Definition (Prisoner’s Dilemma)

The Prisoner’s Dilemma is any game in the following form:

$\begin{array}{c|c|c|} & C & D \\ \hline C & a, a & b, c \\ \hline D & c, b & d, d \\ \hline\end{array}$

With $c>a>d>b$. If both players co-operate (C,C), they both get some payoff $a$. If they both defect (D,D), they both get the second-worst payoff $d$. However, if they miscommunicate, and one player defects and the other player co-operates, then the co-operate gets the worst payoff $d$, and the defector gets the best payoff $c$.

Definition (Games of Pure Competition)

A game of pure competition is a game such that:

• Players have exactly opposed interests
• There must be precisely two players (otherwise there cannot be exactly opposed interests)
• For all action profiles $a \in A$, $u_1(a) + u_2(a) = c$ for some constant $c$ i.e. one player’s outcome is the complement of the other player’s outcome. Often this constant $c=0$, in which case it is called a zero-sum game
• Because of the previous point, we only need to store the utility function for one player, and infer the payoff function for the other player using that

Example (Matching Pennies)

This is a game where one player wants to match the faces on the flipped coins, and the other player wants to mismatch i.e.

$\begin{array}{c|c|c|} & \text { Heads } & \text { Tails } \\ \hline \text { Heads } & 1,-1 & -1,1 \\ \hline \text { Tails } & -1,1 & 1,-1 \\ \hline\end{array}$

This is an example of a zero-sum game of pure competition

Example (Rock-Paper-Scissors)

This is a generalization of the matching pennies example i.e.

$\begin{array}{c|c|c|c|} & \text{Rock} & \text{Paper} & \text{Scissors}\\ \hline \text{Rock} & 0,0 & -1,1 & 1,-1 \\ \hline \text{Paper} & 1, -1 & 0, 0 & -1,1\\ \hline \text{Scissors} & -1,1 & 1,-1 & 0,0 \\\hline\end{array}$

If you pick the same action, then it’s a draw. All other options have a winner/loser. The sum to every case $=0$.

Definition (Games of Pure Co-operation)

In these games:

• Players have the exact same interests
• No conflict: all players want to do the same thing
• $\forall a \in A, ~\forall i,j,~ u_i(a) = u_j(a)$
• They are often written with a single payoff per cell

Example (Co-ordination Game)

Which side of the road should you drive on?

$\begin{array}{c|c|c|} & \text { Left } & \text { Right } \\ \hline \text { Left } & 1,1 & 0,0 \\ \hline \text { Right } & 0,0 & 1,1 \\ \hline\end{array}$

Definition (General Games)

The most interesting games combine elements of co-operation and competition. Consider a husband and wife that want to go watch a movie. The two options are Battle for Armageddon ($B$) and Flowerchild ($F$).

$\begin{array}{c|c|c|} & \text { B } & \text { F } \\ \hline \text { B } & 2,1 & 0,0 \\ \hline \text { F } & 0,0 & 1,2 \\ \hline\end{array}$

Above all, they want to watch a movie, so if they go to different movies, they get the worst payoff. However, the husband would prefer to watch $B$ together, and the wife would prefer to watch $F$.

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