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[Nevermind]

Hey guys!
     I know I'm new to this forum, but I thought I'd mention that there is a neat pin with this card. ITR-ITR-goons-goons-council room-x-y-z-MASQUERADE! As long as you don't play any more discard attacks, you can win by destroying their whole hand/deck. You win. GG. Any questions?

[SirPeebles]

Oh god, I just came across this thread.  I was really happy to see the OP's formulation of the question, but then it immediately devolved into a flurry of misuse of the word "infinite".  I feel physically ill from having been exposed to this.  Please, just stop even using the word "infinite".  It has little if any place in this discussion beyond the formal name of the proposed card.  Try using meaningful and accurate terms, such as "unlimited" or "arbitrarily many".  The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

[soulnet]

Oh god, I just came across this thread.  I was really happy to see the OP's formulation of the question, but then it immediately devolved into a flurry of misuse of the word "infinite".  I feel physically ill from having been exposed to this.  Please, just stop even using the word "infinite".  It has little if any place in this discussion beyond the formal name of the proposed card.  Try using meaningful and accurate terms, such as "unlimited" or "arbitrarily many".  The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

I haven't rechecked the entire thread for specific misuses, but I see no problem in saying something like "have infinite VP", at least if agree that by infinite we mean the "regular countable infinite" or "the only infinite" if you believe finitists. As far as board games go, non-finite is already strange enough, so assuming a finitist universe seems reasonable.

I certainly find having infinite VP less disturbing than transfinite induction.

[SirPeebles]

At no point does anyone ever have a VP total that is not a finite whole number.

[markusin]

At no point does anyone ever have a VP total that is not a finite whole number.

IANAM (I am not a mathematician), but I'm pretty sure Sir Peebles is correct. You can't have infinitely many of something, but you can be bounded by infinity, right?

[Witherweaver]

I don't see a problem.  You play "Infinite Throne Room" on something that gives +VP, and  you are awarded INF VP, where INF is an object that satisfies INF > x for all real x.  What's wrong with that?

[SirPeebles]

I don't see a problem.  You play "Infinite Throne Room" on something that gives +VP, and  you are awarded INF VP, where INF is an object that satisfies INF > x for all real x.  What's wrong with that?

Look at what Infinite Throne Room does.  You choose a card in your hand, let's say Monument, and then you play it repeatedly until a certain condition is satisfied.  If that condition is never satisfied, then you simply don't finish resolving Infinite Throne Room and the game does not halt.  Otherwise, that certain condition is met.  If it is met after 2 plays, then you gained +2 VP and + $4.  If it took 20 plays of Monument, then you gained + 20 VP and + $40.  But there is never a circumstance under which your number of victory tokens is anything but a finite, nonnegative whole number.

[SirPeebles]

At no point does anyone ever have a VP total that is not a finite whole number.

IANAM (I am not a mathematician), but I'm pretty sure Sir Peebles is correct. You can't have infinitely many of something, but you can be bounded by infinity, right?

You can't obtain an infinite number of something by starting with a finite amount and adding a finite number to it a finite number of times.  There may be other things that could lead to infinite quantities, but VP or coin tokens from Infinite Throne Room paired with published Dominion card are not in that list.

[Witherweaver]

I don't see a problem.  You play "Infinite Throne Room" on something that gives +VP, and  you are awarded INF VP, where INF is an object that satisfies INF > x for all real x.  What's wrong with that?

Look at what Infinite Throne Room does.  You choose a card in your hand, let's say Monument, and then you play it repeatedly until a certain condition is satisfied.  If that condition is never satisfied, then you simply don't finish resolving Infinite Throne Room and the game does not halt.  Otherwise, that certain condition is met.  If it is met after 2 plays, then you gained +2 VP and + $4.  If it took 20 plays of Monument, then you gained + 20 VP and + $40.  But there is never a circumstance under which your number of victory tokens is anything but a finite, nonnegative whole number.

I didn't think that was what Infinite Throne Room was doing.  I thought we were defining Infinite Throne Room's action to be that if you play it on something that would never resolve (e.g., something that gives +$ or +VP), you simply define it to be resolved by introducing infinity and take everything to be in the extended reals.

[SirPeebles]

I don't see a problem.  You play "Infinite Throne Room" on something that gives +VP, and  you are awarded INF VP, where INF is an object that satisfies INF > x for all real x.  What's wrong with that?

Look at what Infinite Throne Room does.  You choose a card in your hand, let's say Monument, and then you play it repeatedly until a certain condition is satisfied.  If that condition is never satisfied, then you simply don't finish resolving Infinite Throne Room and the game does not halt.  Otherwise, that certain condition is met.  If it is met after 2 plays, then you gained +2 VP and + $4.  If it took 20 plays of Monument, then you gained + 20 VP and + $40.  But there is never a circumstance under which your number of victory tokens is anything but a finite, nonnegative whole number.

I didn't think that was what Infinite Throne Room was doing.  I thought we were defining Infinite Throne Room's action to be that if you play it on something that would never resolve (e.g., something that gives +$ or +VP), you simply define it to be resolved by introducing infinity and take everything to be in the extended reals.

I interpreted the bulleted list to be (incorrect) clarifications of the card's effect.  If we instead take them as additional rules, then I still object to the use of "infinite VP".  Just say that if you play a card granting VP, then you will win at the end of the game and that VP no longer determines winners.  That's a lot simpler than introducing the arithmetic of extended real numbers.

[Witherweaver]

I don't see a problem.  You play "Infinite Throne Room" on something that gives +VP, and  you are awarded INF VP, where INF is an object that satisfies INF > x for all real x.  What's wrong with that?

Look at what Infinite Throne Room does.  You choose a card in your hand, let's say Monument, and then you play it repeatedly until a certain condition is satisfied.  If that condition is never satisfied, then you simply don't finish resolving Infinite Throne Room and the game does not halt.  Otherwise, that certain condition is met.  If it is met after 2 plays, then you gained +2 VP and + $4.  If it took 20 plays of Monument, then you gained + 20 VP and + $40.  But there is never a circumstance under which your number of victory tokens is anything but a finite, nonnegative whole number.

I didn't think that was what Infinite Throne Room was doing.  I thought we were defining Infinite Throne Room's action to be that if you play it on something that would never resolve (e.g., something that gives +$ or +VP), you simply define it to be resolved by introducing infinity and take everything to be in the extended reals.

I interpreted the bulleted list to be (incorrect) clarifications of the card's effect.  If we instead take them as additional rules, then I still object to the use of "infinite VP".  Just say that if you play a card granting VP, then you will win at the end of the game and that VP no longer determines winners.  That's a lot simpler than introducing the arithmetic of extended real numbers.

It seems to me like the question was, "What rules do we have to assign so that this card can resolve?" 

I don't really see a difference between using the extended reals and defining "you win at the end of the game," "you are unrestrained by $ when you make buys," "you can buy any number of cards," "you can play any number of actions".  I mean, the whole point of the card is making sense of "infinite plays," so using an "infinity" object doesn't seem far fetched.

[SirPeebles]

There's no difference in theory, but come on.  Dominion is a rather approachable card/board game.  Do you really want to rules to reference the extended real numbers?

[Witherweaver]

There's no difference in theory, but come on.  Dominion is a rather approachable card/board game.  Do you really want to rules to reference the extended real numbers?

Well the whole point was a puzzle.. I don't think anyone was going to think of publishing this card.

[soulnet]

I interpreted the card as leaving the game state in the limit of the game states after n=1,2,3,... plays. As long as that does not require an infinite number of decisions, that is a perfectly well determined thing (or not, depending on whether the limit exist). The limit of playing Monument infinitely can be defined rather well as having infinite $ and infinite VP, as specified by the FAQ.

However, the limit of things with decisions require more careful analysis. Infinitely playing Chapel or Courtyard requires you to do infinitely many decisions, and that is a little harder to define. However, if we consider two sets of decisions that yield the same limit as an equivalent class, there are finitely many decisions, because there are finitely many game states. Thus, the game with ITR can stil be regarded as a finite game, where some quantities that previously could only be integer numbers (like the amount of VP or money), can now also be infinite.

The only problem with those rules is having infinite coin tokens, because you can play those, which means you can end up subtracting infinites. But, as long as we do not play with coin tokens, everything sounds sound.

Of course, adding infinitary limits complicates the game a lot, and I would not use this for anything other than a brain teaser. But I don't think using infinite as an object is as problematic as some of you seem to think.

[markusin]

In this case, you can't say you have "infinite VP" because any player who plays ITR-Monument after you did should be able to surpass your lead in VP by playing Monument x + 1 times, where x is the number of times you played Monument with ITR up to that point. However, you can't say (∞ + 1)  > ∞.

[Witherweaver]

In this case, you can't say you have "infinite VP" because any player who plays ITR-Monument after you did should be able to surpass your lead in VP by playing Monument x + 1 times, where x is the number of times you played Monument with ITR up to that point. However, you can't say (∞ + 1)  > ∞.

Huh?  When you play ITR on Monument, the result is INF VP points.  INF is a number such that INF > x for all real numbers x.  If someone else plays ITR on Monument, the result is they get INF VP points.  It is not true that INF > INF (note that INF is not a real number), so you do not beat them, and they do not beat you.  In this case, you can define the result to be a tie.

If you play ITR on Monument, you certainly don't play Monument x times.  That would never resolve.. that's the whole point of defining special rules.  You played it once with ITR, giving INF VP points.

[Witherweaver]

The only problem with those rules is having infinite coin tokens, because you can play those, which means you can end up subtracting infinites. But, as long as we do not play with coin tokens, everything sounds sound.

I don't think this is a problem.  If you play ITR on Baker, you get INF coin tokens.  This just means that you're able to generate $N (by spending N coin tokens) for any real number N every turn thereafter. Since N is finite, you're still left with INF coin tokens.

[markusin]

In this case, you can't say you have "infinite VP" because any player who plays ITR-Monument after you did should be able to surpass your lead in VP by playing Monument x + 1 times, where x is the number of times you played Monument with ITR up to that point. However, you can't say (∞ + 1)  > ∞.

Huh?  When you play ITR on Monument, the result is INF VP points.  INF is a number such that INF > x for all real numbers x.  If someone else plays ITR on Monument, the result is they get INF VP points.  It is not true that INF > INF (note that INF is not a real number), so you do not beat them, and they do not beat you.  In this case, you can define the result to be a tie.

If you play ITR on Monument, you certainly don't play Monument x times.  That would never resolve.. that's the whole point of defining special rules.  You played it once with ITR, giving INF VP points.

Special rules?

...
On right, special rules. They're in the OP.

In the OP, ITR-Monument is said to guarantee that you don't lose while still allowing the card to resolve.

Carry on then.

[SirPeebles]

You play an ITR on a Monument.  The Copper and Curse piles are empty, not a Ruins game.

I play ITR on a Candlestick Maker.  I have a Goons in play and a Trader in hand.  I want to buy Estate repeatedly while revealing Trader.  What happens?  Am I able to share victory?  How do you model this with the extended real numbers?

[Witherweaver]

You play an ITR on a Monument.  The Copper and Curse piles are empty, not a Ruins game.

I play ITR on a Candlestick Maker.  I have a Goons in play and a Trader in hand.  I want to buy Estate repeatedly while revealing Trader.  What happens?  Am I able to share victory?  How do you model this with the extended real numbers?

You could have just played ITR on the Goons; I think the Candlestick Maker is not needed.  But anyway.

I think you get INF VP tokens here, as long as you claim you are revealing Trader each time you buy an Estate.  The first N times, you gain N Silvers, where N is the number of Silvers remaining.  Then (in hypothetical space) you continue to buy Estates, which is fine because the Estate pile is not empty, get the VP token for the buy, fail to gain a Silver (since the pile is empty).  This process has the defined resolution: Silver pile empty, all Silvers gained by you, INF VP to you.

So the game ends in a tie (INF to INF), or a loss if you're first player.

The thing that would be hard to resolve, however, is: Do you have more buys left?  That isn't really defined.  But I don't think it actually has to be.  It only matters if you have more buys left if you want to buy something else.  In which case, you could make any number of (finite) buys (say, to empty piles if there wasn't going to be a three-pile ending), and then resolve the unlimited buy, unlimited Trader reveal scenario. 

[florrat]

The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

The only problem which I have when someone says "you have infinite VP", I'd want to know "What infinity?" Other than that I have no problem with the statement "If you play monument infinitely often, you have infinity coins and infinite VP chips": if "infinity" is replaced by ω (omega, the "smallest infinity" (yes, there are multiple sizes infinity)), this statement makes perfect sense to me.

[soulnet]

The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

The only problem which I have when someone says "you have infinite VP", I'd want to know "What infinity?" Other than that I have no problem with the statement "If you play monument infinitely often, you have infinity coins and infinite VP chips": if "infinity" is replaced by ω (omega, the "smallest infinity" (yes, there are multiple sizes infinity)), this statement makes perfect sense to me.

I think I have addressed this above, with the same thought. BTW, omega should actually be aleph 0. Omega is not the cardinal but the ordinal.

I really do not understand people being fine with things like transfinite induction and not finite with something so relatively simple as a non-finitary game. This is not even the first attempt to define a non-finitary game.

[SirPeebles]

The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

The only problem which I have when someone says "you have infinite VP", I'd want to know "What infinity?" Other than that I have no problem with the statement "If you play monument infinitely often, you have infinity coins and infinite VP chips": if "infinity" is replaced by ω (omega, the "smallest infinity" (yes, there are multiple sizes infinity)), this statement makes perfect sense to me.

Sure, *if* you played a card infinitely often.  But you don't.  Not even with this proposed Infinite Throne Room card.  That is what bothered me.  Also, ω + 1 > ω, whereas 1 + ω = ω, and ω + (-1) is nonsensical as far as I know.  This is not a good start for modeling VP.

[Witherweaver]

The word "infinite" means something, and you hurt it when you say things like "you have infinite VP".

The only problem which I have when someone says "you have infinite VP", I'd want to know "What infinity?" Other than that I have no problem with the statement "If you play monument infinitely often, you have infinity coins and infinite VP chips": if "infinity" is replaced by ω (omega, the "smallest infinity" (yes, there are multiple sizes infinity)), this statement makes perfect sense to me.

I think the extended real line model works better. Arithemtic doesn't work correctly with ordinals (as Peebles said).  Though I think it does work with cardinals, and I think using aleph null would be the same as using the extended reals.  But I don't really know much about this.

Edit: And I think thinking about playing something infinitely often is misleading. Plus what does "infinitely often" mean?  If you "play it once" for every element of the natural numbers, that's different than "playing it once" for every element of the real numbers.

More like when you play infinite throne room and announce its target, the process must resolve.  We therefore must define its action on each card so that the process resolves.  The question is, how to define these things?  The sort of "natural" way would be to think of taking limits and working in a compact set.  "Taking limits" in the sense of.. play it once, what happens?  Play it a second time, what happens?  Play it a third time, etc.  Does this process resolve?  Well, if playing it once means +1 counter, then playing it n times means +n counters, so we might as well take the limit to be +INF counters in the extended reals. 

In general, you would want to define the action of ITR on a target to be the value (game state result) of the limiting process of playing that card sequentially {1,2,3,....}.

[soulnet]

Sure, *if* you played a card infinitely often.  But you don't.  Not even with this proposed Infinite Throne Room card.  That is what bothered me.  Also, ω + 1 > ω, whereas 1 + ω = ω, and ω + (-1) is nonsensical as far as I know.  This is not a good start for modeling VP.

You should not be using \omega, but \aleph_0 instead. Since \aleph_0 = \aleph_0 + 1 = \aleph_0 - 1, and there is nothing with the hability to produce -infinity VP, I see no issue. You really only need \omega is you want to do transfinite induction, but you do not, because there is only one kind of infinite throughout the entire game, so regular limit arithmetic is fine.

The ad-hoc rule that is needed is that decisions are choosing one of the possible equivalence classes of decisions that yield a limiting behavior including the infinites as possible values for VPs, coins, buys and actions. I.e., making decisions that yield oscillating behavior is forbidden.