Last Updated: October 21, 2018

# Blackjack X-Change

## Introduction

Blackjack X-Change is a blackjack variant that lets the player switch any card in his hand for a random card. Depending on the situation, the player will either have to pay to switch a bad card or get paid to switch a good one. It is my understanding that the game may be played on the Caesars online/mobile casino.

## Rules

Blackjack X-Change is based on standard blackjack rules, which I assume that the reader is already familiar with. Following are the particulars and adjustments to conventional blackjack.

1. An infinite number of decks are used.
2. Dealer stands on soft 17.
3. Blackjack pays 3 to 2.
4. Dealer peeks at hole card for blackjack.
5. Player may double on any two cards, including after splitting.
6. Player may split once only.
7. Player may hit and double after splitting aces.
8. The player my exchange any card in his hand, in most situations, for a random card at a price. This option is only offered with four or less cards, 21 or less points, and not after doubling or splitting.
9. If the player exchanges a card, he may not double nor split afterward.
10. If the player has blackjack after exchanging a card, it shall still pay 3 to 2.
11. The game will charge the player to exchange a bad card, known as a "Buy" price, or pay the player to exchange a good one, known as a "Sell" price.
12. Both Buy and Sell prices are dependent on the situation. The game strives to charge the player 2.5% above the fair price to Buy a card and underpay by 2.5% to sell a card, although I find sometimes the percentage is a little higher.
13. If the players buys or sells a card, the cost or price paid is directly added/subtracted from the player's balance, not his bet amount.

## Example

In the file above I bet £20 and have a 3-card total of 13 against a dealer 5. My choices are:

• Hit or stand on the 15 against a 5, as in conventional blackjack.
• Buy a random replacement card for the 6 for £6.67.
• Sell the 2 for a random replacement card for £1.74.
• Buy a random replacement card for the 7 for £4.90.

## Analysis

Assuming the player never exchanges, I show the house edge is 0.37%. This is the ratio of the amount the player can expect to lose to the original wager. The help files of the game claim the rate of return is 99.68%, leaving 0.32% to the house. I suspect the difference is that the "return" from the help files is the ratio of the amount return to the player to the total amount bet, including doubles and splits. In other words, the difference is due to comparing the house edge to the element of risk.

The following table shows the expected value of every situation, if the only options are hitting and standing.

### Expected Values — Hit or StandExpand

Player
Total
Dealer
2
Dealer
3
Dealer
4
Dealer
5
Dealer
6
Dealer
7
Dealer
8
Dealer
9
Dealer
10
Dealer
Ace
Hard 4 -0.114913 -0.082613 -0.049367 -0.012380 0.011130 -0.088279 -0.159334 -0.240666 -0.289198 -0.253077
Hard 5 -0.128216 -0.095310 -0.061479 -0.023979 -0.001186 -0.119447 -0.188093 -0.266615 -0.313412 -0.278575
Hard 6 -0.140759 -0.107291 -0.072917 -0.034916 -0.013006 -0.151933 -0.217242 -0.292641 -0.337749 -0.304147
Hard 7 -0.109183 -0.076583 -0.043022 -0.007271 0.029185 -0.068808 -0.210605 -0.285365 -0.319055 -0.310072
Hard 8 -0.021798 0.008005 0.038784 0.070805 0.114960 0.082207 -0.059898 -0.210186 -0.249375 -0.197029
Hard 9 0.074446 0.101265 0.128981 0.158032 0.196019 0.171868 0.098376 -0.052178 -0.152953 -0.065681
Hard 10 0.182500 0.206088 0.230470 0.256259 0.287795 0.256909 0.197954 0.116530 0.025309 0.081450
Hard 11 0.238351 0.260325 0.283020 0.307350 0.333690 0.292147 0.229982 0.158257 0.119482 0.143001
Hard 12 -0.253390 -0.233691 -0.211063 -0.167193 -0.153699 -0.212848 -0.271575 -0.340013 -0.381043 -0.350540
Hard 13 -0.292784 -0.252250 -0.211063 -0.167193 -0.153699 -0.269073 -0.323605 -0.387155 -0.425254 -0.396930
Hard 14 -0.292784 -0.252250 -0.211063 -0.167193 -0.153699 -0.321282 -0.371919 -0.430930 -0.466307 -0.440007
Hard 15 -0.292784 -0.252250 -0.211063 -0.167193 -0.153699 -0.369762 -0.416782 -0.471578 -0.504428 -0.480006
Hard 16 -0.292784 -0.252250 -0.211063 -0.167193 -0.153699 -0.414779 -0.458440 -0.509322 -0.539826 -0.517149
Hard 17 -0.152975 -0.117216 -0.080573 -0.044941 0.011739 -0.106809 -0.381951 -0.423154 -0.419721 -0.478033
Hard 18 0.121742 0.148300 0.175854 0.199561 0.283444 0.399554 0.105951 -0.183163 -0.178301 -0.100199
Hard 19 0.386305 0.404363 0.423179 0.439512 0.495977 0.615976 0.593854 0.287597 0.063118 0.277636
Hard 20 0.639987 0.650272 0.661050 0.670360 0.703959 0.773227 0.791815 0.758357 0.554538 0.655470
Hard 21 0.882007 0.885300 0.888767 0.891754 0.902837 0.925926 0.930605 0.939176 0.962624 0.922194
Soft 12 0.081836 0.103507 0.126596 0.156482 0.185954 0.165473 0.095115 0.000066 -0.070002 -0.020478
Soft 13 0.046636 0.074119 0.102477 0.133363 0.161693 0.122386 0.054057 -0.037695 -0.104851 -0.057308
Soft 14 0.022392 0.050807 0.080081 0.111894 0.139165 0.079507 0.013277 -0.075163 -0.139467 -0.093874
Soft 15 -0.000121 0.029160 0.059285 0.091960 0.118246 0.037028 -0.027055 -0.112189 -0.173704 -0.130027
Soft 16 -0.021025 0.009059 0.039975 0.073449 0.098821 -0.004890 -0.066795 -0.148644 -0.207441 -0.165637
Soft 17 -0.000491 0.028975 0.059326 0.091189 0.128052 0.053823 -0.072915 -0.149787 -0.196867 -0.179569
Soft 18 0.121742 0.148300 0.175854 0.199561 0.283444 0.399554 0.105951 -0.100744 -0.143808 -0.092935
Soft 19 0.386305 0.404363 0.423179 0.439512 0.495977 0.615976 0.593854 0.287597 0.063118 0.277636
Soft 20 0.639987 0.650272 0.661050 0.670360 0.703959 0.773227 0.791815 0.758357 0.554538 0.655470
Soft 21 0.882007 0.885300 0.888767 0.891754 0.902837 0.925926 0.930605 0.939176 0.962624 0.922194

The following table shows the expected value of every situation, if the only options are hitting, standing, and doubling. Only player totals where doubling might be the best play are shown, otherwise see the hit or stand table above.

### Expected Values — Hit, Stand, or DoubleExpand

Player
Total
Dealer
2
Dealer
3
Dealer
4
Dealer
5
Dealer
6
Dealer
7
Dealer
8
Dealer
9
Dealer
10
Dealer
Ace
Hard 9 0.074446 0.120816 0.181949 0.243057 0.317055 0.171868 0.098376 -0.052178 -0.152953 -0.065681
Hard 10 0.358939 0.409321 0.460940 0.512517 0.575590 0.392412 0.286636 0.144328 0.025309 0.081450
Hard 11 0.470641 0.517795 0.566041 0.614699 0.667380 0.462889 0.350693 0.227783 0.179689 0.143001
Soft 12 0.081836 0.103507 0.126596 0.156482 0.185954 0.165473 0.095115 0.000066 -0.070002 -0.020478
Soft 13 0.046636 0.074119 0.102477 0.133363 0.179748 0.122386 0.054057 -0.037695 -0.104851 -0.057308
Soft 14 0.022392 0.050807 0.080081 0.125954 0.179748 0.079507 0.013277 -0.075163 -0.139467 -0.093874
Soft 15 -0.000121 0.029160 0.059285 0.125954 0.179748 0.037028 -0.027055 -0.112189 -0.173704 -0.130027
Soft 16 -0.021025 0.009059 0.058427 0.125954 0.179748 -0.004890 -0.066795 -0.148644 -0.207441 -0.165637
Soft 17 -0.000491 0.055095 0.118653 0.182378 0.256104 0.053823 -0.072915 -0.149787 -0.196867 -0.179569
Soft 18 0.121742 0.177641 0.237004 0.295225 0.381506 0.399554 0.105951 -0.100744 -0.143808 -0.092935

The following table shows the expected value of every situation, if hitting, standing, doubling, and splitting are all options.

### Expected Values — Hit, Stand, Double, or SplitExpand

Player
Total
Dealer
2
Dealer
3
Dealer
4
Dealer
5
Dealer
6
Dealer
7
Dealer
8
Dealer
9
Dealer
10
Dealer
Ace
2,2 -0.088887 -0.025616 0.042947 0.127250 0.194779 -0.007399 -0.159334 -0.240666 -0.289198 -0.253077
3,3 -0.138164 -0.063866 0.014625 0.102293 0.169420 -0.067760 -0.217242 -0.292641 -0.337749 -0.304147
4,4 -0.021798 0.008005 0.038784 0.080260 0.145957 0.082207 -0.059898 -0.210186 -0.249375 -0.197029
5,5 0.358939 0.409321 0.460940 0.512517 0.575590 0.392412 0.286636 0.144328 0.025309 0.081450
6,6 -0.218637 -0.136678 -0.049560 0.043987 0.107923 -0.212848 -0.271575 -0.340013 -0.381043 -0.350540
7,7 -0.155485 -0.074767 0.010511 0.099965 0.187691 -0.090501 -0.371919 -0.430930 -0.466307 -0.440007
8,8 0.019285 0.086888 0.156567 0.228318 0.325533 0.211530 -0.087582 -0.405400 -0.489488 -0.394058
9,9 0.184629 0.242140 0.301503 0.363348 0.443375 0.399554 0.215323 -0.093660 -0.178301 -0.100199
10,10 0.639987 0.650272 0.661050 0.670360 0.703959 0.773227 0.791815 0.758357 0.554538 0.655470
A,A 0.739275 0.783881 0.834126 0.895723 0.977514 0.914737 0.801496 0.642847 0.528001 0.597239

The following table shows the expected value for every possible situation after exchanging a card. The player totals indicated are after removing the card bought/sold but before considering whatever random card replaces it. The first two rows are for situations where the player a single card of a 10 or ace only, where a blackjack is possible.

### Expected Values after ExchangingExpand

Player
Total
Dealer
2
Dealer
3
Dealer
4
Dealer
5
Dealer
6
Dealer
7
Dealer
8
Dealer
9
Dealer
10
Dealer
Ace
10 only 0.230038 0.253373 0.277488 0.303047 0.333731 0.301068 0.241753 0.159670 0.066645 0.125896
ace only 0.559789 0.576812 0.594448 0.612905 0.639639 0.634007 0.575946 0.493984 0.429347 0.476406
Hard 2 -0.075884 -0.049751 -0.022100 0.013730 0.038883 -0.027257 -0.103162 -0.190047 -0.241998 -0.203354
Hard 3 -0.100523 -0.068876 -0.036261 0.000170 0.024471 -0.057438 -0.130942 -0.215077 -0.265329 -0.227937
Hard 4 -0.114913 -0.082613 -0.049367 -0.012380 0.011130 -0.088279 -0.159334 -0.240666 -0.289198 -0.253077
Hard 5 -0.128216 -0.095310 -0.061479 -0.023979 -0.001186 -0.119447 -0.188093 -0.266615 -0.313412 -0.278575
Hard 6 -0.140759 -0.107291 -0.072917 -0.034916 -0.013006 -0.151933 -0.217242 -0.292641 -0.337749 -0.304147
Hard 7 -0.109183 -0.076583 -0.043022 -0.007271 0.029185 -0.068808 -0.210605 -0.285365 -0.319055 -0.310072
Hard 8 -0.021798 0.008005 0.038784 0.070805 0.114960 0.082207 -0.059898 -0.210186 -0.249375 -0.197029
Hard 9 0.074446 0.101265 0.128981 0.158032 0.196019 0.171868 0.098376 -0.052178 -0.152953 -0.065681
Hard 10 0.182500 0.206088 0.230470 0.256259 0.287795 0.256909 0.197954 0.116530 0.025309 0.081450
Hard 11 0.238351 0.260325 0.283020 0.307350 0.333690 0.292147 0.229982 0.158257 0.119482 0.143001
Hard 12 -0.253390 -0.233691 -0.213537 -0.193271 -0.170526 -0.212848 -0.271575 -0.340013 -0.381043 -0.350540
Hard 13 -0.307791 -0.291210 -0.274224 -0.257333 -0.235626 -0.269073 -0.323605 -0.387155 -0.425254 -0.396930
Hard 14 -0.362192 -0.348729 -0.334911 -0.321395 -0.300726 -0.321282 -0.371919 -0.430930 -0.466307 -0.440007
Hard 15 -0.416594 -0.406249 -0.395599 -0.385457 -0.365826 -0.369762 -0.416782 -0.471578 -0.504428 -0.480006
Hard 16 -0.470995 -0.463768 -0.456286 -0.449520 -0.430927 -0.414779 -0.458440 -0.509322 -0.539826 -0.517149
Hard 17 -0.536151 -0.531674 -0.527011 -0.522986 -0.508753 -0.483486 -0.505983 -0.553695 -0.584463 -0.557300
Hard 18 -0.622439 -0.620005 -0.617462 -0.615260 -0.607479 -0.591144 -0.591056 -0.616528 -0.647671 -0.626515
Hard 19 -0.729077 -0.728033 -0.726937 -0.725991 -0.722554 -0.715450 -0.713660 -0.715574 -0.729449 -0.724795
Hard 20 -0.855230 -0.854977 -0.854710 -0.854480 -0.853628 -0.851852 -0.851492 -0.850833 -0.849029 -0.852139
Soft 12 0.081836 0.103507 0.126596 0.156482 0.185954 0.165473 0.095115 0.000066 -0.070002 -0.020478
Soft 13 0.046636 0.074119 0.102477 0.133363 0.161693 0.122386 0.054057 -0.037695 -0.104851 -0.057308
Soft 14 0.022392 0.050807 0.080081 0.111894 0.139165 0.079507 0.013277 -0.075163 -0.139467 -0.093874
Soft 15 -0.000121 0.029160 0.059285 0.091960 0.118246 0.037028 -0.027055 -0.112189 -0.173704 -0.130027
Soft 16 -0.021025 0.009059 0.039975 0.073449 0.098821 -0.004890 -0.066795 -0.148644 -0.207441 -0.165637
Soft 17 -0.000491 0.028975 0.059326 0.091189 0.128052 0.053823 -0.072915 -0.149787 -0.196867 -0.179569
Soft 18 0.062905 0.090248 0.118502 0.147613 0.190753 0.170676 0.039677 -0.100744 -0.143808 -0.092935
Soft 19 0.123958 0.149340 0.175577 0.202986 0.239799 0.220620 0.152270 0.007893 -0.088096 -0.005743
Soft 20 0.182500 0.206088 0.230470 0.256259 0.287795 0.256909 0.197954 0.116530 0.025309 0.081450
Soft 21 0.238351 0.260325 0.283020 0.307350 0.333690 0.292147 0.229982 0.158257 0.119482 0.143001

Next, let's look at the following example.

My bet amount was £100 and I have a soft 19 against an 6. We should know from basic strategy that if the dealer stands on a soft 17 we never double a soft 19, so we check the first expected value table to see the value of this situation is the produce of 0.495977 and the amount bet, or £100 × 0.495977 = £49.5977.

One option is to sell the ace for £37.16. That would leave us with a hard 8 and a random card. The table directly above shows the expected value of this situation is the product of 0.114960 and the amount bet, which equals £100 × 0.114960 = £ 11.4960. The reduction in expected value, compared to standing on the 19, is £49.5977 - £ 11.4960 = £ 38.1017. We are being offered to sell that ace for £37.16. The ratio of the sell price to the fair price is £ 37.16/£ 38.1017 = 97.53%. In other words, we are getting fair value for the ace, less 2.47%.

Another option is to buy a replacement card for the 8 for £14.76. That would leave us with an ace and a random card. Remember from the rules that if the replacement card is a 10, then the player will be paid the full 3 to 2. The "ace only" row of the table directly above shows the expected value of this situation is the product of 0.639639 and the amount bet, or £100 × 0.639639 = £ 63.9639. The increase in expected value, compared to standing on the 19, is £63.9639 - £ 49.5977 = £ 14.3662. We are being offered to buy a replacement card for the 8 for £ 14.76. The ratio of the buy price to the fair price is £ 14.76/£ 14.3662 = 102.74%. In other words, we are paying 2.74% above the fair price.

## Strategy

Following is my basic strategy for Blackjack X-Change. If some of the soft doubles look different than the standard dealer stands on soft 17 game, it is because of the infinite decks.

As for exchanging cards, the player should never do it, because the 2.5% to 3.0% margin is more than the 0.32% in the base game.

.

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