Game Theory Part 2: Applications To Poker

Mathematics, Poker Concepts, Strategy

April 11, 2008

Last time, I explained what game theory is using examples from roshambo. That’s all well and good, but this isn’t a roshambo strategy site. So let’s look at the poker implications. As we stated, any time there is a strategy rho, you can minimize the worst result you can get by choosing randomly between the options in the head of the rho at some frequency. Thus far, for poker, we’ve only discussed one rho: the big rho of tight play, aggressive play, and calling down. This is certainly an interesting example of a strategy rho, but it’s lousy for a discussion of game theory. The reason is that it’s somewhat of an abstract concept – we haven’t defined what exactly each strategy entails, and therefore it’s impossible to figure out the exact expectation when two strategies meet. This makes solving the associated game theory problem of how frequently you should do each to get a game theoretic optimal result impossible. To that end I want to introduce a new rho: the bluffing and calling on the end rho.

This rho exists between 2 players. Player 1 holds either a weak hand or a strong hand on the last street of betting, and is first to act. Player 2 holds a mediocre hand behind him and has to decide if he will call. Here are the distributions of hands:

Player 1: Weak 30%, Strong 70%

Player 2: Mediocre 100%

For the sake of the example, we’ll say Strong hands beat Mediocre hands which in turn beat Weak hands.

Now, what are the simple strategies here? Well, Player 1 has two simple strategies: bluff his Weak hands, or don’t. Player 2 likewise has two strategies: call with his mediocre hands, or check behind & fold to a bet. This produces a rho with all 4 choices in the head. There are of course tail strategies too, like Player 1 open-folding his Strong hands, but we’ll ignore them as game theory is really only interested in the head of the rho. We should also define that this is a limit game, there are 10 betting units in the pot, and Player 1 will be able to bet 1 unit.

Now, in order to find the optimal strategy, we have to put ourselves in one player’s shoes, and figure out what the results are for each combination of strategies.

  • Player 1 bluffs if weak, Player 2 folds to a bet: Player 1 gets the pot 100% of the time. Expectation 10 units won
  • Player 1 bluffs if weak, Player 2 calls: Player 1 gets the pot + one bet 70% of the time, and loses 1 unit 30% of the time. Expectation 7.4 units won.
  • Player 1 doesn’t bluff, Player 1 folds to a bet: Player 1 gets the pot 70% of the time and loses it 30% of the time in a showdown. Expectation 7 units won.
  • Player 1 doesn’t bluff, Player 1 calls: Player 1 gets the pot plus 1 bet 70% of the time, loses it 30% of the time in a showdown. Expectation 7.7 units.

Notice that this is in fact a rho – each strategy has a counter by the other player that is their best strategy.

Now, for the game theory: if Player 1 is going to make a composite strategy by combining his bluff and don’t-bluff simple strategies, how often should he do each? As with last time, I’ll punt on doing the math in the article, and just tell you that the answer is that Player 1 should bluff on the end 21.2% of the time when he holds a weak hand. I’ll explain how I got that in the next article in the series.

The key point right now, however, is not the exact number. It’s that game theory can be applied to a very realistic problem in poker. We determined the strategies in the head of the rho, and how they faired against each other, and that was enough to derive the game theoretic optimal play for player 1. We’ll talk more about what the answer we got means next time.

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