# Strategies¶

## Motivating example: Strategy for Rock Paper Scissors¶

The game of Rock Paper Scissors is a common parlour game between two players who pick 1 of 3 options simultaneously:

Rock which beats Scissors;

Paper which beats Rock;

Scissors which beats Paper

Thus, this corresponds to a Normal Form Game with:

Two players (\(N=2\)).

The action sets are: \(\mathcal{A}_1=\mathcal{A}_2=\{\text{Rock}, \text{Paper}, \text{Scissors}\}\)

The payoff functions are given by the matrices \(A, B\) where the first row or column corresponds to \(\text{Rock}\), the second to \(\text{Paper}\) and the third to \(\text{Scissors}\).

If we consider two players, assume the row player always chooses \(\text{Paper}\) and the column player randomly chooses from \(\text{Rock}\) and \(\text{Paper}\) (with equal probability) what is the expected outcome of any one game between them?

The expected score of the row player will be: \(1 \times 1/2 + 0 \times 1/2 = 1/2\).

The expected score of the column player will be: \(-1 \times 1/2 + 0 \times 1/2 = -1/2\).

In Game theoretic terms, the behaviours described above are referred to as
**strategies**. **Strategies map information to actions.** In this particular case,
the only available information is the game itself and the actions are
\(\mathcal{A}_1=\mathcal{A}_2\).

## Definition of a strategy in a normal form game¶

A strategy for a player with action set \(\mathcal{A}\) is a probability distribution over elements of \(\mathcal{A}\).

Typically a strategy is denoted by \(\sigma \in [0, 1]^{|\mathcal{A}|}_{\mathbb{R}}\) so that:

Question

For Rock Papoer Scissors:

What is the strategy \(\sigma_r\) that corresponds to the row player’s behaviour of always choosing \(\text{Paper}\)?

What is the strategy \(\sigma_c\) that corresponds to the column player’s behaviour of always randomly choosing between \(\text{Rock}\) and \(\text{Paper}\)?

Answer

\(\sigma_r = (0, 1, 0)\)

\(\sigma_c = (1 / 2, 1 / 2, 0)\)

## Definition of support of a strategy¶

For a given strategy \(\sigma\), the support of \(\sigma\): \(\mathcal{S}(\sigma)\) is the set of actions \(i\in\mathcal{A}\) for which \(\sigma_i > 0\).

Question

For the following strategies \(\sigma\) obtain \(\mathcal{S}(\sigma)\):

\(\sigma = (1, 0, 0)\)

\(\sigma = (1/3, 1/3, 1/3)\)

\(\sigma = (2/5, 0, 3/5)\)

Answer

\(\mathcal{S}(\sigma) = \{1\}\)

\(\mathcal{S}(\sigma) = \{1, 2, 3\}\)

\(\mathcal{S}(\sigma) = \{1, 3\}\)

Note here that as no specific action sets are given the integers are used.

## Strategy spaces for Normal form Games¶

Given a set of actions \(\mathcal{A}\) the space of all strategies \(\mathcal{S}\) is defined as:

## Calculation of expected utilities¶

Considering a game \((A, B) \in \mathbb{R} ^{(m\times n) ^ 2}\), if \(\sigma_r\) and \(\sigma_c\) are the strategies for the row/column player, the expected utilities are:

For the row player: \(u_{r}(\sigma_r, \sigma_c) = \sum_{i=1}^m\sum_{j=1}^nA_{ij}\sigma_{r_i}\sigma_{c_j}\)

For the column player: \(u_{c}(\sigma_r, \sigma_c) = \sum_{i=1}^m\sum_{j=1}^nB_{ij}\sigma_{r_i}\sigma_{c_j}\)

This corresponds to taking the expectation over the probability distributions \(\sigma_r\) and \(\sigma_c\).

Question

For the Rock Papoer Scissors:

What are the expected utilities to both players if \(\sigma_r=(1/3, 0, 2/3)\) and \(\sigma_c=(1/3, 1/3, 1/3)\).

Answer

## Linear algebraic calculation of expected utilities¶

Given a game \((A, B) \in \mathbb{R} ^{(m\times n) ^ 2}\), considering \(\sigma_r\) and \(\sigma_c\) as vectors in \(\mathbb{R}^m\) and \(\mathbb{R}^n\). The expected utilities can be written as the matrix vector product:

For the row player: \(u_{r}(\sigma_r, \sigma_c) = \sigma_r A \sigma_c^T\)

For the column player: \(u_{c}(\sigma_r, \sigma_c) = \sigma_r B \sigma_c^T\)

Question

For Rock Paper Scissors:

Calculate the expected utilities to both players if \(\sigma_r=(1/3, 0, 2/3)\) and \(\sigma_c=(1/3, 1/3, 1/3)\) using a linear algebraic approach.

Answer

## Exercises¶

For the following vectors explain which ones are valid mixed strategy vectors for a strategy set of size 5. If there are not: explain why.

\(\sigma=(1, 0, 0, 0, 0)\)

\(\sigma=(1/4, 1/4, 0, 0, 1/4)\)

\(\sigma=(1/4, 1/4, 1/2, -1/2, 1/2)\)

\(\sigma=(1/4, 1/4, 0, 0, 11/20)\)

\(\sigma=(1/5, 1/5, 1/5, 1/5, 1/5)\)

Obtain the supports for the following strategy vectors:

\(\sigma = (1, 0, 0, 0)\)

\(\sigma = (1/2, 0, 1/2, 0)\)

\(\sigma = (1/4, 1/4, 1/4, 1/4)\)

Calculate the utilities (for both the row and column player) for the following game for the following strategy pairs:

\[\begin{split}A = \begin{pmatrix} 1 & -1\\ -3 & 1\end{pmatrix} \qquad B = \begin{pmatrix} -1 & 2\\ 1 & -1\end{pmatrix}\end{split}\]\(\sigma_r = (.2, .8)\qquad\sigma_c = (.6, .4)\)

\(\sigma_r = (.3, .7)\qquad\sigma_c = (.2, .8)\)

\(\sigma_r = (.9, .1)\qquad\sigma_c = (.5, .5)\)

## Using Nashpy¶

See Calculate utilities for guidance of how to use Nashpy to calculate utilities.