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Guessing game idea
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.
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While one of its charms is that it's quite easy to play, if you want to play it well it's essential to keep track of what you know the other players know, both in order to guess what extra information they might posess and in order to gauge how much your own questioning will reveal.
I think it encapsulates the gameplay elements you're considering, and a few more besides.
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How about:
1) Each ship has a radius of fire equal to the number of surviving segments of the ship (i.e. the aircraft carrier starts with radius 5, reducing to 4,3,2,1 as it gets progressively damages).
2) You can only fire at sectors within the radius of fire of at least one of your ships
3) A player may move any undamaged ship to a new location, but must forgo their following n turns.
That means that each shot gives away a small amount about your own deployment, but not too much. Possibly the player should also declare the size of ship used to make any given shot.
(3) is necessary as (2) means that at least some of the board is unlikely to be within range of any ship. I also intrinsically approve of allowing some degree of manoeuvering.
n would need to be set to discourage moving ships except when absolutely necessary, to avoid things getting too frustrating. Or perhaps only one of the ships should be mobile (the submarine?)
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Also, it'd be important to be able to track the sequence of the other player's shots over time where movement is allowed, and the standard board game set is not really suited to this. A computer implementation would, of course, resolve this one.
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The second objection I think lends itself to the suggestion of having only a subset of the ships be mobile. Say two of them to avoid the game being decided by a single lucky shot.
Possibly the mobile ship(s) should have a lowered radius to make them more detectable - thus there's a real danger to launching a submarine strike. You get one or two shots before you have to move it as you're narrowing down its position too far.
The endgame would come down to a battle between keeping your mobile units out of danger, and having to use them because the rest of your fleet is dead.
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Would it perhaps make sense to limit the mobility to, say, one unit forwards or a 90-degree turn, taken in place of a single shot rather than n turns?
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I don't think limiting speed and direction of movement is viable - the movement is needed because reducing field of fire means you don't cover the whole board and thus can't find the enemy. Slow movement such as you suggest does nothing to alleviate this.
I think that having a small number of mobile ships with a very small fire radius is better than an n turn penalty - it means that you can move it, but as soon as you *use* it, you're vulnerable.
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My first thought was the obvious geek response "You have to rule out questions like 'Are you me OR does your card have red hair? My answer is 'yes''" :)
Also, note, it's not *completely* hopeless in Who's Who. Say 4/5 cards have brown hair and yours does. Doesn't that put you at an advantage to ask, because he might not have, then you've gained more info? Of course, he might then deduce brown hair was your most ordinary feature because that would have this effect the most. But you could bluff.
Also, could there be an interesting endgame? If any information is likely to give the win to the other player, like the three-way gunfight, you might get a standoff where people keep asking oblique questions.
Of course, with a fairly limited set of choices sometimes a game simplifies to make this largely irrelevant and you can find a good strategies almost by trial and error. I remember dad telling a story about a game of bingo played with music instead of numbers, except they always played the same tape, which meant the same card always won :)
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Going back to Cluedo for a moment, it's quite imporant always to choose as randomly as possible between accusations your opponents currently regard as equivalent. A common mistake is to favour asking questions about cards in your hand over asking questions about cards you know form part of the solution. (-8
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I meant... Suppose my analysis was correct in that asking question about the commonest features of your character gave you the most information. And you had a very common hair colour (brown) and a fairly common eye colour (brown), but asked about eye colour, then he might incorrectly deduce you had brown hair, and act on that assumption. Is that not bluff?
Going back to Cluedo for a moment,
Agh. Yes, you're right. I was always hopeless at cluedo :)
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Say you know that, from your opponent's perspective, it is 90% likely you have brown hair, and 60% likely you have brown eyes. You are about to confirm one of those facts, and wish to do so without giving any information about the other one. The two are independent, so overall the probability of both brown eyes and hair is 54%, brown hair only is 36%, brown eyes only 6%, neither 4%.
I think the 36% versus 6% means that when you have both you should six times out of seven exhibit brown hair, and one time out of seven exhibit brown eyes. If you do anything else, your opponent can exploit the systematic bias… if they spot it.
But yes, bluff is an issue apart from such strategising. Bluffing is simply adopting a strategy other than that which your opponent thinks you've adopted. In general, the more subtle the strategies, the more subtle the bluffs. (-8
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