Balderdash

[jack]

I assume it goes without saying that I'm not very good at Balderdash :) However, I was wondering about the rules.

The basic idea (varied slightly in different incarnations) is that one player picks a random word (or film title, or acronym[1]), either by drawing the next card in the pile from the published game, or by flopping a dictionary open in the freestyle version.

She announces it, and then each other player invents a definition, handed to her on a bit of paper. Then she reads them all out, and each player guesses which is correct. You get one point for guessing the correct definition, and one point for each person who guesses yours.

This has the implicit assumption that no-one will know the definition already. Typically, if you know the right definition beforehand, you get N points (where N is 1 or larger), but don't participate in the rest of the round. (One set of rules says the rest of the round is cancelled if several players do.)[2]

The question is, what's the fairest way of doing it? Should players be rewarded for knowing? It's actually barely related to the real point of the game. Giving them a bonus and moving on seems most sensible.

Another option would be that you don't get anything extra; you make up a definition anyway, and just get one extra point for voting for the right one.

Another would be that your definition is entered, and everyone who votes for an equivalent answer to the real one gets a point, but you get a point for anyone who votes for yours instead of the real one.

The things to avoid are: it being an advantage to *not* know the answer, which really seems unfair, and putting too much judgement on the caller. After all, if she doesn't know wha

[1] You know what I mean.

[2] Did someone in fact get it right?

Words are ok, you generally know or not.

Complete the silly law is ok for the opposite reason. They're all made up, so (unless by an immense stroke of luck, you actually really know the answer), you don't get any points for saying something else it's illegal to do on Tuesdays in Cardiff -- after all, there's lots of things -- you have to guess what's on the card.

But we had difficulty with people. Do you have to guess whatever's on the card, however weird? The correct answer for "John Dee" was "invented the crystal ball" and for "Christian Huygens" was "invented the pendulum clock"[3]. Do you get points for saying "British court magician/philosopher" and "Dutch physicist and astronomer"[4]?

Those are possibly less specific, but a whole lot more accurate. They (when we went over to the internet) basically the first sentences of the wikipedia entries.

But if so, how is the caller supposed to know if they're accurate or not? I guessed that John Dee supposedly invented the crystal ball, but I wouldn't done if I didn't know who he was.

[3] Leaving aside the inaccuracy of that.

[4] I wasn't *sure* of either. I knew the scientists I was thinking of existed and had similar names, but I could equally well have been confused with "Jack Dee" and "Hayden Christensen"

Comments

[hairyears.livejournal.com]

Oddly enough, we were playing this in the Pembury last night - the freestyle dictionary version - and we were wondering what the 'dealer' would score.

That is to say, the person who opens the dictionary on that turn of play: they, too, scribble down a definition (in this case, the right one), and read out all the definitions in a random order.

In boxed-set versions of the game using a card deck of obscure words, the rules state that the dealer scores 1 if no-one gets the right answer; our consensus at the Pembury was that a fair score would be n/2 rounded down, where n is the number of players.

Do you have the logical tools to perform a sum-of-all-outcomes analysis and tell us whether this is the correct approach? I suspect that one or two of the postdoctoral algorithmatists around the table did the math in their heads and the theoretical linguist didn't complain... Me, I just do spreadsheets and a bit of VBA and I am somewhat at a loss here.

Note that our game relied on a fair degree of honesty, as we agreed to tell the dealer to choose another word if any of the players knew the correct definition at the start of the turn. This was a deliberate decision to make the game a test of our ability to bluff, mislead, and guess; a test of our vocabulary and powers of recall would require many hundreds of rounds to achieve a meaningful result, as there are very few words in Collins' dictionary that our glittering assembly of linguists, scientists, academics, polymaths and lexicographical geeks don't know.

Doubtless this is different in the Fens, where the only word they know is 'Ug' and they don't know how to spell it.

[cartesiandaemon.livejournal.com]

That is to say, the person who opens the dictionary on that turn of play:

:) I was going to say "dealer" too, but thought it would confuse more people than it helped. That's the rule I heard too, though I didn't try to explain it.

our consensus at the Pembury was that a fair score would be n/2 rounded down, where n is the number of players.

I don't know, but I can guess. (I appreciate the confidence that I might be able to :)) It depends what you seek to achieve. I see two considerations (am I missing more? I've only played about twice...):

#1: Balance the scores out, so if you don't have a whole number of rotations, it's neither an advantage or disadvantage to be dealer more often
#2: Encourage fair play, reading out all the descriptions in the same tone of voice, etc.

I'm going to think aloud a bit, apologies if I ramble...

#2 doesn't seem to make much difference. They have little control, so a *small* reward seems appropriate, one point. Do they choose the word, or just scan for the first sufficiently obscure word on the page? If you were asking any choice, a larger outcome might make sense. Eg. if you wanted words people *might* be able to guess[1].

But so long as they just have to pick something really obscure, I think not scoring anything at all would be fine.

That leaves #2. They should score the average of all the other scores that round, so they don't fall behind/get ahead. Maybe with a one-point swing if no-one guesses it for the above reasons.

The expected number of scores? If no-one guesses the right answer, the average is one point each: there are N non-dealer players, each giving a point to one other player by voting for them. N/N=1.

If n of the players do guess the right answer, suppose they all score one point each. That would be the same total: each player gives out one point, to someone else if they guess wrong, or themselves if they guess right.

In either of those cases, it would be fair to just give one point to the dealer.

What would be a fair score given only if no-one guesses? That would satisfy both criteria. I'd advocate giving the smallest swing possible, but if the average has to be one, then the score when anyone does guess must be less than one, so nought.

You'd have to divide by the chance no-one guesses to get the dealer's average score, and make that equal to 1. If the players are really good, each choice will be random, so there'll be an [(N-1)/N]^N chance no-one guesses it. That's approximately three points for 3-10 non-dealer players, though you could work it out at the time if you wanted.

If the players guess the right answer more often than average, then the dealer would need a higher score when none of them do -- I don't know what the chance of that is.

If the players score *more* than one point for guessing right -- or for writing the right answer in advance -- then it's harder to calculate, even the average score depends on the number of people who guess.

However, if you score 2 points for guessing right, and on average one person per round guesses right, then that's one extra point over everyone, ie. the average is (N+1)/N = 1+1/N, so you could basically forget it.

That was a bit disjointed, hopefully, I'll have time later to rewrite it to be a bit more helpful (I'm going out now). Although, more likely, also hopefully, someone'll come along and correct the answer before I have a chance :)

Briefly:

* Always getting one point is pretty fair
* Always getting three points is fairly fair
* But I'm not certain, I need to check. N/2 isn't very far off, though I don't think is correct. (But I've a horrible feeling I've missed something obvious and that is right after all.) It might well be fairest if people guess the right answer every so often.

How often *do* you find people guess rightly? I seem to recall it fairly rare, none, one or two a round. But don't know if that's normal.

[1] That can be fun, choosing a word you think *one* person would have heard of, etc. Though wouldn't be appropriate here and probably too complicated even if it were. FWIW, I did see a game designed for that sort of judgement, and it was ingenious, though I wasn't particularly good at that either.

[cartesiandaemon.livejournal.com]

OK, sober reanalysis. I'm probably slightly off the mark, I think I need some things cleared up.

* Skipping the round if anyone knows the word is probably very sensible. You *could* do it without, but you'd have to decide a fair reward, and does create procedural difficulties. If only one person did, giving them a slight bonus would probably be reasonable. Although personally I like knowing whether its a game of bluff or not, more than which it is, so prefer your method.

* And conveniently, that makes the maths easier.

* If you don't care about rewarding the dealer for no-one guessing right, but want the score to be fair if you stop half way through, you could just give the dealer an average of the scores got by the other players that round.

* Alternatively, you could give the dealer a constant score, based on the expected average total score per round. This is easy to calculate: each round scores one point per guess, plus one point for every correct guess. If you assume everyone guess right 1/Nth of the time, that's a total score per round of N (for N people) + N*1/N (N people each having a 1/N chance of guessing right.) = N+1. Divide by N to get the average per player, =1+1/N. You could call that 1, have fractional scores, roll a die and give the player a 1 in N chance of one extra point, or something like that, if you cared.

* If you don't care about the score being exactly fair if you stop half way through, you could just give the dealer one point. That's nearly right, but the dealer falls slightly behind whenever anyone guesses the right answer (one person per round, on average).

* If you want both, you must give the dealer no points when someone guesses right, and more points when no-one does, having an average equal to the average score in a round. Ie. if everyone is wrong every 5 rounds, give the dealer 5 times (the average we worked out in bullet point 2).

Except what is 5 really? If you assume the guesses are random, the chance of person failing to guess correctly is N-1 out of N. The chance of N people *each* failing, is that to the power N. [(N-1)/N]^N

This is the number that I said was approximately 1/3, ie every three rounds. Mysteriously, because we're doing maths, if N is large you get an exact answer -- once every e (=2.718...) rounds.

[cartesiandaemon.livejournal.com]

Gah. I'm *still* wrong. The chance of guess correctly is (N-2/N-1)^N, because each person is only guessing amongst the definitions they themselves didn't submit. I think that still gives the same answer though.

[hairyears.livejournal.com]

Yes... I ran in to algebraic errors when I tried to reconcile the two approaches, the chance of any player guessing correctly per round vs the chance of no player guessing correctly per round, but eventually came out with an expected guessing score for each player in each round of (N-2/N-1).

However, that's just the 'guessing' score: points are also awarded for inventing the balderdash that each player guessed at. That made it far more interesting... It's a bit early in the morning for stretching the brain, but I'm chasing down an intuitive belief that the strategy of never giving an answer that someone else gave is sound, because it depresses the other players' 'balderdash' scores.

We are departing from the game as played - and as intended - because I happen to believe in rewarding the dealer for misleading the other players. However, the equal random chance model is a far simpler mathematical exercise.

The maximum potential balderdash score is N-1 (everyone could guess your invented answer); the minimum is zero. Ignoring gameplay, that suggests an expected score per player per round of (N-1)/2 for balderdash, plus (N-2/N-1) for guessing.

A simplified approach to fairness is to give the dealer a 'pass' score equal to this and (N/2) rounded up is a workable approximation... But, if you regard the dealer as a participant who should be rewarded with a score for successfully misdirecting the other players away from the right answer, we need to establish some simple relatiionship between misdirecting N players way from one answer and the expected 'balderdash' score of (N-1)/2.

[cartesiandaemon.livejournal.com]

Something like that, although:

* I didn't quite follow what you meant by guessing score? The score from guessing the correct answer? Wouldn't that be 1/N-1?, N-1 answers which aren't yours, of which one is correct? N-2/N-1 looks like the chance of guessing someone else's answer, isn't that supposed to be covered by their balderdash score?

* "The maximum potential balderdash score is N-1 (everyone could guess your invented answer); the minimum is zero. Ignoring gameplay, that suggests an expected score per player per round of (N-1)/2 for balderdash,"

Right, except the average doesn't work if the guesses are random, because the chance of an N-1 score is quite low.

* We are departing from the game as played

Ah, thank you. I see. Does that mean the dealer actively commenting trying to mislead players? I've no idea what would be fair score in that case.