# Principles of poker [DRAFT]

Author: Christopher Wolff
Email: [e-mail]

## Introduction

The goal of this project is to describe a set of general principles that explain Nash equilibria in 6-max No-Limit Hold'em. We'll focus on heads-up situations for now, but we may address multiway situations in the future.

### Notation and glossary

| Symbol | Description |
| ------------ | ------------------------------ |
| $P$ | Initial size of the pot |
| $S$ | Effective remaining stack size |
| $B$ | Bet size |
| $\text{SPR}$ | Stack-to-pot ratio |
| $\text{EV}$ | Expected value |

### Nash equilibria

In heads-up poker, a Nash equilibrium (NE) is a tuple of two strategies, where neither player can increase their EV by deviating. In other words, if we play an NE strategy, we are guaranteed a minimum amount of EV, no matter what the opponent does. Thus, playing to approximate an NE is a very defensive approach to poker; we try to maximize the worst-case scenario, by assuming that our opponent may know our exact strategy and could perfectly counter it.

## The big questions

### Why should you bet?

Reasons to bet:

1. Deny equity.
2. Polarize.

### What size should you bet?

### Why should you fold?

### Why should you check?

## Basics

### Indifference

At an NE, every hand always takes the action with the maximizes its EV against the opponent's strategy. Mixed action frequencies can only occur when multiple actions have the same EV. If this happens, a player is said to be *indifferent* between those actions.

## Toy games

### Perfect polarization 1-street

Suppose we are in a river situation, where $P_1$ has $n_{\text{value}}$ value hands that would always win at showdown, and air hands that would never win at showdown. We'll assume that the number of air hands is comparatively large. $P_2$ only has bluff catchers that lose against the value hands, but beat all bluffs.

The optimal strategy for $P_2$ is to always check, regardless of the positions. The optimal strategy for $P_1$ is to go all-in with every value hand, and go all-in with $\left(\frac{S}{P + 2S}\right) n_{\text{value}}$ bluffs. Facing an all-in, $P_2$ is indifferent between calling and folding, and thus has an EV of 0. He should call a fraction $\frac{P}{P + S}$ of the time. The plot below shows how $P_2$'s call frequency and $P_1$'s bluff-to-value depend on the SPR.


Note that as the SPR increases, $P_1$'s bluff-to-value ratio approaches 0.5 and $P_2$'s call frequency approaches 0.

### Perfect polarization with traps

Value of traps: how do traps improve EV of bluff catchers?

### Perfect polarization multi-street

Suppose we have the same equity distribution as in the previous scenario, but we have more than one street left to play. How much should $P_1$ bet, how many bluffs should they use, and how frequently should $P_2$ call? We will assume that $P_2$ can only play passively, and never bet or raise themselves.

The answer to the third question is simple. $P_2$ has to make $P_1$'s bluffs indifferent on every street, so $P_2$ should always call with frequency $\frac{P}{P + B}$ when faced with a bet of size $B$. If $P_1$'s bluffs weren't indifferent, they could improve their strategy by bluffing more or less.

The answer to the second question ...

Now to the tricky question. What should $P_1$'s bet sizes be? The answer turns out to be a size called the **geometric** bet size. The geometric bet size $g$ is the multiple of the pot that a player needs to bet on every remaining street such that they will be exactly all-in by the river.
$$
g = \frac{1}{2} \left(\sqrt[1/b]{(P + 2 S) / P} - 1\right)
$$
where $b$ is the number of bets remaining.

Why is this the case?

## Ideas for future principles

- Blockers and unblockers
- Exploitative play
- SPR
- River play: every value hand has a preferred bet size, [0, 1] toy game
- Information hiding
- Vulnerability / equity shifts: the vulnerable nut theorem
- River play: complete [0, 1] toy game solution
- Multi-street [0, 1] toy game?
- Preflop play vs. loose players. Players after the loose player gain EV, but EV for other players stays the same or even decreases. Tighter RFI when acting before loose player. Higher call / 3bet when IP facing loose open.
- Higher postflop EV vs. fish => higher preflop RFI
- More multiway pots => play more hands with nut-potential, e.g. suited Ax and pocket pairs.
- Multi-way play: bet sizing smaller?, need more protection?, shared defense frequencies among players => don't need to defense as much?, nuts matter more?
- Slowplaying -- when and why?
- Betting vs. checkraising? When and why?
- How % of nutted hands in ranges govern bet frequencies / sizes, maybe in relation to SPR
- Air hands: when to use them as bluffs, when to give up
- Ambitious question: what generally affects bet frequencies and sizes?
- When do you not need to meet MDF? E.g. in bet/bet/x spot, sometimes you can x/fold 100% because your bet/bet/bets protect you.
- Safe exploitation -- risk vs. reward when making adjustments