Wednesday, 10 December 2014

Affine Transformations of Payoffs should not affect Decision-making

Behavioural economics has unearthed interesting features of decision-making that sometimes seem to violate rationality assumptions.  These include loss aversion - people's willingness to take risks to avoid losses, but to avoid risks to take certain gains - and the endowment effect, in which people put a higher price on assets they own compared to those they don't.

These findings don't necessarily violate rationality.  They might also support the hypothesis that people have complex objectives comprising many different factors.  For instance, if losing confers psychological or social costs in itself, additional to any material loss, taking risks to avoid losses might be optimal.

But where objectives are well understood - say in business, where (mostly) the objective of a firm is to maximise long-run profits - optimal decision-making won't be influenced by anything other than the outcomes associated with each possible course of action, their relative probabilities, and the organisation's appetite for risk.

One possibly-surprising feature of optimal decisions is that they are robust to affine transformations of payoffs.  What this means is that multiplying all payoffs by the same (positive) factor, or adding or subtracting a constant sum from all payoffs, should make no difference to the optimal decision.  This means that affine transformations make a good test of a proposed decision rule.  If all the outcomes were halved, or doubled, or all worth exactly £1m more than they are, the decision rule should always identify the same choice.  Inter alia, this is one reason that lump sums (like licensing fees or the poll tax) are considered to be the least distortionary types of tax.

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