Consuming Cookies

[Based on a problem by Emily Moore. Reworded by Sam Rebelsky.]

The Smith's have a pair of twins, which they've named One and Two. Mr. Smith regularly bakes cookies for the twins. To help ensure "fair" consumption, he sets up a number of rules for eating cookies.

• The twins must alternate eating, passing the plate between them.
• A twin may eat one or two cookies before passing the plate to the other twin.
• The last twin to eat a cookie must do the dishes.
• The twins must eat all the cookies.

Develop a strategy for eating cookies. Assume that the twins don't like to wash dishes, and that they so dislike washing dishes that they'll sacrifice eating a large number of cookies in order to avoid washing dishes.

Is this at all similar to a game you've previously encountered? How does your strategy for this problem relate to your strategy for that game?