Playing Games in the Transfinite: An Introduction to “Ordinal Chomp”

Chomp is a two-player game which is played as follows: The two players, A and B, start with a “board” which is a chocolate bar divided into small squares. With Player A starting, they take turns choosing a square and eating it together with all squares above and to the right. The catch is that the square at the lower left-hand corner is poisonous, and the player who is forced to eat it loses.

This image from the Wikipedia article shows a typical sequence of moves for a chocolate bar:

At this point, Player A is forced to eat the poisoned square and hence loses the game.

Although the question of what the winning strategies are for this game is very much an open problem, the question of who has a winning strategy is not: On the board, Player B wins (since Player A must eat the poison piece on his first move). But for any other board, Player A has a winning strategy.

To see why, suppose not. Then if Player A’s first move is to eat just the one square in the top right-hand corner, Player B must have a winning response (since we are supposing that Player B has a winning response to any move that Player A makes). But if Player B’s response is winning, then Player A could have simply made that move to start with.

However, suppose we play Chomp not just on boards, but on boards, where and are arbitrary ordinals. The game still makes sense just as before, and will always end in finite time, but Player A no longer wins all of the time (there will no longer be a top right-hand corner square if either or is a limit ordinal).

Scott Huddleston and Jerry Shurman investigated Ordinal Chomp in this paper, and showed that it has a number of interesting properties. I’ll describe a few of them below.

First of all, to make things a bit easier to discuss, we will consider only games where is finite rather than games where is arbitrary. However, everything said will go through fine in that case as well.

Secondly, we will use the following game which is equivalent to Chomp: At any point in the game, the board is described by a non-increasing sequence . On each turn, the appropriate player picks an and a such that and replaces the sequence with . We call this taking a bite of height . Playing on an chocolate bar as described above corresponds to playing with the sequence consisting of ‘s.

For ease of discussion later, we make the convention that any sequence (not just a non-increasing one) describes a game position by stipulating that is the same as the position .

We saw above that Player A wins all non-trivial finite Chomp games, so let’s start by looking at a transfinite chomp game that Player B wins: , or in our new notation. What is Player B’s winning strategy? Well, notice that Player A’s first move has to put the game either in the state for some or for some . In either case, Player B can then move the game into state of the form for some . From a position of that form, whatever Player A does, Player B can again move to a position of that form. Eventually, Player B will move to the position , and Player A will be forced to eat the poison piece.

So, Player B can win at least some transfinite Chomp games, although he still loses a lot of them: for example, he loses all games of the form where . The reason is that in such a game Player A can win by first moving to the position and then using Player B’s winning strategy! Similarly, Player B loses all games for .

In fact, for any ordinal , there is exactly one ordinal such that Player B wins . This is a consequence of what Huddleston and Shurman call the Fundamental Theorem of Transfinite Chomp in the above paper. Another interesting consequence, which illustrates the style of reasoning used in their proof of the Fundamental Theorem, is the following:

Theorem: For any sequence of ordinals, there is exactly one ordinal such that Player B wins the game . (Remember the convention above about sequence which are not necessarily non-increasing.)

Proof: The uniqueness is similar to the argument given above: If Player B has a winning strategy on the game then Player A has a winning strategy on all games where , since Player A can just move to the position and then use Player B’s winning strategy.

The existence is by induction. Fix and suppose that for all where for all and for at least one , we know that there is a such that Player B has a winning strategy.

For each ordinal , let be the set of all obtainable from by taking a bite of height (this term was defined above, if you forgot what it means).

Let . Let be the minimal ordinal not in . I claim that Player B has a winning strategy for . We will show this by showing that, for any move Player A makes, Player B can make a move that we know leads to a winning strategy for B.

So suppose Player A moves to . Either or . First suppose that . This means that Player A moved by taking a bite of height (say) out of by moving to . But by construction, we know that , which means that the player who is to move (Player B in this case) has a winning strategy.

Now suppose . This means by construction of that where for all and for at least one . Thus, if Player B moves to the position he ensures himself a win. .

Note that this proof is constructive. This means that you can actually use it to compute that (as we already know), the unique such that Player B wins is . As a puzzle, you might like to find the unique such that Player B wins (or such that Player B wins or or whatever you like, although the last one will be hard).

The much-more-general Fundamental Theorem of Transfinite Chomp in the paper linked above is also constructive. It allows you, in theory, to compute who will win the -dimensional game (we have been considering -dimensional games) for any ordinals . However, this is quite difficult in practice: according to the Wikipedia article, it is an open question who wins the -dimensional game .

As a final note, in the book Tracking the Automatic Ant, David Gale gives a very nice non-constructive proof that for all , there is a unique such that Player B wins .