In this post I will discuss why Villain doubled after dealing the cards in round 1.
The position in round 1 is this:
There is only one way to get two turnovers. The best play is “fg,bf,bc,jc,jb” giving the following position (I used MS Paintbrush instead of explicitly undoing moves in the Spider Solitaire program).
Now let us compute the chances of improving our minimum guaranteed turnovers after turning the card in Column 10. For simplicity assume each of the 13 ranks are equally likely. Let us assume each rank is worth one happy star if it yields a turnover, 0.5 happy stars if you need the correct suit to get the extra turnover and 0 happy stars if no turnovers regardless of suit. Multiple turnovers are possible. For instance, 1.5 happy stars means you always get one extra turnover, with a chance of a second turnover if you were allowed to call the suit, etc.
Note that if the next card in Column 10 were an Ace or Four then we don’t get an extra turnover since we counterfeited the turnover in column 9. It turns out the only good ranks are Three, Nine, Jack and Queen. That’s only 4 happy stars out of a possible 13. Actually, we can increase that to 4 and a half, since a Jack of Spades gives us a double turnover in columns 8 and 10 to go with our turnover in column 9. Given that we only start with two turnovers, it’s about an even chance we get bad cards in both column 9 and column 10.
The other piece of bad news is there are no “atomic columns”. Note that we were forced to pollute column 6 to guarantee our two turnovers in the first place. If we assign the capital letters A and B to columns 9 and 10 respectively, then it’s hard to find decent plans corresponding to the rest of the alphabet. This also means that we need to turnover every card in column 9 or 10, not just the first to get a fighting chance.
In a nutshell, we have difficult short-term problems and long-term problems to deal with. This is why Villain doubled. If I were in Hero’s shoes, I would have passed the double.
Once upon a time, there lived a Beaver in the Animal Kingdom.
The Beaver had just beat the highest difficulty level of Spider Solitaire – four suits sans undo. He felt he had played well after a difficult start, but it was hard to judge his overall ability at the game. After all, one wins and zero losses does not a large sample size make. And the fact none of his friends displayed any aptitude for the Royal Game certainly didn’t help. So, the Beaver decided to have a chat with his best friend, the Raccoon, who was known for his extensive knowledge of all things mathematics.
“It’s hard to judge your playing strength after one game,” said the Raccoon. “You need to play a large number of games to prove your victory wasn’t just beginner’s luck.”
“Suppose I played 129 games in a row,” replied the Beaver, plucking a three-digit number at random. “Then we can tally up my wins and losses and then we have a much better understanding of where I’m at.”
“Agreed,” replied the Raccoon. “Right now, the only thing we can agree on is you can play a hell of a lot better than I can.”
The Beaver chuckles, and he soon notices Captain Obvious is eager to join in the conversation.
“The only problem is it will take a long time to churn through 129 games,” says Captain Obvious. “Spider GM probably doesn’t wanna hear this but we all have better things to do in our lives than playing the Royal Game all day.”
“True,” says Raccoon. “Very True.”
Hang on, thinks the Raccoon. 129 happens to be a power of two plus one. This has me thinking – what if we can involve powers of two somehow? Let us say some games can be worth more than others. Suppose that each individual game was worth N victory points, where N was a power of two. A series of 129 games is equivalent to “First to 65 wins”. This should speed things up considerably. But Captain Obvious will gleefully point out Spider Solitaire is a game for one player, not two. Hang on (***thinks for a while***) I think I might have something.
“Okay I have an idea,” says Raccoon.
“What is it?” asks the Beaver and Captain Obvious simultaneously.
“Let us pretend Beaver is the protagonist,” says Raccoon. “Only Beaver can move any cards. I am the Antagonist and I am willing Beaver to lose.”
Using a stick, the Raccoon sketches a hypothetical cube with all powers of 2 between 1 and 32.
“Initially, each game is worth 1 Victory Point. If Beaver thinks he has a good position, then he can double the stakes. I must concede 1 VP or agree to play on for 2 VP. Similarly, if I think Beaver has a poor position then I can double the stakes and Beaver has the same choice of refusing or accepting.”
“Sounds interesting,” says Beaver. “But if my game state were really bad, can’t you just double the stakes after every move? That wouldn’t be very interesting”
“That is correct,” replies the Raccoon. “Therefore, I propose another rule: if either side doubles the stakes and the opponent accepts then the opponent has the exclusive right to make the next double.”
“So that means, if I get a poor position, you double, I accept, then I turn the game around, then I can redouble and play for four VP?”
“Quite correct,” replies the Raccoon.
“Wait a minute,” says Captain Obvious. “If first to 65 wins then is it possible to get more than 65 if the doubling cube is more than 1?”
“Yes,” replies the Raccoon. “It doesn’t matter if you’re above 65 or exactly equal to 65. And before you ask, it’s perfectly legit for someone to double near the end of the match regardless of the game state because the math says he has nothing to lose.”
“Just to touch base,” says the rot13(fzneg nff) as he gleefully pokes the rot13(nff) of Captain Obvious, “does that mean only Beaver can moves cards, but both Beaver and Raccoon participate in cube-decisions.”
“That’s correct,” says Raccoon. “Even though I don’t move any cards, I can still participate in evaluating the winning chances of a given game-state. Win-win for everybody since I get a chance to improve my game as well.
This idea proved quite successful, and soon Raccoon was discussing the implications of the doubling cube with his friends, many of whom were also avid mathematicians. They had independently discovered some interesting theory and concepts such as market losers, the Crawford Rule, Jacoby Paradox, Woolsey’s Law for Doubling and so on. Not surprisingly, much of this theory is well-known to expert Backgammon players today.
For the record, the Beaver managed to win 66-42, although that may have been a function of Raccoon’s limited understanding of the Royal Game (and hence sub-optimal decisions with the cube). At least it was a lot better than the 8-65 drubbing that Raccoon received when they reversed the roles of Protagonist/Antagonist. Initially the Raccoon thought the best equaliser for a mediocre player is to play each game at high stakes and hope to get lucky, even if the game state rot13(fhpxrq) since a long match would allow the antagonist to “grind” his way to victory. But the Beaver thought it was better to be aggressive with even marginal advantages – for instance if an intermediate player starts with six guaranteed turnovers or a “good five” then he should immediately double. Then at least he is fighting from a position of strength. If the protagonist thought his chances without a doubling cube were 50-50 then he is probably better off grinding and should hope to win on skill, not luck.
And the less said about Ninja Monkey’s first Match-to-65 and his infamous random move algorithm the better 😊
Well, that didn’t last long. Hero got off to a bad start after Round 1 of Game 1 and decided to go all-in on a lousy hand, reasoning that if Hero’s chances of winning an individual game were less than 50%, then the chances of winning a long match would be rot13(fuvg) regardless.
I can see where Bart/Bug are coming from, but if the “protagonist” believes his chances of winning an individual game are less than 50% I think a better strategy is to be aggressive with even marginal advantages. For instance, Hero could insta-double if the initial game state allowed six guaranteed turnovers or a “good five”. That way, Hero would at least be fighting from a position of strength. If I had to play a 25-point match against Kit Woolsey or Paul Magriel, I would certainly consider a similar strategy – looking for any excuse to double from a position of strength. I would be less sure about accepting/refusing when my opponent doubles, but at least I would avoid any kamikaze redoubles unless the value of the cube is already enough to give opponent a win.
Bart has had a go at analysing the mathematics of an unbalanced match where Hero has, say, a 25% chance of winning a match to 5 and tries to equalise the match to some extent through judicious use of the cube. I will have more to say about this in a future post.
I don’t have much to say about the actual card-play. Hero had a reasonable position after round 0, managing to turn over every face-down card in column 7, albeit without getting the empty column. It’s hard to make bad decisions when we have very few face-up cards, no spaces, and all face-up cards are arranged in descending sequence. But after dealing a less-than-stellar 10 cards in round 1 the situation was already desperate. Rightly or wrongly, Villain immediately doubled the stakes and before we knew it, the number on the D-cube exceeded the number of points required for victory.
When you have a lousy position, the good news is it’s hard to make a mistake because your options are so limited. But we all know what the bad news is. Indeed, I couldn’t find any serious card-play mistakes by Bart or Bug for the entire game. Maybe Hero could have taken an extra turnover in round 3 at the expense of exposing two more Aces, but that’s always easier to say with hindsight. I think it was one of those games we were destined to lose. I’ve had similar games when playing with Microsoft Spider Solitaire and – dare I say it – I have no reason to believe this version is biased.
In the post-mortem, I will discuss in some detail why Villain doubled after we dealt the initial 10 cards in round 1. The TL;DR version basically says “I know from vast experience this is probably not gonna end well”.
Again, another short round. The game state is rather poor – if we don’t get a space in column 5 then there are very few plans corresponding to any letter of the alphabet after ‘A’. And sure enough, we got a dreaded King of Diamonds. At least we managed to clear every face-down card in Column 2 and our Club suit is almost complete. The only bright spot is it’s hard to make a mistake when options are severely limited. Maybe it would have been wiser to decline the initial double and start afresh. At least we know next time if Villain is willing to stake the entire match on a single game then he probably has a very good reason.
With three Eights being dealt simultaneously, I think it would be good to see a few Nines in the next round …
Actual play (3 April, score = 441): eh, ei (Kd) ba, bh, bf (Qh)
Actual play (trivial): bd (3h) gf, jg
Actual play: ???
Spider GM Comments: I’m assuming this completes the round, unless Bart/Bug can justify moves like “gd” or “hg” etc. Also, a rare lapse from IM Bug who forgot to mention the Jack of Clubs in his song lyrics.
Quite an eventful round. Despite the position rot13(fhpxvat), Hero decided to raise the stakes regardless, reasoning that Hero was already in bad shape if the probability of winning an individual hand is significantly less than 50%. With no turning back for either side, it was inevitable that somebody would end up holding a dead 8-cube that has no further effect on the match.
At least Hero managed to avoid the ignominy of not acquiring an empty column for the entire game. The club suit is looking promising and with several Eights unseen, there might be a potential space in Column 5. Still, drawing three Aces when the name of the game is not Canasta was never a good sign. Hero even sacrificed a turnover in column 3 which would have exposed two more Aces. Meanwhile Villain is enjoying the rot13(fuvgfubj), knowing that he has no more cube-decisions to make and can focus entirely on watching Hero suffer.
It’s hard to reach a dead-lost position when there are still 30 cards in the stock, so don’t write off the good guys just yet – especially considering that I’ve seen Hero come back from worse game states. Meanwhile IM Bug continues on his merry busking ways, entertaining his Spider-Solitaire-playing friends with the strains of Eights and Fives (sung to the tune of Lightly Row). And Bug will probably start adding Jacks into the mix as well.
Our first double in the match. I’m not 100% sure if the cube decision of both sides is correct, but we’re here to learn right? 😊 After this game I definitely intend to discuss the rationale behind Villain’s decision to send the cube over.
Hero got a dreadful round with only 2 guaranteed turnovers and failed to improve before being forced to deal the next batch of 10 cards.
Actual play: (Mar 24, score = 484): fg, bf, bc, jc, jb (King of Clubs) if, ic (Two of Clubs)
SpiderGM Comments: Recall there are three basic options: (1) accept, (2) refuse and start a new game (3) refuse but play on, seeing what would have happened at the cost of slowing down the match.
SpiderGM Comments: I swear I didn’t peek at the unseen cards before doubling. That would be impossible since otherwise the score would be less than 475 with a 1-point penalty for every move or undo. In any case, that round didn’t last very long.