Monday, November 23, 2009

Initial Game Positions

Note: it's unlikely there will be a post on Wednesday, as I am traveling to Massachusetts for American Thanksgiving.

We know the following non-constructive result about Hex and Chomp: the first player has a winning strategy.

This only applies to games in the starting position, and for different reasons for both games (although they both go by the "strategy-stealing argument" monker). In Hex, we know that it is better to have more tiles of your color on the board, so it is naturally better to be playing first. In Chomp, we have a very special move from the first board. This move has, as children, all the other children of the initial position. Thus, if you go first, there are two possibilities: either one of those other children is a winning move (make it!) or none are, meaning that the special move is a winning move (make it!).

As mentioned, neither of these are constructive as in neither hint at the best strategy. They just mention that such a strategy exists. We've noticed a similar situation with Atropos (at least as far as brute force searches have allowed so far). A repeating pattern of winnability appears: boards with n open circles along a side have a winning strategy for the first player exactly when n is congruent to 0 or 3 modulo 4. The other two (1 and 2) have a winning strategy for the second player. This may seem a bit convoluted (especially since it's only been shown up through boards of size 11 or so) but it is the same as saying that the first player has a winning strategy when there's an even number of open circles on the initial board, and doesn't when there are an odd number.

In general, Atropos and Hex are difficult to analyze (both PSPACE-complete) and Chomp has resisted all efforts to figure out. Could it be that for many games, the complexity of those initial positions is just inherently easier than a general game state? To some, this is very disconcerting. In some cases, the general state argument is a bit distracting: it could be that a winning strategy exists from that initial position and is easy to describe. Complexity results come from reductions that don't usually say anything about the difficulty of the game in common positions.

Could it be, then, that this easiness comes from the size of information needed to describe those initial positions? In a game of Hex, to describe an intermediate state, you need to list the color (or lack of) for each hexagon. With an initial state, however, you only need to specify the size of the game board.

Have a great Thanksgiving! Also, if you will happen to be in the South-Central Ohio area next week, I will be giving a talk here at Wittenberg on Matchmaker on Thursday (Dec. 3). More info about that soon!