von Neumann and Newman Pokers with Finite Decks

By Tipaluck Krityakierne, Thotsaporn "Aek" Thanatipanonda, and Doron Zeilberger

.pdf    .tex

First Written: July 22, 2024.

John von Neumann and Donald J. Newman proposed, and brilliantly solved, toy models of poker where the cards are drawn from an infinite deck (in fact very infinite, the set of real numbers from 0 to 1). We show the power of symbolic computation by implementing, and experimenting with, these models for finite decks of cards.

# Pictures

John Nash and friends

Also See pictures,

# Maple packages

• FinitePoker.txt, a Maple package for finding pure and mixed Nash Equilbria for poker with a finite number of cards, doing it completely ab initio

• ThreePersonPoker.txt, a Maple package for studying three person poker in the footsteps of John Nash

# Sample Input and Output for FinitePoker.txt

• If you want to see ALL pure Nash equilibria (and one mixed one) for von Neumann poker with number of cards from 2 to 10, and bet sizes from 1 to 5 and one mixed one
the input gives the output.

• If you want to see ALL pure Nash equilibria, and one mixed one, for von Neumann poker with number of cards from 2 to 11, and bet sizes from 1 to 3
the input gives the output.

• If you want to see ALL pure Nash equilibria and one mixed one, for von Neumann poker with number of cards from 2 to 27, and bet size 2 only using suggested strategies
the input gives the output.

• If you want to see ALL pure Nash equilibria and one mixed one, for DJ Newman poker with number of cards from only using suggested strategies
the input gives the output.

• If you want to see ONE MIXED Nash equilibrium for von Neumann poker for all games with the number of cards from 2 to 40 and bet sizes from 1 to 10. TERSE VERSION
the input gives the output.

• If you want to see ONE MIXED Nash equilibrium for von Neumann poker for all games with the number of cards from 2 to 40 and bet sizes from 1 to 10. VERBOSE VERSION
the input gives the output.

• If you want to see ONE MIXED Nash equilibrium for DJ Newman poker for all games with the number of cards from 2 to 14 and maximal bet sizes until "saturation" (where increasing the maximum allowed bet no longer increases the value)
the input gives the output.

# Sample Input and Output for ThreePersonPoker.txt

• If you want to see spelled-out instructions for playing 3-player pokers with various scenarios
the input gives the output.