Can quantum mechanics be combined with game theory?

[Spathi]

I suppose, anybody here knows about the Elitzur–Vaidman bomb tester and the counterfactual definiteness:

https://en.wikipedia.org/wiki/Elitzur–Vaidman_bomb_tester

https://en.wikipedia.org/wiki/Counterfactual_definiteness

I have a question: can this experiment be performed at the level of countries for avoiding a nuclear war? I mean, if a dictator threatens the humanity a nuclear extermination, the states will have a difficult choice: according to all international laws, using nukes against the country of this dictator can be done only in response of his use. However, maybe a superposition can be created of two Earths (two universes) – in the first, the nuclear war has not started, and in the second, this dictator had pushed the button. Like in the Elitzur–Vaidman bomb experiment, the information that the dictator has pressed the button will be accessible in the first universe so the states will have the right to nuke the country of this dictator.

I remember that this forum does not like politics and philosophy, but my question relates physics only and is sufficiently specific and clear. We can discuss it without politics. Besides that, I have a question, what applications of the game theory are mostly useful. My idea is a suggestion of a new science, which combines the quantum mechanics and the game theory; the game theory has important use with the strategy of nuclear wars, and if the nuclear wars are not a good subject for this forum, I can think about combining quantum mechanics with other applications of the game theory.

 

[Frabjous]

Where are you using game theory?

 

[Spathi]

Where are you using game theory?

The game theory is broadly used in planning a strategy for a nuclear war. Two pics from Web googled:

 

[Frabjous]

Your scenario is just game theory using an information source that happens to be quantum. I would’t say that they are combined.

 

[Spathi]

The Game of Chicken illustrates cases where people can win through "rational irrationality":

https://en.wikipedia.org/wiki/Chicken_(game)

This is an explanation of the politics of Kim Jong Un and Putin, and my idea is that in the quantum reality the civilized world can find solutions against this. If it is not allowed to discuss the nuclear war at this forum, I will try to find other applications of the game theory which can be similar.
I suppose my idea does not look crazy, because the EV bomb experiment is "crazy" already and the physicists must know that.

 

[Spathi]

An example of a situation which can be studied by the game theory is the Truel:


https://en.wikipedia.org/wiki/Truel


One example of this game was described in a book of Martin Gardner. The shooters A, B, C have chosen a truel. A never misses, B hits 80% of the time, C hits 50% of the time. In such a game, C has the best chance of winning because he can shoot into the air first while A and B are shooting at each other.

My question is: is it possible to suggest some versions of this truel in which players have access to the Mach–Zehnder interferometerwith bombs, and this can change the probabilities of the outcomes. I mean something like this: player C does not shoot either A or B, but creates a superposition of these two states, and in one of the states also sends a message to B - if you don’t want me to shoot at you next time, do as I ask. I hope my idea can be understood.

 

[pines-demon]

if a dictator threatens the humanity a nuclear extermination, the states will have a difficult choice: according to all international laws, using nukes against the country of this dictator can be done only in response of his use. However, maybe a superposition can be created of two Earths (two universes) – in the first, the nuclear war has not started, and in the second, this dictator had pushed the button. Like in the Elitzur–Vaidman bomb experiment, the information that the dictator has pressed the button will be accessible in the first universe so the states will have the right to nuke the country of this dictator.

Be clearer with the analogy. How the dictators, action of pushing a botton and response of the other countries maps to the Elitzur–Vaidman (EV) experiment?

Does the action of pushing the botton correspond to the EV bomb?

My idea is a suggestion of a new science, which combines the quantum mechanics and the game theory;

This is not new. Quantum game theory exists. Here are some links:

• CHSH game
• Quantum pseudo-telepathy
• Quantum game theory

 

[Spathi]

This is not new. Quantum game theory exists. Here are some links:

• CHSH game
• Quantum pseudo-telepathy
• Quantum game theory

The refs are nice, I'll study all this later, but for now I would like to formulate my idea (in my opinion it is quite simple).
Imagine that it is 2200 year now. A cobalt bomb has been developed that can destroy the entire Earth if used anywhere (i.e., the user will also die). Many countries have this bomb. In the country of Eritrea, dictator A is in power and is threatened by revolution; to keep his power, he decides to attack a neighboring country that has many natural resources.
The humanity is collectively governed by the UN, under the leadership of the Secretary. The Dictator announces to the Secretary that if the UN intervenes in the war, the Dictator will activate the bomb. The dictator expects that the Secretary, choosing between concessions and complete destruction, will choose the first. He has calculated that the probability of activating the Bomb is 1%, and the probability for him to win is 60%; if he does not start a war, with a 50% probability he will be overthrown and judged.
The Secretary can preventively nuke Eritrea, and then the Bomb will not be activated; but moral standards prohibit doing this.
The Dictator has programmed the Bomb so that it would work as he promised; there is also a small chance of the Bomb going off due to random errors and accidents.
It is known that a Bomb can be either true or nude. If the Bomb is true, this gives the Secretary the right to perform a preventive nuclear strike. Now it turns out that the Secretary can create a superposition of two universes: in the first he starts a conventional war and the Dictator activates the Bomb, in the second nothing happens. Is my further logic clear?

 

[Frabjous]

creates a superposition of these two states

it turns out that the Secretary can create a superposition of two universes:

What makes you think this is possible?

 

[Spathi]

What makes you think this is possible?

I suppose, the concept that in the Wigner's friend experiment a superposition of two universes is created, is mainstream in the scientific community? Since the superposition of the Schrödinger's cat occurs due to the superposition of a radioactive decay device or a beamsplitter, the Secretary can use a beamsplitter too for choosing his actions.

Perhaps one more assumption must be added - Dictator A is obliged to use quantum entanglement when interacting with the Secretary. Let's say that Eritrea have signed a pact on the use of quantum entanglement to avoid nuclear criseses, and the Dictator promised to comply with these conditions (if he broke his promises, the Secretary General would have the right to destroy Eritrea).

 

[Frabjous]

I suppose, the concept that in the Wigner's friend experiment a superposition of two universes is created, is mainstream in the scientific community? Since the superposition of the Schrödinger's cat occurs due to the superposition of a radioactive decay device or a beamsplitter, the Secretary can use a beamsplitter too for choosing his actions.

Perhaps one more assumption must be added - Dictator A is obliged to use quantum entanglement when interacting with the Secretary. Let's say that Eritrea have signed a pact on the use of quantum entanglement to avoid nuclear criseses, and the Dictator promised to comply with these conditions (if he broke his promises, the Secretary General would have the right to destroy Eritrea).

Superposition is a not a magic spell. You cannot apply it willy nilly to whatever you want. Look at the bomb tester. It is convertible to a quantum-only problem. Your scenarios are not.

Regardless, game theory is there to help us think through complex situations. I do not see how your scenarios are of interest.

 

[Spathi]

I try to uderstand the article about quantum pseudotelepathy:

Game rules[edit]​

This is a cooperative game featuring two players, Alice and Bob, and a referee. The referee asks Alice to fill in one row, and Bob one column, of a 3×3 table with plus and minus signs. Their answers must respect the following constraints: Alice's row must contain an even number of minus signs, Bob's column must contain an odd number of minus signs, and they both must assign the same sign to the cell where the row and column intersects. If they manage they win, otherwise they lose.

Alice and Bob are allowed to elaborate a strategy together, but crucially are not allowed to communicate after they know which row and column they will need to fill in (as otherwise the game would be trivial).

Classical strategy[edit]​

It is easy to see that if Alice and Bob can come up with a classical strategy where they always win, they can represent it as a 3×3 table encoding their answers. But this is not possible, as the number of minus signs in this hypothetical table would need to be even and odd at the same time: every row must contain an even number of minus signs, making the total number of minus signs even, and every column must contain an odd number of minus signs, making the total number of minus signs odd.

With a bit further analysis one can see that the best possible classical strategy can be represented by a table where each cell now contains both Alice and Bob's answers, that may differ. It is possible to make their answers equal in 8 out of 9 cells, while respecting the parity of Alice's rows and Bob's columns. This implies that if the referee asks for a row and column whose intersection is one of the cells where their answers match they win, and otherwise they lose. Under the usual assumption that the referee asks for them uniformly at random, the best classical winning probability is 8/9.

Currently I don't understand the point. If their task is to simply name three bits each, why can’t they, if the judge tells them the third row and the third column, name the option not based on this picture, but so that everything matches? For example -1,+1,-1 in the third row and +1, +1, -1 in the third column?

 

[jbriggs444]

Currently I don't understand the point. If their task is to simply name three bits each, why can’t they, if the judge tells them the third row and the third column, name the option not based on this picture, but so that everything matches? For example -1,+1,-1 in the third row and +1, +1, -1 in the third column?

It is not clearly specified, but I believe that Alice is only told which row she is to fill in and Bob is only told which column he will fill in.

Alice reads from the agreed-upon 3 x 3 table by looking at the selected row and making the set of three choices listed in that row.

Bob reads from the agreed upon 3 x 3 table by looking at the selected column and making the set of three choices listed in that column.

If the strategy is successful, the resulting configuration (for the selected row and column) must match Alice and Bob's agreed upon table.

But since any column could have been selected, every column must have an odd number of minus signs. The total number of columns is odd, so the total number of minus signs in the agreed-upon table must therefore be odd.

Similarly, since any row could have been selected, every row must have an even number of minus signs. The total number of minus signs in the agreed-upon table must therefore be even.

 

[Spathi]

Now I try to understand the second example from the article, firstly the classical version too.

The players win if �⊕�⊕�=�∨�∨�

, where ∨

indicates OR condition and ⊕

indicates summation of answers modulo 2. In other words, the sum of three answers has to be even if �=�=�=0

. Otherwise, the sum of answers has to be odd.

I can't paste correct symbols here. What means modulo 2 - the remainer of the devision by 2 (0 mod 2 =0, 1 mod 2 =1, 2 mod 2=0, etc)? Then it is unclear for me, why in the tables the strings 0 mod 2, 1 mod 2 are written instead of simply 0 and 1.

 

[nuuskur]

Are you trying to shoehorn the use of game theoretical techniques in the argument? That never goes well. Better formulate your problem in game theoretical terms, the technique follows naturally.

 

[jbriggs444]

Now I try to understand the second example from the article, firstly the classical version too.


I can't paste correct symbols here. What means modulo 2 - the remainer of the devision by 2 (0 mod 2 =0, 1 mod 2 =1, 2 mod 2=0, etc)? Then it is unclear for me, why in the tables the strings 0 mod 2, 1 mod 2 are written instead of simply 0 and 1.

Computer people and mathematical people tend to think about modulus differently.

For computer folks, we think about modulus arithmetic as a function. You put a dividend and a divisor in. You get a remainder out. So "3 mod 2" is a number: The remainder of three upon division by two. Just as you say.

Computer folks tend to have conventions for what happens when the divisor is negative or if the dividend is not an integer. One can find those conventions mentioned in the documentation.


Mathematicians, on the other hand, tend to think about modulus arithmetic in terms of an equivalence. Two numbers are "equivalent" if they have the same remainder when divided by the modulus. A mathematician would write: $$3 \equiv 1 \ (\text{mod 2})$$and might think about the equivalence relation as partitioning the original set, establishing a quotient algebra on the equivalence classes.