Blackjack X-Change

Introduction

Blackjack X-Change offers players the opportunity to swap any card in their hand for a random one. Depending on the context, players may need to pay to exchange a less favorable card or will receive payment for exchanging a more advantageous one. This variant is reportedly playable at the Caesars online or mobile casino.

Rules

Blackjack X-Change follows traditional blackjack rules, which I assume the reader is already familiar with. Here are the specific variations and adjustments made to typical blackjack gameplay.

1. An infinite number of decks are used.
2. Dealer stands on soft 17.
3. Blackjack pays 3 to 2.
4. The dealer checks their hole card to determine if they have a blackjack.
5. Players are permitted to double down with any two cards, including after splitting pairs.
6. Player may split once only.
7. After splitting aces, players have the option to hit and then double down.
8. Players can generally exchange any card from their hand for a random card at a cost, although this is only allowed with four or fewer cards and a point total of 21 or less, and cannot be done after doubling or splitting.
9. Once a player opts to exchange a card, they forfeit the chance to double or split afterward.
10. If a player achieves blackjack after making an exchange, it will still pay out at 3 to 2.
11. The game will charge players to swap a less favorable card, known as a 'Buy' price, while they will be compensated to exchange a desirable card, referred to as a 'Sell' price.
12. Both Buy and Sell prices vary based on the situation. The game aims to charge players around 2.5% over the fair price when buying a card, while offering 2.5% less when selling a card, although this percentage can sometimes exceed that amount.
13. When a player engages in a buy or sell transaction for a card, the amount paid or received directly impacts their balance rather than their wager amount.

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Example

In the previous example, I placed a £20 bet and currently have a 3-card total of 13 against a dealer showing a 5. My potential actions include:

• Standing or hitting on a total of 15 versus a dealer's 5, similar to standard blackjack rules.
• Purchasing a random card to replace the 6 for £6.67.
• Selling the 2 in exchange for a random card for £1.74.
• Buying a replacement card for the 7 at £4.90.

Analysis

Assuming the player neither buys nor sells, I find that the house advantage is 0.37%. This ratio reflects the expected loss in relation to the initial bet. The game's help documents suggest a return rate of 99.68%, which implies that 0.32% is left for the house. I suspect this discrepancy arises because the return mentioned tallies the player's payout against the total amount wagered, including bets made during doubles and splits. Essentially, the difference lies in contrasting the house edge against the level of risk involved.

The upcoming table presents the expected values based on scenarios where the options are limited to hitting and standing.

The next table outlines the expected values when players can choose to hit, stand, or double down. It lists only those situations where doubling is deemed the best option; otherwise, refer to the previous hitting or standing table.

The subsequent table illustrates the expected value across all scenarios where players can hit, stand, double down, or split.

The next table highlights the expected values for every conceivable scenario following a card exchange. The player totals represented are after the removed card from the exchange, but before incorporating the value of the random card that will replace it. The first two rows focus on situations where the player possesses only a single 10 or ace, thus allowing for the possibility of achieving blackjack.

Let’s analyze the example that follows.

My stake was £100, and I currently have a soft 19 against a dealer showing a 6. According to basic strategy, when the dealer stands on soft 17, it is never advisable to double a soft 19. So, I will check the first expected value table, which reveals that the value of this situation is calculated by multiplying 0.495977 with the bet amount, resulting in £100 * 0.495977 = £49.5977.

One option available is to sell the ace for £37.16. This would leave us with a hard 8 and a random card. The table shown above indicates that the expected value in this new situation is derived from multiplying 0.114960 by the amount wagered, equating to £100 * 0.114960 = £11.4960. The decline in expected value, compared to standing on the 19, amounts to £49.5977 - £11.4960 = £38.1017. The offer for selling the ace is £37.16. Consequently, the ratio of the sell price to its fair value stands at £37.16/£38.1017 = 97.53%. This means we receive a justifiable value for the ace, minus 2.47%.

Another alternative is to buy a random card to replace the 8 for £14.76. This would result in holding an ace along with a random card. It’s important to remember that should the replacement card be a 10, the player will earn the full 3 to 2 payout. The 'ace only' row in the previously mentioned table illustrates that the anticipated value here is the product of 0.639639 multiplied by the bet, leading to £100 * 0.639639 = £63.9639. The enhancement in expected value, in contrast to standing on the 19, is £63.9639 - £49.5977 = £14.3662. The replacement card offer for the 8 stands at £14.76, and thus, the ratio of the buy price to the fair market price becomes £14.76/£14.3662 = 102.74%. Hence, we are paying 2.74% more than what would be considered a fair price.

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Below is my recommended strategy for playing Blackjack X-Change. If certain soft doubles appear different from those in standard games where the dealer stands on soft 17, it's because of the use of infinite decks.

Regarding card exchanges, players should generally avoid doing this, as the 2.5% to 3.0% margin exceeds the 0.32% found in the base game.

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