# Capacity to Dominate: Corner Cases

The overall measure $\mathcal{C}_{A \rightarrow B}$ is

$$
\mathcal{C}_{A \rightarrow B} = f(\{\mathcal{C}_{A \rightarrow B}(s) \mid s \in S_A \}),
$$

i.e., some function $f(\cdot)$ of the set of **deviation-specific** capacities

$$
\mathcal{C}_{A \rightarrow B}(s) = \frac{\mathcal{L}_{A \rightarrow B}(s)}{\mathcal{L}_{A \rightarrow A}(s)} = \frac{\text{Interference}}{\text{Arbitrariness}}
$$

for strategy choices $s$.

## The One-Thing-Happens Game

Consider the game where nobody can do anything

| | Do Nothing |
| - | - |
| **Do Nothing** | 0,0 |

This game will give rise to corner cases if we normalize, but I think they can be reasonably "defined away".

In this case, since there are no possible deviations, the set of strategy-specific capacities $\{\mathcal{C}_{A \rightarrow B}(s) \mid s \in S_A\}$ is just the empty set $\{\}$, so that if we pick the max then we can define the overall score as

$$
\mathcal{C}_{R \rightarrow C} = \max\{\} = 0
$$

## The Nobody-Can-Affect-Anything Game

Now there are possible deviations,