Above is my question. Could someone please help me with the first part? I should be ok once I have set up the linear programming problem, but I don't even know what $x_1, x_2 \ \text{and} \ x_3$ are here, let alone how to set up the LP.
I've had a look online for help but can't find any - I would most appreciate some help! :)
Sam (:
On
Here's a start: you want to maximize the expected value. so you want to choose three probabilities (one for each strategy) and you maximize the sum of the probabilities times the payoff. so the first constraint is that the probabilities add to one. (three more making them all positive). For greater detail, see chpater 15 of Linear Programming and Economic Analysis by Dorfman et al, or section 1.26 of Vorobev, Game Theory.
The original question seems to be poorly written (not the op's question) because at first glance you are not maximizing $x_1+x_2+x_3$ as it is stated. To answer I will have to make the following assumption about the convention being used: In a zero-sum game, usually $A$ denotes the payoffs of player 1 (row player) while $-A$ denotes the payoffs of player 2 (column player). Check if the textbook uses this convention.
Let $(p_1,p_2,p_3)$ be a probability distribution over the rows (strategy of player 1) and $(q_1,q_2,q_3)^T$ be a probability distribution over the columns (strategy of player 2).
Given $q$, player 1 chooses $p$ to maximize $p\cdot A\cdot q^T$ subject to $p\ge 0$ and $p\cdot \vec{1}=1$ (because $q$ is a probability).
Conversely, given $p$, player 2 chooses $q$ to maximize $-p \cdot A\cdot q^T$ subject to $q\ge 0$ and $q\cdot \vec{1}=1$ (because $p$ is a probability).