Purely Tactical Games

Some fun math with applicability to real games

Board games are said to have ‘position’ which is about the longer term implications of what’s happening on the board, and ‘tactics’ which is amount immediate consequences. Let’s consider a game which is purely tactical. In this game the two sides alternate picking a bit which is added to a string and after they’ve both moved 64 times the secure hash of the string is calculated and that’s used to pick the winner. I suggest 64 as the number of moves because it’s cryptographic in size, so the initial moves will have unclear meanings and will become clearer towards the end of the game.

The first question to ask about this is what are the chances that the first player to move will have a theoretical win, assuming both sides have unlimited computational capability. It turns out if the chances are greater or less than a certain special value then the probability of one particular side having the win goes up rapidly as you get further from the end of the game. If the win probability is set to exactly that value then the winning chances remain stable as you calculate backwards. Calculating this value is left as an exercise to the reader. The interesting thing is that the value isn’t 50%, in fact it’s fairly partisan, which raises the question of whether the level of advantage for white in Chess is set about right to optimize for it being a tense game.

There are other variants possible. The number of possible plays could be more than 2, or somewhat variable since you might have a chance of making the opponent skip their turn and you go again. This would allow the ‘effective’ fanout to be something non-integer, but it’s an interesting question whether there’s a way for it to be less than 2.

There’s a variant where instead of there being a fixed number of moves in a game after each move the player who just moved has some probability of winning (or losing). It isn’t obvious whether any win probability guarantees a 100% probability that the game is winnable by one side or the other. It seems like that should be a foundational result in computer science but I’m unfamiliar with it.

In practice of course analyzing this sort of game is constrained by computational ability. That can be ‘emulated’ by assuming that the outcomes are truly random and there’s an oracle which can be accessed a set number of times on one’s turn to say who wins/whether a player wins in a given position. There are a lot of variants possible based on the amount of queries the sides have and whether there’s optionality in it and whether you can think on the opponent’s time. It feels like optimal play is slightly randomized. Intuitively if one player has more thinking time than the other then the weaker player needs to mix things up a bit so their analysis isn’t just a subset of what the opponent is seeing. But this is a wild guess. Real analysis of this sort of game would be very interesting.