choice can be understood as that choice society would make by aggregating all the individual choices specified by its members. An individual choice would be specified as a preference ranking over all possible alternatives whether they be candidates standing for election or alternative economic states. A social welfare function would then determine society’s choice or the social choice as a function of the individuals input data.

Arrow lays down five “rational and ethical criteria” which any social welfare function would have to meet (according to him) and then proceeds to demonstrate that no such social welfare function exists. The problem is one of his “rational and ethical criteria” is neither rational nor ethical. That criterion is “Independence of Irrelevant Alternatives.” This criterion reduces the voters to making binary decisions between each possible pair of alternatives. Arrow uses the example of a slate of candidates standing for election. “Suppose that an election is held, with a certain number of candidates in the field, each individual filing his list of preferences, and then one of the candidates dies. Surely the social choice should be made by taking each of the individuals’ preference lists, blotting out completely the dead candidate’s name, and considering only the orderings of the remaining names in going through the procedure of determining a winner.”

Sounds reasonable enough. The problem is that each individual might order his list differently if one candidate is eliminated. Arrow forbids this to happen. Say there is an election among candidates A, B and Hitler. A voter might rank A and B tied for first and Hitler last on the grounds that any candidate is preferable to Hitler. Then if Hitler dropped out of the race and the individual were repolled, his preference between A and B might then emerge. For example, given that Hitler is not in the race, the individual might prefer A to B or vice versa. In other words a tie might become an actual preference. Therefore, the problem, if the individuals cannot be repolled, is to estimate (probabilistically) how the individuals would have voted had they been repolled. But Arrow simply wants to apply the same procedure as was applied to the original set of candidates and force each individual not to change any of his or her rankings. This is not rational and, since it limits each individual’s freedom of expression, it’s not ethical either.

Arrow’s mathematical analysis is very impressive, but the same essential result known as the “paradox of voting” was discovered in the eighteenth century by the French philosopher Condorcet. Arrow’s work is just an uber-mathematization of Condorcet’s result. In fact much simpler proofs of this result can be found than Arrow’s elaborate one. See my web page “A Short Proof of Arrow’s Theorem.”

Arrow also does not allow ties as social choices. In fact he confuses the concepts of indifference and tie. An individual can be indifferent between 2 or more candidates. He can so indicate this in his preference list. Society, on the other hand, might have a tie among whole preference lists themselves. In a simple election between 2 candidates in which the voters simply vote for A or B, society could have a tie between those candidates. When whole preference list are involved as voting inputs, society would have to provide for the eventuality that there could be a tie among preference lists. Arrow doesn’t do this.

In conclusion, Arrow defines the problem in such a narrow fashion that his result seems almost a tautology. A deeper understanding of the nature of the problem could lead to a different result: that social choice is indeed possible.