# von Neumann Poker Analysis

## The Game

Imagine two people are playing a, greatly simplified, poker game:

Each player makes a $1.00 wager (ante) to start the game.

Each player is then ‘dealt’ a number between 0.0 – 1.0 (continuous), to represent their ‘hand’. Players can see their own number, but not their opponent’s.

*Fold*,

*Check*(stand), or

*Raise*(Doubling the wager to $2.00).

*Call*(if the first player raised),

*Check*, or

*Fold*.

*What is each player’s optimal strategy?*

## First Level Thinking

__never__a good idea for the first player to fold! Ever! No matter how low their score there’s a chance they could win, and they’ve already paid the wager. If they fold, they are guaranteed to lose the $1.00. If they stay there’s a chance they could win.

__never__fold if the first player did not raise.

*Check*.

*low*quality hands that the first player should bluff (not medium quality). Why? Well, when a player bluffs they typically expect to lose if they are called (so it does not depend on quality; you can bluff with anything), but with a medium hand, quality matters, and so one is better off checking.

## Next Level Thinking

__and__their understanding of how the second player will play. The first player has to assume the second player will play optimally (after all, if the second player doesn’t play optimally, it’s even better for the first player!) Based on this, the first player will act accordingly to maximize their chances.

## Nash Equilibrium

For optimal play by both players we can describe the solution to this game as something that has reached a Nash Equilibrium. (Named after the late brilliant mathematician John Forbes Nash Jr who won the Nobel prize for some of his work). A Nash Equilibrium is a solution in which both players can gain no benefit from changing their strategy.

A very commonly quoted example of a Nash Equilibrium is The prisoner's dilemma.

*A simple test to determine if a Nash Equilibrium exists is for all players to reveal their play strategy. If, knowing all this, no player elects to change their strategy, then a Nash Equilibrium is proven.*

## von Neumann

__above__this threshold value is

*Called*. Any hand with a value

__below__the threshold value is

*Folded*. (As we will see below, the value of the threshold depends on the level of the raise). A very simple strategy.

- If they have a high hand, they will want to raise. If the second player folds, they’ve bought the pot. If the second player calls, they stand a good chance of winning a larger pot.
- If they have a mediocre hand, they will simply check (no use risking a double).
- If they have a very poor hand, they should bluff (what do they have to lose?) There’s a chance that this will cause the second player to fold (if their hand is below the threshold), that otherwise would have come to showdown with both players checking.

__knows__that player one bluffs sometimes, a raise on a bluff is indistinguishable from a raise on a high hand, and if they call, they risk butting up against a high hand at an increased stake. What von Neumann showed is that the first player’s bluff threshold is at a level of indifference for the second player in that there is no advantage to calling on these. If the first players bluff threshold was artificially higher, there would be added incentive for the second player to call more often in the chance of catching the first player in a bluff.

## Math

**B**dollars. So, in our example above: B=1. The initially ante is $1, and to double the bet is an additional $1 (B=1).

_{T}> 4/10 (or 0.4), otherwise fold.

*when to raise with good hand*is shown to be:

_{TH}> 7/10. Anything higher than this and it's an automatic raise.

*when to bluff*is:

_{TL}< 1/10. Any hand below this value, and it's in the first player's best interest to bluff and raise.

## Simple game with a single double down

*Check*, otherwise

*Double*.

*Call*if raised, otherwise

*Fold*.

*Check*if checked to.

## Game Decision Tree

## Graph

__essential__part of P1's strategy to maximize their return.

## Charting

*really*good hand. At the extreme limit, the second player will never call a raise. Why risk it?

## Optimal Betting

## Non-continuous Game

## The Price is Right

If you enjoyed this article, you might like reading about the optimal strategy if you ever get selected to be a contestant on the game show *The Price is Right*, and make it through the to the Showcase final and need to spin the big wheel.

In the final spin, the last player to go has nothing to lose and can spin either to win, or bust trying, but the first player to spin has to consider what the optimal stopping score would be.