# Stephan WehnerBlog and Homepage

## May 10, 2010

### Rock, Paper, Scissors with Handicap

Filed under: rps — sw @ 12:12 pm

So I was playing Rock, Paper, Scissors with my daughter the other day. (It was her idea; I didn’t tell her about the game, my previous blog post about Rock, Paper, Scissors, nor about the new rpsmatch.com site). She won, and I lost a few rounds. Then I noticed: she wasn’t using “Paper”!

So I  thought of this variation on the game: one player is not allowed to use one of the weapons. All other rules stay the same.

To make it more interesting, the other player would not know which weapon is not allowed. How to do that? The handicapped player chooses one card from 3, each card showing a different weapon. The player is not allowed to use that weapon. Only after the winner is determined is the card shown to the other player for verification. If the handicapped player cheated by using the forbidden weapon, then they lost!

I doubt this change gives the non-handicapped player an advantage.  But you never know. The number of draws should decrease, though. Then again, maybe not. Would be nice to figure it out (mathematically)!

## 6 Comments »

1. Number of draws should stay the same. In a normal game each player would choose each weapon with 1/3 frequency so that there would be nine interactions each with a 1/9 probability. Three of these would be draws or 1/3 of the time you would get a draw. When one weapon is deleted then there are six interactions each with 1/6 probability (any departure from an even distribution can be taken advantage of by the other party) and 2 of these six are draws for 1/3 chance of a draw. If both sides deleted the same weapon then there are four interactions of which two are a tie for a 1/2 chance of a draw. If each player had a different weapon deleted then the game is no longer symmetric and one player will win 1/2 the time, 1/4 will draw and the other player will win 1/4 of the time. The dominant strategy for both players remains randomly choosing the available weapons.

Comment by Sean Murphy — August 7, 2010 @ 11:38 pm

2. Sean – how about after several rounds? The “forbidden” weapon would become known to both players with some likelihood, at which point the player with no handicap has a clear strategy to avoid losing. (Playing for 3 rounds is a kind of standard for the game) — Stephan

Comment by sw — August 8, 2010 @ 2:26 pm

3. If the handicap persists then the game degenerates into a 2×2 where the handicapped player wins 1/4, ties 1/4 and loses 1/2.

Comment by Sean Murphy — August 8, 2010 @ 8:06 pm

4. Three rounds probably wouldn’t be long enough to safely guess the handicap for the third round.

Comment by Sean Murphy — August 8, 2010 @ 8:08 pm

5. Once the handicapped player has used both allowed weapons, the non-handicapped will know the handicapped weapon. Then the non-handicapped player knows with which weapon they cannot lose: the weapon that the handicap-weapon would beat. (Example: If the handicapped player cannot choose Rock, then the non-handicapped player can safely choose Scissors)

That’s only two rounds, if the handicapped player chooses to switch between their two weapons.

Comment by sw — August 8, 2010 @ 9:55 pm

6. My friend competed in a paper, rock scissors tournament at a bar and noticed people have personal predisposition to one of the three choices generally associated with their personality which he could pick up semi-reliably like a poker read. I didn’t believe him until I saw him win a couple tournaments several weekends in a row. I only went for the free beer initially but its actually fairly amazing to see in action.

Comment by CC — August 11, 2010 @ 9:37 am