Yes, this is your first hint that my blog will be all over the place thematically, LOL. Future posts may be about poker, tennis, politics, music, random things that annoy me, or even esoteric intellectual/philosophical questions of epistemology or the like. This one is about a cheesy game show, though it does involve some discrete mathematics (probability calculations) and psychological “risk of ruin” issues that economists like to study.
[UPDATE: Those unfamiliar with the game should read the explanation here.]
A recent contestant (with the ironic surname Einstein, I kid you not) accepted an offer for $23,000 with six cases left (five on the stage and one she is holding), including the $500,000 (top amount), plus one containing $25,000, and four small amounts of $1,000 or less. If she passed up the offer, she would have had to open two more cases before getting another offer. This is right at the point just before when the offers usually improve markedly, in terms of what percentage of the average expected value (EV) is offered. (To get the EV, you just add up all the amounts left and divide by the number of cases left.)
In this situation, her EV was about $88,000; so the offer was barely over 25% of her EV. And for some context if you don’t watch the show, $23,000 is a generic offer you might get really early in the game (she had passed up $24,000 a couple offers earlier). If you go one more round, your offer’s going to be more like 70% of your EV; the round after that (when there are two cases on the stage and one with the contestant), over 80%; and then the final offer (which you have to either take, or take what’s in your own case) is for some reason always just a titch over EV, like 101%.
And the show makes you go through the whole “what if” scenario, opening the cases in the order you think you would have opened them if you kept playing. It turned out this contestant actually had the half million in her case, so as she went along, her next (hypothetical) offer was $84,000, the one after that $139,000, and the final offer was $251,000.
Now of course she couldn’t know she actually had the big money in her case. But in this scenario she definitely should have gone one more round. She only would have really badly screwed herself if she had eliminated precisely the top two amounts. I’m not sure exactly how the math is affected by the fact that she couldn’t choose to eliminate her own case, but I’m thinking it doesn’t matter, based on poker (you don’t know what the other players have, but you compute your odds of hitting your flush as if the deck the dealer draws from includes the cards that have been dealt out to players). And if indeed it doesn’t matter, then as long as her own case is a mystery to her, here are the odds of various scenarios (not bothering with the specifics of the four negligible amounts) after playing one more round:
(1) Worst case. The chance of eliminating the $500,000 and the $25,000 with two picks is 2/6 * 1/5, which is 1/15, or less than 7%.
(2) Best case, keeping both the $500,000 and $25,000 in play is 4/6 * 3/5, or 40%.
(3) Knocking the $25,000 out but keeping the $500K in play has a 27% chance of occurring.
(4) Likewise, knocking out the half mil but keeping the $25K also has a 27% chance of happening.
Scenario (3) is what actually would have happened, which would have meant an offer of $84K which she probably would have been right to take since she had no “safety net” of more than one significant number left. Had the best case scenario happened, the offer would have been slightly more, but it would have been a little more tempting to continue (but still a reasonable, safe move to stop).
Scenario (4) would have hurt, of course, but she would still get an offer of about $4,000 which she could take as a consolation prize, or she could gamble and continue with that $25K still up there.
Einstein carried “lucky dice” with her and handed them to Howie (his disgust was evident–he is a germaphobe like I am). I wonder then if she would have made the same choice had it been a dice game, with the following proposition:
“You can keep $23,000 or make one roll with two dice. If you get a three or less, you get five hundred bucks. If you roll a six or an eight, you get $4,000. If you roll a four, five, seven (most common roll), nine, ten, eleven, or twelve, you get $87,000. Deal or No Deal?”
I couldn’t make the dice odds fit the game odds exactly, so I erred toward actually decreasing slightly (down to under 64%) the chance of hitting one of the two scenarios where the offer gets increased substantially. I still suspect though that she would have passed up the offer had it been framed this way!