## The “rho” Nature of Game Strategies

### Games (general), Strategy

July 27, 2007

If you haven’t read up on what a rho is, first familiarize yourself here.

Now, I said that I would eventually explain what that experiment has to do with poker, but that’s still one post away. Before I get to that I need to discuss rhos in the context of game play in general, namely the types of strategies that a player can adopt when playing a game.

First, I should define what I mean by a “strategy” or more specifically, a “simple strategy”:

A simple strategy for a game is defined by the following information: a list of all possible scenarios you can encounter in the game, and a choice of play for each scenario.

For example, for tic-tac-toe, it would be a list of all possible boards, and where you would choose to place your next mark.

Now, consider: for the vast majority of games, there are a finite number of scenarios you can find yourself in, and a finite number of move options you have in any given scenario. Therefore there are a finite number of possible simple strategies you can adopt. That number may be very big depending on the game in question, but the key is that it’s finite.

Ok, so we’ve got all these strategies. What does it have to do with a rho? Well, imagine that you took a piece of paper like in our experiment, divided it in two columns, numbered the lines, and on each line in the first column wrote down one of the simple strategies. It might have to be a very big piece of paper, but you should agree that this is at least in theory possible because of the finite number of strategies. Now, on each line, in the second column, you put the line number of the strategy that is the best possible counter to the strategy in the first column. We then start drawing out our rhos starting from various lines, just as before.

I contend that these pictures are the most elegant description of the strategies in a game. Any strategy that appears only in the tail of a rho isn’t really any good. You can improve on it by simply moving to a strategy farther up the tail. But one you get into the head, or cycle, you can’t say that any one strategy is, in an objective sense, any better than any other. Their merit depends entirely on what strategy your opponent selects.

The rhos with degenerate structure are worth discussing at this point since they can occur when maping out the strategies in a game. Any time you get a rho with no tail, you’ve got a game where there are no “bad” strategies – rock is as good as paper is as good as scisors. It depends entirely on what your opponent chooses. In contrast, any time you get a rho with a single node as the cycle, you’ve got a game that can only be played correctly 1 way. Tic-tac-toe or something similar. In other words, game that only have one strategy in the head portion of the rho are lousy games.

We’ll start talking about the structure of the game of poker in the next post.