# Is either player at a disadvantage when cards "aren't' shuffled well enough"?

Perhaps I should be asking this on one of the math sites rather than here, but I reckon cardplayers would know: when my wife and I play Skip-Bo and things aren't going her way, she claims that I "didn't shuffle the cards well (long) enough."

While it may be true that I only shuffled the cards a few times (rather than dozens), my assertion is that it doesn't really matter, as far as who has the advantage - we're both playing with the same "badly shuffled" cards, so it's an "even playing field."

Am I wrong? Is either player at a disadvantage with insufficienlty shuffled cards?

I am even tempted to not shuffle at all between game 1 and game 2 to see how that turns out.

• Are you looking for proof that shuffling poorly provides no advantage to a player? If so, then you have not provided enough information to answer the question. What was the state of the deck before you shuffled it? What process do you use to shuffle? How do you deal? What game are we playing? Do both players have equal knowledge of the state of the deck before it gets shuffled? Do both players have equal knowledge of the shuffling process (e.g., is it possible for the shuffler to know that a card that was on the bottom or top stayed on the bottom or top, without the opponent knowing)? May 13, 2016 at 17:37
• The state of the deck is that there are many cards in this order before shuffing: 12, 11, 10, ...1 (but there could also be "wild cards" (SkipBo) anywhere in there). I shuffle the normal way, with my two hands, especially my thumbs. Dealing is done by alternating cards from the top: one for her, one for me, until both have 30; then the same for another 5 each. As mentioned, the game is SkipBo (which I highly recommend - it's just complex enough to be interesting, but "anyone" can learn it pretty quickly). Both players have equal knowledge. May 13, 2016 at 17:41
• @Rainbolt I would be curious to know how any of your questions would actually change an answer that you would give. It seems to me that adding information for your question 2, 3, and 4 would make the question too specific to an individual player. I also feel that 6 doesn't add useful information and 7 would imply cheating. That leaves your question 5, but the OP is curious about shuffling in general, and an answer to this question could apply to other games (e.g.: Spite & Malice). May 13, 2016 at 18:07
• @Rainbolt As such, I think it is perfectly clear what the OP is asking: In a game that uses a shared deck, do bad shuffling practices inadvertently hurt one player more than another? May 13, 2016 at 18:07
• With Skip-Bo in particular, a game would go faster due to poor shuffling, as you would be more likely to draw a series of cards in ascending order, leading to more plays per turn and more options. This could give a slight advantage to the first player because that player goes first. (In other words, if both players could go out on turn 8, player 1 will go out first and still win). May 16, 2016 at 19:23

As long as neither player has information about the order of cards in a deck of cards, no player can gain an advantage from a "bad" shuffle.

A "bad" shuffle could be defined as a shuffle that does not, by a reasonable standard, erase all information remaining from the end of the previous game, most notably the order of cards. This does not mean that there is no recognizable order to the cards, only that neither player knows in what order the deck is, including after a player has seen a part of the deck.

If a particular game produces long streaks of ordered cards on the table, and a subsequent shuffle before the next game does not break up that order, then there is the potential that information remains to be gained in the next game, and a possible advantage for one player. The amount of shuffling needed to prevent that depends on the technique used and the amount of order that is present (e.g. how large are the blocks of ordered cards the game typically produces).

As for your problem with the half-serious allegations of negligience or even cheating by your wife: That question is not likely to be resolved with better shuffling alone, it's about venting her frustration after losing - she is not going to complain about a bad shuffle if she wins. Simply offer to let her shuffle the deck whenever she wants, or at least let her cut the deck before the game starts. This involves her in the shuffling process and she has no more reason to take insufficient shuffling as an excuse.

• Truth be told, she wins more often than I do (probably 60-65% of the time) - and it seems to me that when she wins, she has "better" cards than I do - but in reality I'm sure that playing style/philosophy has a lot to do with it. May 13, 2016 at 17:24
• If both players are using the same strategy, then I agree that there is no advantage either way from a "bad" shuffle, but if individual players have specific biases (that could be suboptimal for a fully randomized deck), advantages could be had for specific kinds of "bad" shuffles. For example, if one shows a bias towards going for a flush in the game of poker, badly distributed suits could be an advantage over a player that prefers going for a straight. May 13, 2016 at 19:12
• Note that for certain games, poor shuffling can give a considerable advantage to one side: consider the simplest game where each player draws a single card, and the higher card wins (and then the pair is placed in a 'trash' heap) the heap will be ordered with win-lose pairs. If it is later poorly shuffled and cards are dealt to each player, one will get all the 'win' cards and the other all the 'lose' is a given unshuffled streak. Similar phenomena can happen in much more complex games... May 17, 2016 at 8:55
• @G0BLiN: but that does not in fact give any advantage to a player unless he a) knows which side is the winner and b) can somehow take advantage of the knowledge. A biased (or even two-headed) coin can still give an even chance - so long as the person calling does not know of the bias (and can therefore be assumed to call heads or tails with equal probability). May 19, 2016 at 12:30

There are certainly games in which a bad shuffle can benefit one player. I remember playing Ligretto, also called Dutch Blitz, against my teenage nieces who were winning far more frequently than even teenage reflexes would indicate likely.

In Ligretto the cards end up stacked in order, and there is a distinct advantage to having low cards on the top of your playing pile. It turned out that my nieces were giving their own cards only the most perfunctory shuffle possible, essentially just turning them over.

• This is a bit different from the OP's question, since the game does not involve a shared deck. A poorly shuffled deck may fare well against a well shuffled deck, but if both players are using the same poorly shuffled deck, it won't matter. Aug 30, 2018 at 19:25

## Your wife is confusing 'random' with 'balanced'

Human beings, in general, have a terrible understanding of probability.

There's a rather popular probability demo that teachers use, where they'll give half their class coins and ask them to flip the coin 100 times and record the results. The other half the class is instructed to attempt to fool the teacher by creating a sequence of fake results for 100 flips.

When the teachers look at the anonymous sequences, they can usually identify them correctly about 90% of the time. The secret is that the fake sequences almost never include runs of more than four or five heads or tails in a row, while the randomly generated ones almost always do.

Randomness does not mean "evenly distributed". It means that sometimes you'll get an even distribution, and sometimes you'll get horribly unbalanced clumps. At random.

The answer to this question is going to depend on how you are shuffling the cards. If you are doing a standard riffle shuffle then as long as you can do a decent shuffle you should not need to do it more then 7 times. In most games shuffling the deck 3-4 time is enough to provide the randomness that the game needs. It should also be noted that even if you get streaks of cards that is not a sign that the deck is not random enough. It is easy to forget that random can still produce streaks that do not appear random.

Back to the original question, can a bad shuffle put a player at a disadvantage? Yes, but so can a good shuffle as the issue will be that one player will be at a disadvantage if they don't get the cards they need but the other players do.

Last thing to remember is that while you want to perform a good shuffle there comes a point where you have shuffled to much and can undo the work you have done.

Seven Shuffles
This talks about randomizing a deck of cards

Shuffling
Wiki article with different shuffling techniques

How many times should you shuffle a deck of card
Really detailed paper about math of shuffling a deck of cards.

Perfect Shuffles With the perfect shuffle, cut the deck in half and interlace every other card it takes only 8 shuffles to restore the deck the original order

Since I don’t know Skip-Bo, I will give a more general answer:

In deck-building games (like dominion) it is absolutely possible that a player benefits from a bad shuffle of their individual deck, because good cards, once close together, will tend to stay close and are more likely to be on their hand at the same turn, making for some really good and some almost useless hands which allows to buy extremely valuable things one on turn and forfeit the other, compared to many medium hands if well shuffled which leads to many purchases of less valuable things. If you feel that another player profits from bad shuffling, there is an easy solution for that: Let the player's individual decks always be shuffled by their right neighbor instead of by themselves.

But even in games with only one deck for all, bad shuffling can mean a different advantage to players: In trick-taking card games high trumps as well as cards from the same suit are often in one trick. A bad shuffle means a higher probability that any one player has a superior hand. But in trick-taking games, a superior hand is more valuable when you are the starting player (usually left of the dealer) and can control what suit others have to follow on the first (few) trick(s). Therefore the player sitting left of the one player who shuffles less well than the others has an advantage on the long run. As a consequence, Skat, a traditional German trick-taking game, has a rule for tournaments - if one player is not shuffling by hand but uses a machine (which will generally shuffle more thoroughly), all players on that table are required to use the machine. Of course, in casual games (and even competitive ones outside official tournaments) this slight imbalance is usually ignored.

There is no need for card runs to be long or for one person to have deck knowledge for a "shuffle" to be unfair for one player.

Consider the case where high cards are placed on top of low cards during game play. If these alternations are then flipped over and dealt, the high cards will now be in the even-numbered positions and will be predominantly dealt to dealer of a two player game.

One reason for the tradition of having non-dealer cut is so that any simple inversion such as this is (nearly) randomized as to which player it will favour.

Slide shuffles are notorious for not breaking up runs efficiently - but in addition if the deck has not been fully squared first the individual slides will tend to be full tricks. This can mean that in terms of breaking up tricks of two cards, a slide shuffle may not be a shuffle at all. Card sharps know all these tricks and many more.

The standard in hand-dealt Contract Bridge (a single-deck game) is for a minimum of seven riffle shuffles followed by a cut by non-dealer's side - though even this is subject to manipulation as eight perfect riffle shuffles restore a deck to its original order. To be seen as scrupulously honest, make one or two of your riffle shuffles obviously imperfect.