Which is Better?

Back when I was in graduate school studying economics, professors would often compare Model A versus Model B, where the two models were identical except for one specific difference. The professor then addressed a situation where the two models would yield different results, and so we learned what feature of the models led to what kind of results.

The real world isn’t like that, of course. Virtually any two things you could want to compare would differ in far more than one area. Still, it’s a useful type of exercise. Sometimes you can extrapolate the results of this kind of exercise and get real world conclusion, and today we’re going to try to do just that.

Assume there are two must-be-there-to-win drawings at different casinos at 7 p.m. next Friday. The casinos are similar in size with equivalent games and slot clubs, 10 winners sharing $xxxxx, the prize structure is identical, and it requires $500 coin-in to earn each drawing ticket. You know from experience that you can play enough so that you have a reasonable chance at being called.

The only relevant difference is that at Casino A, the first person called (and is present within 90 seconds) gets the top prize, the second person gets the second prize, etc. At Casino B, all ten winners pick an envelope so the order you were called is of no importance. Which structure do you prefer?

Before we can answer that, assume the actual number of drawing tickets from the players drawn this particular night, sorted from highest to lowest rather than in the actual order selected, is as follows:

1,003
127
83
60
51
47
31
28
19
2

If you’re the guy with 1,003 tickets, meaning you played more than $500,000 for this drawing and nobody else played as much as $70,000, it’s much better for you to compete in Casino A. You’re not guaranteed to be called first. In fact, before the drawing, you aren’t a lock to be called at all. But most of the time that you play this much you’re going to be picked early on and your prizes will be bigger on average at Casino A.

For similar reasons, if you are the guy with two tickets, you were extremely fortunate to be called at all. There were likely a few hundred entrants with fewer than ten tickets and you were the lucky one who was chosen this time. On those rare occasions when you do get chosen, you’re more likely to be one of the last ones drawn than one of the first ones. For you, Casino B represents your better option.

We can extrapolate from this. If you tend to play more than average, the better the Casino A structure benefits you. Conversely, if you tend to play less than average, the better Casino B structure benefits you.

Anyone who has paid attention to lots of drawings knows that occasionally the guy with one or two tickets wins the big prize and the guy with the most tickets more than occasionally gets completely shut out. There is a large random element to drawings.

While the preceding paragraph is undoubtedly true, don’t make the mistake of concluding that how many tickets you have in the drum is irrelevant. The more tickets you have, the better your chances are to win. You just need to recognize that having a better chance doesn’t mean you’re 100% certain to win, and having a slim chance doesn’t mean no chance at all.

Also, for those in the Casino B drawing:  The first guy gets his choice of all ten envelopes, the second guy gets to pick from the remaining nine, all the way down to the tenth guy getting whatever is left. Usually by the time the last guy gets his remaining envelope, the best prize is already gone. It can seem at that point like it would have been far better to be chosen first and have all ten envelopes to choose among.

And that is an illusion. Assuming the envelopes are indistinguishable from each other (a typical condition, but not one that’s guaranteed to be true every time), there is no advantage to going first. The last-drawn guy has the same 1-in-10 chance of getting the biggest prize as the first-drawn guy does.

As a final factor to consider I want to look at how well you’ve done at one of these casinos recently. If you’ve won first prize twice in the past three months at one of the casinos, then the other casino is a better option for the next few months. Do not fall into the trap of believing “Casino A is my lucky casino because I win more there.” A far bigger concern is casinos can and do remove your welcome when you win frequently.

Players can argue until they’re blue in the face that these kinds of things shouldn’t matter. But in the real world they do. So, act accordingly.