probabilities in "Set for Life"

On Friday, July 20, on ABC, a program called Set For Life premiered; another episode aired last night.

It's July. It's a prime-time game show. As you may have guessed, this is the stupidest game show ever. By "stupidest" I don't mean "the show is a bad idea" -- although I think it is, and it probably won't last long, nobody ever went broke by betting on the stupidity of the American public. But what I mean is that it involves absolutely no skill.

It reminds me of Deal or No Deal, in that no skill is involved except the still of knowing when to stop. In fact, the New York Times compared this show to Deal or No Deal back in October, writing: "On “Deal or No Deal,” Mr. Mandel does not even pretend that skill is involved. He says upfront that his game is about “giving away a ton of money,” not winning a ton of money."

Well, this is in the same vein. Here's how it works. First, there's a round that I didn't see, in which somehow it is determined how much the player will win per month; I don't know how they do this. (The Wikipedia article implies it doesn't air; people on various game show forums, like this one, seem to think that this part is probably more interesting than what actually airs. I agree, because what actually airs is less interesting than staring out my window. To be fair, staring out my window is pretty interesting.) Then, in the second round, it's determined how long the player will receive this monthly amount for. There are fifteen lights embedded in the stage; eleven are white, four are red. The player picks a light, and if it's white they go one step "up the ladder"; if it's red they go one step "down the ladder". After picking a white light, the player can elect to get out of the game and keep the money; after a red light, they are not allowed to make this choice. If they pick all four red lights, they go home with no money. The ladder contains the following time amounts:

zero, 1 month, 6 months, 1 year, 2 years, 3 years, 5 years, 10 years, 15 years, 20 years, 25 years, "set for life" (40 years).

(It's not clear to me what happens if you pick a red light on the first step. This turns out not to matter in my analysis.)

They have someone in the audience (in the first episode it was the contestant's nephew) giving "advice" on which lights to pick, and there's the pretense that some level of "skill" is involved in picking which lights are going to be white and which are going to be red, but there's actually no pattern at all. So it's really just a game of chicken. There's something of a morality play embedded in this show -- sometimes greed is good, but sometimes it can backfire and you get screwed over.

(The show's gimmick is that there's someone in an "isolation pod" and they can decide to end the game as well, but the person outside doesn't know when that's happened; the person in the isolation pod can choose to stop the game; but I'll ignore this.)

At least with Who Wants To Be A Millionaire?, which started this whole modern game-show craze, some knowledge was required. What I like about the new shows is the amount of drama that's invested in them -- each "decision" to keep going comes with dramatic music, as if it Really Matters. As if the players had any sort of control over the game. But for some reason the format of uncovering lights that have already been set works a lot better, on TV, than if there were just a random number generator simulating the light picking. For example, if there were four red lights left and seven white lights left, it would say "you win" with probability 7/11 and "you lose" with probability 4/11.

The first question that comes to mind -- what is the probability that a player wins the 40 years of monthly checks, given that they don't decide to "chicken out" at some mearlier step? To win this, the player has to pick all eleven white lights and no red lights, The probability of this is 1/C(15,4) = 1/1365.

Now, what's the right "strategy" for this game, in terms of trying to maximize your expected winnings? There are, at any point after you've picked a white light, two things you can do -- stop or keep going. If you "keep going", then we want to know what the probability of you being at various points on the ladder after Let's say, for example, that you've found six white lights and two red lights so far, so you're at the "2 years" position on the ladder, and five white lights and two red lights remain. You pick a light at random. The probability is 5/7 that it's white, and so now you end up with "3 years" winnings, 2/7 * 5/6 = 10/42 that you pick a red light and then a white light and end up back where you began, and 2/7 * 1/6 = 2/42 that you pick two red lights and lose everything. So the expected winnings (in years) are 3(5/7) + 2(10/42), which is greater than 2, so it's a good bet.

This same sort of reasoning works for any combination of lights. If you've so far picked zero to seven white lights and zero red lights, it turns out to be the right move to keep going. If you've picked eight white lights and zero red lights -- thus being at the "15 years" level -- staying or going are both equally good. If you've picked nine white lights and zero red lights, always stay; there are only two white lights and four red lights left! Similarly:

• if one red light has already been picked, and eight or less white lights have been picked, it makes sense to keep going; if nine or more white lights have been picked, stop.


• if two red lights have been picked, go if nine white lights or less have been picked; stop if ten white lights have been picked.


• if three red lights have been picked, keep going if seven or less white lights have been picked; if eight or nine have been picked, staying and going are equally valuable; if ten white lights have been picked, stop.

The usual caveat applies; calculating "expected value" is kind of meaningless when you only get to play once. People tend to be risk-averse with their winnings, so I'd expect to see people stopping before this strategy dictates it. For example, if two red lights and nine white lights have already been picked (leaving two of each), I don't see people taking the one-in-six risk of picking both of those red lights and losing it all.