Game theory question: iSketch

[Stereolab]

I don't know if any of you have played, but there is a fun game at www.isketch.net. It's like Pictionary, etc. where you draw a picture and others have to guess the word or phrase it represents. It's quite fun.

My question is regarding optimal strategy. Virtually all the players on the website believe that the winning strategy at all times is to get many other players to guess your word. Even the rules seem to indicate that, but it is clearly not the case. Here are the rules:

<b>Artist:
The artist receives 10 points for the first correct guess.
One (1) point for each additional guess is awarded, up to a maximum of 5, and a total earning potential of 15 points per round.

Guesser:
The first guesser is awarded 10 points.
The 2nd guesser is awarded 9 points, 3rd guesser gets 8 points and so on, with a minimum of 5 points.

The scoring system ensures a big payout for guessing quickly, drawing effectively and getting as many correct guessers as you can. </b>

Clearly you do not want more than five people to guess your word. (Let's say this is a ten-person game.) If everyone gets it, four other players get points while you get no bonus. That is not ideal under any circumstance.

(While you obviously cannot determine exactly how many people will get your word, you get a feel for how detailed/accurate a picture is needed for a certain number of correct answers.)

In many games, I will work on a fairly abstract picture until someone gets it, then stop drawing. This puts me ten points above just about everyone else in the room. In many cases it is not worth putting extra people in the pack just for one additional point. It is somewhat amusing to see other players insist I am not doing what it takes to win, as they are simultaneously losing to me.

Of course, the above means that someone else has as many points as I do, and if all players had the same skill level and played in that fashion, the expected outcome would be a tie.

ANYWAY, my question is whether there is an optimal strategy / expected equilibrium for this game. Note: Assume the game lasts ten rounds, with each player having a turn to draw.

Finally, most players on the site are playing for fun and don't really care about the score. I don't take the thing too seriously, and I go with the flow if everyone wants a shot at a good picture. But--in any game with points, you deserve to be able to play to win. That is what I am trying to figure out what to do.

Thoughts?

 

[CapelDodger]

If everyone shared your strategy - which I think is sound - winning comes down to superior guessing. The depiction points on each round are a given - they go to the artist - but the guessing points are up for grabs.

This assumes that the other players don't collude in refusing to guess at your depiction. Because you're not playing in the spirit of the game, or because you're not doing it right but the bastard's still winning ,or whatever. That would mean they wouldn't themselves converge on your strategy and would therefore be playing sub-optimum strategies, which is fine and dandy for you anyway.

It's a shame there's no money involved.

 

[Art Vandelay]

If you don't mind, I'd like to redefine some terminology. Let's refer to a "turn" as an opportunity for a particular player to draw a picture, and a "round" as a sequence of turns in which each player gets one turn. If there are, say, ten rounds, then each points can be considered to have approximently the same utility, so in a game with n players, if you expect the other players to get about the same number of points, then you break even if all of you get one point. In such a case, you get 1 point, and "not you" gets n-1 points, so that suggests that a point to you is worth n-1 points not to you.

According to the assumptions that I've presented so far, then, you get the equivalent of 10(n-1) NYP (Not-You Points) for the first correct answer, but lose 10 NYP to whoever gets it. Since n-1 is presumably greater than one, that's profitable. One additional guess (I'll use the term "additional guess" to refer to anything beyond the first) gets you just n-1 NYP, while the guesser gets 9. So if n-1<9, that's not good. In other words, if there are fewer than ten players, you don't want exactly two players to guess it. But if there are two additional guesses, then you get a total of 2(n-1) NYP, while losing 17. Now you want 2(n-1)>17, n-1>8 n>9.

Continuing, I found that if there are more than six players, you should try for all 15 points. If there are six or fewer, you should try for just one.

Of course, if you want exactly one correct guess, you might want to stop before there's a guess, since people may take a while to figure out what it is.

And this isn't really addressing your question. With only one round, we can't make the assumption that all points have the same utility. The analysis becomes much more complicated. It's simplified a bit by the assumption that the other players are of equal skill, so that the results of the other turns can be modeled as random variables. Also, the analysis is probably going to take into account what the current scores are (is this public knowledge?).

But let's say that your turn is first, so you have no information about other scores to worry about. The simplest game would be two players, but that would be rather trivial. The simplest game with any real game theory would be three players. Here, you have three possible outcomes: no guesses, one guess, or two. Clearly the first is the worst, so that just leaves the question of which of the other two are best. Let's say that only the second player guesses it correctly. Then when he draws, you could try to guess. If you get it first, you both will be tied at 20. AT this point, there's no way that the third player can win. One his own turn, he can get at most 11 points, which is not enough. He could get a total of 20 points by guessing on this turn, but then the second player has 21 points. The only way that you can win is if the second player doesn't guess, and you guess first on the third turn. Now, let's go back to the second turn. If the third player gets it first, then you will be tied for last place with 10, and the second player will have 20. You now can't possibly win.

So it's not looking like all that great of a strategy.

Another example where the best strategy isn't in the spirit of the game is a game where everyone presents purported definitions of a word. If someone thinks that your fake definition is right, you get a point. If you guess the correct one, you get two. So you can take a shot at two points (and probably give someone else a point) or guess your own definition, guaranteeing yourself a point.

 

[Stereolab]

Tied for best album of all time with Music for the Masses, I'd say.

 

[Stereolab]

<i>Also, the analysis is probably going to take into account what the current scores are (is this public knowledge?). </i>

Yes it is.

 

[EGarrett]

I just played it, I see what you mean.

It took me a game or two to get the hang of it, then I placed second three times in a row then tied for first and stopped playing.