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Bob and Hilal fall in a prison. If both/none of them confess that they stole the money, they will both stay 11 months in prison. If one of them confesses but the other does not, the one who confesses will be free, the one who doesn't will stay for 12 months.

What is the Nash equilibrium in this game?

My thinking: 11 is almost equal to 12, so they both confess and the Nash equilibrium is that "both confess and stay in prison for 11 months."

I think it is almost clear. What do you think? Thanks for helpful comments. ( I am really in jail and scared. Father please help me. NOBODY IS REALLY HELPING ME.)

Bob Dobbs
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2 Answers2

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We will have Equilibrium when Both B & H converge to a Single State.

If B has to act depending on what H chooses AND H has to act on depending on what B chooses , then Situation is a little Complicated & we may not have the Equilibrium.

In Current Case (Case 1) , we can see (Pictorially) that B & H have choices independent of the other :

B & H

I have made the Penalty Values $X,Y,Z$ to high-light a Point.

Let B not confess , then he thinks he can either get a Penalty of $X=11$ or a Penalty of $Y=12$.

In Either Case , he can get a lesser Penalty by confessing : He will either change the Penalty of $X=11$ to $0$ or change the Penalty of $Y=12$ to $Z=11$.

This is independent of what H chooses.

Likewise , B will confess.

This Situation occurs not because $11$ & $12$ are close , but because of these two or three conditions , listed in Different Wordings :

(1) $Z < Y$ (here it is $11<12$ but it can even be $11.99<12$ or $0.00011<12$)
(2) $X > 0$
(3) All "Arrows" are Pointing to Same "Stable" Choice.
(4) There are no Saddle Points.

If the Situation had been (Case 2) $Z>Y$ , then the "Arrows" may look like this :

B # H

Both want the State which is beneficial to them , but that Depends on what the other choses !

When analysing the other Situation where $Z > Y$ , we should consider :
(Case 2A) $Y < Z < X$ (not confessing gives biggest Penalty : never confess ?)
(Case 2B) $Y < X < Z$ (confessing gives Penalty which is somewhere between Extremes : what action to take ?)
(Case 2C) $X < Y < Z$ (confessing gives least Penalty : always confess ?)
There may be Saddle Points, no Equilibrium , Etc.

Prem
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    Nice game theory diagrams. Thanks. – Bob Dobbs Nov 05 '22 at 08:37
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    I am trying to get related references to add to the Post , but most online articles are about Zero-Sum (here , it is not Zero-Sum) with Payoffs (here , it is Penalty) , hence not getting Exact Match ! – Prem Nov 05 '22 at 13:50
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Both confessing is the Nash equilibrium in this game because confessing is always better for you than not confessing. It is assumed you are completely selfish, so you don't care what happens to the other guy. The fact that $11$ is close to $12$ is irrelevant, it is still less than $12$ and you would rather not stay in prison any longer than necessary.
The conclusion would be the same if the difference between the two was one microsecond rather than one month (this is mathematics, not psychology - an actual person's response might be different).

Robert Israel
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