Well, it sorta seems like any game where you can reliably force a draw is inherently broken for the purpose of this discussion. So, let's use Hex as the example here.
Why is a forced draw broken but a forced win not broken? What if we calleda draw in chess a "small win for black" instead of a draw? Does the presence or absence of a forced draw somehow affect the skill cap?
If you are playing the optimal strategy as player 1, you can do no better than to play that strategy. There's nothing about a weak opponent that is worth exploiting. A win is a win.
That's true but I think that's peculiar to hex rather than indicative of perfect information games in general. For example consider just being the second player in hex, knowing the equilibrium play won't help you because EVERY strategy is in equilibrium for the second player, however there is still room for the second player to do better than just playing random moves (if they are not playing against an ideal opponent.
Well, I guess it's the same as Checkers, then, were the opt-vs-opt states is a draw, and the only way to win is to play sub-optimally and draw your opponent out into sub-optimal play. In Checkers, you'd be hoping that they would be trying to exploit you for better than a draw, and thus play sub-optimally, and thereby open themselves up to exploitation.
It's not quite the same. Call a strategy co-optimal if no equilibrium strategy "beats" it. In a game with mixed equilibria any pure strategy which is mixed between in an equilibrium strategy is co-optimal. These are the sub-optimal strategies you'd want to play to entice an equilibrium player to dance in donkeyspace, as if they don't they don't get any benefit from your "mistakes". In a game like checkers the co-optimal (but non-equilibrium) strategies still exist but look very different, they are strategies which only go wrong once the opponent has already gone wrong. As a result they can't be used to draw an opponent out into donkeyspace, because unless they go out there voluntarily they'll never know you're playing co-opimtally rather than optimally, and once they do make such a mistake you are best off playing optimally from then on.
]]>If you are playing the optimal strategy as player 1, you can do no better than to play that strategy. There's nothing about a weak opponent that is worth exploiting. A win is a win.
I agree that non-binary win states are part of the picture here.
But with the tribute, two Nash Eq CM players will both lose money. The only way to win money is to draw your opponent out into sub-optimal play through sub-optimal play.
Well, I guess it's the same as Checkers, then, were the opt-vs-opt states is a draw, and the only way to win is to play sub-optimally and draw your opponent out into sub-optimal play. In Checkers, you'd be hoping that they would be trying to exploit you for better than a draw, and thus play sub-optimally, and thereby open themselves up to exploitation.
]]>It's interesting to think about how there's no skill cap in CM as compared to perfect information games.
In those game, a perfect strategy exists, and after you master that, you can get no better. Hypothetically, of course, because for most games beyond tic-tac-toe, the perfect strategy is unknown.
Here, even if you discover and play the Nash equilibrium strategy (and one does exist), you still are not maximizing the amount you win against a non-equilibrium player, so you can STILL get better. It seems like the dance in donkeyspace can recurse into infinite layers of complexity, as you develop a strategy to exploit the strategy that is supposed to exploit the strategy that your opponent thinks you are using, etc.
This picture isn't qutie accurate. Even in a perfect information game a player can still sometimes improve on their equilibrium strategy against a poor player. For example, you pick an equilibrium strategy for checkers. A poor player memorises the main line of your strategy and can get a draw every time, until you change your strategy (maybe even to a non-equilibrium one) where they won't know what they are doing and lose. I'd argue that the reason that poker players for example, focus much more on explouiting weak play than chess, checkers or go players (and these players do look at playing sub-optimally to exploit weak opponents, Emmanuel Lasker was famous for making "psychological moves", and there are opening traps etc.), is the increased granularity of the result. Even in go results are scored generally as win-loss, and there's no benefit to winning by more points, whereas in poker you are given a monetary reward which can be much smaller or larger. As a result a chess player has no need to play differently against a weak player as a win is still a win, might as well assume they're going to see that 10-move winning combination, even though they usually can't see two moves ahead. In poker if you assume your opponent is going to play well then you won't win as much money from them. So in perfect information games there is still room for getting better at exploiting weaker players by playing off equilibrium strategies, just like imperfect information games.
i think the difference between perfect and imperfect information games in this ballpark is that in imperfect information games equilibrium strategies are typically mixed strategies. Games with mixed equilibria do have interesting properties which arguably adds a level of skilll not found in other games. For instance, there's actually no particular strategy against which mixing has any benefit at all (every pure strategy has a calculable EV against any given strategy, so against that strategy just pick any of the pure strategies which maximise your EV against that strategy, no need to mix them). When you mix then it must be because you think your opponent will guess your strategy and counter it, but of course if you can predict what strategy she will guess you have then you can just counter her counter etc. Mixing only makes sense if you think they can model your thought process [including your model of their thought process!] better than you can model them. So picking the equilibrium strategy is sort of acknowledging you can't win the modelling war (in poker they call it a "levelling" war). This modelling war is an element of skill not captured by the nash equilibrium, and cannot be present in perfect information games where both players know an equilibrium strategy.
]]>It's interesting to think about how there's no skill cap in CM as compared to perfect information games.
In those game, a perfect strategy exists, and after you master that, you can get no better. Hypothetically, of course, because for most games beyond tic-tac-toe, the perfect strategy is unknown.
Here, even if you discover and play the Nash equilibrium strategy (and one does exist), you still are not maximizing the amount you win against a non-equilibrium player, so you can STILL get better. It seems like the dance in donkeyspace can recurse into infinite layers of complexity, as you develop a strategy to exploit the strategy that is supposed to exploit the strategy that your opponent thinks you are using, etc.
]]>The only argument for the game to have any aspect of chance is, given two players of significantly high skill levels, does the favour of the board determine the victor?
I believe the answer is no because I believe CM is and ideal skill based game (comparable to chess, go shogi), and there is no cap on the amount of skill one could have.
Post pick-2 forced win scenarios do exist in CM, these aren’t invisible to the opposing player if they can deduce the possible outcomes (of the game) and know the odds of you having one (of the forced wins).
Now Im curious, In boards containing post turn-2 forced win scenarios are the potential games resulting in these (% of 720) equal for both players? (does one player have a higher chance of getting a forced win?)
]]>So, in theory, the coin flip would make the game subject to chance, if both players knew the perfect strategy that we know exists. In that case, whoever played first would use that perfect strategy and win.
In practice, because the perfect strategy is unknown, the coin flip does not make the game subject to chance.
]]>frank wrote:(I haven't tried the game yet because I'm a tad fearful of putting my CC info into a portable app. Sorry if my suggestion doesn't actually make sense for it.)
This thread might help put you at ease:
Yes, thanks, Dan. That was an interesting read, to boot.
@jason: Yeah, it is a little overly complex; I was only thinking about the legal question (and maybe a little bit about shoehorning in auctions, because they are cool).
My reading of the legal quotes I've seen in this thread is that, yes, the procedure
1) Pay stake
2) Flip coin to play first
3) Play Hex to win the pot
*would* constitute a game subject to chance. The procedure sounds very much like gambling, since step 3 degenerates to "first player wins" (or should, based on the strategy-stealing argument you mention, anyway).
This procedure is analogous to what happens in your game, right? You pay before you see the random handicap you've been given. It's not the asymmetry that is the problem per se, but that the players have paid in before seeing it.
It's pretty clear that the layout of the board that may favor one player is not a significant factor in the outcome of the game. On a given round, the layout of the board is not going to tilt the game in the favor of a beginner over an expert.
You could permit this randomness into your game, yeah, but then you'd have to fight the prosecutor's claim that it is a "significant" factor. It may be pretty clearly not, but who knows how that can be shown to a judge's satisfaction. You have to show that it's like flipping a coin to play white in chess. If you think you can do it, then I guess you're on pretty secure footing against the line of legal attack we're talking about in this thread.
The board-twice-in-a-row thing could also do the trick, as you say, so long as the second round is always reached and is arguably just as valuable no matter what happened in the first round. Again, an argument is required. I think rearranging the sequence of events so that players can't be construed as paying for a chance (i.e., something determined by "Nature") at a favorable handicap would put you on unassailable footing, but it might entail compromising your vision for the game...
Anyway, thanks for inviting me to try the game! I've just played my first match and look forward to trying it out some more.
]]>Therefore, even if chance is not the predominant factor in the outcome of a game, but was still a significant factor, the game could still be deemed to be a "game subject to chance" under a plain reading of the Act.
See page 19 here:
http://www.scribd.com/doc/7916861/UIGEA … g-11-12-08
It's pretty clear that the layout of the board that may favor one player is not a significant factor in the outcome of the game. On a given round, the layout of the board is not going to tilt the game in the favor of a beginner over an expert.
In Poker, the beginner could get dealt AA and flop AA4. Thus, chance is a significant factor in Poker.
]]>Yeah, there's no proof that chess is a forced win for white. That's my hunch, though. We have non-constructive proofs for games like Hex through the strategy-stealing argument. Of course, Hex cannot end in a draw. That's sort of a fundamental flaw of Chess, though, eh? But maybe that's part of its beauty...
But anyway, we'd have to chose player 1 in Hex through some mechanism, and that player can force a win. If we flipped a coin to pick who plays first, I don't think that would make the game subject to chance.
]]>(I haven't tried the game yet because I'm a tad fearful of putting my CC info into a portable app. Sorry if my suggestion doesn't actually make sense for it.)
This thread might help put you at ease:
]]>Just issuing a new board each round gives the advantage to both players in a way that approaches a perfect 50/50 split in the limit.
Yes but so long as you don't pick out some players for a special advantage some way this will be true of any random element.
Isn't chess a forced win for white?
We're not likely to know any time soon. We don't even know that black doesn't have a forced win, though most players assume that they don't (see this short piece by a friend of mine giving some thoughts on it if you're interested: http://www.csse.monash.edu.au/~dld/Publ … Q_p158.jpg, http://www.csse.monash.edu.au/~dld/Publ … _p159.jpg). My money is is on chess being a draw though, white scores about 54-56% at the highest level which is much closer than to 50% than 100%, the vast majority of theorists think that and it generally feels much more like mistakes were made when black loses then when there's a draw. It's very rare to see a game where black lost but it seems they couldn't have improved their play but this is fairly common with drawn games. This Wikipedia page has lots of interesting stuff on this http://en.wikipedia.org/wiki/First-move … e_in_chess. The chess example is still good though. White seems to have an inherent advantage but chess would surely be a paradigm example of a pure skill game. In tournaments though there are always some systems in place to mitigate this effect and it generally involves very little randomness.
]]>1) Have players agree on and pay initial stakes.
2) Show both players the board.
3) Each player chooses a preferred side (rows or columns) and how much they are willing to pay to play on it.
4) The player with the higher bid gets their preferred side and adds the lower bid to the pot.
5) Play proceeds as usual.
You can have a different board every time, generated in any way.
-----------------
Some related ideas: Steps 1 and 2 could be reversed. You could circulate boards for some brief time, collecting data on bids.
By the way, white being able to force a win in chess is news to me. I mean, no one solved the game by backward induction yet, right?
(I haven't tried the game yet because I'm a tad fearful of putting my CC info into a portable app. Sorry if my suggestion doesn't actually make sense for it.)
]]>Rotating the board and presenting it again would give the advantage to both players with a perfect, 50/50 split. Though it seems like the first/second to play from a given perspective on a given board STILL has a different experience. You know, we already picked on this board once. Won't that change the information we have on the second pick? Also, our stacks will be different, as you point out.
Just issuing a new board each round gives the advantage to both players in a way that approaches a perfect 50/50 split in the limit. For now, it's okay as-is, I think.
Isn't chess a forced win for white?
Don't we flip a coin to decide who plays white? Oh, we flip a coin to decide who gets to pick their color first.
But that doesn't make it a game subject to chance.
Clearly, the advantage for one player on a given board is strictly less than the advantage for white in Chess (where white can force a win, but the advantaged player in CM cannot force a win).
]]>A simple solution:
If a player leaves the game before they commit to any decisions then both players get their coin back.
In theory players should just keep refusing until they get what they're sure is a favourable board in this case. In practice personally I'd be refusing probably at least 50% of boards.
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