4.4: What Makes a Prisoner's Dilemma?

Last updated
Save as PDF

Page ID: 82778

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In this section, we give a mathematical description of Prisoner's Dilemma and compare it to some similar games.

The Class-wide Prisoner's Dilemma game we played in Section 4.3 has the payoff matrix given in Table \(4.4.1\) for each pair of players.

Table \(4.4.1\): A Class-wide Prisoner's Dilemma.
		Player 2
		Cooperate	Defect
Player 1	Cooperate	\((3, 3)\)	\((0, 5)\)
Player 1	Defect	\((5, 0)\)	\((1, 1)\)

We can classify each of the values for the payoffs as follows:

Reward for Mutual Cooperation: \(R=3.\)
Punishment for Defecting: \(P=1.\)
Temptation to Defect: \(T=5.\)
Sucker's Payoff: \(S=0.\)

In order for a game to be a variation of Prisoner's Dilemma it must satisfy two conditions:

\(T>R>P>S\)
\(\dfrac{(T+S)}{2} \lt R\)

Let's apply this description of Prisoner's Dilemma to a few games we've seen. We can use the conditions to check if a game is really a Prisoner's Dilemma.

Exercise 4.4.1 : Description of Conditions

Describe each of the conditions (A) and (B) in words.

Hint: \((T+S)/2\) is the average of \(T\) and \(S\text{.}\)

Exercise 4.4.2 : The Conditions of Classwide Prisoner's Dilemma

Show that the two conditions hold for the Class-wide Prisoner's Dilemma.

Exercise 4.4.3 : The Conditions For Prisoner's Dilemma

Recall the matrix for Prisoner's Dilemma from Example \(4.2.2\).

Table \(4.4.2\): Prisoner's Dilemma (again).
		Prisoner 2
		Confess	Don't Confess
Prisoner 1	Confess	\((8, 8)\)	\((0.25, 10)\)
Prisoner 1	Don't Confess	\((10, 0.25)\)	\((1, 1)\)

Determine \(R, P, T,\) and \(S\) for this game. Be careful: think about what cooperating versus defecting should mean. Show the conditions for Prisoner's Dilemma are satisfied.

Hint: Time in jail is bad, so the bigger the number, the worse you do; thus, it might be helpful to think of the payoffs as negatives.

Exercise 4.4.4 : The Conditions For Chicken

Recall the matrix for Chicken from Example \(4.2.3\).

Table \(4.4.3\): Chicken (again).
		Driver 2
		Swerve	Straight
Driver 1	Swerve	\((0, 0)\)	\((-1, 10)\)
Driver 1	Straight	\((10, -1)\)	\((-100, -100)\)

Determine \(R, P, T,\) and \(S\) for this game. Again, think about what cooperating and defecting mean in this game. Determine if the conditions for Prisoner's Dilemma are satisfied. If not, which condition(s) fail?

Exercise 4.4.5 : The Conditions on Another Game

Consider the game:

\(\begin{array} & & C & D\\ C & (3,3) & (0,50)\\ D & (50,0) & (.01,.01) \end{array}\)

Determine \(R, P, T,\) and \(S\) for this game. Determine if the conditions for Prisoner's Dilemma are satisfied. If not, which condition(s) fail?

Exercise 4.4.6 : A Little More Practice

Consider the game:

\(\begin{array} & & C & D\\ C & (1000,1000) & (0,100)\\ D & (100,0) & (100,100) \end{array}\)

Determine \(R, P, T,\) and \(S\) for this game. Determine if the conditions for Prisoner's Dilemma are satisfied. If not, which condition(s) fail?

Exercise 4.4.7 : Compare the Games

The games in Exercise \(4.4.4\), Exercise \(4.4.5\), and Exercise \(4.4.6\) are not true Prisoner's Dilemmas. For each game, how do the changes in payoffs affect how you play? In particular, in Prisoner's Dilemma, a player will generally choose to defect. This results in a non-optimal payoff for each player. Is this still true in Exercise \(4.4.4\), Exercise \(4.4.5\), and Exercise \(4.4.6\)? If possible, use the changes in the conditions (A) and (B) to help explain any differences in how one should play.

We can now define defection as the idea that if everyone did it, things would be worse for everyone. Yet, if only one (or a small) number did it, life would be sweeter for that individual. We can define cooperation as the act of resisting temptation for the betterment of all players.

Exercise 4.4.8 : Example From Real Life

Give an example of defection and cooperation from real life.