Rabin's calibration theorem

In microeconomics and decision theory, Rabin's calibration theorem (also known as Rabin's paradox or Rabin's critique) is a theoretical result related to the calibration of risk aversion within expected-utility theory. In intuitive terms, it shows that an expected-utility-maximizer who is moderately risk averse over small-stake gambles must show implausibly high risk aversion over high stakes. It is seen as critique of how the classical model of diminishing marginal utility of wealth can deal with representing risk-averse behavior over money within expected-utility theory.

The result was first shown by Matthew Rabin in 2000.^[1] It has since been extended to non-expected-utility models of choice under uncertainty.^{[2] [3]}

Example

Consider an expected-utility decision-maker with wealth level ${\displaystyle w}$ and concave Bernoulli utility function ${\displaystyle u}$. Imagine that she rejects the following lottery:

${\displaystyle L={\begin{cases}{\text{win }}\$125{\text{ with }}50\%{\text{ chance,}}\\{\text{lose }}\$100{\text{ with }}50\%{\text{ chance.}}\end{cases}}}$

This implies that

${\displaystyle 0.5u(w+125)+0.5u(w-100)<u(w)}$

${\displaystyle \iff u(w+125)-u(w)<u(w)-u(w-100)}$

${\displaystyle \iff {\frac {u(w+125)-u(w)}{125}}<{\frac {100}{125}}{\frac {u(w)-u(w-100)}{100}}.}$

Since ${\displaystyle u}$ is concave, we have

${\displaystyle u'(w+125)<{\frac {u(w+125)-u(w)}{125}}<{\frac {100}{125}}{\frac {u(w)-u(w-100)}{100}}<{\frac {100}{125}}u'(w-100).}$

This means, therefore, that on each interval ${\displaystyle [w-100,w+125]}$ of length ${\displaystyle 225}$ for which the above holds (i.e., the lottery is rejected), increasing wealth by ${\displaystyle \$225}$ decreases marginal utility ${\displaystyle u'}$ by a factor of at least ${\displaystyle {\frac {100}{125}}=0.8}$. This would imply that receiving ${\displaystyle \$1000}$ would lead to marginal utility being reduced to ${\displaystyle 0.8^{\frac {1000}{225}}\approx 37\%}$ of its current value, and receiving ${\displaystyle \$10000}$ would lead to it being reduced to ${\displaystyle 0.8^{\frac {10000}{225}}\approx 0.005\%}$.

For example, suppose that the decision maker rejects the above lottery ${\displaystyle L}$ for all wealth levels ${\displaystyle w\leq \$300,000}$. This will imply that, at wealth level ${\displaystyle w=\$290,000}$, she rejects the following lottery:^[1]

${\displaystyle L'={\begin{cases}{\text{win }}\$160{\text{ billion with }}50\%{\text{ chance,}}\\{\text{lose }}\$1000{\text{ with }}50\%{\text{ chance.}}\end{cases}}}$

Such rejection is intuitvely absurd and empirically counterfactual.^{[4] [5]}

Theorem

Below is a simplified version of Rabin's original theorem.^[6]

Theorem: suppose an expected-utility maximizer has Bernoulli utility function ${\displaystyle u:\mathbb {R} \to [-\infty ,\infty )}$ which is increasing and concave. Suppose that there exists an interval ${\displaystyle W\subseteq \mathbb {R} }$ and money values ${\displaystyle g>l>0}$ such that, for all ${\displaystyle w\in W}$, we have: 1) ${\displaystyle u(w)>-\infty }$, and 2) the decision maker rejects the binary lottery ${\displaystyle L=(w-l,0.5;w+g,0.5)}$. Then for any integers ${\displaystyle m,k}$ such that ${\displaystyle [w-kg,w+mg]\subseteq W}$, we have

${\displaystyle u(w+mg)-u(w)\leq {\frac {1-\left({\frac {l}{g}}\right)^{m}}{1-{\frac {l}{g}}}}[u(w+g)-u(w)]<}$

${\displaystyle <{\frac {\left({\frac {g}{gl}}\right)^{k}-1}{1-{\frac {l}{g}}}}[u(w+g)-u(w)]\leq u(w)-u(w-kg)}$

Corollary: Under the assumptions of the Theorem above, an expected-utility maximizer rejects the lottery ${\displaystyle L'=(w+mg,0.5;w-kg,0.5)}$ for any integers ${\displaystyle m,k}$ such that ${\displaystyle [w-kg,w+mg]\subseteq W}$.

Extensions

Zvi Safra and Uzi Segal extended Rabin's calibration results to any model which has a Gâteaux differentiable utility ${\displaystyle V}$ from lotteries over final wealth. This includes many non-expected-utility models, such as rank-dependent expected utility or Gul's disappointment aversion.^[2]

James C. Cox and others have also shown that any utility representation which is noninear in either probabilities or payoffs suffers from similar calibrational problems.^[3]