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Hold 'Em Challenge

Introduction

Rules

The 'Hold 'em Challenge' is an intriguing poker-themed video game I've come across at the Mirage and Caesars Palace. Here are the guidelines to play.

  1. The game utilizes a standard deck of 52 cards.
  2. Once the player places a bet and hits the deal button, three hands consisting of two cards each are dealt face up.
  3. The player is required to select one of these three hands as their own. This choice will form the initial two cards of the player's hand, while the remaining two pairs will function as the first cards for two distinct opponents.
  4. Next, five community cards are revealed face up on the table.
  5. The objective of the game is to create the best 5-card poker hand by combining the player's 2 cards with the 5 community cards. Similarly, the game will assess the best hands for the two opponents.
  6. The payout for the player is determined based on the final hand and the accompanying pay table provided below.

The pay table listed is the only one I am aware of, and all payouts are made on a 'for one' basis, meaning the player does not receive their original wager back, even when winning.

Hold 'em Challenge Pay Table

Event Pays
On board royal flush 2000
Royal flush 100
Straight flush 25
Four of a kind 10
Full house 3
Flush 3
Straight 3
Three of a kind 2
Two pair 2
Jacks or better 2
Low pair 1
Garbage 1
Tie 1
Loss 0

Return

The table provided illustrates the frequency of each outcome based on a random simulation, with the bottom right cell indicating a return rate of 98.61%, translating to a house edge of 1.39%.

Hold 'em Challenge Return Table

Event Pays Observations Probability Return
Royal on board 2000 172295 0.000002 0.003071
Royal Flush 100 7345422 0.000065 0.006546
Straight Flush 25 28941374 0.000258 0.006448
Four of a kind 10 244685854 0.00218 0.021804
Full House 3 3024857400 0.026955 0.080865
Flush 3 3143432932 0.028012 0.084035
Straight 3 3604889822 0.032124 0.096372
Three of a kind 2 4541020679 0.040466 0.080932
Two pair 2 16958238827 0.151119 0.302237
Pair: J-A 2 10519745821 0.093744 0.187487
Pair: 2-10 1 8297624269 0.073942 0.073942
Ace high or less 1 1805881455 0.016093 0.016093
Tie 1 2950191571 0.02629 0.02629
Loss 0 57091119243 0.508751 0
Total   112218146964 1 0.986122

Strategy

While I've made attempts, I haven't been able to pinpoint an effective strategy for this game. A strategy primarily focused on evaluating the strength of various 2-card combinations fell short by about 2% of the optimal approach. I suspect this shortcoming is due to the significance of certain factors. penalty cards The number of possible combinations in this game is astronomical (418,597,840,861,200), making a comprehensive analysis take approximately 220 years on my computer. Instead, I conducted a random simulation of 81,866 initial hands resulting in a total of 112,218,146,964 hands. For each starting hand, the program evaluated all (46,5) = 1,370,754 combinations of community cards and selected the hand with the best expected return. This simulation required 22.3 seconds for each initial hand, accumulating to 21 days of processing time.

Methodology

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