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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 $L$.

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    $\begingroup$ This doesn't seem to be about mathematics, in any case I'll tell you why I play. It is true that lottery is $-\text{EV}$ for the player, financially. But the $\text{Life-EV}$ is huge. If I win, I don't just win money, I win the possibility a lot of the things I want to do. $\endgroup$ Commented Feb 28, 2014 at 20:24
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    $\begingroup$ This is a question about psychology, not mathematics. $\endgroup$ Commented Feb 28, 2014 at 20:24
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    $\begingroup$ My probability lecturer plays on the lottery. He even said: "you should play, you have a better chance of getting rich by playing than by working". But he's biased, he won $\ddot \smile$ $\endgroup$ Commented Feb 28, 2014 at 20:32
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    $\begingroup$ The mathematical question is, obviously, "do there exist mathematical arguments (probably different from expected value) for risk-averse persons to play lotteries". @amWhy and close voters. $\endgroup$ Commented Feb 28, 2014 at 20:57
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    $\begingroup$ This is a perfectly sensible mathematical question. Not off-topic. One possible answer: who's to say people are risk-averse when it comes to the lottery? Risk-aversion means having a concave vNM-Morgenstern utility function, i.e. marginal utility decreases w.r.t. wealth. People buying lotteries need not have this attitude. The question should be re-opened. I am sure the economists around here have something to say about this. $\endgroup$ Commented Mar 1, 2014 at 6:03

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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.

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    $\begingroup$ In real life, people do not care about the expected value and that's I think is the point. $\endgroup$ Commented Feb 28, 2014 at 21:46
  • $\begingroup$ This answer states that a sensible answers exists without actually giving them. In what sensible specification would someone use a lottery for risk diversification? $\endgroup$ Commented Mar 1, 2014 at 8:04
  • $\begingroup$ "This answer states that sensible answers exist without actually giving them".// The third paragraph mentions that a rational investor could evaluate the expected value as positive under certain playing strategies (wait for large enough jackpots, play unpopular numbers). At the extreme there are syndicates of investors who buy millions of tickets, sometimes the full set of possible bets when payoffs are high enough. This already makes it untrue that "no sensible answer was given" or that such betting is impossible under economists' definitions of risk aversion. @MichaelGreinecker $\endgroup$ Commented Mar 3, 2014 at 20:56
  • $\begingroup$ "use a lottery for risk diversification?" // The portfolio diversification accomplished by lottery gambling is not a form of risk management, it is diversification of opportunity. Risk averse individuals who do not believe themselves capable of becoming independently wealthy without lotteries can rationally make small bets to allow that as a remote possibility, and they can and apparently do compare lottery opportunities using expected value. Syndicates are the extreme, but bets on large jackpots based on the estimated expected payoff are quite common in people who do not usually gamble. $\endgroup$ Commented Mar 3, 2014 at 21:05
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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...)

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