Today’s column isn’t specifically about video poker, but it concerns a gambling situation that video poker players encounter in a casino. And most video poker players don’t care whether a $1,000 prize comes from a jackpot, a casino drawing, or a tournament.
Assume a casino is giving away money. If your name is called, you get to spin a fair wheel, where the possible prizes range between $5 and $1,000 in the following ratios:
| Number of | Prize |
| Occurrences | Amount |
| 1 | $1,000 |
| 1 | $500 |
| 4 | $250 |
| 4 | $100 |
| 5 | $50 |
| 10 | $20 |
| 15 | $10 |
| 20 | $5 |
If you add this all up, you’ll see there are 60 slots on the wheel, and the sum of the prizes adds up to $3,600. This makes the spin worth $60 on average ($3.600 / 60 slots = $60 / slot). A few of the prizes are quite a bit larger than the average, and three-fourths of the prizes are $5, $10, or $20. To make the problem more interesting, assume the casino offers a $50 buyout. This is less than the $60 average, of course, but it’s a lot better than 45 out of the 60 prize and equal to another five out of the 60 prizes.
If somehow you were in the enviable position of being allowed to spin the wheel 400 times, you’d be a fool to take the buyout of $20,000 (400 * $50 = $20,000) rather than the average of $24,000 ($400 * $60 = $24,000) that would come if you spun the wheel every time. Spinning 400 times is close enough to the “long run” that you figure to hit the $1,000 and $500 enough times to make spinning pay off more than the buyout. It doesn’t HAVE to work out this way, of course, but the odds are in your favor.
The more interesting case is if the spin is “maximum once per person.” Now if we choose to spin and end up with a lousy $5, we have forever lost the $45 we could have gotten from the guaranteed $50. We will never get it back from this promotion simply because we wouldn’t be allowed to spin again, so the results could never average out. In this case, is it better to take the guaranteed $50 or spin for the prize with a bigger average (but a significant probability for a smaller result)?
To my way of thinking, whether we get the opportunity once or 400 times is not an important distinction. I believe spinning is correct in either case. All of us have MANY gambles, and we are NEVER in balance in all of them. To the smart gambler, we take the advantage every time we can and trust/hope that it all balances out in the end.
If the numbers were “large” (which is personally defined), then it can certainly make sense to take the “bird in the hand”. For example, if we were guaranteed $5 million or could spin the wheel and get an average of $6 million, I would take the $5 million in a heartbeat.
Even though the math is the same, $5 million is such a potentially life-changing amount that there is no way I can feel comfortable gambling with it. But $50? For me that’s pocket change and I’m going with the math.
It’s possible that $50 is not ‘pocket change’ to someone in this position. If that’s a large amount to you, by all means take the sure thing if that will make you feel better.
Also, please note that I’m stipulating that the wheel is fair — meaning each of the 60 positions are equally likely to come up. In the real world, that’s assuming away part of the problem. You have to use your judgment here. In Nevada casinos, I’m going to assume the wheel is fair. I’m not sure I’m going to make the same assumption everywhere.
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$5 million aka the Certainty Equivalent
The concept you’re trying to explain is marginal utility. $50 is not that much more useful to a person than $5, so the pain from turning down the $50 and then getting the $5 would be small. Now, if it were, say, a buyout of $50,000, an EV of $60,000, and a worst-prize scenario of $5,000, most people would probably take the sure thing, because the marginal gain ($10,000, from going ahead and spinning) has relatively little value/utility compared to that of the $45,000 that the player might lose by spinning. Add a couple or three zeros to this amount, and the decision is even easier. If we were talking $6/5/0.5 million, you’d basically be asking someone to gamble $4.5 million for the chance of getting $1 million more. In such situations, EV goes out the window–when a loss of X magnitude hurts MUCH more than a gain of the same magnitude feels good.
You see this is in the insurance market. Most insurance is a terrible bet–but people are risk-averse. You also see (saw?) it on that profoundly stupid game show, Deal or No Deal, where the banker never offered a good deal (until the very end of the game) because he knew people were risk-averse. Of course, the dispassionate gambler always shoots for the highest EV. But at some point, bankroll considerations intervene.
I absolutely like this concept of effektive casino marketing. Although most Wheel spins are virtual nowadays, it usually attracts the people. From what I see therefore the Problem is how to calculate the real value of such a spin if the Wheels are only virtual (kiosk game). I have never seen or heard a Person winning the 1 million points prize at the A-May-zing May promotions that all Boyd casinos were Advertising. Usually what I got was a Food voucher or a few points. Still, Overall the value was higher than the average Investment I had for “earning” the 5 points that were necessary to Play that game. Which is probably not the Kind of Wheel spin Bob was talking about, but more or less it’s the same idea behind.
This article is just like being back in Sheen Kassouf’s class on decision theory, only we were pricing insurance.
Maybe week1 in a lower division class.
The late Professor Sheen Kassouf taught at University of California, Irvine and was Dr. Ed Thorp’s co-author on “Beat the Market,” among other works. There is a lot of heavy mathematics in that book. My article does not require such high-powered math to comprehend.