Would a risk-averse agent ever accept gambles with negative expected value?

Question

Consider a risk-averse agent (whose utility for money is strictly concave) that maximizes expected utility. Would such agent ever a accept a gamble whose expected value is negative? (E.g., think of state sponsored-lotteries Lotto 649, or Atlantic lotto, etc.)

More formally, consider an agent with a utility function $u$ that is increasing and concave, e.g., $u(x) = \sqrt{x}$. Define a lottery $L$, with probability $\alpha$ for a low state $x_l$ and probability $1-\alpha$ for a high state $x_h$, that has negative expectation, i.e., $E[L]<0$. Assume initial wealth $W$ that is then higher $E[L]$. We say the agent will accept the lottery $L$ iff her expected utility from this lottery $E_u[L]$ is higher then her utility without the lottery. The question is: given that $E[L]$ is negative, can we say that the agent with a concave utility function $u$ will never accept the lottery $L$.

Answer 1

For risk-averse people with many good alternatives for spending small sums of money, an occasional lottery play is portfolio diversification.

For poor people or ones without good alternative micro-investments (and, typically, many bad options), there are all sorts of reasons why saving one more coin is not necessarily more appealing than using it sometimes to purchase a lottery ticket.

Expected value is a meaningless metric for the lotteries with low odds and low entry costs. The positive part of the expectation would usually take thousands of lifetimes to realize, and the negative total can be mitigated or maybe even reversed (the analysis is complicated) by playing selectively when the jackpot is large.

One of the more famous Berkeley mathematicians (Chern?) had a Ph.D student who won millions of USD in a lottery and donated some of the money to the department. It is hard to say how many such windfalls might have been lost by math departments that dutifully taught students never to invest for negative expected returns, but it is food for thought.

Answer 2

Using the decision theory (i.e. micro-econ 101) framework:

Define the operator $E_u(.)$ as the expected utility from a lottery. That is, for lottery $L$ with probability $\alpha$ for state $X_l$ and probability $(1-\alpha)$ for state $X_h$: $$E_u(L) = \alpha u(X_l) + (1-\alpha) u(X_h).$$

We say that an agent is risk averse if her utility function $u$ is concave. By definition, a concave function $f$ satisfies the property $$ f(\alpha X_l + (1-\alpha) X_h) \geq \alpha f(X_l) + (1-\alpha) f(X_h). $$

We can use the above property on the concave function $u$, noting that $\alpha X_l + (1-\alpha) X_h = E[L]$. That is, $$ u(\alpha X_l + (1-\alpha) X_h) = u(E[L]) \geq \alpha u(X_l) + (1-\alpha) u(X_h). $$

Lastly, remember that we are considering whether the agent will want to participate in a lottery with the property $E[L]<W$, i.e. a lottery where the agent loses in expectation compared to not participating (where $W$ is initial wealth).

If $E[L]<W$, and $u$ is increasing and concave, then $u(E[L])<u(W)$. Putting it all together we get $$ u(W) > u(E[L]) \geq \alpha u(X_l) + (1-\alpha) u(X_h)=E_u(L)$$ which means that a risk-averse agent will never pay to participate in a lottery with a negative expected value (but may participate if she is being paid...)