Jim Fedako sent in this Wall Street Journal column, written by one Dan Ariely, a “Professor of Psychology and Behavioral Economics.”
A lady wrote Ariely asking for economic party games. Ariely suggested this one:
Give each of your guests a quarter and ask them to predict whether it will land heads or tails, but they should keep that prediction to themselves. Also tell them that a correct forecast gets them a drink, while a wrong one gets them nothing.
Then ask each guest to toss the coin and tell you if their guess was right. If more than half of your guests “predicted correctly,” you’ll know that as a group they are less than honest. For each 1% of “correct predictions” above 50% you can tell that 2% more of the guests are dishonest. (If you get 70% you will know that 40% are dishonest.) Also, observe if the amount of dishonesty increases with more drinking. Mazel tov, and let me know how it turns out!
Let’s see how useful these rules are.
Regular readers have had it pounded into their heads that probability is always conditional: we proceed from fixed evidence and deduce its logical relation to some proposition of interest. The proposition here is some number of individuals guessing correctly on coin flips.
What is our evidence? The standard bit about coins plus what we know about a group of thirsty bored people. Coin evidence: two-sided object, just one side of which is H, the other T, which when flipped shows only one. Given that evidence, the probability of an H is 1/2, etc. That’s also the probability of guessing correctly, assuming just the coin evidence.
If there were one party guest, the probability is thus 1/2 she’ll guess right. Obviously 100% of the guests claimed accuracy, and we can score the game using Ariely’s rules. Take the percentage of guests who predicted accurately over 50% and multiply this percentage by 2%. (He gave the example of 70% correct guesses, which is 20% over 50%, and 20% x 2% = 40% dishonest guests.)
Since 100% of the guests claimed accuracy, our example has 50% above 50%, thus “you can tell” 2% x 50% = 100% of the guests are cheating. Harsh! You’d toss your invitee out on her ear before she could even take a sip.
If there were two guests, the probability both honestly shout “Down the hatch!” is 25%. How? Well, both could guess wrong, the first one right with the second wrong, the first wrong with the second right, or both right. 25% chance for the last, as promised. Suppose both were honestly right. We again have 100% correct answers, making another 50% above 50%. According to Ariely, we can tell 2% x 50% or 100% “of the guests are dishonest.” Tough game! Seems we’re inviting people over for the express purpose of calling them liars.
Now suppose just one guest (of two) claimed he was right. We have 0% over 50%, or 2% x 0% = 0% dishonest guests. But the gentlemen who claimed accuracy, or even both guests, easily could have been lying. The second who said she guessed incorrectly might have been a teetotaler wanting to be friendly. Or the second could have guessed incorrectly, and so did the first but he really needed a drink. Who knows?
If you had 10 guests and 6 claimed accuracy, then (with an excesses of 10%) 2% x 10% = 20% of your guests, or two of them, are labeled liars. Yet there is a 21% chance 6 people would guess correctly using just the coin information. Saying there are 2 liars with such a high chance of that many correct guesses is pretty brutal.
Ariely’s rules, in other words, are fractured.
So let’s think of workable games. I suggest two.
(1) Invite economists to use their favorite theory to make accurate predictions of any kind, three times successively. Those who fail must resign their posts, those who succeed are re-entered into the game and must continue playing until they are booted or they retire.
(2) Have guests be contestants in your own version of Monty Hall. Use cards: two number cards as “empty” doors and an Ace as the prize. Either reward your guests with a drink for (eventually) picking correctly, or punish them with one for picking incorrectly (if you think drinking is a sin).
Update In this original version I misspelled, in two different ways (not a record), Ariely’s name. I beg his pardon.
Update Mr Ariely was kind enough to respond to me via email, where he said he had in mind a party with a very large number of guests. This was my reply:
I supposed that’s what you meant, but it’s still wrong, unfortunately.
If you had 100 guests there’s a 7.8% chance 51 guess correctly (and truthfully). But the rules say 1% x 2% = 2% of the guests, or 2 of them, are certainly lying. Just can’t get there from here.
Worse, the more people there are the more the situation resembles the one with just two guests, where both forecasted incorrectly but where one said he was right. In that case the rules say nobody cheated. But one did.
The more guests there are the easier it is to cheat and not be accused of cheating, too. You just wait until you see how many people said they were right, and as long as this number isn’t going to make 50 or so, you can lie (if you had to) and never be accused.
There’s no fixing the game, either. Suppose all 100 guests said they answered correctly. Suspicious, of course, but since there is a positive chance this could happen, you can’t claim (with certainty) *anybody* lied. All you could do is glare at the group and say, “The chance that all of you are telling the truth is only 10^-30!”
But then some wag will retort, “Rare things happen.” To which there is no reply.
There might be a way to make a logic game of this, but my head is still fuzzy from jet lag and I can’t think of it.
Also, apologies for (originally) misspelling your name!