Consider the game below where we will only focus on the column player. The greek letters $\beta$, $\gamma$ and $\delta$ denote the probabilities assigned to each pure strategy Left, Center and Right.
Left | Center | Right | |
---|---|---|---|
Up | 3, 5 | 7, 3 | 1, -1 |
Middle | -2, 1 | 1, 4 | 9, 0 |
Down | -5, 2 | 6, 8 | 4, 5 |
$\beta$ | $\gamma$ | $\delta$ |
i. $\gamma =0$
ii. $\gamma =1/2,\delta =1/2$
iii. $\beta =1/4,\gamma =3/4,\delta =0$
iv. $\beta =1$
v. $\beta =0,\gamma =1/4,\delta =1/3$
A mixed strategy gives information about the probability with which each pure strategy is played. It is well-defined if each of its components is between zero and one and if all of its components add up to one. In this case, the requirements are $\beta$, $\gamma$, $\delta$ $\in [0,1]$ and $\beta + \gamma + \delta=1$.
Which of the above are well-defined mixed strategies?