2.1 Strategic Form Game

Strategic (Normal) Form Game

The strategic form representation of a game is used to illustrate simultaneous decisions by the decision-makers and is specified by three finite sets:

1. A set of players $N$ with a single player denoted as $i\in N$.
2. A set of strategies $S_i$ for each player $i\in N$ s. t. the strategy profile $s=\{s_i,s_{-i}\}$ denotes all chosen strategies for each player, and the set of all strategy profiles is $S=S_1\times \dots \times S_n$.
3. A set of payoff functions $u_i(s)\in \mathbb{R}$ that denote the payoff received by player $i\in N$ when the strategy $s\in S_i$ is played.

A strategic form game need not have a winner, and rational play consists of choosing the strategy that maximizes utility for the given player.

Examples

Rock-Paper-Scissors

We can create a strategic form representation of the classic game of rock paper scissors. It consists of:

• Two players: $A$ and $B$ independently decide their hand gesture.
• Strategies: $S=\{\text{Rock},\text{Paper},\text{Scissors}\}$.
• Payoffs: Winner wins $1.

We can model these rules into a game matrix, containing all of the game outcomes:

\begin{document}
 
\begin{tikzpicture}
   % Draw the vertical lines
   \draw (0, 0) -- (0, -4.5);
   \draw (1.5, 0) -- (1.5, -4.5);
   \draw (3, 0) -- (3, -4.5);
   \draw (4.5, 0) -- (4.5, -4.5);
 
   % Draw the horizontal lines
   \draw (0, 0) -- (4.5, 0);
   \draw (0, -1.5) -- (4.5, -1.5);
   \draw (0, -3) -- (4.5, -3);
   \draw (0, -4.5) -- (4.5, -4.5);
 
   % Add labels for players
   \node at (2.25, 1) {Player 2};
   \node[rotate=90] at (-1, -2.25) {Player 1};
 
   % Add labels for Player 2's choices
   \node at (0.75, 0.25) {Rock};
   \node at (2.25, 0.25) {Paper};
   \node at (3.75, 0.25) {Scissors};
 
   % Add labels for Player 1's choices
   \node[rotate=90] at (-0.25, -0.75) {Rock};
   \node[rotate=90] at (-0.25, -2.25) {Paper};
   \node[rotate=90] at (-0.25, -3.75) {Scissors};
 
   % Add payoffs for each cell
   \node at (0.75, -0.75) {(0, 0)};
   \node at (2.25, -0.75) {(-1, 1)};
   \node at (3.75, -0.75) {(1, -1)};
 
   \node at (0.75, -2.25) {(1, -1)};
   \node at (2.25, -2.25) {(0, 0)};
   \node at (3.75, -2.25) {(-1, 1)};
 
   \node at (0.75, -3.75) {(-1, 1)};
   \node at (2.25, -3.75) {(1, -1)};
   \node at (3.75, -3.75) {(0, 0)};
\end{tikzpicture}
 
\end{document}

\begin{document}
 
\begin{tikzpicture}
    % Draw the vertical lines
    \draw (0, 0) -- (0, -4.5);
    \draw (1.5, 0) -- (1.5, -4.5);
    \draw (3, 0) -- (3, -4.5);
    \draw (4.5, 0) -- (4.5, -4.5);
 
    % Draw the horizontal lines
    \draw (0, 0) -- (4.5, 0);
    \draw (0, -1.5) -- (4.5, -1.5);
    \draw (0, -3) -- (4.5, -3);
    \draw (0, -4.5) -- (4.5, -4.5);
 
    % Add labels for players
    \node at (2.25, 1) {Player 2};
    \node[rotate=90] at (-1, -2.25) {Player 1};
 
    % Add labels for Player 2's choices
    \node at (0.75, 0.25) {Rock};
    \node at (2.25, 0.25) {Paper};
    \node at (3.75, 0.25) {Scissors};
 
    % Add labels for Player 1's choices
    \node[rotate=90] at (-0.25, -0.75) {Rock};
    \node[rotate=90] at (-0.25, -2.25) {Paper};
    \node[rotate=90] at (-0.25, -3.75) {Scissors};
 
    % Add payoffs for each cell
    \node at (0.75, -0.75) {(0, 0)};
    \node at (2.25, -0.75) {(-1, 1)};
    \node at (3.75, -0.75) {(1, -1)};
 
    \node at (0.75, -2.25) {(1, -1)};
    \node at (2.25, -2.25) {(0, 0)};
    \node at (3.75, -2.25) {(-1, 1)};
 
    \node at (0.75, -3.75) {(-1, 1)};
    \node at (2.25, -3.75) {(1, -1)};
    \node at (3.75, -3.75) {(0, 0)};
\end{tikzpicture}
 
\end{document}