Game Matrices

Game Matrices#

While other representations exist, this course will be primarily focused on solving matrix games. The textbook demonstrates with the game of Matching Pennies. We will set up a matrix game that will demonstrate:

  • Payoffs

  • Payoff Vectors

  • Matrix Setup

Example Game#

Matching Pennies

Each player has two choices: Heads (H) or Tails (T). When their choices match, Player 1 wins $1 from Player 2. If they don’t match, Player 1 loses $1 to Player 2.

The game matrix is represented as follows:

\[\begin{split}\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{c|cc} & H & T \\\hline H & (1,-1) & (-1,1) \\ T & (-1,1) & (1,-1) \end{array}\end{array}\end{split}\]

For instruction on the DSM equation editor that produces this matrix, see the Dollar Sign Mathematics page and scroll to the bottom.

Payoffs#

The outcomes are shown as ordered pairs where the payoff to Rose is the \(x\)-coordinate and the payoff to Colin is the \(y\)-coordinate. If Colin plays strategy \(H\) while Rose plays \(T\) the payoffs are given as shown below:

  • Rose wins $-1

  • Colin wins $1

Preferred Payoffs#

If a player has a great strategy that always wins more than her other strategies, she will plays it. Remember our Rational Play Principle. However, as in Match the Pennies, the player may have to randomize their selection to choose different strategies in different iterations of the game.

We have two types of solutions:

  • Pure Strategy Solutions (PSS) where a single strategy is played 100% of the time.

  • Mixed Strategy Solutions (MSS) where two or more strategies are played.

However, as our textbook points out, it can be tricky at times to see whether the game has a PSS or an MSS.

Zero Sum Matrix Games#

In a zero sum game, we can identify We will introduce some notation:

  1. \(v_R\) indicates the value of the game for Rose.

  2. \(v_C\) indicated value for Colin.

For zero sum games, we have the following identity:

\[v_R = -v_C\]

Colin’s payoff is the opposite of Rose’s payoff. This leads to alternative form of representation in the game matrix where we show only Rose’s payoffs:

\[\begin{split}\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{c|rr} & H & T \\\hline H & 1 & -1 \\ T & -1 & 1 \end{array}\end{array}\end{split}\]

In this format, we find Colin’s payoff by taking the opposite of Rose’s payoff. It simplifies the game matrix.

Minimizing and Maximizing#

Rose is choosing strategies in this simplified game matrix that will maximize her payoffs. However, Colin wants the opposite. He is minimizing his payoffs in this simplified matrix.

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