Jason (jcreed) wrote,

Was talking to Marty Zinkevich about the stuff he's working on. He mentioned a neat deterministic two-person game where the players make moves in sequence, not simultaneously, which nonetheless has probabalistic equilibria. I was quite surprised. Here's my attempt at putting a story to it:

A pair of fraternal twins, a brother and sister, named Wanda and Tuomas, have grown up, gone through high school and college together, graduated, etc. W landed a high-powered high-paying management job, and Tuomas watches TV and plays video games all day in his parent's basement.

Some days they write letters to each other. No sibling will ever write two letters to the other until the first has gotten a reply. In this way, we can imagine that it's always someone's 'turn' to write the next letter. We can assume that at most one letter per day is written, and that a letter reaches its destination and is read on the same day it is sent.

So the four possibilities for any given day are: Tuomas writes a letter and Wanda reads it; Wanda writes a letter and Tuomas reads it; It's Tuomas's turn but he doesn't write a letter; It's Wanda's turn but she doesn't write a letter.

Now the mail carrier in Tuomas's neighborhood is really hot, and he likes to flirt with her. So his payoff for the event of Wanda sending him a letter is, say, 3. If it's Tuomas's turn to send a letter but he doesn't, then he gets to sit around all day and play video games, but the fact that he has a letter to send gives him a faint sense of purpose in his life, so his payoff is 1. If he does send a letter, like, that takes effort, man. Payoff zero. Also, if it's Wanda's turn and she procrastinates, Tuomas's payoff is also zero, because he gets depressed that his sister is too busy for poor old him.

In point of fact Wanda is pretty busy, and she likes to relax with a nice trip around the green. If it's Tuomas's turn and he doesn't send a letter, then she plays golf in the evening with a free conscience and an empty inbox, and gets payoff 3. If it's her turn, though, and she skips out on replying, then her nagging guilt makes her game suffer: payoff 1. On the other hand, actually reading or writing a letter throws off her whole schedule, and she doesn't even have enough time for the front nine: if any correspondence takes place on a given day, Wanda gets payoff zero.

The cute thing is that if both agents want to maximize their average payoff, the Nash equilibrium strategy is to flip a coin each day, and write a letter on heads. Or at least this is what Marty claimed. Assuming I transcribed the example right.

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