1)

The "question" posed by the referee is a three-bit string taken from the set $\{000, 011, 101, 110\}$. The task of Alice, Bob and Charlie is then to give the correct "global" answer for any given input. At every turn, each of them can answer with "$0$" or with "$1$".

Correct here means that it must satisfy a specific set of winning conditions. More specifically, if the referee says $000$, then A&B&C win if and only if the binary sum of their answers is $0$. For example, the answer $000$ wins, and so does the answer $110$. On the other hand, the answer $010$ is wrong and makes them lose.

When on the other hand the "question" (that is, again, the given bitstring) is one of the others, say $110$, then the winning conditions are inverted, so that for example $000$ loses and $010$ wins.

So again, the round proceed as follows:

1. Referee chooses one of the four possible words $000$, $011$, $101$, $110$.
2. The first bit of the chosen word is given to Alice, the second to Bob, and the third to Charlie.
3. Alice, Bob and Charlie each answer with either $0$ or $1$ (the choice of $0$ or $1$ here is just conventional, other references use $+1$ and $-1$, or just two different generic symbols).
4. Alice, Bob and Charlie win or lose depending on whether they gave the correct answer or not, according to a specific "truth table".

2)

Why use that specific rule to decide whether they win or lose the game? Simply because it works! In other words, it turns out (and this is exactly what is being explained in the text) that with that particular choice of winning condition one can show easily that any classical strategy cannot do better than a specific amount, while quantum mechanics allows to do better.

The arbitrariness of the rule is not important here, what it shows is that there exist circumstances in which QM allows to win a game better than what is possible with only classical rules.

3)

I do not fully understand what you write, but the reason only those particular four words are given by the referee is, again, because it results in a game that displays the characteristics we are looking for. You can change the game by allowing the referee to give addition question words, and then see if you get the same classical/quantum performances.

4)

Maybe it would help to completely change the notation?

Let us say that the referee has a bunch of apples ($0$) and bananas ($1$). At every round it distributes them among Alice Bob and Charlie, but not in a random fashion: he either gives apples to everyone, or an apple to someone and bananas to the others.

Alice Bob and Charlie answer by each giving back, say, a mango ($0$) or a papaya ($1$). The winning conditions are as follows:

1. If everyone got an apple, then they must give back either zero or two papayas.
2. If someone (that is, two people) got a banana, then they must give back either one or three papayas.

This is of course totally equivalent to the version with $0$s and $1$s. The only difference is that when using $0$s and $1$s (or $+1$ and $-1$) it can be easier to express some of the winning rules, using logical operators and such.

1)

The "question" posed by the referee is a three-bit string taken from the set $\{000, 011, 101, 110\}$. The task of Alice, Bob and Charlie is then to give the correct "global" answer for any given input. At every turn, each of them can answer with "$0$" or with "$1$".

Correct here means that it must satisfy a specific set of winning conditions. More specifically, if the referee says $000$, then A&B&C win if and only if the binary sum of their answers is $0$. For example, the answer $000$ wins, and so does the answer $110$. On the other hand, the answer $010$ is wrong and makes them lose.

When on the other hand the "question" (that is, again, the given bitstring) is one of the others, say $110$, then the winning conditions are inverted, so that for example $000$ loses and $010$ wins.

So again, the round proceeds as follows:

1. Referee chooses one of the four possible words $000$, $011$, $101$, $110$.
2. The first bit of the chosen word is given to Alice, the second to Bob, and the third to Charlie.
3. Alice, Bob and Charlie each answer with either $0$ or $1$ (the choice of $0$ or $1$ here is just conventional, other references use $+1$ and $-1$, or just two different generic symbols).
4. Alice, Bob and Charlie win or lose depending on whether they gave the correct answer or not, according to a specific "truth table".

2)

Why use that specific rule to decide whether they win or lose the game? Simply because it works! In other words, it turns out (and this is exactly what is being explained in the text) that with that particular choice of winning condition one can show easily that any classical strategy cannot do better than a specific amount, while quantum mechanics allows to do better.

The arbitrariness of the rule is not important here, what it shows is that there exist circumstances in which QM allows to win a game better than what is possible with only classical rules.

3)

I do not fully understand what you write, but the reason only those particular four words are given by the referee is, again, because it results in a game that displays the characteristics we are looking for. You can change the game by allowing the referee to give addition question words, and then see if you get the same classical/quantum performances.

4)

Maybe it would help to completely change the notation?

Let us say that the referee has a bunch of apples ($0$) and bananas ($1$). At every round it distributes them among Alice Bob and Charlie, but not in a random fashion: he either gives apples to everyone, or an apple to someone and bananas to the others.

Alice Bob and Charlie answer by each giving back, say, a mango ($0$) or a papaya ($1$). The winning conditions are as follows:

1. If everyone got an apple, then they must give back either zero or two papayas.
2. If someone (that is, two people) got a banana, then they must give back either one or three papayas.

This is of course totally equivalent to the version with $0$s and $1$s. The only difference is that when using $0$s and $1$s (or $+1$ and $-1$) it can be easier to express some of the winning rules, using logical operators and such.

1)

The "question" posed by the referee is a three-bit string taken from the set $\{000, 011, 101, 110\}$. The task of Alice, Bob and Charlie is then to give the correct "global" answer for any given input. At every turn, each of them can answer with "$0$" or with "$1$".

Correct here means that it must satisfy a specific set of winning conditions. More specifically, if the referee says $000$, then A&B&C win if and only if the binary sum of their answers is $0$. For example, the answer $000$ wins, and so does the answer $110$. On the other hand, the answer $010$ is wrong and makes them lose.

When on the other hand the "question" (that is, again, the given bitstring) is one of the others, say $110$, then the winning conditions are inverted, so that for example $000$ loses and $010$ wins.

So again, the round proceed as follows:

1. Referee chooses one of the four possible words $000$, $011$, $101$, $110$.
2. The first bit of the chosen word is given to Alice, the second to Bob, and the third to Charlie.
3. Alice, Bob and Charlie each answer with either $0$ or $1$ (the choice of $0$ or $1$ here is just conventional, other references use $+1$ and $-1$, or just two different generic symbols).
4. Alice, Bob and Charlie win or lose depending on whether they gave the correct answer or not, according to a specific "truth table".

2)

Why use that specific rule to decide whether they win or lose the game? Simply because it works! In other words, it turns out (and this is exactly what is being explained in the text) that with that particular choice of winning condition one can show easily that any classical strategy cannot do better than a specific amount, while quantum mechanics allows to do better.

The arbitrariness of the rule is not important here, what it shows is that there exist circumstances in which QM allows to win a game better than what is possible with only classical rules.

3)

I do not fully understand what you write, but the reason only those particular four words are given by the referee is, again, because it results in a game that displays the characteristics we are looking for. You can change the game by allowing the referee to give addition question words, and then see if you get the same classical/quantum performances.

1)

The "question" posed by the referee is a three-bit string taken from the set $\{000, 011, 101, 110\}$. The task of Alice, Bob and Charlie is then to give the correct "global" answer for any given input. At every turn, each of them can answer with "$0$" or with "$1$".

Correct here means that it must satisfy a specific set of winning conditions. More specifically, if the referee says $000$, then A&B&C win if and only if the binary sum of their answers is $0$. For example, the answer $000$ wins, and so does the answer $110$. On the other hand, the answer $010$ is wrong and makes them lose.

When on the other hand the "question" (that is, again, the given bitstring) is one of the others, say $110$, then the winning conditions are inverted, so that for example $000$ loses and $010$ wins.

So again, the round proceed as follows:

1. Referee chooses one of the four possible words $000$, $011$, $101$, $110$.
2. The first bit of the chosen word is given to Alice, the second to Bob, and the third to Charlie.
3. Alice, Bob and Charlie each answer with either $0$ or $1$ (the choice of $0$ or $1$ here is just conventional, other references use $+1$ and $-1$, or just two different generic symbols).
4. Alice, Bob and Charlie win or lose depending on whether they gave the correct answer or not, according to a specific "truth table".

2)

Why use that specific rule to decide whether they win or lose the game? Simply because it works! In other words, it turns out (and this is exactly what is being explained in the text) that with that particular choice of winning condition one can show easily that any classical strategy cannot do better than a specific amount, while quantum mechanics allows to do better.

The arbitrariness of the rule is not important here, what it shows is that there exist circumstances in which QM allows to win a game better than what is possible with only classical rules.

3)

I do not fully understand what you write, but the reason only those particular four words are given by the referee is, again, because it results in a game that displays the characteristics we are looking for. You can change the game by allowing the referee to give addition question words, and then see if you get the same classical/quantum performances.

4)

Maybe it would help to completely change the notation?

Let us say that the referee has a bunch of apples ($0$) and bananas ($1$). At every round it distributes them among Alice Bob and Charlie, but not in a random fashion: he either gives apples to everyone, or an apple to someone and bananas to the others.

Alice Bob and Charlie answer by each giving back, say, a mango ($0$) or a papaya ($1$). The winning conditions are as follows:

1. If everyone got an apple, then they must give back either zero or two papayas.
2. If someone (that is, two people) got a banana, then they must give back either one or three papayas.

This is of course totally equivalent to the version with $0$s and $1$s. The only difference is that when using $0$s and $1$s (or $+1$ and $-1$) it can be easier to express some of the winning rules, using logical operators and such.