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Subject: How many shuffles? rss

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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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Tedious Shuffling
A question I've pondered in between decks is how many shuffles are enough. I sleeve my cards, and there're 162 cards in Up Front, so shuffling them is a bit more troublesome than shuffling a standard deck after a game of Set Back.

The Problem
There are, I think, two main structures that can be imposed upon a random deck during play: (1) Clumping, as after a game of Go Fish, where all the twos are stacked together, and all the Jacks, etc., and (2) Ordering, where sequences are formed within the deck, say after a game of Solitaire, where you get 2-3-4-5-6-7, etc.

Up Front doesn't suffer much from ordering, in my experience, but it definitely suffers from clumping. And I see two offenders: (a) after-game cleanup and (b) play-induced order.

(a) After-game cleanup is by far the worse of the two, for at the end of the game you typically have just a few cards lying on the table--the scenario-defined cower cards (say, 5 buildings cards) and the last terrain/movement cards played by your groups (say 3 to 5 more terrain cards and a movement card or two). The most natural thing to do at that point is to grab up all of these cards and toss them onto the discard stack. But in doing that, you create a clump of up to a dozen terrain cards, including maybe six or seven of the eight buildings cards in the deck.

(b) Play-induced order creates much smaller clumps in the discard pile: typically just a wire card plus a movement card, or a stream card plus a ford card, or a fire card plus a concealed card, or a terrain card plus a movement card.*

So, how much effort does it take to remove this clumping?

The Solution
(a) I think the best thing you can do is to not drop the clean-up cards straight into the discard pile. Just take them and slide them randomly into the deck, one here, one there. Simply doing that reduces the number of shuffles you'll need to sufficiently randomize the deck.

(b) So how many shuffles does it take to disperse the latter-mentioned two-card clumping? I use a 6-deck shuffler I got from Amazon for $11. (It works very well, and as the cards are sleeved I'm not worried about wear on the cards.) Assume for a moment that it interleaves the cards perfectly, one card from the left deck, then one card from the right deck. After a single shuffle, the two clumped cards will be separated by 1 randomly chosen card. After two shuffles, by 3 cards. Three shuffles, 7 cards. Four shuffles, 15 cards. Five shuffles, 31 cards. Six shuffles, 63 cards.

Note also that, in Up Front, cards are always being "skipped" by being drawn for RNCs and RPCs, perhaps having an equal chance of ending up in a player's hand, as much as not. So if two cards are sufficiently separated, say by a dozen cards, there's a good chance one will end up in a player's hand, and the other flipped for an RNC/RPC--this will also break up clumping.

Recommendation
For most games, parceling out the clean-up cards and then making five riffle shuffles should suffice. I could even see four riffle shuffles sufficing.

If, however, you shuffle sleeved cards by chopping them in, and if you don't parcel out the clean-up cards, even twenty or so shuffles might not provide the same disorder.**

----------------------

* - Some further analysis:

(A) In the deck there are 48 fire cards (1 in every 3 cards) and about 15 concealed cards (1 in every 11 cards). So every concealed card is likely to find itself close to another fire card in any event: two to three shuffles will restore the average separation.

(B) The other three cited examples deal with a rare card (Wire, Stream, or a specific Terrain card) and the more numerous Movement card (about 25 in the deck, or 1 in every 6 to 7 cards).
--In shuffling, if you separate the original terrain card and the original movement card by only 7 other cards (which is what happens after three shuffles), on average you'll have three movement cards within +/-7 cards of the original terrain card instead of the expected two movement cards. A fifty-percent increase in movement cards within a 14-card span. That's a noticeable effect.
--After four shuffles, there'll be 5 Movement cards within +/-14 cards of the terrain card, instead of the expected 4 Movement cards--much less noticeable: twenty-five percent more movement cards within a 28-card span.
--After five shuffles, there'll be 9 Movement cards within +/-28 cards of the terrain card, instead of the expected 8 Movement cards--and that shouldn't raise an eyebrow: twelve-percent more movement cards within a 56-card span.

-----------------

** - Chop shuffling of sleeved cards:

When I chop shuffle a triple deck of sleeved cards I encounter three problems:

(1) Often I can't handle the entire deck, so a portion of it is sitting idle. (Granted, the other player can be shuffling the other half of the deck, occasionally trading partial decks back and forth.)

(2) Often I'll get a clump of ten to twenty cards at one end of the deck that doesn't get disturbed in a given chop shuffle.

(3) A one-to-one interleave rarely happens: I'll often get two to four cards sticking together.

To counter (2) and (3), I would probably need two to five chop shuffles to guarantee that I've separated every pair of cards, which one shuffle by a mechanical riffle shuffler is likely to achieve.

The upshot is that I wouldn't trust fewer than ten chop shuffles (of the entire deck), and would prefer more than twenty chop shuffles to feel comfortable with the resulting randomness. Thirty is probably what I would aim for (thirty for each half of the deck, assuming two chop shufflers). True, a single chop shuffle is quicker than a single feed through a riffle machine, but I think you'll need to make three to five chop shuffles to match the "guaranteed" separation that's achieved across the entire deck with a single pass through a riffler.

[I also have to factor in the time it takes to pick up dropped cards, something I always seem to need to do whether chop shuffling or loading the riffler.]
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Definitely a physicist.
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Spyros Gkiouzepas
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Impressive.

Useful too...

I think you just made me suffer a little more...

With OCD!
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Edward Kendrick
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A useful dissertation, Bill, and I agree with your analysis.

We tend to "unclump" the infamous Stream/Movement combination when the victim gets out of the Stream and the cards are returned to the discard pile, by putting them at different heights in the pile.

And at the end of the game I divide the deck into a dozen or so heaps, none with a Terrain card on top, and then put one of the outstanding Terrain cards on top of each heap - that helps to ensure that you don't get two Terrain cards together. Of course, when we shuffle, our carefully structured disorder is potentially reduced to non-chaos ...!
 
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Isaac Citrom
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Bill, in some casinos at some tables they use a single deck of cards. But, the point still stands even with more than one deck. If you've ever seen them hand-shuffle cards, there is a strict method they use in order to sufficiently randomize the reshuffled deck. I'm sure there is a demo of it somewhere on YouTube.

So, using such a correct shuffling method, I think there may not be a need to bother with the cards other than the shuffle.
.
 
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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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isaacc wrote:
So, using such a correct shuffling method, I think there may not be a need to bother with the cards other than the shuffle.

It would be interesting to observe their shuffling pattern--thank you for that recommendation. In most casino games, however, I suspect there isn't much order being introduced into the deck: in Black Jack, hardly any at all; in Poker, not much more than that. My primary interest, though, was to consider how little shuffling I could get away with, engaging in that eternal pursuit of civilized man, the substituting of brainpower for dexterity.
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Richard Pardoe
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Smash62bill wrote:
Assume for a moment that it interleaves the cards perfectly, one card from the left deck, then one card from the right deck. After a single shuffle, the two clumped cards will be separated by 1 randomly chosen card. After two shuffles, by 3 cards. Three shuffles, 7 cards. Four shuffles, 15 cards. Five shuffles, 31 cards. Six shuffles, 63 cards.

Not necessarily. Remember a deck of 52 cards will return to its original order after 8 perfect out-shuffles (ie, the top and bottom cards remain the top and bottom card after the perfect interleave.)
 
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Zak
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There is a good gametech segment on this on the dice tower. Apparently 7 shuffles is the optimum number to randomize a deck keeping in mind a riffle shuffle is almost never perfect.

Zak
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Keegan Fink
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I always thought the ideal number was 42
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Marty M
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Good grief.
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Rob Rob
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I agree, riffle shuffling once or twice won't break up a deck sequence 100%. Not so easy with a 162 card action deck but with more manageable decks I will often do what I call a "pile shuffle." I'll repetitively deal out seven cards, left to right, left to right, into seven stacks then stack them back up, left to right. That breaks up any potential for the above described sequencing.
 
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Chandler Braswell
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While I primarily deal craps and try to avoid card games like the plague that they are, the multideck shuffle I have used in a few casinos might help. To explain this is a bit complicated but I think this is clear. First the two technical terms used: 1) Riffle: separate the group of roughly one deck (app 52 cards) into two roughly even parts and lift the corners with your thumbs as you drop them into a single deck. Basically a traditional shuffle. 2) Strip: hold the roughly one deck loosely in one hand while using the other hand to pull off three-to-ten cards off the top followed by three-to-ten cards off the bottom to create a new pile. Now the shuffle:
1) separate the cards into two roughly equal stacks called A and B, place the stacks far enough apart that you can work between them.
2) take app half a deck from A and half a deck from B and stack them front of you.
3) Riffle, strip, Riffle this new stack and then place it a bit aways from you into new stack C.
4) Take app half a deck from A and half a deck from C and Riffle, strip, Riffle. Placing these handled cards on top of stack C.
5) Repeat step 4 for C and B.
6) Repeat 4 and 5 until all cards are in stack C.
7) Clear you hands and go on break because snapper (blackjack, 21) SUCKS!

A little dry because not sure how to do pictures on my phone but should work wonders.
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Mark J
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or just shuffle the pile three times and call it good enough.
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Keegan Fink
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For all the games I play, I try to alternate successive rounds of using pile shuffling and overhand shuffling. After about 4-6 rounds of each, I call it a day. I have found that pile shuffling alone tends to lead to "ordering" as described by the OP, and doesn't seems to do an efficient job of breaking patterns or reduce clumping.

Pile, overhand, cut the deck. Pile, overhand, cut. Rinse & repeat.

I don't use card sleeves (hate the feel), and I refuse to rifle shuffle cards as it tends to really lead to quick wear and tear.
 
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Richard Irving
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Since an Up Front deck has so many cards, the most common way is for both players to shuffle about half the deck each and then occasionally trade half of their halves a few times.

I won't claim it is a perfect method, but it seems to be sufficient to break up bunches.
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David Millette
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rri1 wrote:
Since an Up Front deck has so many cards, the most common way is for both players to shuffle about half the deck each and then occasionally trade half of their halves a few times.

I won't claim it is a perfect method, but it seems to be sufficient to break up bunches.


I like this idea...
 
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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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DiploGuy wrote:
or just shuffle the pile three times and call it good enough.

But is it?

That was my question: how few shuffles will give me a sufficiently randomized deck?

My conclusions:
(1) Distribute the end-of-game terrain cards into the deck, and
(2) Riffle 4 or 5 times, or Chop shuffle 20 to 30 times.

elirlandes wrote:
Good grief.

Yeah, you got me, I went deep into the math--but hey, I enjoy math! I also enjoy games. And in this thread I got to enjoy both. And now I also know how many times to shuffle. Win-win-win.
 
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Aaron Bredon
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Smash62bill wrote:

That was my question: how few shuffles will give me a sufficiently randomized deck?


My general rule is start doubling 2,4,8,16,32,64,128,256,512,etc.
When I reach a number larger than twice the number of cards in the deck, the deck is sufficiently shuffled for most use. This means that 2 cards that started adjacent would have been separated by over half the deck at least once and probably twice.

This requires using a decent riffle like shuffle. When using an overhand shuffle, you would need to alternate with a pile shuffle and repeat about twice to get 1 riffle equivalent.

If the number I have reached is < half the number of cards in the deck, 2 cards that started adjacent will be about that many cards apart.

I actually looked into the math once, and it turns out that this is roughly when the increase in randomization from each shuffle drops to a low number and the randomization is pretty good.

This means for the Up Front deck, you would have to shuffle about 8-9 times to get a good shuffle.
 
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Nathaniel GOUSSET
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But arranging for cards that start adjacent to be separated is not adequatly randomising a card deck.

In a truly randomised deck, some cards should stay adjacent. Heck, a truly randomised deck could have a chance that all the cards stay in the same order they started with.
 
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Aaron Bredon
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IKerensky wrote:
But arranging for cards that start adjacent to be separated is not adequatly randomising a card deck.

In a truly randomised deck, some cards should stay adjacent. Heck, a truly randomised deck could have a chance that all the cards stay in the same order they started with.


And unless you are using a true random number generator to decide how to shuffle, you can't get a truly random deck.

Each time you shuffle, you increase the relative randomness of the deck. At some point, the randomness increase becomes small enough that we can consider the deck to be randomized for our purpose. For most purposes with a riffle shuffle, that is 1-2 times after you reach the size of the deck in the doubling sequence. At that point, a card that started at the top of the deck could be anywhere in the deck with relatively similar probabilities, and you cannot predict where another card is based on their starting positions.
 
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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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abredon wrote:
For most purposes with a riffle shuffle, that is 1-2 times after you reach the size of the deck in the doubling sequence. At that point, a card that started at the top of the deck could be anywhere in the deck with relatively similar probabilities, and you cannot predict where another card is based on their starting positions.

I think, in general, you have a good argument.

I, however, would prefer not to have to riffle shuffle the deck 8 to 9 times each time the tray empties, and I think that with Up Front you don't really need to. Reasons:

(1) Assuming you begin play with a roughly randomized deck, playing the game won't introduce much clumping or sequencing into the deck.

(2) Cards are used for several purposes: RNCs, RPCs, or Face value. Depending on the purpose, there are usually many other cards in the deck with the same or very similar values: movement cards, say; or Fire 4, 5 or 6 cards; or red RNC 4 cards; etc. Rarely, therefore, do you have to require that a single card be anywhere within the deck: it usually suffices merely that a similar card (for whatever specific purpose you're considering) is likely to be found anywhere. After a typical play, I believe four or five riffle shuffles will accomplish that.

Sure, eight or nine riffle shuffles would be better. And fifty-three better yet, of course. But with each shuffle, the value of the next ensuing shuffle diminishes in utility (in terms of making the deck more random). So you have to draw a line somewhere.

For me, I think breaking up clumps is all that is needed, and for that, four or five shuffles should suffice. If your requirement, contrarily, is that card #135, say, has to have an equal chance of being anywhere in then deck, then you're probably right that eight or nine shuffles are needed.
 
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Mark J
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Smash62bill wrote:
DiploGuy wrote:
or just shuffle the pile three times and call it good enough.

But is it?

That was my question: how few shuffles will give me a sufficiently randomized deck?



What's the definition of a sufficiently randomized deck? And if you were able to come up with one why is it necessary to achieve in this game?

This isn't a casino card game where all the players have the same goal every single hand.

Is a sufficiently randomized deck of UF mean that every fire card is at equal distance from each other? That the red 6s are equally balanced in the deck? That the 1-5 RPNs that are mostly red numbers are spread fairly throughout the deck? That the 0 RPCs are evenly distributed?

What if on a players turn he attempts to entrench 3 groups and he draws 3 straight 0s. Is that bad shuffling? But what if instead of entrenching he just refilled his hand and drew a movement, fire and rally, all RPN 0? Now would that be good shuffling?

So I don't see a way you can define a sufficiently randomized deck in UF.

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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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DiploGuy wrote:
What's the definition of a sufficiently randomized deck?

That's what we've been discussing. My definition is that clumps that are created during play should be adequately dispersed. Another definition that's been mentioned is that any given card has a uniform probability of appearing anywhere within the shuffled deck. Those are two valid definitions.

And it should be obvious, but yes, you can have the three similar cards (three Wire cards, for instance) appear in a row in a randomized deck. And I don't think that anyone has argued here that your example of three 0 RNCs appearing in a row would constitute an unrandomized deck.

My point is that if, (a) during one deck cycle, the three Wire cards appear in a row, that (b) after the ensuing shuffle, it should be very unlikely that those three Wire cards are still in close proximity. That's what I want to achieve in shuffling the decks, breaking up clumps that existed after reaching the end of the previous deck.

Your earlier suggestion of "just shuffle it three times and play" seems insufficient, for after three riffle shuffles, those three successive Wire cards will now likely all be found within a 17-card stretch, giving an observant player useful knowledge about the deck due to an inadequate shuffle. That's one of the things I wish to avoid. (I don't mind if--after a good series of shuffles--those three Wire cards again end up in a small sub-region of the deck; I just don't want a proximity in one play-through of the deck to dictate a proximity in the next play-through.)
 
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Edward Kendrick
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There's been some analysis here of the effectiveness of riffle and chop shuffles. How about the informal kind of shuffle where you spread all the cards out face down on the table, churn them round a bit, then sweep them together again? Not very well-defined I know, but that would probably be quite good at breaking up clumps.
 
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Bill K
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. . . then came the king's son, wounded, sore bestead, and weaponless, and saw the broken sword, hilt-buried in the dry and trodden sand, and ran and snatched it, and with battle shout lifted afresh, he hewed his enemy down . . .
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See how the flowers of the field grow. They do not labor or spin. Yet I tell you that not even Solomon in all his splendor was dressed like one of these. If that is how God clothes the grass of the field . . . will he not much more clothe you? (NIV)
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An interesting question. Let's compare the effectiveness of the various shuffling techniques regarding . . .
(a) speed
(b) breaking up clumps
(c) breaking up sequences (in my view not a big issue for Up Front)
(d) shifting cards into other regions of the deck

Riffling
Speed: good
Clumps: very good
Sequences: poor
Shifting: poor

Clumps are broken up slowly at first, but the effect is geometric. Sequences aren't reordered until they fall across a deck split, but they are slowly moved apart. Regarding the goal of shifting a card's relative position within the deck, cards located near the top or bottom of the deck don't move quickly: riffling should therefore probably be intermixed with other techniques--even simple cutting should improve this aspect.

Chopping
Speed: good
Clumps: good
Sequences: poor
Shifting: poor

Very similar to riffling, but quicker with sleeved cards than riffling would be, but also less efficient, perhaps requiring two to four chops to match one riffle.

Cutting
Speed: excellent
Clumps: very poor
Sequences: very poor
Shifting: good

Often used just before play begins. Probably would be very effective when occasionally done on the half-decks just before a riffle or a chop.

Multiple Cut - cutting the deck into several piles, then randomly re-stacking.
Speed: very good
Clumps: very poor
Sequences: poor (unless the sequence falls across a deck split)
Shifting: very good

Pile Deal - dealing out the cards into half a dozen stacks, then re-stacking.
Speed: extremely slow
Clumps: excellent
Sequences: excellent
Shifting: very good

Smear - smearing the cards across the table then re-grouping them into a stack.
Speed: medium (depends on how far apart you smear them)
Clumps: poor (unless you separate them enough so that adjacent cards are no longer overlapped, and so they don't often end up close to the cards they started close to)
Sequences: poor (unless you separate them enough so that adjacent cards are no longer overlapped)
Shifting: good

Strip - stripping off the top and bottom three-to-ten cards into a new pile followed by a riffle shuffle.
Speed: quick
Clumps: like one riffle
Sequences: like one riffle
Shifting: good for the cards at the top and the bottom which tend to hover there with simple riffle shuffles.

-------------

A key, I suspect, is to combine various techniques during a single reshuffling of the deck, thereby mitigating the deficiencies of any given technique, for instance including a strip or a cut during a series of riffles.

Another thing to consider--if your cards are unsleeved--is the wear and tear resulting from the various techniques. Riffle shuffling is easy on the cards, I find, unless using a mechanical shuffler. I cringe when other people chop-shuffle my unsleeved cards (and when they give them to their two-year-old to suck on). Smearing would probably also make me stir in my chair.
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