Utility Function

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A utility or objective function defines a value that a particular model seeks to optimize. It is a way to quantify an individual's preferences over some domain of choices. Here we define a utility function for agents based on their presumed desire for both absolute and relative power.

Motivation

Per Axiom 4, agents generally want to increase their power in an absolute sense. They want their well-being, capital, and capacity to affect events to rise whenever possible. For example, given the two power structures below, the focal agent (highlighted in red) would prefer the structure on the left, in which it is stronger:

Utility BW 1.png

Similarly, agents do not merely want to amass as much power as possible. They also are preoccupied with how much power everyone else has and they generally prefer other agents to be relatively weak. For instance:

Utility BW 2.png

Along the same lines, agents want their competitors to be weak and divided, rather than strong and united. For example, an agent would generally prefer to have five competitors with one unit of power each instead of one competitor with five units of power:

Utility BW 3.png

In short, part of an agent's fantasy is to be very large, with tiny competitors, or to have 100% of the market share of power:

Utility BW 4.png

But again, agents don't want to dominate so much that they can't grow in an absolute sense. These two flavors of greed — absolute and relative — are sometimes in tension, such as when one agent can grow in absolute terms only by allowing another agent to grow even more, or when one agent can dominate another in a relative sense only by suffering a reduction in absolute power.

Derivation and Definition

We can encode these conflicting objectives into the utility function u, which is derived as follows. We approximate the combination of absolute and relative greed with:

[math]\displaystyle{ \mathbf{u}_{i} \approx \frac{\mathbf{s}_{i}^2}{\displaystyle\sum_{j=1}^{n} \mathbf{s}_{j}^2} \sqrt{\mathbf{s}_{i}} }[/math]

where [math]\displaystyle{ \mathbf{u}_{i} }[/math] is the utility to the i th agent. The first component on the right side of this equation reflects dominance: it is the ratio of an agent's size squared to the total of all the agents' sizes squared. This component embodies the idea that the smaller and more divided one's competition, the better off one is. The second component on the right side of the equation provides an incentive for absolute growth. The square root of size is used in order to reflect the diminishing marginal utility of acquiring power, which would mean that one additional unit of power is more valuable to a small agent than to a large one. This is a common facet of economic utility functions.

The equation above is equivalent to:

[math]\displaystyle{ \mathbf{u}_{i} \approx \frac{\mathbf{s}_{i}^{2.5}}{\displaystyle\sum_{j=1}^{n} \mathbf{s}_{j}^2}. }[/math]

But we don't want to assume that 2.5 is the correct exponent. To give ourselves more flexibility, we replace it with the exponent α and posit that 2 ≤ α ≤ 3. The utility function can then be expressed as:

[math]\displaystyle{ \mathbf{u}_{i} = \frac{\mathbf{s}_{i}^\alpha}{\displaystyle\sum_{j=1}^{n} \mathbf{s}_{j}^2}. }[/math]

As α decreases, relative power is incentivized and agents become more apt to use violence to cut other agents down to size, so they can hoard market share. As α increases, they're more prone to pursue absolute growth, which requires mutual constructive action.

Implemented in code, the utility function takes the size vector as an input and returns a vector representing the utility of all of the agents:

Utility[s_] := s^α/Total[s^2]

In this function, α can be a vector in which each agent has its own preference for absolute versus relative power. Allowing α to vary for each agent might better reflect the heterogeneity of the international system, in which some agents are said to have "revisionist" intentions (i.e. they are eager to seek power for themselves).

Behavior

To get a feel for how the utility function appraises different situations, here are the size and utility vectors of a few small power structures, where α=2.25:

Utility vectors.png

It's worth noting that this is not how utility is typically defined by international relations theorists. For one thing, power is not always measured numerically, partly due to longstanding problems in defining exactly what it is, and partly because IR has historically been a qualitative discipline. But even when power has been quantitatively assessed, other sets of assumptions have been applied. For example, Mearsheimer (2014) asserts that states in the international system seek to maximize their "market share" or percentage of total power, or s/Total[s]. While apparently reasonable on the surface, such a metric would be unable to distinguish between the following two scenarios, in which the focal agent has the same market share:

Utility market share.png

Clearly, however, the focal agent would prefer to be in the scenario on the left, as the utility function recognizes. Even though there has been much ink spilled in the debate over whether nation states are motivated to seek absolute versus relative gains (see Powell 1991), we are not aware of a mathematical expression having been devised to mediate between those two preferences.

This utility function not only puts a definite value on those preferences, but also provides a way to identify equivalent outcomes. For example, consider the size vectors [math]\displaystyle{ \mathbf{s}_{A} }[/math] = {1.1, 1.1} and [math]\displaystyle{ \mathbf{s}_{B} }[/math] = {1, 0.976}, where α=2.25. Agent #1 has the same utility in both of these situations. It can grow by 10% along with agent #2, or reduce #2's size by around 2.4%, and be equally happy either way. Because destruction is easier to accomplish than cooperation — due to Axiom 2 and because it can be done unilaterally — agents often turn to it as a way to satisfy their utility.

The utility function has a few other desirable properties worth mentioning. First, agents that are the same size have the same payoff, dead agents have a payoff of zero, and the largest agents will have the largest payoff. Adding agents with a size of zero to the population doesn't affect the payoffs to the existing agents. Further, when there are two agents whose sizes differ by a constant amount, their payoffs will tend to converge as their sizes increase by the same amount. For example, the payoffs to agents with s = {100, 101} will be closer together than those with s = {1, 2}. Finally, the utility function is smooth and well-behaved, except when all agents have a size of zero, in which case there's no one left to care.

Utility as function of s.png

Probably there are other utility functions that could be devised with properties similar to this one. For example, why not multiply the factors for absolute and relative power, as opposed to adding them linearly or combining them with weighted exponents, as in a Cobb-Douglas production function?

However, this function seems fairly simple and elegant enough, and is basically fit for purpose. It is conceptually similar to the Herfindahl-Hirschman Index (HHI), a standard measure of competition within a given market which can be computed from a size vector using Total[(s/Total[s])^2].



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