# Two-Player Model: Axiomatic Proof

## Setup

* *Game*: Let $\Gamma$ be a game with players $i$ and $j$
* *Strategies*: $i$ and $j$ choose from strategy sets $S_i$ and $S_j$, respectively.
* *Outcomes*: Any vector of choices $\mathbf{s} = (s_i, s_j) \in S_i \times S_j$ gives rise to an outcome $o(\mathbf{s})$, that is, $o: S_i \times S_j \mapsto O$.
* *Utilities*: Players have utility functions $u_i(o)$ and $u_j(o)$ defined over these outcomes. For simplicity, though, we write $u_i(\mathbf{s})$ and $u_j(\mathbf{s})$ as shorthand for $u_i(o(\mathbf{s}))$ and $u_j(o(\mathbf{s}))$, the players' utilities from the outcome $o(\mathbf{s})$ induced by a strategy profile $\mathbf{s}$.
* *Equilibria*: Although any solution concept could be used, for concreteness we assume that the **Nash equilibrium** is the chosen solution concept, and furthermore that there exists a unique such equilibrium[^unique]. Thus, under the standard definition of Nash equilibrium, there is some particular Nash equilibrium strategy profile $s^* = (s^*_i, s^*_j)$ at which neither player can increase their utility by unilaterally deviating. Crucially, for linking with the republicanism literature, we treat this strategy profile $s^*$ and the outcome $o^*$ that it induces as the "non-arbitrary" profile/outcome.
* *Counterfactuals*: Given this definition of the non-arbitrary choices and non-arbitrary outcomes, as those associated with the Nash equilibrium, our measure hinges on considering the range of **counterfactual** strategy profiles $\widetilde{s}(i, s'_i) = (s'_i, s^*_j)$ which result from player $i$ unilaterally deviating from their equilibrium strategy choice $s^*_i$ to an alternative choice $s'_i$.

## Definitions

### Interference and Arbitrariness

Define the **interference** $\mathcal{I}_{i \rightarrow j}(s'_i)$ that $i$ inflicts on $j$ via the deviation to $s'_i$ as a normalized version of this loss that $i$ inflicts on $j$ via this deviation:

$$
\mathcal{I}_{i \rightarrow j}(s'_i) = \frac{u_j(s^*) - u_j(\widetilde{s}(i,s'_i))}{\max_{s_i \in S_i}\{u_j(\widetilde{s}(i,s_i))\} - \min_{s' \in S_i}\{u_j(\widetilde{s}(i,s_i))\}}
$$

It is important to note that the max and min in the denominator are both computed over the **full set of strategies** for player $i$, $S_i$---that is, **including** their equilibrium strategy choice $s^*_i$ (in addition to the strategies $s'_i$ that $i$ can **deviate to** from their equilibrium choice $s^*_i$).

Similarly, we can define the **arbitrariness** $\mathcal{A}_{i \rightarrow j}(s'_i)$ of $i$'s deviation to $s'_i$ as a normalized version of the "self-loss" (i.e., cost) that $i$ inflicts on themselves by deviating to $s'_i$:

$$
\mathcal{A}_{i \rightarrow j}(s'_i) = \frac{
u_i(s^*) - u_i(\widetilde{s}(i,s'_i))
}{
\max_{s_i \in S_i}\{u_i(\widetilde{s}(i,s_i))\} - \min_{s_i \in S_i}\{u_i(\widetilde{s}(i,s_i))\}
}
$$

### Capacity

Now we can define the **capacity** of $i$ to arbitrarily interfere with $j$ by unilaterally deviating from $s^*_i$ to $s'_i$ as

$$
\mathcal{C}_{i \rightarrow j}(s'_i) = \frac{
\mathcal{I}_{i \rightarrow j}(s'_i)
}{
\mathcal{A}_{i \rightarrow j}(s'_i).
}
$$

Next we need a way to **aggregate** the set of **strategy-specific capacities** $\mathcal{C}_{i \rightarrow j}(s'_i)$ into a single overall capacity $\mathcal{C}_{i \rightarrow j}$ for $i$ relative to $j$. For this, we take the **maximum** of the strategy-specific capacities^[max] (while keeping in mind that one could use a variety of aggregation functions: the sum of capacities or the mean capacity, for example, or the expected value of a probability distribution over the set of possible unilateral deviations):

$$
\mathcal{C}_{i \rightarrow j} = \max_{s_i \in S_i \setminus \{s^*_i\}}\{\mathcal{C}_{i \rightarrow j}(s_i)\}
$$

### Balance of Capacities

Now, finally, we can define the most general measure, the **balance of capacities** between player $i$ and player $j$, as

$$
\mathcal{B}_{i \rightarrow j} = \mathcal{C}_{i \rightarrow j} - \mathcal{C}_{j \rightarrow i}.
$$

It is this balance measure that we now prove is the unique measure satisfying a set of axioms that a reasonable measure of domination should satisfy.

## Axioms

**A1 (Dictators)**: If a player $i$ is the dictator in $\Gamma$, then

$$
\mathcal{C}_{i \rightarrow j} = \infty, \; \mathcal{C}_{j \rightarrow i} = 0,\text{ and }\mathcal{B}_{i \rightarrow j} = \infty
$$

**A2 (Null Player)**: If a player $j$ is a null player in $\Gamma$ then

$$
\mathcal{C}_{j \rightarrow i} = 0\text{ and }\mathcal{B}_{i \rightarrow j} \leq 0
$$

**A3 (Compound Games)**:

**A4 (Equal Gains)**: If the maximum loss[^max] that $i$ can inflict on $j$ by deviating from $s^*_i$ is the same as the maximum loss that $j$ can inflict on $i$ by deviating from $s^*_j$, then

$$
\mathcal{C}_{i \rightarrow j} = \mathcal{C}_{j \rightarrow i}\text{ and }\mathcal{B}_{i \rightarrow j} = \mathcal{B}_{j \rightarrow i} = 0.
$$

## Theorem

**Theorem**: $\mathcal{B}_{i \rightarrow j} = \mathcal{C}_{i \rightarrow j} - \mathcal{C}_{j \rightarrow i}$ is the unique measure satisfying A1 through A4.

## Proof

Let $\Gamma$ be a game, and assume WLOG that $j$'s capacity to arbitrarily interfere with $i$ in $\Gamma$ weakly exceeds $i$'s capacity to arbitrarily interfere with $j$:

$$
\mathcal{C}_{j \rightarrow i} \geq \mathcal{C}_{i \rightarrow j}.
$$

Let $\Gamma'$ be a game with the same players, outcomes, and strategy choices, but where $i$ is a dictator and $j$ is a null player.

## Appendix

We don't need to define loss to define any of the other terms, but I think of Interference and Arbitrariness as just normalized versions of the loss, so I'm including that here for context:

### Loss

Given this setup, define the **loss** inflicted by $i$ on $j$ via $i$'s deviation from $s^*_i$ to $s'_i$ (a deviation which induces the new strategy profile $\widetilde{s}$ and its associated outcome $o(\widetilde{s})$ which we suppress in the notation) as

$$
\mathcal{L}_{i \rightarrow j}(s'_i) = u_j(s^*) - u_j(\widetilde{s}).
$$

Importantly, this loss function can be used to evaluate the loss that a player inflicts "on themselves"---in other words, the **cost** that they incur---as a result of the deviation:

$$
\mathcal{L}_{i \rightarrow i}(s'_i) = u_i(s^*) - u_i(\widetilde{s})
$$

[^max]: We don't necessarily need to keep the max, it could be a more general function $f(\{s \in S \mid s \neq s^*_i\})$

[^unique]: Assuming uniqueness of the Nash equilibrium shouldn't be an issue since the measure "automatically" extends to games with multiple equilibria if one can model the likelihoods of reaching each equilibrium as a probability distribution.