Quintessential
A game is quintessential if it presents the most basic implementation of a particular mechanism and theme. 'Group penalty' is an elusive theme with a terrible name. In its previous incarnations - Star et al - it was a mix of static and dynamic connection, static in the sense of connecting to the edges, dynamic in the connecting of groups. Symple, instead of counting the number of edge-cells a group touches, simply counts the size of the group. Thus it replaces the static connection theme by 'territory' (i.e. groupsize), while the dynamic connection theme remains intact.
The Symple move protocol
Say you have to fill half a Go board, some 180 points, with stones, in a number of turns, following this procedure:
- Put one stone on a vacant point, not connected to a like colored group, thereby creating a new group, or ...
- ... grow any or all groups already on the board by one stone (a single stone being a group by definition).
What's the fastest way?
| # single placement | # grow all | # points | # turns |
| 1 | 179 | 180 | 180 |
| 2 | 89 | 180 | 91 |
| 3 | 59 | 180 | 62 |
| ... | ... | ... | ... |
| 45 | 3 | 180 | 48 |
| 60 | 2 | 180 | 62 |
| 90 | 1 | 180 | 91 |
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| # single placement | # grow all | # points | # turns |
| 11 | 16 | 187 | 27 |
| 12 | 14 | 180 | 26 |
| 13 | 13 | 182 | 26 |
| 14 | 12 | 182 | 26 |
| 15 | 11 | 180 | 26 |
| 16 | 11 | 192 | 27 |
| 17 | 10 | 187 | 27 |
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Clearly the moment to start growing lies, and comfortably broad for that matter, around the root of the number of points required.
The first move advantage
White moves first and has a clear advantage: If he grows first, he has the initiative with the same number of groups. If black grows first, white can follow suit with one group more. In the course of the next 10 to 15 turns this amounts to 10 to 15 points extra, which is more than black can get out of his initiative.
The balancing mechanism: trading move order advantage for limited growth
Move order advantage constitutes a basic a-symmetry in almost any two player abstract game. It may be taken for granted (Chess, Draughts), compensated for (Go), or negotiated by means of a pie (Hex).
Of these, the compensation given in Go is most pragmatic: a means to an end based on concensus, with a certain disregard for style. Taking it for granted is no option in Symple and a pie isn't applicable: there may be better and worse opening moves, but is there one so bad as to be rejected? The first move still gives the advantages mentioned above: either the initiative in growing, or an extra group.
"If the system is sound, the rule will be there". That's a deep truth I've relied on throughout my career as a game inventor. Symple's balancing rule is another clear example. It uses its own mechanism to negotiate the advantage. Normally players on their turn may either start a new group or grow all their groups present on the board. Here's the conditional exception:
- If, and only if, neither player has grown yet, then black may use both the above options, growth and placement, in the same turn.
So both players have their finger on the switch in terms of trading the move order advantage against the growth of a limited number of groups by the opponent. Let's first look at this from white's position:
- If he grows on his second move, he will have one group of two stones and black will have one stone, black to move. For black this is almost as good as having the first move without any compensation for his opponent. So white must wait if he wants his compensation for terminating black's prerogative to grow. But how long?
Now let's first look at this from black's position:
- If he uses both options on his second move, he will have one stone and one group of two stones and white will have two stones, white to move. For white this is almost as good as having the first move without any compensation for his opponent. So black must wait if he wants the advantage of his prerogative to grow. But how long?
See the beauty?
Setting the group penalty
Symple's theme is 'group penalty', so its hardly amazing that the extend of the penalty should matter. If Symple is to be drawless, P must be even. But how large should P be?
That's up to the players themselves. The applet allows P to be set fom 2-12 and keeps track of resulting groupvalues during a game, displaying the total for each player. The score will initially be negative for both players and even may remain negative, depending on the value of P and the number of groups in the final position.
The value of P and its effect on tension
The amount of 'tension' in a game of Symple can be set by changing the value of P. The main strategic idea is that groups should grow and cooperate to carve out pockets of vacant territory that are safe from voluntary invasion. Strategic advantage lies not only in the number of stones, but also in the territory they manage to secure.
The question regarding placement or growth is: what can a new group add to the score. It starts out at a value 1-P and must grow to a number equal to half the penalty to become 'neutral'. If the group penalty is set at 6, then a group of three stones will have a value of '-3'. At the same time it takes 3 points from the opponent (because if you don't take it, he will). So creating a new groups would seem to be disadvantageous or at most useless if it cannot grow beyond half the penalty. Set the penalty at 2 and you can always invade at no cost. But set it at 10 and you must grow it to 5 to 'play even'. Fail to do so and the group will cost you more than it delivers.
The higher the group penalty, the higher the risk of creating new groups, the higher the tension, especially in the endgame, where either or both players may be forced to create new groups in the absence of growing options.
Yet more sliding: a gradual reversal of objective
The initial arguments concerning creating a new group or growing, all implicitly go from the premiss that more separate groups and thus more opportunities for growth are good. And that goes a long way, but not all the way. Reducing the number of groups, which can be done by connecting them, has two consequences:
- The options for growth are reduced by the same number.
- The penalty is reduced with P times that number.
At some point the penalty reduction obtained by connecting will become higher than the growth potential of the individual groups. This is another 'sliding' process that should be taken into account towards the endgame: keep options to connect open and/or prevent the opponent to do so.
Finite and drawless - so who has the advantage?
Symple is a finite abstract perfect-information zero-sum game and as such completely determined. That means that the truth of any position - in this case a white win or a black win - is locked in the gametree. To determine which of the two it is, it might as well be locked in a black hole.
Hex, without the pie rule, is a proven win for the first player. Checkers and Awari are proven draws. In Chess you can't prove anything, but the arguments that white has an 'advantage' (note the quotes) are questioned by few. Few, too, doubt that Draughts is a determined draw, although it is not proven.
What makes Symple different is that you can't even argue one way or the other, because as long as no growth has taken place, both players have the option to trade move order advantage against limited growth. Given the sheer size of the number of positions to consider - all possible positions up to the first player growing - there will never be any extensive exploration of one particular opening.
Is Symple a great game?
"Not according to J. Mark Thompson's criteria in his leading article Defining the abstract - criteria I happen to agree with" I wrote in a previous version of this article, and I based my judgement on the fact that "it has no capture and thus misses the drama associated with it.".
That was wrong. It was based on my assumption that Symple was a fairly 'hot' game: you'd always want to move as long as the score could be increased. So I had allowed for a pass as a legal 'move' for the moment this no longer happened to be the case. Invasions were volontary, and if unprospective one could simply leave vacant territory surrounded by the opponent to him.
It was this unobtrusive mail by Luis Bolaños Mures, the inventor of Yodd and Xodd that pointed to a problem:
"One rules question, though: is passing allowed? I'm just asking because I've seen some passes played in the recorded games, even though the rules don't seem to allow it. Of course, if passing is allowed, trivial draws are possible, so I guess it isn't."
Players could legally agree to a draw by passing at an equal count. That's not exactly in the game's spirit, but neither is it much of a problem: don't allow a pass at an even score.
But I routinely considered compulsory movement in the sense that a player must (instead of 'may') either place an isolated single, or grow all of his live groups. To my surprise this minute change turned out to have deep consequences for the endgame. Whereas the main consideration regarding invasions used to be whether they could be advantageous, they now should be regarded in terms of whether they could be forced. If the board fills up, there may come a point where it has become impossible to grow because all a player's groups are fully enclosed. In that case, instead of simply leaving vacant territory held by the opponent to him, the player is now forced to invade, and be penalized for it. In other words, where Symple used to suffer from a a certain lack of drama, compulsory placement turns this around in a rather dramatic fashion, with a sharp increase of tension towards the endgame (in a balanced game of course - unbalanced implicitly games aren't interesting in terms of 'tension').
So where I've always argued that Symple lacked the drama associated with the really great games, this minute change would appear to increase that drama. One shouldn't make too much of it though: being forced to invade makes a difference of 2-P (it costs the opponent 1 point, it costs the player 1-P points) and the placed stone may well have growing options that bring it to 'neutral' or more. On the other hand an implicit tactics like connecting 2 groups brings P points and may cost the opponent P points on top of that, if the connection happens to be a simultaneous cut. Connecting and cutting still seem to have have more impact than being forced to invade.
Is Symple a programmable?
Symple's move protocol brings with it some new obstacles regarding programmability:
- The choice between placing a single stone and growing all groups doesn't align smoothly with a search based on random play-outs.
- The branch density isn't of this world. Positions would likely have to be divided in local sub-sections to counter that aspect.
- Balanced games become increasingly tense in the endgame, if and when connections and the forced creation of groups become an issue. Humans can plan ahead towards that stage, bots cannot.
The parameter P affects not so much the nature of a Monte-Carlo evaluation but the spreading of the initial moves.
We promote Symple as 'the next AI challenge' for abstract games because of its simplicity. AI programmers should be provided with essential games, and Symple is quintessential. So we think Symple is the perfect game to consider for the abstract games AI community in the years to come. Because if they can't crack it, humans can do something computers can't, and the question "how do humans do it?" remains a mystery.
Some preliminary thoughts by an expert
Allow us to cite some preliminary thoughts on the subject by Timo Ewalds, programmer of the Castro Havannah bot, whose MSc thesis at the University of Alberta is titled Playing and Solving Havannah:
It is likely quite a bit harder than Havannah due to the large amount of moves. It's not quite clear yet how to represent multi-moves like this. Even still, it is quite doable to make the programs only consider a smaller subset of the moves. The true branching factor is something like the factorial of the number of groups, but if each can be considered independently, that makes it much smaller. Even if neighbouring pairs in a Voronoi diagram need to be considered, that's still less than N2, which is big but not absurdly huge. Go uses quite a few learned patterns to suggest good moves, as most of them are bad at any given time. There isn't a large set of games to learn from here, but I'd guess that patterns will work similarly well in Symple.
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Voronoi diagrams are not that slow to generate, and while they wouldn't be very accurate, they would give a fairly good approximation. They would show which groups are next to each other and may be worth joining and suggesting which empty areas are big and worth attacking.
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Basic UCT only uses the outcome of the random game to add experience to the tree. RAVE is based on the realisation that the win is made up of good moves even if they were chosen randomly, and so gives a bonus to making those winning moves earlier in the tree. It is good at finding moves that are good on average and encouraging them to be explored earlier. This works great in Go and Havannah and other games where moves made later on are valid earlier, and the order in which they are made has little relevance.
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There certainly are many general purpose game playing algorithms, but it'll be a long time before they are good enough to play at human level on these harder games like Go, Havannah, Symple or Sygo. To be fair though, humans don't use a general algorithm either. We also learn game specific strategies, tactics and patterns. There are few programs that continually learn through playing more games the way humans do, but that may come one day too.
Here's the (online) story of Symple's discovery, as recorded shortly after.
Symple © MindSports and Benedikt Rosenau
Java applet © Ed van Zon
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