# Parameters

## Parameters and Settings

### Parameters

The parameters used in the model are summarized in the chart below:

Symbol Parameter Range Axiom
β Constructive multiplier β > 1 1
μ Destructive multiplier μ > β 2
λ Decay multiplier λ < β 3
α Utility exponent 2 ≤ α ≤ 3 4
σ Coefficient of social inertia 0 < ρ < 1 5
δ Discount rate 0 < δ < 1 6

### Settings

Settings are not intrinsic to the model, but provide levers of control over its implementation in code. They are:

Symbol Setting Range
ρ Self-allocation percentage 0 < ρ ≤ 1
k Number of steps used in PrinceRank calculations k > 0

### Preferred Values

Our simulations generally use parameters in the neighborhood of the following, which were not chosen for any special reason other than that they tend to give rise to reasonably intuitive behavior. Experiments suggest that varying β and μ do not significantly impact the ordering of agents' PrinceRank values.

β=2; μ=3; λ=1; α=2.25; ρ=0.98;

Experiments also suggest that the value of the discount rate δ does not have a significant impact on the preference structure of the model. We set it and k to the following in order to maximize the amount of variability in PrinceRank vectors. This minimizes the computation that must be done (by keeping k low) while making sure that the agents' PrinceRank values are not so similar that it is difficult to distinguish among them.

δ=0.85; k=10;

The coefficient of social inertia σ is typically not needed because most of our simulations use ternary tactics with a Hamming distance of 1.

### Setting α as a vector

The parameter α can be set as a vector, giving each agent its own preference setting. This is done by appending a key-value pair to a power structure object:

<|"s" → s, "T" → T, "α" → {2.1, 2.6, 2.35}|>

This "α" key will override the global setting for α.

## Theoretical Approaches to μ

Here we consider theoretical approaches to defining the destructive multiplier μ. Elsewhere we explore the economics of destruction using real world quantities.

### Expected Destruction

Suppose you send a warrior to attack an opposing group of warriors and that there is a 50% chance that she will kill the first opponent she encounters before moving on to fight the next one. If she doesn't kill the opponent, then she will be killed. On average, how many opponents can we expect her to kill? The answer is given by the infinite series:

$\displaystyle{ \frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \left({\frac 12}\right)^n = 1. }$

In other words, under these conditions, the expected destruction from using 1 unit of power (the warrior) is equal to 1 (warrior). In quantitative realism, this is equivalent to saying that μ = 1.

What happens when the warrior's likelihood of success is something other than 50%? Let that likelihood be p. The expected destruction is then:

$\displaystyle{ \mu = \sum_{n=1}^\infty p^n. }$

Intuitively, μ is higher when it is more likely that a unit of power will destroy a unit of the opponent's power. Since μ > 1 (see above), in the model p would be expected to exceed 50%. The warrior in our thought experiment would need an edge over her opponents or have to destroy some assets during her attack. Plotting μ as a function of p, we get the graph below.

In short, the destructive multiplier μ is related to the likelihood that one unit of power will cause a unit of destruction.

### Relation to the Offense-Defense Balance

In international relations theory, the offense-defense balance (see Glaser 1998) reflects whether offense or defense has the advantage in a given relationship between two countries — for example, whether it's easier to conquer territory or to defend it. When defense has the relative advantage, there is a deterrent to aggression, and vice versa. Glaser defines the offense-defense balance (ODB) as the ratio of the cost of attacking forces required to prevail in a conflict to the cost of the defender’s forces. This ratio may be different for each dyad of states, and it varies by direction, i.e. which state is the attacking one. Using Glaser's definition, we can connect the offense-defense balance to μ:

$\displaystyle{ \textrm{ODB} = \frac{\textrm{cost}_{\textrm{attack}}}{\textrm{cost}_{\textrm{defense}}} = \frac{1}{μ}. }$

For example, if it costs state A $10 to prevail against state B, which spends$100 on defense, $\displaystyle{ \textrm{ODB}_{AB}=0.1 }$ and $\displaystyle{ μ_{AB} = 10 }$.

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