Mulligan Poker

Introduction

Three Card Mulligan is a variation of Three Card Poker in which the player may trade his first hand for a second one if he chooses. The game was invented and patented by yours truly and marketed by Shuffle Entertainment. In January, 2013, the game made its debut at the Cowboys Casino in Calgary, Alberta.

Rules

1. A single 52-card deck is used.
2. The poker-ranking hand order is as follows:
• Straight flush
• Three of a kind
• Straight
• Flush
• Pair
• Ace high and less
• Play starts with the player making an Ante bet. The player may optionally make a Pair Plus bet.
• The dealer shall give the player and himself three cards each. The dealer's cards are dealt face down.
• The player has the choice to stand or take a Mulligan.
• If the player chooses to take a Mulligan, then he must make a Mulligan bet equal to his Ante bet and exchange his original three cards for three new cards.
• If the player takes a Mulligan, then he may make a new Pair Plus bet based on the Mulligan hand.
• The dealer will turn over his cards.
• If the dealer has a king high or higher, then he will stand. Otherwise, if the dealer has a queen high or less, then the dealer will take a Mulligan, discarding his hand for three new cards.
• The dealer will compare his own hand to the player's hand, and the higher hand wins. If the dealer has the higher hand, then the player will lose the Ante and, if made, the Mulligan wager. If the player has the higher hand, then both Ante and Mulligan wagers will pay according to the ante and raise pay table below. A tie will result in a push.
• The Pair Plus wager(s) shall pay according to the pay table below.
• Ante and Mulligan Pay Table

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

Pair Plus Pay Table

Player Original Hand Pays
Straight flush 40 to 1
Three of a kind 30 to 1
Straight 6 to 1
Flush 3 to 1
Pair 1 to 1
All other Loss

Ante Analysis

The following table shows the probability and return for all possible outcomes of the Ante bet in Three Card Mulligan. The lower right cell shows a house edge of 3.57%.

Three Card Mulligan Return Table

Event Mulligan Pays Probability Return
Straight flush Yes 12 0.001075 0.012894
Three of a kind Yes 8 0.001160 0.009284
Straight flush No 6 0.002169 0.013014
Three of a kind No 4 0.002341 0.009365
Straight Yes 3 0.015675 0.047025
Less than straight Yes 2 0.169427 0.338855
Straight No 1.5 0.031575 0.047363
Less than straight No 1 0.286811 0.286811
Tie No 0 0.000608 0.000000
Tie Yes 0 0.000500 0.000000
Loss No -1 0.176960 -0.176960
Loss Yes -2 0.311698 -0.623396
Total 1.000000 -0.035744

The player will make the Mulligan wager 50.0% of the time. The average wager is thus 1.5 units, for an Element of Risk of 2.38%.

Pair Plus Analysis

The following table shows the probability and return for all possible outcomes of the initial Pair Plus bet in Three Card Mulligan. The lower right cell shows a house edge of 7.28%.

Pair Plus Analysis — Initial Hand

Event Pays Combinations Probability Return
Straight flush 40 48 0.002172 0.086878
Three of a kind 30 52 0.002353 0.070588
Straight 6 720 0.032579 0.195475
Flush 3 1096 0.049593 0.148778
Pair 1 3744 0.169412 0.169412
Loser -1 16440 0.743891 -0.743891
Total 22100 1.000000 -0.072760

The next table shows the probability and return for all possible outcomes of the subsequent Pair Plus bet in Three Card Mulligan, assuming the player follows optimal strategy. The lower right cell shows a house edge of 7.66%. The reason it is different from the original Pair Plus bet is the effect of removal of the discarded cards.

Pair Plus Analysis — Mulligan Hand

Event Pays Probability Return
Straight flush 40 0.002155 0.086192
Three of a kind 30 0.002335 0.070036
Straight 6 0.032363 0.194179
Flush 3 0.049547 0.148641
Pair 1 0.169018 0.169018
Loser -1 0.744582 -0.744582
Total 1.000000 -0.076515

Strategy

The player should stand with K-10 or higher; otherwise, take a Mulligan.