Independence of irrelevant alternatives

In social choice theory, Arrow’s ‘independence of irrelevant alternatives’ (IIA) is one of the conditions in Arrow’s impossibility theorem, which states that it is impossible to aggregate individual rank-order preferences satisfying IIA in addition to certain other reasonable conditions.

Arrow defines IIA as: The social preferences between alternatives x and y depend only on the individual preferences between x and y.  In other words, preferences for x or y should not be changed by the inclusion of z, i.e., z is irrelevant to the choice between x and y.

IIA can be illustrated by providing the following practical example of a violation of this condition.  At an ice-cream shop, a customer is given the choice of chocolate and vanilla ice-cream.  The customer orders the chocolate ice-cream.  The shop assistant then says that they also have some strawberry ice-cream not on display, at which point the customer says ‘In that case I’ll have the vanilla ice-cream.’

The point of IIA is that availability of the strawberry ice-cream in the above example is irrelevant to the choice between chocolate and vanilla ice-cream.  By extension, the addition of a third candidate to a voting ballot paper is irrelevant to the choice between the first two candidates.

The IIA condition works when voters act rationally in accordance with it.  A voting procedure that satisfies IIA is much less open to manipulation by strategic voting or by agenda setting.

On the other hand, experiments by social theorists such as Amos Tversky, Daniel Kahneman, and others have shown that human behaviour rarely adheres to IIA in practice.  People irrationally take into account irrelevant alternatives.  An implication of this finding is the verification of Arrow’s impossibility theorem, at least as far as IIA is concerned.

References

Kenneth J. Arrow, 1951, 2nd ed., 1963. Social Choice and Individual Values, Yale University Press.

Russell Hardin, “Public Choice Versus Democracy” in David Copp, Jean Hampton, and John E. Roemer (eds) The Idea of Democracy, Cambridge University Press, Cambridge, 1993, pp.157-172.