# When does gambling reduce risk?

Suppose that you face risk. It is obvious that taking gambles whose outcomes are negatively correlated with the outcomes of your other gambles can reduce your overall risk ('hedging'). My question, however, concerns uncorrelated gambles - can these reduce your overall risk, and if so when?

Edit: to be clear, I am asking about when you can reducing your risk by taking additional uncorrelated gambles, not simply by replacing some gambles in your portfolio with others.

In standard cases, you cannot generally reduce risk by gambling with uncorrelated gambles. With that said, a lot of your question depends entirely upon how you start defining terms.

Let me begin by defining risk as being exposed to uncertainty. If you are not exposed to any uncertainty at the moment, then adding uncorrelated gambles to any outcome present in your life merely adds risk. This definition is important because it often generates a completely opposite result from defining risk in terms of exposure to a chance of loss.

Imagine you went out today to play the Mega Millions Lotto with once chance in forty-five million of winning. There is almost no risk in that transaction as there is almost no uncertainty. You are going to lose. It is not risky at all. It is nearly a risk-free asset. It is a risk-free asset whose expected value is less than a penny, but it is nearly risk-free. This is why states love offering lotteries.

Now let us change our definition to risk as being exposed to a chance of loss. In that case, a lottery ticket is very risky. Indeed, the exposure is nearly the full dollar and the chance of an upside looks pretty grim. With that definition of risk, voluntary exposure makes little sense.

It is the word voluntary that is the key. Now let us make this game a little more extreme and involuntary. To set this up consider the film, Phone Booth. In the film, the lead character has been using the last phone booth in New York to contact his mistress behind his wife's back. A sniper takes up a location where he can call the lead character through the phone booth and control him. If he fails to do what he is told, he will be killed. He has to confess his betrayal. There is risk present in that the killer may kill him anyway.

Consider any circumstance where an involuntary outcome will happen in the future unless a gamble is taken. The gamble is uncorrelated with the outcome but will prevent the outcome. An example of this is any consumption deferral to avoid starving in the future, such as saving or investing.

The firm's outcomes are uncorrelated with your life expectancy unless you happen to run the company. If you take the gamble, then your life is correlated with the firm's well-being, but theirs is not correlated with yours. This is known as asymmetric association. One of the best measures of this is Somers' D. If you take no gambles then you will run out of resources when you can no longer work. If you take gambles, uncorrelated with your long-run well-being, then you may live to be very old.

Now imagine that you must gamble involuntarily, then you can improve your rate of return and reduce your risk by using the Kelly Criterion. The Kelly Criterion would increase your utility provided that the return was anticipated to be positive on every gamble. Otherwise, if all gambles yield a negative return, then you should only gamble if the probability of loss from gambling is less than the probability of loss from fire or theft of cash.

As @DaveHarris points out, it would be nice to define your concepts. Risk is misused quite often.

That having been said, diversification, placing several smaller uncorrelated bets (or buying several financial assets with uncorrelated returns) instead of placing all your money on a single bet is a fairly simple way of reducing one's risk according to most definitions. It is straightforward to show that if $X_i$ are i.i.d. random variables then $$var\left(\sum_{i=1}^n \frac{X_i}{n} \right) = \frac{var(X_i)}{n} < var(X_i)$$ for all $n>1$, $var(X_i)>0$. If $X_i$ represents your winnings then by placing $n$ small bets instead of just one big one you can greatly reduce the variance of your winnings. This piece of wisdom is so well known there is even a proverb for it: Don't place all your eggs in one basket.

In case $$E\left(\sum_{i=1}^n \frac{X_i}{n} \right) = E(X_i)$$ holds (this is true for a lot of gambles, e.g. roulette or blackjack) a risk averse player would always prefer several smaller gambles to one big one.

• Thanks for taking the time to answer. If you look at the question, however, you will see that it was not about replacing your gambles with new gambles (which allows you to diversify) - rather, it was about reducing your overall risk by taking on additional gambles. – afreelunch Jun 17 '18 at 16:43
• Re your first sentence: I deliberately did not define risk (or specify the choice set) since I wanted to leave some room for interpretation. However, if that is too open ended for you, you can consider gambling over just one good, your wealth, and define risk in terms of second order stochastic dominance. – afreelunch Jun 17 '18 at 16:45
• @afreelunch Re your first comment: This is not clear at all, perhaps edit your question to make it clearer? – Giskard Jun 17 '18 at 17:01
• Also there is a mistake in your answer: for the expression for the variance of the average, the denominator should be $n$ not $n^2$. – afreelunch Jun 29 '18 at 15:17