Ask your favorite family member who is bad at math if they will take the following bet. (As a word to the wise, I would play with pennies, or M&Ms, or my personal favorite: chocolate covered espresso beans. Just make it something small just in case they look up Jensen’s Inequality and get mad. Or in case you lose, because there is some chance you will. The odds are in your favor in the long term, though, so if you do, find yourself another sucker. )
Here is what you say:
“I will pay you in chocolate covered espresso beans to roll a die, as long as you pay me based on your winnings. Since I know you like candy, let’s make it the square of the number. The average number rolled should be 3.5, and that squared is 12.25. Since you are my _____ (boss, little sister, etc.) and I want to be generous, and besides, we don’t want to quarter candies, I’ll pay you 13 beans for each roll. You just have to give me the square of what you roll. So if you roll a 2, you pay me 4 candies out of the 13. Do we have a deal?”
Try to get them to roll at least 10 times if you can. The more the better, because the long term odds are in your favor. Below I’ve graphed the outcome of one person playing another. Each starts out with 50 chocolate covered espresso beans. Red is the House (you) and blue is your favorite coworker who is bad at math. See how many candies you could win if you let your friend have 100 rolls of the dice?
The trick here is that you’re paying the square of the average roll (13ish), but the average of the squared rolls is higher! It is just over 15. Basically, you get so many more beans when you roll high that it more than makes up for all the lower rolls where you are playing your friend. This game illustrates, as I allude to above, a form of Jensen’s Inequality. What is unequal? The square of the averages is not equal to the average of the squares.
What if we are measuring something other than rolling dice? The insects I wrote about in the last post essentially play this game against the environment. Imagine little bugs picking one of the ephemeral bedrock pools Galen studies, and laying its eggs there. Many of them are small, and dry up and the eggs bake. Sorry, bugs. This round you pay me more than I pay you.
But…. When they do survive, and they hit a pool with few competitors and few predators and all of their hundreds or thousands or hundreds of thousands of offspring grow up and move out into their own pool and lay their own eggs, they win big. Kind of like rolling a six. This was well described in Peter Chesson and Robert Warner’s 1981 paper in the journal American Naturalist. This works for plants, too. Little seedlings wait in the baking Arizona earth as part of a seed bank, waiting for a winter thunderhead to explode above them, moisten the soil, and sound the starting gun for them to bust out of their seed covers and grow and reproduce like crazy. I’m not saying this happens for every species. Just that it can, and has been demonstrated for some, like in Anna Sears and Peter Chesson’s 2007 paper in Ecology.
The environmental variation is not the whole story. Bugs (and plants) play this kind of game against one another, too, essentially each one rolling their own die, paying the other. That gets more complicated.
Now go win yourself some chocolate covered espresso beans. With the resulting caffeine buzz, write a comment about how many people fell for it.
[Note: huge thanks to Simon for feedback on the game design!!]