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3. Lemon Tart or Ice Cream?
Irrelevant Alternatives

The question:

If you have to decide between two alternatives, can the sudden availability of a third option make you change your mind about the first two? Could the addition of an irrelevant alternative, that you don’t even care about, change the ranking of the other options?

The paradox:

The answer, surprisingly, is yes. To illustrate, let me paraphrase an anecdote that is told about the Columbia University philosopher Sydney Morgenbesser: the professor was in a restaurant and was ready for desert. “We have apple pie and lemon tarte,” the waiter told him. Morgenbesser ordered lemon tarte. After a few moments the waiter returned and announced that there wass also ice cream. "In that case I will have the apple pie,” Morgenbesser retorted.

With his world-famous famous sense of humor, Morgenbesser hit on a profound paradox. An option that has no chance of being chosen, may, purely because of its existence, reverse a decision.


Something like this just should not happen. The availability of unloved ice cream on the menu should not influence the choice between the other options. Yet occasionally it does happen and, not so long ago, it has changed the course of history.

In the US presidential elections of 2000, the contest was between George Bush and Al Gore. It was a close race and nationwide Al Gore won the popular vote with 50,999,897 votes, as against Bush’s 50,456,002. However, it is the Electoral College that decides who becomes president and the real showdown was in Florida whose 25 delegates go to the winner. Bush won Florida’s delegates, and with it the presidency, by a hair’s breadth, with 2,912,790 votes against Gore’s 2,912,253. Gore missed the top job by 537 votes.

But there was a third candidate, the nominee of the Green Party, Ralph Nader. Nationwide, he received no more than 2,882,955 votes which rendered him totally irrelevant. But, crucially, in the State of Florida he garnered 97,488 votes, and that turned out to be decisive. Had Nader not run, most of the Greens would certainly have opted for Al Gore, a vociferous defender of the environment. Surely, he would have garnered 537 votes more than Bush from among Nader’s voters. But by entering the race, Nader handed the presidency to George Bush. He was the game changer. He was Morgenbesser’s ice cream.

* * *

In the mid 1940s, the Hungarian John von Neumann, the most renowned mathematician of the twentieth century, and the Austrian economist Oskar Morgenstern were working on their Theory of Games and Economic Behavior which would usher in what would come to be known as game theory. (See also the chapter on the Prisoner’s Dilemma///??)

As theoreticians, they analyzed their subject matter in a rigorous manner. No wishy-washy arguments, no hand-waving, nothing but clear definitions, proofs and conclusions. They specified several characteristics to which the players of a game must adhere. In fact, they elevated these characteristics to axioms, i.e., to postulates about human behavior that are so obviously true and self-evident that they require no proof. Take axiom 1: players faced with the choice between A and B must either prefer A to B, or B to A, or be indifferent between the two. Or axiom 2: if A is preferred to B, and B to C, then A must also be preferred to C.

* * *

Just a few years later, a graduate student in New York, Kenneth Arrow was busy with his doctoral thesis at Columbia University. The subject was whether the tastes of many individuals can be aggregated to make social choices.[1] Is it possible to “construct a procedure for passing from individual tastes to a pattern of social decision-making”?

It’s not simple. For starters, the tastes of different people cannot be compared. One person’s love of bananas is not comparable to another’s love for apples. In the same manner, a ballot cast for candidate A does not indicate whether the voter prefers candidate A intensly to candidate B or just a little bit.

One mechanism that comes to mind is majority voting. What could be fairer than that? But as we saw in the chapter on the Condorcet Paradox, there is a serious problem with majority voting….apart from the Ralph Nader situation and Morgenbesser’s ice cream: majority voting may result in cycles. So the question becomes: is there an aggregation procedure that describes the preferences of an entire population fairly, but without producing cycles?

Like von Neumann and Morgenstern, Arrow built a mathematical model. To make it rigorous, he specified some conditions that the aggregation procedure must fulfill. These conditions are so reasonable that he too elevated them to axioms. For example: if all voters prefer A to B, then the aggregation should reflect this choice. Or: when one individual decides to change her vote, and rank a candidate lower than previously, the aggregated ranking of this candidate must not become higher. Or: a single voter must not impose his or her choice on the social welfare function. After all, this would be akin to a dictatorship.


How can the paradox of irrelevant alternatives be resolved? And what does it have to do with von Neumann and Morgenstern’s game theory and Arrow’s social choice function? Did they somehow resolve the conundrum?

Actually, they didn’t. They hit upon the paradox of irrelvant alternatives like on a brick wall. There was no way around it, it could not be explained away. So, instead of declaring the paradox a bug, they made it a feature! They incorporated the fact that choices must be independent of irrelevant alternatives as a precondition into their models. The Independence of Irrelevant Alternatives (IIA) was made one of the axioms.

But if IIA is one of the preconditions of a fair election, and it is violated, then what? In a path-breaking theorem, Arrow proved that no fair election procedure that fulfills all the axioms and does not produce Condorcet Cycles can exist. Something has to give.

Usually a violation of the IIA is tolerated as, for example, in the American elections in 2000. But if we want to hold on to IIA, one of the other axioms must be relaxed. The only one that comes to mind is the one which says that a single voter must not impose his or her choice on society. And so we have the utterly depressing conclusion: either we allow a nincompoop to screw up our election, or we accept a dictatorship!

Technical supplement:

In Arrow’s model the axiom takes the form “The social preferences between alternatives x and y depend only on the individual preferences between x and y.” The fact that in the real world Ralph Nader did change the ranking between George Bush and Al Gore, and the availability of ice cream did make Sidney Morgenbesser change his mind about apple pie versus lemon tarte, makes no difference to mathematicans, neither to Arrow nor to Morgenstern and von Neumann. At least the model is correct and self-consistent. The main thing is that the paradox is explained away; real-world relevancy is secondary.



[1] which was later published as Social Choice and Individual Values, ////

Corrections, comments, observations:    

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