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Three Card 2nd Chance Poker

Introduction

Three Card 2nd Chance is a unique poker game that I came up with and secured a patent for (patent number 7,735,830 ). The concept of the 'Best Hand Bonus' side bet originated from WagerWorks. This game maintains a similar framework to Five-Card Mulligan Poker, which has been offered at Odds On Internet casinos since May 2007. The main feature of this game is that players can replace their initial hand if they're dissatisfied, simply by increasing their wager. The dealer also has the opportunity to swap their cards under specific conditions. Subsequently, the highest-ranking hand wins, and all winning hands pay out at least at even odds. WagerWorks holds the exclusive rights to provide Three Card 2nd Chance for online casinos. I am open to discussions with 'brick and mortar' casinos that would like to introduce this game.

Rules

  1. A standard deck of 52 playing cards is utilized. All hands are evaluated based on traditional poker scoring, although the hierarchy of hands is modified for three-card combinations.
  2. The game commences with the player placing an Ante bet, with the option to also make a 'Best Hand Bonus' wager. The maximum allowable wager for the Best Hand Bonus cannot exceed two times the Ante bet.
  3. The dealer will deal three cards to the player and three cards to themselves. The player's cards are dealt face up while the dealer's cards are kept face down.
  4. Players can either decide to keep their initial hand as is or opt to exchange it for a new three-card hand.
  5. If the player opts to exchange their hand, they must discard their current cards and place a Raise bet that matches the Ante bet.
  6. Following this, the dealer will reveal their cards.
  7. If the dealer's hand is king high or better, they will stand.
  8. Conversely, if the dealer has a queen high or lower, they will discard their current hand and receive three new cards.
  9. The dealer then compares their hand against the player's hand to determine the winner. Should the dealer have the stronger hand, the player forfeits both their Ante and Raise bets. Conversely, if the player has the superior hand, both the Ante and Raise bets will be settled according to the respective payout tables outlined below. In the case of a tie, the bets are returned.

    Ante and Raise Pay Table

    Hand Pays
    Straight flush 6 to 1
    Three of a kind 4 to 1
    Straight 3 to 2
    All other 1 to 1

  10. Payouts for the Best Hand Bonus bet are determined based on the stronger hand between the player and dealer, detailed as follows.

    Best Hand Bonus Pay Table

    Hand Pays
    Three aces 35 to 1
    Straight flush 15 to 1
    Three of a kind 12 to 1
    Straight 3 to 1
    Flush 2 to 1
    All others Loss


    Screenshot courtesy of WagerWorks

Strategy

For players only making the Ante bet, the best strategy would be to exchange cards for K87 or lower, while standing with K92 or higher.

For players who make the maximum possible Best Hand Bonus bet, the most favorable strategy would involve switching with AQJ or lower and standing with AK2 or higher.

Among the two betting strategies, opting for the maximum Best Hand Bonus bet tends to yield a lower house edge overall.

Ante Only Analysis

The following table illustrates the potential outcomes when a player places only the Ante bet and chooses to switch cards with K87 or lower.

Ante Only Return Table — Swap with K87 or Lower

Event Bet Pays Probability Return
Player wins with straight flush 2 12 0.001041 0.012497
Player wins with three of a kind 2 8 0.001125 0.009002
Player wins with straight flush 1 6 0.002176 0.013055
Player wins with three of a kind 1 4 0.002349 0.009395
Player wins with straight 2 3 0.015199 0.045598
Player wins with straight 1 1.5 0.031641 0.047461
Player wins with flush 2 2 0.021846 0.043693
Player wins with pair 2 2 0.061861 0.123722
Player wins with nothing 2 2 0.080141 0.160282
Player wins with flush 1 1 0.045172 0.045172
Player wins with pair 1 1 0.127912 0.127912
Player wins with nothing 1 1 0.117501 0.117501
Player ties 2 0 0.000482 0
Player ties 1 0 0.000637 0
Player loses 1 -1 0.189183 -0.189183
Player loses 2 -2 0.301734 -0.603468
Total 1 -0.03736

The bottom right cell indicates a house edge of 3.736%. Players will opt to raise their bet 48.34% of the time, resulting in an average final wager of 1.4834 units. This signifies the Element of Risk 3.376%/1.4834 = 2.518%.

2 to 1 Ratio of Side Bet to Ante Analysis

When players choose to place a Best Hand Bonus bet, they should adopt a more proactive approach regarding their card exchanges. The next table provides insights into the Ante wager outcomes when players change cards with AQJ or lower, which is the ideal tactic for maximizing the Best Hand Bonus side bet.

Ante Only Return Table — Exchange with AQJ or Lower

Event Bet Pays Probability Return
Player wins with straight flush 2 12 0.001553 0.018641
Player wins with three of a kind 2 8 0.001669 0.013354
Player wins with straight flush 1 6 0.002174 0.013046
Player wins with three of a kind 1 4 0.002345 0.009379
Player wins with straight 2 3 0.022650 0.067950
Player wins with flush 2 2 0.032376 0.064751
Player wins with straight 1 1.5 0.031611 0.047416
Player wins with pair 2 2 0.091599 0.183197
Player wins with nothing 2 2 0.118533 0.237066
Player wins with flush 1 1 0.045192 0.045192
Player wins with pair 1 1 0.127938 0.127938
Player wins with nothing 1 1 0.016185 0.016185
Player ties 1 0 0.000170 0
Player ties 2 0 0.000704 0
Player loses 1 -1 0.057368 -0.057368
Player loses 2 -2 0.447933 -0.895867
Total 1 -0.109119

The subsequent table lays out the various outcomes associated with the Best Hand Bonus bet when players choose to switch cards with AQJ or lower.

Best Hand Bonus Return Table — Exchange with AQJ or Lower

Event Pays Probability Return
Three aces 35 0.000569 0.019902
Straight flush 15 0.006811 0.102171
Three 2-K 12 0.006783 0.081396
Straight 3 0.099023 0.297068
Flush 2 0.140714 0.281428
Loser -1 0.746100 -0.746100
Total 1 0.035865

The Ante wager table indicates a player return rate of -10.912%. Meanwhile, the Best Hand Bonus table reflects a player return of 3.587%. When averaged appropriately, the overall return for players is (1/3)×-0.109119 + (2/3)×0.035865 = -0.012463. Consequently, the cumulative house edge is 1.246%. Players will raise approximately 71.7% of the time, leading to an Element of Risk at 1.006%.

Methodology

This evaluation was conducted through random simulation. Given the potential for both players and dealers to exchange cards, the total number of possible card combinations is combin (52,3) × combin(49,3) × combin(46,3) × combin(43,3) = 76,277,828,779,152,000, a calculation that would require years to complete.