Ask the Slot Expert: Statistical Decision Making at video poker and game shows

Statistical Decision Making is the technique of choosing the option that has the highest expected value (EV) when you have to make a decision. I learned the technique in the Operations Research class I took in college.

When you have to make a business decision, you look at how much each option could be worth and the probability of achieving that outcome. For example, you might be trying to decide whether to convert your buffet to a high limit slots salon. Making up numbers, you figure there is a 30% chance the salon will earn $10,000/day, a 50% chance $7,000/day, and a 20% chance $5,000/day. The EV of the earnings from the salon is 0.3(10,000) + 0.5(7,000) + 0.2(5,000) = 3,000 + 3,500 + 1,000 = 7,500. So the EV of the salon is $7,500/day.

We don't need to calculate the EV of the buffet option because it is the status quo. Do we expect the slot salon to earn more per day than the buffet?

Apparently, the buffet was a loser in almost all locals casinos, so they have been replaced by high limit salons or restaurants or something. When one option has a negative EV, any option with a greater EV -- even if it's still negative ("At least we're losing less money.") is preferable.

I think an implicit assumption in using this technique is that we'll be able to use it over and over again. One decision may not turn out as well as we had hoped, but if we keep choosing the option with the highest EV, we'll have achieved the best result possible with the hands we were dealt.

(Was that too obvious a segue to video poker?)

Statistical Decision Making works perfectly when playing video poker. Even though you may not get the best outcome possible when playing a hand, you'll get that hand again many times in the future. If you hold the combination of cards with the highest EV, you'll achieve the best result possible from that dealt hand.

The ability to use the technique multiple times is key. What about using the technique when you have to make a decision on a game show? I'll let John Grochowski explain why he didn't follow the math when he was a contestant on Who Wants To Be a Millionaire?

Millionaire is not a long-run situation; it's a one-shot deal. Once you're in the hot seat, you can never go back.... There's no long run for the probabilities to balance out.

John had reached the $250,000 question and had used his 50/50 on it. He had a 50% chance of winning (at least) $250,000 and a 50% chance of winning $32,000 (which is guaranteed once you answer that question correctly). The EV of answering is 0.5(250,000) + 0.5(32,000) = 125,000 + 16,000 = 141,000. The EV of walking away is $125,000.

The math says that answering the question has the higher EV, but John took the $125,000 he had won so far and stopped.

You either are correct for $250,000, or wrong and drop back to $32,000.

If he could have played this decision over and over again, he would have achieved the EV of $141,000. But there was no opportunity to repeat this decision.

There's not much difference between an EV of $125,000 and an EV of $141,000. When you consider that the first EV is really a value and guaranteed and the second is not, the guaranteed amount looks even better. There might be a way to quantify that a dollar you already have is worth more than a dollar you might get, but I don't know how to do it.

In John's decision, the dollar amounts did accurately reflect the value of each option to him. In this next one, maybe it's the probabilities.

In the end game of the new game show Snake Oil, contestants have to pick the fake product in a group of products. Contestants can choose the number products from which they'll have to choose. The more products, the more they can win. Because there is always one and only one fake product, contestants are really choosing how many real products they'll have to eliminate.

If they choose two products, they can win $10,000. At three products, $25,000; four $50,000; and five, $100,000. Most players choose five products. Is that the best choice?

Let's calculate the EVs assuming that we are picking the fake product at random. (That is the case most of the time, but once I actually had one of the real products presented in the game.) If the product you identify as fake is real, you don't win anything.

Half of $10,000 is $5,000; one-third of $25,000 is about $8,000; one-quarter of $50,000 is $12,5000; and one-fifth of $100,000 is $20,000. Going for five products has the highest EV, so it seems like the best choice.

None of the options involve guaranteed money, so a dollar in the two-product option is worth just as much as a dollar in the five-product option.

But consider this? Maybe a 50/50 chance of winning $10,000 is worth more to someone than a 20% chance of winning $100,000. The contestants get to play this game once and only once. Maybe it's better to settle for a higher chance at a lower amount.

I guess we can't really change the probabilities -- if you have two products, you have a 50/50 chance of being right and there's no justification for making that higher. I don't know how to quantify the fact that this is a once-in-a-lifetime decision and that the $5,000 EV of the two-product choice may actually be worth more to contestants than the $20,000 EV of the five-product choice because they have a higher probability of winning when they choose two products. I'm sure the producers prefer when contestants go for the big amount. It's not really exciting to see someone win (or not win) $10,000.

Maybe the best way to look at this decision is by ranking the choices by probability of winning something, anything, and ignoring how much could be won.

As long as I'm talking about game shows, I have a few random rants to get off my chest.

Has Wheel of Fortune (WOF) gotten cheap? For a few years, the minimum you could win if you got the puzzle in the end game correct kept increasing with the season number. Now Wheel is on its 41st season, but the minimum is still $40,000.

Jeopardy is supposed to increase the prize money for the second and third place finishers in regular game play, but we haven't seen it yet because of all the tournaments.

And I know WOF contestants are supposed to be play along and be high energy, but is there really any reason to be excited and jump up and down when Pat opens the bonus round envelope and they win the minimum?

America's Got Talent (AGT) stopped asking the judges what they were hoping to see in the new season a few years ago. How many different ways can you say "something we haven't seen before"? I just wish they'd stop asking contestants what they would do with $1,000,000. The winner doesn't necessarily get $1,000,000. The only way to get $1,000,000 is to take a 40-year annuity and bank a whopping $25,000 per year.

Winners get something less than a mil if they take the "cash value of the annuity" lump sum payment instead. It's still a good chunk of change, but not a million dollars.

Survivor and The Amazing Race pay their million-dollar prizes in a lump sum. AGT should either do the same or lower the prize to an amount they can pay in a lump sum. I'll admit that the real prize, though, is the exposure that they acts get, win or lose.

Another NBC show, The Wall, has the option to pay winners with an annuity, but it's only 20 years. Winners on The Wall have the potential to win millions of dollars, so there is some justification for this annuity.

Speaking of AGT, Howie Mandel. I came across a game show he hosted last year, Bullsh*t The Game Show, streaming on Netflix. Contestants have a chance to win $1,000,000 even if they don't know the correct answers to any of the trivia questions. The main contestant moves up the money ladder either by answering a question correctly or convincing at least one of the three challengers that the answer they gave is correct. Contestants could answer every question incorrectly and still win $1,000,000 because they convinced the challengers that they answered correctly. Hence, bullsh*t.

Here's the money ladder:

$1K $10K $25K $50K $75K $100K $250K $500k $750K $1M

There is a nice symmetry to the amounts, but I think the producers made a mistake with the upper amounts.

Most money-ladder game shows (e.g., Millionaire) have guaranteed amounts. Bullsh*t gives the players two opportunities to guarantee amounts called locks. Players choose when they want to lock an amount. That's the minimum they will leave with if they lose later in the game.

The first lock is available after the contestant wins at the $1K level. The player can wait and use it at a higher amount. Once the first lock is used, the second isn't available for another two questions. Players tend to lock at $25K and $100K.

Let's say you've made it to the $500K question. If your lock is at $100K, would you be willing to risk falling back to it to gain an extra $250K? Similarly, if you were at $750K, would you risk $650K for an extra $250K? Unless you locked in over $100K, it seems to me that the downside is too big relative to the upside for the last two or three questions.

Nevertheless, some contestants do risk a big loss and play the last few questions.

If you would like to see more non-smoking areas on slot floors in Las Vegas, please sign my petition on change.org.