I'm finally getting around to reading "Play Optimal Poker" by @thinkingpoker. POP is a poker strategy guide that puts game theory at the center of its discussion.

So, I thought I'd start a twitter thread where I put my thoughts as a read along.

Caveat: I'm a mediocre game theorist and even worse poker player, so take my observations with the grain of salt they deserve. But I do know both areas well enough to feel confident starting a twitter thread... so there's that.

The book starts off by discussing an important contrast, which is often overlooked, in how people discuss poker strategy. Most poker strategies guides do NOT really use game theory, they are strictly speaking, using decision theory only.

They start from the assumption that your opponent is not playing optimally, and that your goal is to exploit their non-optimal play. Game theoretic concepts, like equilibrium, rarely come up.

This book means to be different -- and that's why I'm interested!

Chapter 1 starts with a very nice and clear introduction to some of the basic concepts of game theory. It does so with a few classic game theory games (pure coordination, battle of the sexes, and a version of matching pennies).

Matching pennies allows for a discussion of zero sum games and mixed strategies -- something I'm assuming will be central to the rest of the book.

I think the introduction to game theory is very nice and clear, and should be accessible to most poker players. It's well done.

The chapter discusses why implementing mixed strategies is hard; we are bad at randomizing. In particular, we tend to "alternate" too much. That's why I recommend, in repeated rock-paper-scissors, that you play what would have lost last round.

I'm happy to see this discussed.

My only concern comes in the section titled "indifference means giving your opponent no good options." The section is a defense for why one might want to play one's part of a mixed strategy Nash equilibrium.

The title of the section is correct. By making one's opponent indifferent, one is giving them no "good" option. But parts of that section seem to imply that playing a mixed strategy makes it more likely that your opponent will make a mistake.

I don't think this is right. When you make your opponent indifferent, it's true they have no "good" option, but they also have no "bad" option either. At least none from among the set they are indifferent over.

We'll see if this becomes critical later...

This is where I've stopped for today. I'll add more as I read further.

Onto chapter 2! I really liked this chapter because it focused on using a simplified version of poker to illustrate concepts both in game theory and poker.

Many of the "classic" poker strategy guides eschew using simplified examples. But I think that's a mistake.

POP isn't the first strategy book to use simplified examples, but it's one of relatively few. (The other that puts them front and center -- that I know -- is Chen and Ankenman's _Mathematics of Poker_, which is also a great book.)

Although other authors (especially those associated with 2+2 publishing) will occasionally use simple examples to illustrate a point. This is part of the reason I was drawn to those books when I started playing.

The central example for this chapter is a simplified version of Kuhn's Poker. Here's the wikipedia page for
Kuhn's poker:

https://en.wikipedia.org/wiki/Kuhn_poker 

First introduced in this paper, I think:

https://books.google.com/books?hl=en&lr=&id=OVfQCwAAQBAJ&oi=fnd&pg=PA97&dq=kuhn+simplified+two-person+poker&ots=2Km9eR3h5-&sig=Ukc0eOsZ9DZUGMGIFUh1uei5A8Y#v=onepage&q=kuhn%20simplified%20two-person%20poker&f=false

It's also basically the model analyzed in the first part of this very nice paper:

https://pubsonline.informs.org/doi/abs/10.1287/mnsc.17.12.b764

(POP attributes it to _Mathematics of poker_ which is the first poker strategy guide -- that I know -- which used this example.)

In the book, @thinkingpoker uses this example really adeptly to illustrate some concepts from game theory like: (1) under what conditions one would like to mix and (2) what is the value of a game to each player.

Also, he uses the game to illustrate important poker strategy concepts like: (1) the relationship between pot size and optimal bluffing frequency and (2) the impact of different bet sizes on a game.

The chapter admirably distinguishes between two arguments for randomization that are conflated in the poker strategy literature: (1) mix it up now to prevent yourself from being learnable in the future and (2) mix to prevent you from potentially being exploitable *on this hand.*

The second one is the focus of this chapter, since a one-shot game is being analyzed. And it's nice to see the reasoning carefully disentangled.

Overall, I really liked chapter 2 of the book. By putting a simple example front-and-center and then expanding, it really illustrates how game theorists approach poker (starting simple and building up). I really like this style, and I'm looking forward to reading more.