I had a thought yesterday about guessing games.

It was brought on by seeing Benedict playing with a children's guessing game in the Carlton (which I think was called ‘Who's Who’ but might have been some trivial variant on that general theme). The basic idea of the game is that two players each choose one element from a small fixed set, and then take turns trying to guess the other player's choice by asking questions which narrow down the possibilities. In this particular case the concrete instantiation of this concept is that the elements of the set are pictures of people's faces, and the permitted questions involve things like hair colour, gender, facial features and so on; but that's not the important aspect.

It struck me that this game concept, as it stands, is essentially a race: there's no interaction between the two players' strategies. You're just trying to streamline your guessing process so that it completes as fast as possible, and the other player is trying to do the same. Hence, there's no reason you have to interleave your guesses at all: you could just as easily have one player set up a problem for the other one to guess, then reverse roles for the next problem and compare the number of guesses required; indeed, Mastermind already works this way.

So it occurred to me that the game might become more interesting if you could make the two guessing processes interdependent in some way. It's a well known phenomenon that in many situations asking a question gives away information; so I wondered if it might work to formalise this phenomenon as part of the game rules. Suppose that you played a guessing game of this type, with the additional rule that every time you asked a question about the opponent's hidden information you were also required to reveal what the answer to the same question would be if it were asked of you.

Then, suddenly, the emphasis would shift away from trying to construct questions which got you to the answer fastest, and more towards trying to think up questions which gave away as little as possible about your hidden state while finding out as much as possible about the opponent's. At the simplest level this would just be a matter of keeping in mind the relative sizes of the possibility sets (which elements have I ruled out for my opponent's secret? which elements can he not yet have ruled out for mine? what question reduces the former by more than the latter?), but after that you get into more interesting types of reasoning such as ‘if my opponent chose to reveal this rather than anything else, his secret probably wasn't that because in that case he could have given less away by playing the other’, and probably bluff creeps in somewhere as well.

When I thought of this yesterday, I was staring at a Who's Who set, so I was thinking mostly in terms of Who's Who; and I concluded at the time that the idea probably wouldn't work too well, because the only way a question could give away more for one person than another would be if the properties you could ask about weren't orthogonal (if, for example, more of the glasses-wearers had red hair than any other colour), which would mean good strategy was critically dependent on detailed knowledge of the particular data set used in the game, which feels unsatisfying. But having thought about it a bit more today, it strikes me that the concept might work much better in other guessing games, such as Mastermind (in which you can ask questions that aren't mutually orthogonal, even though the initial set of possible secrets itself is) or possibly Battleships (in which you give a lot more away by firing a shot which damages yourself than by firing one which doesn't; so your challenge is to figure out where the opposing ships are without scoring any hits on your own, and – if there turns out to be any overlap – to save your final few self-damaging shots until it's too late for the opponent to make good use of the information they reveal).

It sounds like a fun idea on paper, if a bit brain-bending; also it strikes me that someone must have thought of it, or something like it, before. If not (and perhaps even if so), I wonder if it's worth a try.