Do you know when to quit a casino session?

Casino aficionados routinely accept risks on series of adverse wagers they wouldn't consider as isolated bets. The casino and isolated cases differ in that the former involve cumulative effects players think can be profitable if they know when to quit. But, as Hamlet said, "there's the rub."

To some players, knowing when to quit means continuing as long as they're winning and stopping when losses start building. This exit-on-loss strategy ostensibly keeps rewards and limits penalties, yielding long-term profits by outweighing numerous small losses with sporadic large wins. To other bettors, knowing when to quit entails cashing out with moderate gains but continuing while losing in the hopes of a turn-around. This exit-on-gain approach is presumably beneficial because extended runs of failures are rare, and numerous small wins more than offset occasional large losses.

Relatively few solid citizens who start a casino visit or session with either of these exit strategies actually follow through. Instead, they typically switch between modes during the action. Nicholas Barberis of Yale University has hypothesized that the initial and mid-course strategy choices can be explained in terms of the "cumulative prospect theory" of managing risks.

One aspect of prospect theory bearing on choosing exit strategies is that individuals act on qualitative impressions of chance rather than actual values of probability. In particular, the theory holds that low probability events are generally considered more realizable than is actually the case, while moderate and high probability phenomena are regarded as less likely.
Another pertinent feature of prospect theory is that the relationship between the actual and perceived value – the utility – of a win or a loss is not constant. On the positive side, as actual value increases, utility grows but at a declining rate. This can be pictured as analogous to the law of diminishing returns. A $1,000 increase of profit from $1,000 to $2,000 is typically not viewed as desirable as the same amount going from zero to $1,000. Casino players therefore become more risk-averse as their profits rise. The opposite holds in the negative domain. People show greater sensitivity to losses than gains. The impact of losses escalates rather than declines with the actual amount involved. A $1,000 loss is bad, but a $2,000 loss is more than twice as dismal. This leads to increasing risk tolerance, in that mounting losses encourage players to continue chancing a turn-around rather than simply accepting defeat.

Under prospect theory, players who enter a casino intending to follow an exit-on-loss strategy often base their initial plans on an overestimate of a low-probability major profit. If they achieve early success, the appeal of greater gains starts to diminish relative to the amount they've already earned. Further, while players overestimate the small chance of the string of net successes needed to reach some high profit level, in deciding at a particular point whether to quit or continue, the probability of winning the next bet is likely to be moderate and accordingly underestimated. Together, these factors provide motivation to switch to an exit-on-gain strategy and quit.

Players who start intending to exit-on-win but find themselves with an early profit have a tendency to continue and seek greater earnings than they had originally envisioned. Because of the moderate amounts usually selected as targets by these players, they're low enough along the curve of diminishing returns so further gains seem attractive. They're accordingly risk-tolerant under these conditions.

Players with either strategy often get into trouble if losses begin to accumulate. The escalating impact of the loss and the increase in risk tolerance induce them to continue gambling. Probability is also a factor. Players in this position don't normally consider the consequences of a continued string of losses – a low probability event that would be discouraging because it would be overestimated. Rather, they think in terms of another loss on the next bet, an event with a moderate probability that they're likely to underestimate and imagine to be less than it actually is.

One of the enigmas of gambling is that you can't determine what would happen in a game after you quit. You may have made money at craps when a table was hot, then quit when it seemed to turn cold. If you go back and watch the same table get hot again, do you kick yourself for leaving prematurely? In fact, you altered the future of the game when you left – there was a delay in moving the dice when you "colored-up," the dice were passed to subsequent shooters at later times than had you stayed and were presented, picked-up, and thrown differently than they would have been. So you have to accept not second-guess your decision. Of course, sometimes quitting isn't a choice. You might go bust or have to meet someone for dinner. When it is a decision, do you quit on the spur of the moment or because you had a strategy that reflected your belief that you knew when to do so? Here's how the poet, Sumner A Ingmark, generalized this question.
Is a rule good because of its stringency,
Or because of the way it allows for contingency?