---
title: "Pettit, Axioms and Theorems of Republican Liberty"
---

# Axioms and Theorems of Republican Liberty

Let $S_i$ be the set of strategies available to an agent $i$. Keeping in mind that our measure of the **absolute** capacity of an agent $A$ to dominate another agent $B$ is given by

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

and similarly for the absolute capacity of $B$ to dominate $A$.

This also gives us a measure of the **relative** capacity of $A$ to dominate $B$, that is, the **difference** between $A$'s capacity to dominate $B$ and $B$'s capacity to dominate $A$:

$$
\mathcal{R}_{A \rightarrow B} = \mathcal{C}_{A \rightarrow B} - \mathcal{C}_{B \rightarrow A}
$$

As described in Schmidt (2018) ("Domination without Inequality? Mutual Domination, Republicanism, and Gun Control"), it is important to keep these two measures separate, as (e.g.) one can have a **zero-dominance** case, where $\mathcal{C}_{A \rightarrow B} = \mathcal{C}_{B \rightarrow A} = 0$, and a **balanced-dominance** case where $\mathcal{C}_{A \rightarrow B} = \mathcal{C}_{B \rightarrow A}$.

## Axioms

### Axiom 1: The reality of personal choice

Represented in our model as the choice set available to each agent

### Axiom 2: The possibility of alien control

Represented in our model as the **interference**, the numerator of the arbitrary-interference capacity

### Axiom 3: The positionality of alien control

This comes out of our model if we consider $\mathcal{R}_{A \rightarrow B}$, the **difference** between $\mathcal{C}_{A \rightarrow B}$ and $\mathcal{C}_{B \rightarrow A}$.

## Theorems

### Theorem 1: Alien control may materialize with interference

This is satisfied by our model since $\mathcal{C}_{A \rightarrow B}$ increases as interference (the numerator) increases

## Theorem 2: Alien control may materialize without interference

This arises from the fact that our measure takes into account the **possible** deviations, away from the Nash equilibrium (which is what we can expect to "actually" occur under normal circumstances).

Meaning, more straightforwardly: increasing the damage that some player can do by deviating, while keeping the Nash eq payoffs the same, will increase the capacity score.

Consider the game $G$ with payoff matrix

| | Do Nothing |
| - | - |
| **Do Nothing** | **0**,**0** |
| **Intervene** | -1, **-5** |

Here the Nash equilibrium is $s = (\textsf{Do Nothing}, \textsf{Do Nothing})$, but the row player $R$ can arbitrarily choose to deviate to $\textsf{Intervene}$, inflicting a loss of 5 utils to the column player $C$ while only paying a cost of 1 util. This makes the capacity score for the row player

$$
\mathcal{C}_{R \rightarrow C} = \frac{\text{Interference}}{\text{Arbitrariness}} = \frac{5}{1} = 5.
$$

Now consider the modified game $G'$ with payoff matrix

| | Do Nothing |
| - | - |
| **Do Nothing** | **0**,**0** |
| **Intervene** | -1, **-10** |

$G'$ is the same as $G$ except that now the row player inflicts a loss of 10 utils rather than 5 utils if they deviate. This makes the row player's capacity, in this game,

$$
\mathcal{C}_{R \rightarrow C}(G') = \frac{10}{1} = 10.
$$

The row player $R$'s capacity to interfere with $C$ (their "alien control" over $C$) has doubled, despite the fact that $C$'s payoffs in the actual (Nash) outcome remain the same.