A game is essential if it presents the most basic implementation of a particular combination of mechanism and objective. '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
A 'group' consists of orthogonally connected like colored stones, a single stone being a group by definition. Now say you have to fill half a Go board, some 180 points, with stones, in a number of turns, following this protocol:
- Put one stone on a vacant point, not connected to a like colored group, thereby creating a new group, or ...
- ... grow all groups already on the board by one stone.
What then is the fastest way?
- If you place 12 singles and grow them 14 times you get to 180 in 26 turns.
- If you place 15 singles and grow them 11 times you get to 180 in 26 turns.
Placing 13 or 14 gives similar results, anything outside that range is increasingly slower.
In this example, groups are considered to remain isolated. This is indeed the strategy in the initial stages of the game and the middle game, but eventually things change. At some point, depending on the height of the penalty, connecting may win more by reducing one's penalty and/or increasing the opponent's, than it loses from the reduction of growing options. Another reason for slowing down growth is to avoid forced placements inside the opponent's territory, and the inherent penalty. Game play so far suggests that from 10 initial placements onwards, growth becomes tempting. It is a crude approximation though, and specific positions may well suggest a different approach.
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 turn order advantage for limited growth
Turn order advantage constitutes a basic a-symmetry in almost any two player abstract game. It may be taken for granted (Chess, Draughts), compensated for (komi in Go), or negotiated by means of a swap (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 swap 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 one conditional exception:
- If, and only if, neither player has grown yet, then black may use both the above options, growth followed by placement, in the same turn.
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 on the board 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 compensation for terminating Black's prerogative. But how long?
Now let's look at this from Black's position:
- If he grows and places a single 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?
Thus both players have their finger on the switch in terms of trading the turn order advantage against the growth of a limited number of groups by the opponent. See the beauty?
Setting the group penalty
Symple's theme includes '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.
In the initial stages, a higher penalty may lead to placements that are nearer to the center, for two reasons:
- Closer to the center means closer to each other and more chances to connect.
- Closer to the center means that regions along the edge are accordingly larger, but then, they can be because a high penalty makes invasions less attractive.
This conditional shift of the tension area is a general idea that does not, of course, stand in the way of any specific game's tactics to push it in a different direction. Symple has no trouble accommodating players who try to push the envelope to carve out new strategies.
Placement or growth
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 even play even. 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. But to do so the player must also forfeit a turn of growth. So creating a new group would seem to be disadvantageous or at most useless if it cannot grow large enough to add positively to the endscore. Set the penalty at 2 and you can always invade at no other cost than forfeiting growth. Which, I remind you, may be involuntary, so if it is set at 10 and you're out of growing options and thus forced to place a single, then you're in trouble and the new 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 a number equal to the reduction in groups.
- 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. For all practical purposes it might as well be locked in a black hole.
Hex, without the swap rule, is a proven win for the first player. Checkers and Awari are proven draws. In Chess you can't prove anything yet, but the arguments that White has an 'advantage' are questioned by few. The truth however is that it is either a decision or a draw and no such thing as an 'advantage' exists in the game tree. 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 turn 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 most likely never be any extensive exploration of one particular opening.
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 the Monte Carlo method.
- The branch density isn't of this world. Positions would likely have to be divided in local sub-sections to counter that aspect.
Regarding a deterministic approach with alpha-beta pruning, the finding of an effective evaluation that keeps the height of the penalty into account may prove no easy task. For a Monte-Carlo approach this should be less of a hurdle, but the multi move character may make the method harder to implement.
At the same time, huge advances have been made in programming abstract strategy games. New games that are made 'inherently hard to program', will be inherently hard to master for humans too. If the AI world sees a challenge worth pursuing, and it is no longer guaranteed that programming an abstract would at all qualify for that, then it will most likely succeed.
Symple © MindSports and Benedikt Rosenau
Java applet © Ed van Zon