Rock-Paper-Scissors Can Be Weird

post by winwonce · 2022-12-28T23:12:11.329Z · LW · GW · 3 comments

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3 comments

Let's consider a symmetrical, two-player, zero-sum game, where each player must simultaneously choose one of three moves: A, B, or C. After each player choses their move, a winner is randomly selected with odds based on what the two players selected. (Note that because its symmetrical, if both players choose the same strategy, it is guaranteed to be a 50/50 coin flip for who wins).

In the simplest case, one of the moves wins more than 50% of the time against both of the other two moves. In this case, the best strategy is to always select this move.

Now let's consider a rock-paper-scissors type game. Say A beats B with probability 0.6, B beats C with probability 0.6, and C beats A with probability 0.6. In this game, the best strategy is to chose each of the moves A, B, and C with probability 1/3. So far, so good.

Now let's improve move C by making it beat move A with probability 1, leaving the other two fixed (so A beats B 60% of the time, and B beats C 60% of the time). What does the optimal play look like now?

Even if you don't want to fully compute it, notice what your intuition is telling you should happen.

As it turns out, the optimal strategy is to play move A 1/7 of the time, move B 5/7 of the time, and move C 1/7 of the time. Against this strategy, move A wins (0.5)(1/7) + (0.6)(5/7) + (0)(1/7) = 1/2 of the time, move B wins (0.4)(1/7) + (0.5)(5/7) + (0.6)(1/7) = 1/2 of the time, and move C wins (1)(1/7) + (0.4)(5/7) + 0.5(1/7) = 1/2 of the time, so there is no way to beat this strategy: it is optimal.

This was really surprising to me! We improved move C, and somehow people want to play is less. Is there a name for this effect?

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comment by Dagon · 2022-12-29T02:07:45.178Z · LW(p) · GW(p)

Cool result!  I expected it would make you chose B more often, but thought it would also make you choose C more often and almost never A.  I absolutely did not expect the symmetry between C and A, nor the magnitude of difference between B and the others.

In this game, the best strategy is to chose each of the moves A, B, and C with probability 1/3.

Not necessarily.  If your opponent is easier for you to predict or manipulate than you are for them, you should psyche them out, and pick the move that beats their most likely move.

Which is similar to the reason that "improving" C makes it less attractive to play.  It's by far the most attractive move to make in a non-predictive environment, so the best one to attack.  

comment by Vitor · 2022-12-29T03:03:18.480Z · LW(p) · GW(p)

The name I have in my head for this is "zones of control". In board- and videogames, sometimes a unit explicitly has an effect on tiles adjacent to its own. I expanded the term from there to include related phenomena, for example where the mere existence of strategy X blocks strategy Y from ever being played, even if X itself is almost never played either. X is in some sense providing "cover fire", not achieving anything directly, but pinning down another strategy in the process.

This case doesn't match that intuition exactly, but it's in the same neighborhood.

comment by Ericf · 2022-12-29T02:34:38.415Z · LW(p) · GW(p)

See also https://www.sirlin.net/articles/designing-yomi for more on this phenomena