Let’s say you lived the life of a gambler and now, dead, you are standing at the gates of hell with your two best friends who were also sinners. The devil, a lonely guy, lets you have the opportunity to go to heaven if you all play a game with him. This is his/her game:
Satan paints either a red or black (his fav. colors) dot on each of your foreheads so none of you can see your own dot’s color. You’re all required to sit in a triangle facing each other and you all are also telepathically denied from speaking, gesturing, or signaling to each other in any way. The only thing you can do is telepathically tell the devil what color you think you have on your own forehead. You may only guess one of three things: “red”, “black”, or “I don’t know”.
Depending on your three guesses, the devil will send you up to heaven. If anyone guesses incorrectly, you all go to hell. If all three of you guess “I don’t know”, you all go to hell. If all the guesses are either correct or “I don’t know”, then you all go to heaven.
Before the game starts, you and your friends are allowed to plan a guessing strategy which maximizes your odds of going to heaven.
For example, one strategy might be to have one of you always guess “I don’t know” and the other two always guess red. In this case the outcomes would be as follows:
The eight rows are the eight combinations of possible dot arrangements; the final column is the outcome after satan evaluates your guesses. Green is heaven. With this strategy you and your friends will get into heaven 1/4 of the time.
Can you come up with a better strategy?
Can you come up with a strategy that gives you and your friends a better than 50% chance of getting into heaven?
Spoiler alert
It turns out there is a strategy that will get you into heaven 6/8=75% of the time. The trick behind it is really interesting, which I will explain after giving the strategy and results.
The strategy is: say I don’t know if the colors on your friends’ foreheads are different, and if they’re the same, guess the opposite color. The following gives the results in each scenario:
Make sure you understand why each person guessed as they did and that you understand the results in the last column.
If you look closely, you will see that each person individually is wrong half the time they guess. There is no way to get above 50% individually given that you know nothing about your own dot. The trick is that they all say the wrong thing at the same time. All three guessing wrong is no worse than just one wrong so aligning the incorrect guesses improved the odds of getting into heaven.
My question is if there is a way to create a trading strategy based on this “paradox”. Even if you cannot improve past 50% odds, can you make it so that when you are wrong it is redundant and therefore irrelevant?
One interesting extension my professor hinted at was that the strategy can be modified to work for groups larger than 3 and that with a larger group the odds of getting into heaven approaches 100%.
