Game Theory Interface
This is a page trying to smooth the path of go players who would like access to the combinatorial game theory (rather than those coming in the opposite direction).
Table of contents |
Minimax play
This corresponds to what go players would recognise as proper or honest play (effectively honte).^{[1]} Both sides do the best not to make mistakes that can be punished by the other side (as measured by the final score): so no overplays.
This is taken for granted in game theory in general, unless players are consciously applying random strategies. This connects with one attitude for handicap play in go: White may assume that Black is (unconsciously) applying a random strategy rather than best play.
In a combinatorial game with perfect information, how can it possibly be good for a player to consciously apply random strategies? - Migeru
If there is no pure strategy available a player must choose from several plays. This could mean, for example, choosing between three strategies in propotions of 1:1:1, or 33.3% chance of choosing each strategy. For 1:1 a coin flip would do. erislover
That's not really the point. When players are of different levels, the weaker player does take certain decisions in a way that appears random to the stronger player. Charles Matthews
That is clear, I'm referring to the qualification "unless players are consciously applying random strategies". - Migeru
Randomness is useful in some games. I think that most examples of this would be games with incomplete information. In poker, a certain amount of randomness introduced into a bluffing strategy can be effective. The trick is to be random enough, but not too random. I don't think that the random play is useful when applied to go though. Tyler
See also Rock-Paper-Scissors, where you try to guess your opponent's move. If you play randomly, your opponent can't guess, and you (on average) come to a draw. In the rock-paper-scissors programming competition, it was a common strategy for programs to play a certain amount of their 1000-move match as random moves to throw off the opponent's analysis, while gaining useful insight from the responses. Doviende
Minimax play corresponds therefore to seeking 'saddle points' or 'pivots' in pay-offs. It also corresponds to the logicians' idea that games in general model alternation of existential and universal quantifiers ("whatever she played I had a move such that whatever she played I had a move such that ...").
Given a tree-representation of a finite game with numbers as leaves, it is routine to calculate the minimax result. One works from the leaves towards the root, simply choosing the maximum (resp. minimum) of the leaf numbers at each level and making that a new leaf.
See also Minimax.
Ending
In CGT the ending condition at foundational level is ''if you can't play you lose". For example, this applies to the normal version of Nim. This sorts out finite games, but allows infinite games where no one wins. For go players that's like not having a superko rule in operation to force the end of the game.
- Interestingly, this also applies in the ancient Chinese rules. - Migeru
One should be aware, however, that in the translation into CGT terms of go positions, that ending condition is masked. It hasn't gone away: scoring comes down to adding up numbers and within the translation of numbers the same ending condition still operates.
If one pushes too hard on that door, though, one finds one is talking about go with a group tax. This is slightly murky, and, more to the point, not required.
[1] Bill: The concept of honte does not come close to defining perfect play. Plays that are called honte are not bad plays, but are not necessarily best.