Why isn't the best case classical solution to the CHSH game 100%? [closed]

Question

[Edit 2] I would prefer to just forget that I had ever asked this question (because I was so wrong it's embarrassing), but for the sake of people who possibly make the same mistake I did, I'll try to fix the question so it accurately represents what I was trying to ask. In the end, it wasn't a constraint in the original theory that prevented the use of my strategy, it was that my "100% chance to win" was just a mathematical goof on my part, and the actual win chance for my strategy was 75%, just like it was supposed to be.

First off, please forgive my ignorance when it comes to physics and Quantum Mechanics, I'm just a programmer that likes to pretend he understands science.

There is an experiment that is used to prove that there are no "hidden variables" at work in quantum weirdness, but that particles in a superposition of states really do collapse only at the moment of measurement.

The experiment uses a game known as the "CHSH game" (summarized at http://www.americanscientist.org/issues/num2/2014/4/quantum-randomness/1) which is basically

With cooperating players "Alice" and "Bob":

1. After agreeing on a strategy they are separated and cannot communicate with each other in any way
2. They are each given a sealed envelope that randomly contains either a red card or a blue card
3. After being separated, they each open the envelope and must then raise either one or two fingers
4. They win if both cards are red and they raise a different number of fingers, or either card is blue and they raise the same number of fingers.

In the game, the best classical strategy for winning is for the players to ignore their cards and both raise one finger, which gives a 75% chance of winning. However, if they each have one of a pair of entangled particles and measure them slightly differently depending on which card they chose, they can increase their chance of winning to ~85.4%, which is only possible because the method the first person used to measure their particle changed the results that the other person would get when they measured their particle. The difference in the two chances of winning is known as Bell's Inequality, and is used as proof that there's actual quantum weirdness going on.

But it seems to me that the best classical strategy for the game (as summarized above) isn't to always raise one finger, but for Bob to always raise 2 fingers, and for Alice to raise 1 finger when her card is red, and 2 fingers when her card is blue. This results in a 100% chance to win.

This is such a simple strategy that there must be some condition in Bell's theory that prevents it from being used. Or perhaps the summary of the CHSH game in the article linked above skipped over some well-known context to the game that prevented it's use.

So what are the conditions in Bell's theory that prevent this 100% win strategy from being used in the CHSH game?

Answer 1

If Bob's card is blue and Alice's is red, then your strategy leads to a loss. (One card is blue, but they raise different numbers of fingers.) So your strategy matches the best possible 75% win rate but doesn't beat it.

Answer 2

Here are all possibilities:

| C1 | C2 | F1 | F2 | O |
| -- | -- | -- | -- | - |
|  R |  R |  1 |  1 | L |
|  R |  R |  1 |  2 | W |
|  R |  R |  2 |  1 | W |
|  R |  R |  2 |  2 | L |
|  R |  B |  1 |  1 | W |
|  R |  B |  1 |  2 | L |
|  R |  B |  2 |  1 | L |
|  R |  B |  2 |  2 | W |
|  B |  R |  1 |  1 | W |
|  B |  R |  1 |  2 | L |
|  B |  R |  2 |  1 | L |
|  B |  R |  2 |  2 | W |
|  B |  B |  1 |  1 | W |
|  B |  B |  1 |  2 | L |
|  B |  B |  2 |  1 | L |
|  B |  B |  2 |  2 | W |

Legend:

* C1 = card 1 (Red / Blue)
* C2 = card 2 (Red / Blue)
* F1 = fingers 1 (1 or 2 fingers)
* F2 = fingers 2 (1 or 2 fingers)
* O = outcome (Win / Loss)

The strategy of "always raise 1 finger" (independent of the color of your card) has a 75% probability of winning:

| C1 | C2 | F1 | F2 | O |
| -- | -- | -- | -- | - |
|  R |  R |  1 |  1 | L |
|  R |  B |  1 |  1 | W |
|  B |  R |  1 |  1 | W |
|  B |  B |  1 |  1 | W |