Yut revisited: strategy musings

After playing many rounds of Yut on different apps, I see some interesting patterns.  I only found one article which tried to do a game theory analysis of it, and it broke it down into a one-dimensional line of 21 spaces, saying that the addition of the decision points on the corners makes the math too complex.  

The main goal of the article was to see if always bumping was a better winning strategy than always stacking ... only to find less than 1% difference in the outcomes.  So, does that mean that the rules are very well balanced, or that random factors outweigh any strategy?  Maybe those two cancel out, and the choice of paths is more tactical.  How much weaker would bumping a piece be if you didn't get that extra roll?  One of the apps has an AI where it is likely to run one piece all the way around the board before bringing in other pieces, even if it ends up on the long path, which as you play it is clearly a terrible strategy, easily beaten.  But what IS the best strategy?  And can there be a real strategy when there are potentially huge open-ended rolls that can outweigh 6 or 8 regular moves?

So, it's a simple, casual game that has enough complexity or surprises to keep it fun.  Although I did print out some paper boards and made some yut sticks out of twigs from the yard, it is just so convenient testing the game variations in apps that we simply don't play the physical version.  However, the apps do tend to lull the players into a routine, where key moves get spoiled by not thinking about them for a few seconds before clicking.

The details which make the game worth playing: the varying paths around the board with their critical corners where you can switch paths, the option to stack your pieces and move them together at the risk of having them all bumped together, and the open ended die rolls where you roll the all first then choose how to use the numbers.  The optimal path has 12 spaces from start to finish, but if you have to go all the way around, it is 21 spaces.  Most of the rolls and moves are trivial, but there are a few that can change the course of the game; these are usually those multiple rolls where more decisions are available.  In some, you miss an opportunity to bump other players off the board.  In others you might have two die rolls to use but you are so focused on bumping a piece with one roll that the other roll leaves your stack of three on the long path, costing you many moves later on.

The most common roll is a two, as the overall odds are the same as flipping four coins: the number of ways of getting 0(5),1,2,3,4 heads is 1, 4, 6, 4, 1 (the good old binomial theorem), and with a total of 16 outcomes the odds are: 1 or 3 = 1/4, 2 = 3/8, 4 or 5 = 1/16.  For a "back yut" roll worth -1, that is represented by one yut stick having the special symbol on the flat side, so in that case, the odds of rolling a -1 is 1/16, bringing the odds of a 1 down to 3/16.  One app had options to have 1, 2 or 3 back yuts, so you would adjust the odds accordingly.

Since you get additional rolls on a 4 or a 5, the actual totals vary more than you might forst expect.  In fact, an initial roll of 5 5 5 3 4 -- although very rare -- could score three pieces all the way around in one move -- stack the three 5s in the top right corner, move them to the center with the 3, bring them home with the 4.  The open-ended roll is a good equalizer.  Likewise, a 5 3 4 can bring one piece all the way home in one move, which I have actually done.  I have been in many games where you are a piece or two ahead, or maybe up by 10 spaces and feeling confident, only to have the opponent roll a combo which gives them 10 or 15 moves AND bumps one of your pieces back home.

If you're playing with the optional -1 roll ("back yut"), it seems like it would always be a bad thing.  But if you overshoot one of the turning points by one space, that -1 lets you back up and hit that space.  As I joked one time last night, "Oh, I missed the off ramp.  So I backed up and got it."  Also, if you are on space 1 and get a -1, two of the apps let you back up to space 0, and then hit the finish line with any regular roll on your next turn.  I could not find whether this is a bug or a known game option.  I'm sure it has come up millions of time over the centuries of actual games.  Also, if you are worried about the piece right behind you, it can be pretty comical to get an unexpected -1 and kick it off the board.

Just now, we had a game where Anne had pieces at 3 and 4, and I rolled a 5 and -1.  So I put a fresh piece on 5, moved it back one to bump off her piece on 4, after which I rolled another -1 and bumped her piece on three as well, finally rolling a two and ending up cleanly on the corner number 5 space.  Fun stuff.




In the opening, those first five spaces are a real battle, where many pieces get bumped.  I ran some 6 player games in one app and it was almost unplayable because it was so hard to survive the first few moves without being sent back home.  Obviously, space 5 is critical, since if you land there by exact count you can choose the short path.  Also, if there is a piece behind you and you start down the short path, you are no longer threatened by the piece behind you, since it would have to reach space five then come after you again on the short path on its next turn, giving you an extra turn to get out of its range, if it can hit the corner at all.

Once a few pieces have been scored, the game takes on a different feel of trying to get on optimal paths and racing to the end.  At this point, expect about half of the pieces would be past space 5, so there are fewer captures, and usually one or two stacked pieces in the mix.  

What would be the weight/value of each move?  This would be instrumental in programming an AI to make the best choices.  Let's go with 1 point per space moved ... I would weigh space 5 as a 9 and space 10 and 23 as a 4 in terms of their importance for skipping longer routes.  Keeping in mind that the above article found the effect of always stacking and always bumping the same, let's think about these:

- the actual harm done to an opponent by bumping would be the number of pieces bumped times the number of spaces of progress they have made.  Having two pieces bumped from space two only eliminates 4 spaces worth of progress, but boy, I had a stack of three bumped from two spaces before they would have reached home, and it more or less negated my entire game.  Since it's basically a racing game, the spaces lost by the opponent should add to the weight of the move.

- the benefit of stacking is harder to pin down, since you can't multiply by future progress along paths not yet taken.  Stacking two pieces on space 4, maybe you could hit space 5 and take the short 7 moves to home, saving the 7 moves the second piece would have taken on its own.  But if you missed space 5 and 10 and had to go the full 16 spaces around, you would have saved 15 spaces.  And you have to subtract the risk of not making it at all due to being bumped.

- yet the article shows no difference in winning percentage between these two moves.

Just out of curiosity, I was wondering how the spaces would be numbered for recording a game.  Probably 1-20 around the outside, then from 5 there would be 21, 22, 23 (center), 24, 25, and from top left to bottom right 26, 27, skip the center, 28 to 29.  As I did when recording Puluc (another race game with captures and throwing sticks), you can put the dice rolled first, then the spaces moved: like "1. [4 1] 4, 4-5" to say you rolled a four and a one, moved one piece to space 4 then from 4 to 5.  Again, this would probably be overkill for actual games, but maybe there is a championship match somewhere, or a Yut club that does record games.  My mental exercises of this sort are all towards AI programming.  If the weights of the moves are not enough, there should be a way of recording games, and an algorithm to weight the moves based on actual master play.

As for the history, the huge variation of stories and concepts would make sense if it started as a casual game, then got tangled up in different levels of divination and philosophizing, with some of those associations bleeding back into the common idea of the game.  I will try to compile a list of all the stories and theories in a future thread.

We had another classic/funny ending this weekend.  I had scored one piece and had a stack of three on the next-to-last space coming down from the center.  Anne had her full family of 4 that went the long way and were now one space from finishing on that other path.  With two being the most common roll, it was likely that the next roll would be the end.  Instead, I rolled a one, moved to the very last space; she rolled a one and sent my whole stack back home.  So it went from being dead even to me being 36 moves behind in a snap.  But the unpredictable bits are what keeps it fun.