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: 𝒜1 = 𝒜2 = {Rock, Paper, Scissors}
- The payoff functions are given by the matrices A, B where the first row or column corresponds to Rock, the second to Paper and the third to Scissors.
If we consider two players, assume the row player always chooses Paper and the column player randomly chooses from Rock and 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 × 1/2 + 0 × 1/2 = 1/2.
- The expected score of the column player will be: − 1 × 1/2 + 0 × 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 𝒜1 = 𝒜2.
A strategy for a player with action set 𝒜 is a probability distribution over elements of 𝒜.
Typically a strategy is denoted by σ ∈ [0, 1]ℝ|𝒜| so that:
Question
For Rock Papoer Scissors <motivating-example-strategy-for-rps>
:
- What is the strategy σr that corresponds to the row player's behaviour of always choosing Paper?
- What is the strategy σc that corresponds to the column player's behaviour of always randomly choosing between Rock and Paper?
Answer
- :math:sigma_r = (0, 1, 0)
- :math:sigma_c = (1 / 2, 1 / 2, 0)
For a given strategy σ, the support of σ: 𝒮(σ) is the set of actions i ∈ 𝒜 for which σi > 0.
Question
For the following strategies σ obtain 𝒮(σ):
- σ = (1, 0, 0)
- σ = (1/3, 1/3, 1/3)
- σ = (2/5, 0, 3/5)
Answer
- :math:mathcal{S}(sigma) = {1}
- :math:mathcal{S}(sigma) = {1, 2, 3}
- :math:mathcal{S}(sigma) = {1, 3}
Note here that as no specific action sets are given the integers are used.
Given a set of actions 𝒜 the space of all strategies 𝒮 is defined as:
Considering a game (A, B) ∈ ℝ(m × n)2, if σr and σc 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 σr and σc.
Question
For the Rock Papoer Scissors <motivating-example-strategy-for-rps>
:
What are the expected utilities to both players if σr = (1/3, 0, 2/3) and σc = (1/3, 1/3, 1/3).
Answer
Given a game (A, B) ∈ ℝ(m × n)2, considering σr and σc as vectors in ℝm and ℝn. The expected utilities can be written as the matrix vector product:
- For the row player: ur(σr, σc) = σrAσcT
- For the column player: uc(σr, σc) = σrBσcT
Question
For Rock Paper Scissors <motivating-example-strategy-for-rps>
:
Calculate the expected utilities to both players if σr = (1/3, 0, 2/3) and σc = (1/3, 1/3, 1/3) using a linear algebraic approach.
Answer
- For the following vectors explain which ones are valid strategy vectors for an action set of size 5. If there are not: explain why.
- σ = (1, 0, 0, 0, 0)
- σ = (1/4, 1/4, 0, 0, 1/4)
- σ = (1/4, 1/4, 1/2, − 1/2, 1/2)
- σ = (1/4, 1/4, 0, 0, 11/20)
- σ = (1/5, 1/5, 1/5, 1/5, 1/5)
- Obtain the supports for the following strategy vectors:
- σ = (1, 0, 0, 0)
- σ = (1/2, 0, 1/2, 0)
- σ = (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:
$$\begin{aligned} A = \begin{pmatrix} 1 & -1\\ -3 & 1\end{pmatrix} \qquad B = \begin{pmatrix} -1 & 2\\ 1 & -1\end{pmatrix} \end{aligned}$$- σr = (.2, .8) σc = (.6, .4)
- σr = (.3, .7) σc = (.2, .8)
- σr = (.9, .1) σc = (.5, .5)
See how-to-calculate-utilities
for guidance of how to use Nashpy to calculate utilities.