Three of the same card on the flop...
- Thread starter Cloud Nine
- Start date
I think it is a fairly general, maybe close to universal, feature of good/interesting games that they have what I call the paper-scissors-stone characteristic, namely that there is no single right/best way to play because your best strategy depends on the strategy chosen by your opponent(s). In other words, no matter what you do, there is some strategy available which will beat it.
-Ww
-Ww
That's pretty immediate, right?
proof
'spose there is an interesting game that had the property that one strategy could not be beaten by any other strategy. All players (maybe we have to restrict to first movers in turn based games) would play this strategy because the worse case scenario is a tie. If it is possible to force a tie against this strategy, all games will end in a tie (tic-tac-toe) otherwise they always end in a loss (that take away coins game). Since each game either ends in a predictable loss or tie, the game is not interesting. Contradiction!
Therefore (Ww's proposition), for every interesting game and every (known*) strategy for playing the game there exists a (known*) counter-strategy which will defeat the first (the Ww criterion).
(The converse, I'm sure, does not hold)
To me an important characteristics of an interesting game are minimal (and generally symetric) rules that none-the-less are sufficient to establish the Ww criterion.
* - If someone were to figure out a theoretically unbeatable chess strategy, I betcha interest would drop off quickly.
proof
'spose there is an interesting game that had the property that one strategy could not be beaten by any other strategy. All players (maybe we have to restrict to first movers in turn based games) would play this strategy because the worse case scenario is a tie. If it is possible to force a tie against this strategy, all games will end in a tie (tic-tac-toe) otherwise they always end in a loss (that take away coins game). Since each game either ends in a predictable loss or tie, the game is not interesting. Contradiction!
Therefore (Ww's proposition), for every interesting game and every (known*) strategy for playing the game there exists a (known*) counter-strategy which will defeat the first (the Ww criterion).
(The converse, I'm sure, does not hold)
To me an important characteristics of an interesting game are minimal (and generally symetric) rules that none-the-less are sufficient to establish the Ww criterion.
* - If someone were to figure out a theoretically unbeatable chess strategy, I betcha interest would drop off quickly.
Right, at least for games whose outcome is determined entirely by the strategy or "moves" chosen by the players.
People often find games in which other factors play a major role quite interesting/entertaining, even if optimum strategies exist and are well known. Examples would include gambling games, where random events influence the outcome, and sports, where skill/ability in executing chosen strategies are very important. In fact, in gambling people very often knowingly choose non-optimum strategies in order to increase the importance of random factors. An obvious comment, I know, but when did that ever stop me.
Such an optimum chess strategy is certain to exist (its existence can be proven, in other words) although it may well never be discovered and quite possibly could never be learned by a human being if it were.
-Ww
People often find games in which other factors play a major role quite interesting/entertaining, even if optimum strategies exist and are well known. Examples would include gambling games, where random events influence the outcome, and sports, where skill/ability in executing chosen strategies are very important. In fact, in gambling people very often knowingly choose non-optimum strategies in order to increase the importance of random factors. An obvious comment, I know, but when did that ever stop me.
Such an optimum chess strategy is certain to exist (its existence can be proven, in other words) although it may well never be discovered and quite possibly could never be learned by a human being if it were.
-Ww
Re Chance: I'm not sure if there is a provable 'optimal' strategy in the sense that you defined* in games of chance
* - Namely a strategy that when adopted can not be beaten
Re Chess: I've never seen such a 'proof' for the existence of optimal strategy. If such a thing exists I suppose it follows from finiteness of moves.
Makes you remember why turn of the century guys were so down on existence proofs.
* - Namely a strategy that when adopted can not be beaten
Re Chess: I've never seen such a 'proof' for the existence of optimal strategy. If such a thing exists I suppose it follows from finiteness of moves.
Makes you remember why turn of the century guys were so down on existence proofs.
I think it would be hard (read impossible*) to prove that a specific strategy was optimal, but it might be easier to prove that there exists an optimal stratgey (although it might not be unique).
I think I can do it if we allow an additional 'cieling rule' that all chess games must be decided or declared a draw before a certain finite number of moves have been completed.
The cieling rule isn't 'official', but it almost is, defacto. Among human players there certain time realities that enter into the equation. I doubt any chess game has gone on for more than 10,000 moves.
And there are existing cieling-type rules such as endgame limiting rules, rules about repeated sequences of moves, and then the clock - so it's not entirely out of the spirit of chess competition.
* - Mostly because chess, as a game, is really inelegant. The rules are far too complicated.
I think I can do it if we allow an additional 'cieling rule' that all chess games must be decided or declared a draw before a certain finite number of moves have been completed.
The cieling rule isn't 'official', but it almost is, defacto. Among human players there certain time realities that enter into the equation. I doubt any chess game has gone on for more than 10,000 moves.
And there are existing cieling-type rules such as endgame limiting rules, rules about repeated sequences of moves, and then the clock - so it's not entirely out of the spirit of chess competition.
* - Mostly because chess, as a game, is really inelegant. The rules are far too complicated.
Zermelo's theorem: optimal chess strategy
See
http://www.math.ucla.edu/~blasius/167.1.02s/handout1.html
for the proof (Google is god-like!). Note the corollary for chess.
Many chess players feel that (ii) is the correct answer on intuitive grounds, but it is basically a guess.
Note also that this proof does not involve the details of the rules of chess very much but only the fact that it is one of a broad class of games (and possible games) that can be represented by a finite "decision tree" with certain properties, etc...just as jm surmised.
Also, this proof has relatively little to do with how computers actually play chess today (though they do search and evaluate such decision trees).
There is a long standing jest in the chess world based on Zermelo's theorem to the effect that the ultimate "chess problem" (like those sometimes published in newspapers) should show the starting set-up for a chess game and say "White to mate in 73 moves" (or however many).
-Ww
See
http://www.math.ucla.edu/~blasius/167.1.02s/handout1.html
for the proof (Google is god-like!). Note the corollary for chess.
Many chess players feel that (ii) is the correct answer on intuitive grounds, but it is basically a guess.
Note also that this proof does not involve the details of the rules of chess very much but only the fact that it is one of a broad class of games (and possible games) that can be represented by a finite "decision tree" with certain properties, etc...just as jm surmised.
Also, this proof has relatively little to do with how computers actually play chess today (though they do search and evaluate such decision trees).
There is a long standing jest in the chess world based on Zermelo's theorem to the effect that the ultimate "chess problem" (like those sometimes published in newspapers) should show the starting set-up for a chess game and say "White to mate in 73 moves" (or however many).
-Ww
Right, you have to replace "cannot be beaten" with something like "cannot be beaten on average", in other words a strategy that has the highest achievable expectation value of all available. However, in games of chance, you will frequently see people knowingly adopt a sub-optimal strategy in order to increase the influence of chance events...presumably because they find the operation of chance to be engaging/interesting/entertaining in its own right.
-Ww
-Ww
In the (not too profound/interesting) sense that there is no single, well defined expectation value for any strategy because people value the outcome by a complex criteria that trades off monetary reward with unpredictablility and thus excitement/drama in some complicated way that varies from one individual to another, I guess you mean. Right?
Anyway, ain't Zermelo's theorem way kewl?
-Ww
Anyway, ain't Zermelo's theorem way kewl?
-Ww
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By "nervous", do you mean that you suspect/worry that it may be possible to "prove" false theorems via transfinite induction?
But in any case, high (official tournament) level chess games are usually subject to explicit ceiling rules (e.g., a stalemate is declared if neither side captures in pieces in a sequence of N consecutive moves, where N is usuallyl something like 50 or a 100), in addition to the implicit limits imposed by practical realities like finite human or cosmic lifetimes.
In any case, if the only escape from Zermelo's theorem in chess were to play a game with a transfinite number of moves, it would make chess less, not more, interesting I'd think.
-Ww
But in any case, high (official tournament) level chess games are usually subject to explicit ceiling rules (e.g., a stalemate is declared if neither side captures in pieces in a sequence of N consecutive moves, where N is usuallyl something like 50 or a 100), in addition to the implicit limits imposed by practical realities like finite human or cosmic lifetimes.
In any case, if the only escape from Zermelo's theorem in chess were to play a game with a transfinite number of moves, it would make chess less, not more, interesting I'd think.
-Ww
Therorem? more like Zermelo's Tautology.
"Corollary. For chess, one of the following three possibilities holds:
(i) White has a strategy which always wins.
(ii) White has a strategy which always at least draws, but no strategy as in (i).
(iii) Black has a strategy which always wins.
No one knows which of the above is actually the case! "
In other words "we really can't say anything about this".
"Corollary. For chess, one of the following three possibilities holds:
(i) White has a strategy which always wins.
(ii) White has a strategy which always at least draws, but no strategy as in (i).
(iii) Black has a strategy which always wins.
No one knows which of the above is actually the case! "
In other words "we really can't say anything about this".