## Section4.4What Makes a Prisoner's Dilemma?

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.

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:

1. $T>R>P>S$

2. $(T+S)/2 \lt R$

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.

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

Hint

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

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

Recall the matrix for Prisoner's Dilemma from Example 4.2.7.

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.

Recall the matrix for Chicken from Example 4.2.16.

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?

Consider the game:

\begin{equation*} \begin{matrix}\amp C\amp D\\ C\amp (3, 3) \amp (0, 50)\\ D \amp (50, 0) \amp (.01, .01) \end{matrix} \end{equation*}

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?

Consider the game:

\begin{equation*} \begin{matrix}\amp C\amp D\\ C\amp (1000, 1000) \amp (0, 100)\\ D \amp (100, 0) \amp (100, 100) \end{matrix} \end{equation*}

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?

The games in Exercise 4.4.6, Exercise 4.4.8, and Exercise 4.4.9 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 4.4.6, Exercise 4.4.8, and Exercise 4.4.9? 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.

Give an example of defection and cooperation from real life.