Games Within Games

A few days ago I ran into the ubiquitous Bib Ladder during an intermission at the Vancouver Opera. Naturally our conversation drifted to poker. ``Say, professor, I've seen a few references to game theory in poker discussions and I'm wondering what it's all about.''

``An interesting question, Bib. There are people who try to build what are called mathematical models of a variety of objects or processes. The models might be systems of equations which describe some phenomenon, or some other kind of structure. They then perform manipulations of the models and hope this gives accurate information about the real objects.

In the case of game theory, the models attempt to capture the essence of whatever game is under discussion. Also, there is a wide interpretation of what constitutes a game. For example, one could view a commercial fishery as a game where one opponent is the fishing fleet and the other opponent is nature. Mathematical models are powerful tools because of the flexibility with which they can be applied.

There are two primary objects to model in game theory. One is the game itself and, as I indicated above, the notion of a game can be very general. The other is the notion of a strategy. Once you have the game modelled and the notion of strategy modelled most people are interested in trying to determine an optimal strategy. Of course, each opponent is trying to optimize her outcome. Some of the mathematics involved is very sophisticated because optimization problems normally are difficult.

So what about poker, you say? One element of strategies in game theory is randomization. In many cases people introduce schemes for randomizing aspects of their strategies. This is what people mostly mean when referring to game theory in poker. It is not particularly sophisticated.

Let me tell you one such scheme I used. A few years back when I first ventured into the social clubs in the Vancouver area, many of the clubs had the players taking turns dealing and choosing the game from a small list of games. I discovered that a large number of people were choosing to deal high-low draw poker. It did not take me long to observe that all the players would not draw a card when dealt three-of-a-kind. It is a good hand for high and by not drawing a card they are trying to indicate they have a low hand for deception purposes. What I decided to do when dealt three-of-a-kind was to create a random process for how many cards I would draw.

Suppose my hand was 2-7-7-7-9. I add together the two and the nine obtaining 11. I then subtract one because one of my cards is of bigger rank than the trips. This gives me 10 which I divide by three obtaining a remainder of one. I then would draw one card.

What I wanted was a random process which gave me equal probability of 1/3 for drawing either zero, one or two cards. So in general I add the ranks of the two non-trips cards, subtract the number of cards of rank larger than the trips, divide by three and draw the number of cards equal to the remainder. The reason for subtracting the number of cards with rank larger than the rank of the trips is to make the probability equal to 1/3 for each of the three possibilities.''

``I don't quickly see how to verify the 1/3 being correct, professor, but I see one problem with your scheme. I have played in many draw poker games where a two-card draw would stick out like a sore thumb.''

``You are perfectly correct, Bib. The beauty of the previous scheme is how it can be adapted to the texture of the game. Suppose I want to have 2/3 probability of drawing no cards and 1/3 probability drawing one card. Then I draw no cards if the remainder is zero or one, and draw one card if the remainder is two. Or I could divide by 2 to get probabilities of 1/2 and 1/2. I could divide by 4 if I want to use probabilities such as 1/4 and 3/4. Finally, even if I tell my opponents the randomization process I am using, they gain no information about my hand - although they may gain some information about me.''

``Alright, I concede that it is better than I first thought. Are you doing stuff like this in hold'em too?''

``Yes, I use some randomization in hold'em, but as I hear them calling us back for the next act, I will have to leave you at this point. By the way, why are there so few good tenors?''