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Subject: Is it possible to have a pure abstract strategy game where no side is favored and it doesn't draw? rss

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christian freeling
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docreason wrote:
That looks like it is akin to Ricochet Robots in that you race to figure out a puzzle. I am curious if it is possible to do a game with offense and defense, where you modify the environment and pieces to block someone, rather than race to be first to solve it.
It's a party game, why modify the environment
 
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christianF wrote:
docreason wrote:
That looks like it is akin to Ricochet Robots in that you race to figure out a puzzle. I am curious if it is possible to do a game with offense and defense, where you modify the environment and pieces to block someone, rather than race to be first to solve it.
It's a party game, why modify the environment

Same with Ricochet Robots. But I don't consider a puzzle game, like that, to be a strategy game. Race to figure out how to optimize a situation, requires problem solving, but not strategy. My idea is to think of a strategy game along the lines of what is being discussed.
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christian freeling
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docreason wrote:
christianF wrote:
docreason wrote:
That looks like it is akin to Ricochet Robots in that you race to figure out a puzzle. I am curious if it is possible to do a game with offense and defense, where you modify the environment and pieces to block someone, rather than race to be first to solve it.
It's a party game, why modify the environment

Same with Ricochet Robots. But I don't consider a puzzle game, like that, to be a strategy game. Race to figure out how to optimize a situation, requires problem solving, but not strategy. My idea is to think of a strategy game along the lines of what is being discussed.
It was a joke, doc
 
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christianF wrote:
docreason wrote:
christianF wrote:
docreason wrote:
That looks like it is akin to Ricochet Robots in that you race to figure out a puzzle. I am curious if it is possible to do a game with offense and defense, where you modify the environment and pieces to block someone, rather than race to be first to solve it.
It's a party game, why modify the environment

Same with Ricochet Robots. But I don't consider a puzzle game, like that, to be a strategy game. Race to figure out how to optimize a situation, requires problem solving, but not strategy. My idea is to think of a strategy game along the lines of what is being discussed.
It was a joke, doc

Thing is that BGG classifications, both games belong here, and one could argue that they are a way to do it. With this in mind, I decided to clarify.

Well, on the note of time and so on, I am wondering if anyone would want to do an abstract strategy game like Clock Blockers:



Pretty much you have pieces that move based upon how they moved in a prior game, and bring them into your game, to complete objectives to win.
 
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Matteo Perlini
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Quote:
Is it possible to have a pure abstract strategy game where no side is favored and it doesn't draw?

As many people have already said, this should be impossible. But maybe there is one exception.

Maybe someone remembers my holy grail, a game where finding the perfect play is an uncomputable problem. My particular interpretation of this goal was a game with a infinite branching factor (and finite number of turn). So I designed INFINITO... but I still don't know if I reached my goal, probably not.

Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
 
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epicurus wrote:
As many people have already said, this should be impossible. But maybe there is one exception.

Maybe someone remembers my holy grail, a game where finding the perfect play is an uncomputable problem. My particular interpretation of this goal was a game with a infinite branching factor (and finite number of turn). So I designer INFINITO... but I still don't know if I reached my goal, probably not.

Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?
 
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russ wrote:
epicurus wrote:
As many people have already said, this should be impossible. But maybe there is one exception.
[...]
Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?
Hi Russ.
I don't know, Russ, I should ponder more about this problem. But talking about "guaranteed win" seems to me to assume a finite set, a set you can, at least in principle, explore in all its vastness... but with a infinite set of moves "guaranteed win" becomes terribly elusive.

p.s. In "Winning Ways", one of the axioms of the theorem is "There are several, usually finitely many, positions [...]". Maybe the theorem implicitly assume a finite number of moves. As said, I have to study more carefully the theorem.
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epicurus wrote:
russ wrote:
epicurus wrote:
As many people have already said, this should be impossible. But maybe there is one exception.
[...]
Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?
Hi Russ.
I don't know, Russ, I should ponder more about this problem. But talking about "guaranteed win" seems to me to assume a finite set, a set you can, at least in principle, explore in all its vastness... but with a infinite set of moves "guaranteed win" becomes terribly elusive.
But you personally don't have to search all the infinite paths of the tree for the optimal path to exist.

Existence and construction are 2 different things. It's perfectly normal to mathematically prove things with non-constructive proofs (unless you're in the tiny minority of constructivist mathematicians.)

E.g. somewhat similarly, we can easily prove mathematically that all prime numbers greater than 2 are odd, even though there are infinitely many prime numbers, so you can't actually look at and verify each one.
 
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russ wrote:
epicurus wrote:
As many people have already said, this should be impossible. But maybe there is one exception.

Maybe someone remembers my holy grail, a game where finding the perfect play is an uncomputable problem. My particular interpretation of this goal was a game with a infinite branching factor (and finite number of turn). So I designer INFINITO... but I still don't know if I reached my goal, probably not.

Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?

The idea I was going for is to make the manipulation of the game environment completely under control, so the decision tree is unstable and not fixed. In this, there wouldn't be any luck to it, because players completely control. But, it is possible one could argue that one couldn't fully see the decision tree from the start, because it could change, based on what players do in the game. I did bring up the GIPF series as in rules hacking, because whether a certain rule is added or not would depend on whether or not a player would win another game.
 
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docreason wrote:
russ wrote:
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?

The idea I was going for is to make the manipulation of the game environment completely under control, so the decision tree is unstable and not fixed. In this, there wouldn't be any luck to it, because players completely control. But, it is possible one could argue that one couldn't fully see the decision tree from the start, because it could change, based on what players do in the game. I did bring up the GIPF series as in rules hacking, because whether a certain rule is added or not would depend on whether or not a player would win another game.
As long as some formally definable process exists for changing the rules, there's still one fixed large game tree. You can view it as a smaller tree which is changing, but mathematically there's still formally a single huge tree which represents all the myriad ways the game could proceed, including the "rule changes". The "rule changes" themselves are simply decisions the players make which determine what the legal moves are from the following states... just as "normal moves" do. The game tree doesn't care about the subjective/emotional distinction humans might make about "normal moves" and "rule changes".
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russ wrote:
docreason wrote:
russ wrote:
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?

The idea I was going for is to make the manipulation of the game environment completely under control, so the decision tree is unstable and not fixed. In this, there wouldn't be any luck to it, because players completely control. But, it is possible one could argue that one couldn't fully see the decision tree from the start, because it could change, based on what players do in the game. I did bring up the GIPF series as in rules hacking, because whether a certain rule is added or not would depend on whether or not a player would win another game.
As long as some formally definable process exists for changing the rules, there's still one fixed large game tree. You can view it as a smaller tree which is changing, but mathematically there's still formally a single huge tree which represents all the myriad ways the game could proceed, including the "rule changes". The "rule changes" themselves are simply decisions the players make which determine what the legal moves are from the following states... just as "normal moves" do. The game tree doesn't care about the subjective/emotional distinction humans might make about "normal moves" and "rule changes".


The decision tree isn't fixed in the normal sense. Its state depends on players being able to play something completely different. Depending on their competency, the decision tree can completely change.

Would you also argue the decision tree would be fixed, if rules changes were implemented based upon how well players shot a basketball? You can't say the decision tree here is fixed at all in this regard.
 
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docreason wrote:
The decision tree isn't fixed in the normal sense. Its state depends on players being able to play something completely different. Depending on their competency, the decision tree can completely change.
Sorry, but I sincerely maintain that the decision tree is fixed. It's a subjective illusion that it's "changing". If "rule changes" are the results of legal player decisions, then "rule changes" are part of the formal decision tree.

The decision tree is no more "changing" than the decision tree of chess "changes" when you move your king, making it no longer legal for you to castle.

It seems like you're possibly conflating the concept of "the current set of legal moves at this specific physical game board position" with "the entire game tree of all possible future states of the game".

Quote:
Would you also argue the decision tree would be fixed, if rules changes were implemented based upon how well players shot a basketball? You can't say the decision tree here is fixed at all in this regard.
Agreed, but then it wouldn't really be an "abstract strategy game" any more, so it's rather a moot point for purposes of the thread's question, isn't it?

E.g. if, in Chess, you had to score a basketball shot successfully to capture a piece, would you still call chess an abstract strategy game? (I wouldn't.) Or if you had to tell a joke and make your opponent laugh out loud to capture a piece?

Or to take the famous popular example: is Chess Boxing an abstract strategy game? (I don't think so.)
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Ok, let me go back to what actually functions on a practical level. I could end up agreeing that the decision tree could be fixed, maybe, BUT it is next to impossible to calculate what it looks like. This is because the paths are decided by factors outside of the game, and not even able to be computed. While the tree is shaped entirely by human interaction, and skill, whether or not parts of the tree is active, is, at best, probabilistic.

In regards to the other parts, it relates to what is being discussed, in regards to the issue of a side having an advantage and there not being draws, and what impacts this being so. It is in the abstract genre forum, because this is a grail of sorts by the abstract games community.
 
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docreason wrote:
I could end up agreeing that the decision tree could be fixed, maybe, BUT it is next to impossible to calculate what it looks like. This is because the paths are decided by factors outside of the game, and not even able to be computed.
If applicability of these factors is formalized in the rules, how can they be 'outside the game'?

docreason wrote:
In regards to the other parts, it relates to what is being discussed, in regards to the issue of a side having an advantage and there not being draws, and what impacts this being so. It is in the abstract genre forum, because this is a grail of sorts by the abstract games community.
Please have a look at Symple then. It is drawless and has a sophisticated high resolution turn order balancing mechanism embedded in its move protocol. The game is subject of course to Zermelo's theorem, but for all practical purposes it would seem as close as one can get to 'perfect balance' (BGG entry).
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docreason wrote:
Ok, let me go back to what actually functions on a practical level. I could end up agreeing that the decision tree could be fixed, maybe, BUT it is next to impossible to calculate what it looks like.
If your real goal is to use "rule-changing" as a way to make the tree practically incomputable as a way of not favoring either side, then there's no need for "rule-changing" anyway. There already exist plenty of drawless games whose trees are practically incomputable, and which in practice show no obvious advantage for one side or the other (even though in theory one side has an advantage.)

(It might also be worth noting that "rule-changing" is only one of various ways to achieve massive or infinite branching, making brute force computation/solution practically impossible. E.g. one could define various abstract strategy games in which players pick an arbitrary integer each turn for some purpose, and voila, the tree is too bushy to calculate.)

Quote:
This is because the paths are decided by factors outside of the game, and not even able to be computed.
If they're decided by factors "outside the game", then the players' legal moves are not formally defined (after all, "changing a rule" is just another kind of move in the game), and I would be dubious of calling the game an "abstract strategy game".

Can you give a more concrete example of the kind of "rule changing" paradigm you have in mind?

If you're imagining wacky stuff like "at any time, a player can challenge their opponent to a basketball game for the right to change the rules" or subjective fuzzy rules such as "whoever has the most beautiful face gains 3 points each turn" or (probably but not always) objectively measurable "external" things like "whoever holds their breath the longest gains a bonus point" or "whoever last bought a bottle of water gain a bonus point", then obviously it's not an abstract strategy game any more. (FWIW I'd probably consider it more of a party game or "experimental game-like activity".)

If you're imagining more formal "reasonable" stuff like adding rules which restrict the number of pieces of various sorts on the board to meet various mathematical constraints (e.g. being able to concrete rules such as "each turn a player can add a rule such as "the sum of red pieces plus blue pieces must be a prime number"), then that is all formally definable and representable as part of the game tree (albeit a much bushier game tree than in conventional "non-rule-changing" games, indeed probably with infinite branching depending on the specific allowed grammar of "rule changes", but combinatorial game theory and the fundamental theorem doesn't care about whether a game has finite or infinite branching.)

Quote:
In regards to the other parts, it relates to what is being discussed, in regards to the issue of a side having an advantage and there not being draws, and what impacts this being so. It is in the abstract genre forum, because this is a grail of sorts by the abstract games community.
In practical terms, such "grail" games already exist. There are plenty of drawless games which don't obviously favor one player in practice.

In theoretical terms, such "grail" games are mathematically impossible, so it's certainly no grail of mine, no more than I would seriously desire a perpetual motion machine, or a compression algorithm which is guaranteed to make any file smaller, or an empty set with 3 elements ...
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epicurus wrote:
russ wrote:
epicurus wrote:
As many people have already said, this should be impossible. But maybe there is one exception.
[...]
Anyway, if it is possible to design a game with an infinite branching factor, I think the answer to you question is yes.
But nothing about the basic theorem of combinatorial games assumes or requires that the game tree be computable or finite. So why would the game tree being infinite make any difference?
Hi Russ.
I don't know, Russ, I should ponder more about this problem. But talking about "guaranteed win" seems to me to assume a finite set, a set you can, at least in principle, explore in all its vastness... but with a infinite set of moves "guaranteed win" becomes terribly elusive.
Hi Russ, my reflection is finished.

You are right, it doesn't make any difference with a infinite branching factor game. The optimal strategy could be uncomputable, but it is there. The only difference is a practical one: a game with an infinite branching factor could be infinitesimally unfair.

Anyway, I agree with you in the discussion with Richard, too. If you define a way to change the rules, you are not really changing the rules. You are changing the applicable rules (analogy: phenotype), but the totality of rules (analogy: genotype) is unchanged.
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A weird new research article about encoding a Turing machine into a Magic: The Gathering position and making one player's win equivalent to whether the Turing machine halts (and thus uncomputable) made me think of these old discussions about games in which players can have an infinite number of possible options on a move.

MTG is the most complex game links to:
https://arxiv.org/pdf/1904.09828.pdf

the MtG article wrote:
This construction establishes that Magic: The Gathering is the most computationally complex real-world game known in the literature. In addition to showing that optimal strategic play in Magic is non-computable, it also shows that merely evaluating the deterministic consequences of past moves in Magic is non-computable. The full complexity of optimal strategic play remains an open question, as do many other computational aspects of Magic. For example, a player appears to have infinitely many moves available to them from some board states of Magic. Whether or not there exists a real-world game of Magic in which a player has infinitely many meaningfully different moves available to them has the potentially to highly impact the way we understand and model games as a form of computation.

Indeed, this result raises several interesting philosophical questions about games as a form of computation. Some authors, such as Demaine and Hearn [9], have sought a formal framework for modelling games that is strictly sub-Turing. Unlike the open-world, non-strategic games in which Turing machines have been constructed before, Magic: The Gathering is unambiguously a two-player strategic game like such models attempt to represent. Therefore this result shows that any sub-Turing model is necessarily inadequate to capture all games. Quite the opposite: it seems likely that a super-Turing model of games would be necessary to explain Magic.

I had no idea MtG was that complicated.

Also interesting that in some of these old threads, I used a simple example of picking the biggest number to win, as an example of an infinite game with nonetheless simplistic strategy. It turns out something like this can apparently appear in MtG:

the MtG article wrote:
Consider the following situation: both players control Lich , Transcendence, and Laboratory Maniac. One player then casts Menacing Ogre. The net effect of this play is the “Who Can Name the Bigger Number” game – whoever picks the biggest number wins on the spot.
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russ wrote:
A weird new research article about encoding a Turing machine into a Magic: The Gathering position and making one player's win equivalent to whether the Turing machine halts (and thus uncomputable) made me think of these old discussions about games in which players can have an infinite number of possible options on a move.

MTG is the most complex game links to:
https://arxiv.org/pdf/1904.09828.pdf

the MtG article wrote:
This construction establishes that Magic: The Gathering is the most computationally complex real-world game known in the literature. In addition to showing that optimal strategic play in Magic is non-computable, it also shows that merely evaluating the deterministic consequences of past moves in Magic is non-computable. The full complexity of optimal strategic play remains an open question, as do many other computational aspects of Magic. For example, a player appears to have infinitely many moves available to them from some board states of Magic. Whether or not there exists a real-world game of Magic in which a player has infinitely many meaningfully different moves available to them has the potentially to highly impact the way we understand and model games as a form of computation.

Indeed, this result raises several interesting philosophical questions about games as a form of computation. Some authors, such as Demaine and Hearn [9], have sought a formal framework for modelling games that is strictly sub-Turing. Unlike the open-world, non-strategic games in which Turing machines have been constructed before, Magic: The Gathering is unambiguously a two-player strategic game like such models attempt to represent. Therefore this result shows that any sub-Turing model is necessarily inadequate to capture all games. Quite the opposite: it seems likely that a super-Turing model of games would be necessary to explain Magic.

I had no idea MtG was that complicated.
Hi Russ, thank you for bringing this article to my attention. This is a super-interesting topic for me.

I'm not a MtG player and I could not imagine all of this too. surprise

But reading the article I could notice some cards that can stand for algorithm structures: selection structure (i.e. if-then-else) and especially repetition structure (i.e. while, for). So all this can, at least in part, makes sense to me now.

russ wrote:
Also interesting that in some of these old threads, I used a simple example of picking the biggest number to win, as an example of an infinite game with nonetheless simplistic strategy. It turns out something like this can apparently appear in MtG:

the MtG article wrote:
Consider the following situation: both players control Lich , Transcendence, and Laboratory Maniac. One player then casts Menacing Ogre. The net effect of this play is the “Who Can Name the Bigger Number” game – whoever picks the biggest number wins on the spot.
I noticed.
Even if I don't fully understand why they mentioned this part in the article.

Anyway, as you can easily notice, your simple example is the basis of Infinito. But in Infinito picking a big number has a price.


There is an obscure part in the article:
the MtG article wrote:
Therefore this result shows that any sub-Turing model is necessarily inadequate to capture all games. Quite the opposite: it seems likely that a super-Turing model of games would be necessary to explain Magic.
Are they referring to the exotic hyper-computation models?

Anyway, this topic reminded me an old project of mine, contemporary with the development of Infinito but never finished. It was about an uncomputable abstract game too. I'm probably going to resume this project.
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epicurus wrote:
There is an obscure part in the article:
Much of it was obscure for me!

I still have not read it all carefully in detail. As far as I understand, the idea is that they managed to make a position which might terminate (in which case one player wins) or which might never terminate (in which case the game is a draw), and that it is uncomputable whether the position will terminate. I.e. the best you can do is simply keep playing it out, and if it stops, cool, you know who won, and if it keeps going, then you don't know if it might stop later or not.

Which raises a completely different practical question: can something like this ever happen in a "real world" Magic game? What happens officially in a tournament if some crazy long sequence starts happening and doesn't seem to be going to stop, yet it is not some clear simple cycle which the players can prove will stop? How do the players or tournament officials know what to do in this weird situation?

I guess that in reality, such strange positions never really happen in "real life".
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slashing wrote:
Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?

Yes, it means just that.

The optimum strategy in RPS is to play randomly. Thus it not only isn't a pure abstract game in theory, it behaves entirely unlike one in practice.

(Just to avoid a nitpick, random - equiprobable, independent when iterated - is optimum for a definition of optimum. If you know your opponent doesn't play optimally you can improve on it. But a suboptimal opponent isn't how games are usually analysed.)

Edit: This and my next post failed to notice this was a necroed thread. But I'll leave them.
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russ wrote:
slashing wrote:
Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?
A pure abstract strategy game consists of alternating turns. (Of course it's trivially easy to define fair games with simultaneous turns.)

Also RPS is not guaranteed to terminate, though of course it probably will terminate within a few rounds if the players are not intentionally making it last forever.

It's not guaranteed to terminate, but it will terminate with probability 1. (And no, that is not a contradiction.)
 
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russ wrote:
epicurus wrote:
There is an obscure part in the article:
Much of it was obscure for me!
For me too, of course.

The (presumed) reference to super-turing models is at a much higher level, though.

To overcome the limit of the turing model, you have some absolutely crazy options, including:
- a machine that can compute real numbers
- a machine that can execute super-task (completing infinitely many steps in finite time, inspired by Zeno's paradoxes)
- a machine working in orbit around a rotating black hole

As said, madness at its purest. laugh

russ wrote:
I still have not read it all carefully in detail. As far as I understand, the idea is that they managed to make a position which might terminate (in which case one player wins) or which might never terminate (in which case the game is a draw), and that it is uncomputable whether the position will terminate. I.e. the best you can do is simply keep playing it out, and if it stops, cool, you know who won, and if it keeps going, then you don't know if it might stop later or not.
Yes and this has been made possible by the cards that include loops.

russ wrote:
Which raises a completely different practical question: can something like this ever happen in a "real world" Magic game? What happens officially in a tournament if some crazy long sequence starts happening and doesn't seem to be going to stop, yet it is not some clear simple cycle which the players can prove will stop? How do the players or tournament officials know what to do in this weird situation?

I guess that in reality, such strange positions never really happen in "real life".

In the article they discuss the real life consequences:

the MtG article wrote: wrote:

While there are practical difficulties involved with correctly setting up the necessary board state, such as running out of space on your table, a sufficiently tenacious player could setup and execute this construction in a real-world tournament game of Magic: The Gathering. An example 60-card deck that is capable of executing this construction on the first turn of the game and which is legal in the competitive Legacy format can be seen in Table III.With the correct draw, the deck uses Ancient Tomb and three Lotus Petals to play Grim Monolith and Power Arti-fact and generate unlimited colourless mana, at which point Staff of Domination draws the rest of the deck and Gem-stone Array generates unlimited coloured mana. The deck casts most of the permanents immediately, and uses Stolen Identity to make token copies of them (using Memnarch first on the enchantments like Cloak of Invisibility). The tape is initialised with Riptide Replicator and Capsize. Djinn Illuminatus or Reito Lantern allow repeated casting of the text-modification cards, as well as Reality Ripple which sets the phase of the Rotlung Reanimators and Donate which gives most permanents to Bob. Once everything is set up,Steely Resolveis cast, and then Karn Liberated and Capsizeare used to exile all setup permanents and all cards from Bob’s hand, eventually exiling Capsize and Karn Liberated themselves. Now no player has any remaining choices except to let the Turing machine execute.

In addition to the Comprehensive Rules, play at sanctioned Magic: The Gathering tournaments is also governed by the Tournament Rules. Some of these rules, most notably the ones involving slow play, may effect an individual’s ability to successfully execute the combo due to concerns about the sheer amount of time it would take to manually move the tokens around to simulate a computation on a Turing machine. This would not be a concern for two agents with sufficiently high computational power, as the Tournament Rules also provide a mechanism called “shortcuts” for players to skip carrying out laborious loops if both players agree on the game state at the beginning and the end of the shortcut.
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christian freeling
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russ wrote:
I guess that in reality, such strange positions never really happen in "real life".
I'm an alien to magic but isn't the very point of abstract games that it's never really like real life? It's boxed conflict that can easily be put aside (well, by most players anyway).
 
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Russ Williams
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christianF wrote:
russ wrote:
I guess that in reality, such strange positions never really happen in "real life".
I'm an alien to magic but isn't the very point of abstract games that it's never really like real life? It's boxed conflict that can easily be put aside (well, by most players anyway).
I meant that such a situation won't appear in a real life game played by ordinary players who are simply playing to win as usual, instead of playing to intentionally create the weird situation.

(Similar to e.g. saying a triple ko rarely appears in "real life" games of Go, even though it's easy to artificially construct a triple ko if you want to.)


(Also, FWIW Magic is not an abstract game.)
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russ wrote:
(Also, FWIW Magic is not an abstract game.)
I realise that. But the thread is about abstract strategy games.
 
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