Odds of winning (Klondike) solitaire

Most people are familiar with the draw three, unlimited pass version of solitaire common on computers and elsewhere. Since there are probably more possible hands than grains of sand, the winning odds might be found by simply playing many games - and of course many millions of games have been played.

This would not really give the odds since many games which are theoretically winnable do not win. Sometimes this is because of a “moderate” mistake - you do not see a transfer so miss an opportunity, instead taking a card from the deck. Often it is because of an “unknowable” mistake, the low card you need to build a foundation is hidden, but you cannot know in which pile.

My questions:

  1. What percent of games do people actually win?

  2. What percentage of games are winnable by making no “moderate” errors - always seeing available plays, playing with a sensible process, but not having knowledge of the “unknowable”.

  3. What assumptions are made when saying 79% or 95% or whatever games are theoretically winnable. What is the proper number? How can they know?

(I have reviewed previous threads on this topic since 2014 and think my question is different, but of course I am often wrong.)

Let’s move this to the Game Room.

Colibri
General Questions Moderator

As to 3-card unlmited pass Klondike, wiki says the winning probability has not been, and cannot be, precisely calculated.

In a nutshell, Klondike is a partial knowledge came of decision. In games with total knowledge, e.g. all cards dealt face up & exposed, a smart enough person / algorithm could play the game optimally and know before they began whether any given deal of cards is winnable or unwinnable. For cards dealt face down, that decision becomes logically impossible. e.g. You can’t know a priori, whether playing to expose what’s under pile 1 before pile 2 is better or worse.

Not sure I buy that the probability cannot be precisely calculated. The best line of play can’t always be determined in a given game state, but you still assign a probability to winning.

An example of a game without certain strategy is that you put one card face down and guess whether it’s red or black before flipping it over. This game has a precisely calculable winning probability, though: 50%.

According to this paper, there’s a relatively simple-to-code strategy that wins about 35% of games.

It might not be easy to calculate probability, but it can certainly be estimated - probably very well. A computer could be programmed to determine if a game is theoretically winnable, which is where these estimates (which vary) seem to come from. That’s only part of my question.

Interesting paper - thanks for unearthing that.

I had a solitaire book as a kid that gave the odds of winning what they considered the original rules of Klondike Solitaire to be 1 in 30. They didn’t explain how they determined the odds, though. Possibly just a bunch of simulations runs.

In the original rules, you drew cards one at a time from the draw pile, and you could never refresh the draw. Only the topmost card of the discard pile was available to you. Once you drew a new card, it blocked you from using the card currently face-up on the discard pile. You also could not do partial tableau moves. You either move the entire stack of face-up cards in a tableau column, or you can only move the top most card of the stack (i.e. the bottom-most column card). You also can’t pull a card out of the foundation once you place it there.

Most people play the Canfield draw variant, where you draw three at a time and can take the top card of those three, followed by the second most card after playing the top card, and finally the last card of the three if you played the other two. You then repeat the process and refresh the draw as many times as you want. That, plus partial tableau moves increases odds of winning significantly as the paper linked to above points out.

On a side note, this book also had Canfield solitaire pegged at a 1 in 30 chance. Canfield has a far more restrictive tableau than Klondike which is why its draw method is far more forgiving than original Klondike.

I also remember it had Accordion solitaire at a 1 in 100 chance of success. But as the book noted, 1 in 100 was the highest odds they had for anything. The authors suspected that winning Accordion was probably significantly worse than 1%.

They also had Golf solitaire, which I recall was hard to win.

Then there were a handful of solitaire games in the book that had like 7 out 8 chances of winning. I think 7 of 8 was the best odds they gave anything, but it’s been decades since I read that book.

It’s not theoretically impossible to calculate odds of winning. It might be extremely hard to calculate, but it’s possible. In any situation of incomplete information, while it’s not usually possible to determine the best move with certainty, it is possible to determine what move has the highest probability of eventually leading to a win. The ideal strategy, then, is the one that always picks the move that has the highest probability of leading to a win. And the ideal probability of winning can then be found by checking that ideal strategy against every one of the finite list of possible shuffles.

Another way of looking at it - Windows Solitaire has (or had) a ‘Vegas’ scoring variant, whereby you would be ‘charged’ $52 for dealing the layout, but you win back $5 for each card you put up to the winning line (i.e. moving an Ace from the layout to the top row gains you 5, and so on). It is obvious that if the winning expectation is better than 1 in 5 (20%), the player would make money from the house in the long run. Then you have to adjust that percentage downwards a bit, to account for the fact that in most layouts, you will win back a few without actually completing the game. I don’t know what the average number of cards put up per game is, but it can be anything from 0 to 52 (a completed game).

I do know that my winning percentage on this variant (where three cards were dealt from the deck at a time, you could only use the top card (until it was played, then you could use the next), and you could run through the deck three times before it was frozen) was around 12%. I’m going to guess that of the games I didn’t win, the average number of cards I put up was around 4. So in 1 in 8 games I was +$208 ($260 - $52) and in 7 in 8 games I was -$32, on that basis. Over 8 games therefore I would expect to lose around $16 ($208 - (7 x $32)). This is consistent with my ongoing ‘score’ which would tend to slowly but steadily deteriorate over time.

Now, what I don’t know is whether any casinos in Vegas actually tabled this game with those payouts. I suspect they used to, but a long time ago realised the rate of return was really poor (they have to pay a dealer to deal to one person in order to net $16 an hour before any other costs - if we also assume 8 hands per hour). Plus, I am probably not the most skilled solitaire player out there, I know I often make mistakes - in a casino, only the skilled players are likely to want to play, reducing the casino’s take further. Of course, in the computer age I see no reason why the casino couldn’t put the game on screen, and perhaps they do.

Anyway, the only concrete answer I have for the OP is to their first question, which is about 12% for me - and I think with a little concentration it might be improved to about 15%, but beyond that, with the ruleset mentioned above, would probably be impossible. And I certainly came across many layouts that were unwinnable even with knowledge of where the hidden cards were (some versions allowed you to undo, and/or replay the same deal), because (say) both black sixes were at the bottom of a stack that you could never get to, due to the distribution of the other cards. Or even that none of the top cards you are dealt can be used on the layout, so you never make progress.

I don’t know what the other answers are, but I’d be surprised if someone somewhere hasn’t calculated them.

There appears to be a website where you can gamble on Solitaire with real money. I’m not linking it because I have no idea if that falls afoul of rules, but it’s easy enough to find.

I haven’t been able to find specific payout info, but it looks like it’s non-linear. That is, it’s not just $x per card in the final piles, it’s some sliding scale payout with a big bonus for actual completion. An interesting question would be whether their payout amounts are calculated to provide a profit against the average player or against an optimal AI player.

Thanks. I would think the answer to the last question would have to be the latter, otherwise someone would probably build one and take them for all they could. There’s probably not a huge difference between a competent human player and a good AI in Solitaire, so it’s not like they’d have to make the game ridiculously hard for human players to achieve this.

Such things have happened. I agree that there’s probably little difference between an optimal AI and a good human.

As to any single game, yes. A good human will play real close to optimal.

IMO …

But that’s hard to sustain over dozens or hundreds of games. Over that longer term, the human will play X% less well than the optimal AI. Which really need not be strictly “AI”; given Klondike’s simple nature and small amount of visible info, it’s just an algorithm, more akin to that for playing optimal poker or blackjack than Chess or Go. The X may not be a huge number, 5 or 10, but it won’t be zero either.

So the vig / payouts / odds / buy-in / whatever needs to be set high enough to defeat the algorithms, which means it’ll beat the tar off even skilled human play if the humans are trying to grind out a living playing lots of games.

Obviously one of the most important features the casino needs to implement is something that slows the allowed rate of play to human levels. If a machine opponent was allowed to play, e.g., 1000 moves per second, if the casino ever did have a glitch they’d be cleaned out quickly.

That rate limiting needs to be implemented across both breadth and depth. Playing 1000 games simultaneously at 1 move per second and playing 1 game at 1000 moves per second are roughly the same effort for the opponent and roughly the same risk for the casino.

Interesting stuff.