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The Glutton can easily win simply by paying attention to the order in which the chocolates were added, and consuming them in that order. If only finitely many chocolates are added at each stage, the Chocolatier should simply make sure to consume them before moving on to the chocolates that were added at later stages.

The Glutton can easily win simply by paying attention to the order in which the chocolates were added, and consuming them in that order. If only finitely many chocolates are added at each stage, the Glutton should simply make sure to consume them before moving on to the chocolates that were added at later stages.

Let us call it the Chocolatier's game. There are two players, the Chocolatier and the Glutton. To begin play, the Chocolatier serves up finitely many unique and exquisite chocolate creations on a platter, and then the Glutton chooses one of them to eat. Play continues — at each stage the Chocolatier adds finitely many additional chocolates to the platter, and the Glutton consumes one of them. The chocolates accumulate on the platter as play progresses (except for those that have been eaten).

The Glutton can easily win simply by paying attention to the order in which the chocolates were added, and consuming them in that order. If only finitely many chocolates are added at each stage, he should simply make sure to consume them before moving on to the chocolates that were added at later stages.

It makes a more interesting game to say that the Chocolatier loses if chocolate types are ever repeated. And for this version of the game, there are some interesting things to say.

Let me remark that in the version of the game where we do not allow the Chocolatier to repeat chocolate types (as opposed to saying that this is allowed, but cause him to lose), then we should really include the list of already-consumed chocolates as part of the position, and this complicates the arguments above. I suspect but do not yet know that the Glutton can have no winning strategy that depends only these more general kinds of positions in the uncountable case.

Let us call it the Chocolatier's game. There are two players, the Chocolatier and the Glutton. To begin play, the Chocolatier serves up finitely many unique and exquisite chocolate creations on a platter, and then the Glutton chooses one of them to eat. Play continues — at each stage the Chocolatier adds finitely many additional chocolates to the platter, and the Glutton consumes one of those available. The excess uneaten chocolates accumulate on the platter as play progresses.

The Glutton can easily win simply by paying attention to the order in which the chocolates were added, and consuming them in that order. If only finitely many chocolates are added at each stage, the Chocolatier should simply make sure to consume them before moving on to the chocolates that were added at later stages.

Perhaps it makes a more interesting game, however, to say that the Chocolatier loses if chocolate types are ever repeated. And for this version of the game, there are some interesting things to say.

Let me remark that in the version of the game where we do not allow the Chocolatier to repeat chocolate types (as opposed to saying that this is allowed, but causes the Chocolatier to lose), then we should really include the list of already-consumed chocolates as part of the position, and this complicates the arguments above. (I asked a followup question about this at The Chocolatier's game.) I suspect but do not yet know that the Glutton can have no winning strategy that depends only on these more general kinds of positions in the uncountable case.

I don't know about any game attributed to Specker, but here is a simple game with your desired features.

In one sense, it is easy to see that there can be no such winning strategy for the Glutton. If the Glutton will choose a particular chocolate from a given assortment, then let the Chocolatier simply replace it on the next move, and again on all subsequent moves. If the strategy does not know the history, then the Glutton will choose it again every time, and the other chocolates will never be eaten.

But it would probably be more satisfactory to insist that the Chocolatier always place new chocolate types, so that replacement isn't possible. For this version of the game, there are some interesting things to say.

Let me remark that in the version of the game where we do not allow the Chocolatier to repeat chocolate types, then we should really include the list of already-consumed chocolates as part of the position, and this complicates the arguments above. I suspect but do not yet know that the Glutton can have no winning strategy that depends only these more general kinds of positions in the uncountable case.

I don't know about the game attributed to Specker, but here is a simple game with your desired features.

In one sense, it is easy to see that there can be no such winning strategy for the Glutton. If the Glutton will choose a particular chocolate from a given assortment, then let the Chocolatier simply replace it with an identical chocolate type on the next move, and again on all subsequent moves. If the strategy does not know the history, then the Glutton will choose it again every time, and the other chocolates will never be eaten.

It makes a more interesting game to say that the Chocolatier loses if chocolate types are ever repeated. And for this version of the game, there are some interesting things to say.

Let me remark that in the version of the game where we do not allow the Chocolatier to repeat chocolate types (as opposed to saying that this is allowed, but cause him to lose), then we should really include the list of already-consumed chocolates as part of the position, and this complicates the arguments above. I suspect but do not yet know that the Glutton can have no winning strategy that depends only these more general kinds of positions in the uncountable case.