OT - Game strategy puzzle

[QuikSand]

Here's a newly-invented game, with some simple rules. For those among us who like to gamble a bit, perhaps you'll find it interesting.


Two players. Each simultaneously does two things: raises either one or two fingers, and speaks the number of fingers that he believes the other person will raise.

(For purposes of the puzzle, let's assume this is a perfectly fair game -- nobody makes any gains by slightly delaying his prediction, or his finger-raising... the game is just as it seems)

If both players correctly guess the other's number of fingers, it's a "push" and nobody wins.
If both players are incorrect, it's also a push.

But, if one player is right and the other player is wrong, then the correct player wins $1 from the incorrect player for every finger showing from that round (which will be either 2, 3, or 4).

Seems like a fair game, just a fun way to gamble a litle bit over blind luck, right?

Or is it?

Puzzle: Is there a way to "beat" this game, and tilt the odds in your favor? If so, describe how you would do so, and how much you'd expect to win against a player who hasn't figured out the angle.

edit - Puzzle continues in response below...

[albionmoonlight]

If I always put up two fingers and always guess two fingers and he is playing random, then

1/4 of the time, he will put up one finger and guess 1 finger (PUSH)

1/4 of the time, he will put up one finger and guess 2 fingers (he wins $3)

1/4 of the time, he will put up two fingers and guess 2 fingers (PUSH)

1/4 of the time, he will put up two fingers and guess 1 finger (I win $4)

If he is too dumb to figure out what I am doing, then I should expect to be up a dollar for every four rounds that we play.

[Silver Owl]

If you always hold up 1 finger and always call out 2 fingers you can only lose $2 per round but, can win $3. Assuming that the other person is 50-50 on 1 finger or 2 you should make more than you lose.

[cuervo72]

Well, my first inclination would be to always put up one finger, and always say 'two'. That way:

He flashes 2 fingers, guesses 1 - push
He flashes 2 fingers, guesses 2 - $3
He flashes 1 finger, guesses 1 - -$2
He flashes 1 finger, guesses 2 - push

You should be ahead in the long run, supposing his distribution is even. And that he doesn't catch on to what you're doing, which seems doubtful. Let's see what happens if you always display 2 fingers while guessing 1:

He flashes 2 fingers, guesses 1 - push
He flashes 2 fingers, guesses 2 - -$4
He flashes 1 finger, guesses 1 - $2
He flashes 1 finger, guesses 2 - push

Well, that obviously won't work. I'm not going to analyze the 1/1 and 2/2 combos, but maybe if you just never display 2 fingers while guessing 1?

[QuikSand]

Originally Posted by albionmoonlight

If I always put up two fingers and always guess two fingers and he is playing random, then

1/4 of the time, he will put up one finger and guess 1 finger (PUSH)

1/4 of the time, he will put up one finger and guess 2 fingers (he wins $3)

1/4 of the time, he will put up two fingers and guess 2 fingers (PUSH)

1/4 of the time, he will put up two fingers and guess 1 finger (I win $4)

If he is too dumb to figure out what I am doing, then I should expect to be up a dollar for every four rounds that we play.

Nice summary, well done.

Now for the hard part. If, as you suggest above, your opponent figures out what you're up to, and is smart enough to come up with the same "optimal" strategy on his own... what should you do?

[TRO]

Would Albion's strategy of lose 3, win 4 or Owl/cuervo's strategy of lose 2, win 3 be better?

On the surface, in a completely even format, they both seem like they would be +1 for every 4 rounds.

But I would think there is more to it. My inclination would be that the 2/3 format would be better as the ratio of amount gained to amount lost is much better.

[QuikSand]

Originally Posted by TRO

My inclination would be that the 2/3 format would be better....

That's usually a good instinct with my puzzles.