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Read Time: 10 minutes

Background Story

Jill and Jack have robbed a bank together and have unfortunately been caught.
The DA’s goal is to maximize conviction time.Upon entering the DA’s office, suspects are separated and isolated to prevent corroboration of falsified testimony.
To help achieve their goal, the DA often offers “plea bargain” deals to suspects in an attempt to extract corroborating information.
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With no additional testimony, assume the DA has enough evidence in the form of eyewitness testimony to convict for 6 years total (say, 3 for Jack and 3 for Jill)
With additional testimony, the DA is able to convict for a full 10 years (split between Jack and Jill, again) which is clearly the worst case scenario for the family.

To demonstrate the payoff for each of these situations we'll use the payoff table on the right.

Jack Rats(R), Jill Quiet(Q)

If Jack “rats” on Jack, and Jill stays quiet, Jack gets 1 yr, Jill gets the rem. 9 yrs.

It's important to note here that the first digit in the matrix corresponds to the payoff of jack. The second digit corresponds to the payoff associated with Jill.

Jack Quiet(q), Jill Rats(r)

If Jill “rats” on Jack, and jack stays quiet, Jill gets 1 yr, Jack gets the rem. 9 yrs.

Same here as well, the firsts number responds to Jack(-9) and the second number responds to Jill(-1).

Jack Rats(R), Jill rats(r)

If they both “rat,” they each get 5 years.

Jack quiet(q), Jill Quiet(Q)

If they both stay “quiet,” the DA can only convict them each for 3 years.

Game Theory

Part 1: Filling out the Matrix
J
A
C
K
-3
-3
Jill
Quiet
Rat
Q
R
-1
-9
-9
-1
-5
-5

What Now!

We now begin "Game Theory". We have to look at each of the pairs in the matrix, and determine which location will be preferred for both

If Jill is quite...

First let's just assume Jill has just gotten a disease, her to be able to talk at all. She will be quiet...

This means now we will be only looking at the Quiet column, from Jill.
Jack can still choose stay quite as well or rat jill out. In this situation, which of the pairs (Q,Q) or (R,Q) will make Jack better off.

(Quiet,Quiet)= -3 for Jack
(Rat,Quiet) =  -1 for Jack

If Jill rats...

Now let's assume Jill is going to rat no matter what. Let's say she has a panic attack and she spills everything about Jack ratting him out.

Again just like before we will have jill at the rat location. Jack will try to determine between staying Quit or Ratting Jill out as well.

(Quiet, Rat)= -9 for Jack
(, Rat)= -5 for Jack

So what will Jack do?

Jill will Rat if Jack is quiet. Jill will Rat if Jack Rats.

Jack Rats No Matter What

Interpretation of this observation

Since“Rat” gives Jack a higher payoff in comparison to “Quiet” for every possible action Jill can take, he should never play “Quiet,” if he is a rational payoff (utility) maximizer.

As a result of all of this, we now know Jack will only play Rat.
Let's see what Jill would play.

What does rationality mean?

Game Theory

Part 2: Modelling Jack's Decision
J
A
C
K
-3
-3
Jill
Quiet
Rat
Q
R
-1
-9
-9
-1
-5
-5
Jack Rats if Jill stays Quiet
Jack will Rat if Jill Rats

What will Jill Do

Let's see now how Jill will respond to the choices Jack makes.

If Jack is quite...

Again we begin by assuming Jack has the terrible disease that makes it impossible to talk. Jack can only be quiet.

This means we will be only looking at the Quiet column, from Jack. Jill can still choose to stay quite as well or rat jill out. In this situation, which of the pairs (Q,Q) or (Q,R) will make Jill better off.

(Quiet, Quiet)= -3 for Jill
(Quiet, Rats)= -1 for Jill

If Jack Rats

Now let's assume Jill is going to rat no matter what. Let's say she has a panic attack and she spills everything about Jack ratting him out.

Again just like before we will have jill at the rat location. Jack will try to determine between staying Quit or Ratting Jill out as well.

(Rat, Quiet)= -9 for Jill
(Rat, Rat)= -5 for Jill

So what will Jill do?

Jill will Rat if Jack is quiet. Jill will Rat if Jack Rats.

Jill Rats No Matter What

Interpretation of this observation

Since “Rat” gives Jack a higher payoff in comparison to “Quiet” for every possible action Jill can take, he should never play “Quiet,” if he is a rational payoff (utility) maximizer.

As a result of all of this, we now know Jack will only play Rat.
Let's see what Jill would play.

Equilibrium

As you can see the two columns end up intersecting at one point. That point is the 5,5 payoff where they both rat. That is the be

What does rationality mean?

Game Theory

Part 3: Modelling Jill's Decision
J
A
C
K
-3
-3
Jill
Quiet
Rat
Q
R
-1
-9
-9
-1
-5
-5
Jill Rats if Jack stays Quiet
Jill will Rat if Jack Rats