Coordination
The next day in Cookie Monster’s Fungeon, the rules of the game are different. Generous as he is, he’s identified your and Bartholomew’s favorite cookies: you crave butter cookies, and Bartholomew yearns for the snickerdoodle.
The rules of the game are simple; both you and Bartholomew must pick between butter cookies or snickerdoodles. If you both pick the same cookie, you will both receive one of that cookie. However, if you pick different cookies, neither of you will receive anything for today. Both of you would prefer to eat your favorite cookie, but would rather eat any cookie than no cookie.
Construct a payoff matrix that represents this game and find the game’s Nash equilibria.
You draw the following matrix:
| Butter Cookie | Snickerdoodle | |
|---|---|---|
| Butter Cookie |
2, 1
|
0, 0
|
| Snickerdoodle |
0, 0
|
1, 2
|
For our current purposes, the exact payoffs don’t matter, as they don’t affect the player preferences. Just make sure that for each player, receiving their favorite cookie rewards a higher payoff than the other cookie, which rewards a higher payoff than receiving no cookie (and both cases in which the player receives no cookie should reward the same payoff).
The Nash equilibria of this game are when both players choose the same cookie. For these outcomes, neither of you can strictly benefit from deviating because you would receive no cookies.
Which cookie will you choose?
Snickerdoodle. How considerate. Unfortunately for you, Bartholomew picked the butter cookie. Looks like you’ll be going hungry today.
Neither strategy is strictly dominated, so it’s impossible to tell with 100% certainty what Bartholomew’s plan was beforehand. If only there were a way to coordinate. Actually, why didn’t you? You should have just talked to him. How foolish of you not to consider options that weren’t presented!
From this game, you discover two new facts about Nash equilibria:
- A game can have multiple Nash equilibria.
- It may be helpful to collude with other players to achieve a Nash equilibrium. Such games are known as coordination games.
In game theory, this game is known as the Battle of the Sexes (or Bach or Stravinsky, another acronym for "BoS"), where the premise is coordinating a date.
Coordination games may come in the form of colluding to determine a Nash Equilibrium (like BoS). They may also come in the form of colluding to settle on one Nash equilibrium over another.
For example, in the Stag Hunt game, a group of