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Power structures are not static in time. There are two processes at work that make them evolve. First, agent relationships cause power to be transferred, and these transfers make the agents stronger or weaker. Second, agents continually alter their relationships to respond to the state of the power structure, in a ceaseless quest to improve their position within it. Since a power structure is a graph, the first process relates to how vertices change in size, and the second describes how edges change.

These two processes boil down to seven core principles that describe how power structures change over time. Six of these principles are fundamental assumptions, or axioms. One of the principles is a heuristic made for convenience. Each axiom is associated with a model parameter that specifies the degree to which the axiom has its intended effect. The parameters allow the axioms' qualitative statements to be quantified.

Principles of Vertex Change

The first three axioms describe how the vertices of a power structure change in size as a result of agent interactions.

Axiom 1: Effect of constructive action

Constructive action is an expenditure of power that increases the power of another agent by more than the amount expended. Constructive action among states might entail trading goods or services, forming a military alliance, supporting an ally with troops and weapons, paying tribute, providing disaster relief, engaging in cultural exchange, or giving other forms of assistance.

As a result of a constructive action, the acting agent loses power and the receiving agent gains power. For instance, in a transfer of one unit of constructive power, the acting agent will lose one unit and the receiving agent will gain more than one unit. When agents cooperate by reciprocating constructive exchanges, they make each other more powerful, with each becoming better off than they were before the exchange. When one agent allocates a positive amount of power to another, the amount received by the other agent is increased by a factor of β, where β > 1. The parameter β, called the constructive multiplier, reflects the idea that, regardless of what has been given, it has more value to the recipient than it did to the giver. β can be thought of as an imaginary interest rate.

The effect of this axiom is that when two agents both cooperate by acting constructively towards each other, they both increase slightly in strength as a result of the cooperation. Consider a sale of oil between two countries. We can assume that to the purchaser, the oil was more valuable than the money it paid, otherwise it wouldn't have purchased it. Likewise, to the seller, the money must have been more valuable than the oil, or else it would not have put the oil on the market. The transaction served to increase the well-being of both parties. In the case of economic transactions, this increase is a manifestation of the division of labor and the benefits of exchange, and similar justifications can be put forward for other types of transactions. Power is not zero sum: agents can work together to create more of it.

Constructive power is expressed as a positive number.

Axiom 2: Effect of destructive action

Destructive action is an expenditure of power that decreases the power of another agent, with more impact than constructive action. Destructive action entails the use of violence or the imposition of unwanted consequences. It includes actions like military assault, siege, bombardment, killing, destruction of property and infrastructure, and terrorism. Both the acting agent and the receiving agent lose power as a result of destructive action. Since it is easier to harm another agent than to help them, and easier to destroy value than it is to create it, destructive action has a greater impact than constructive action. For example, it costs much less to raze a city than it does to build (or rebuild) it, and it's much easier to harm someone than to help them to the same degree.

When a unit of destructive power is transferred, the acting agent loses one unit and the receiving agent loses more than they would have gained had the transfer been constructive. Like constructive action, destructive action is subject to a multiplier, μ, the destructive multiplier, such that μ > β. That is, when an agent uses power negatively, for every unit of power it expends, it causes a reduction in the recipient's power by μ. Destructive power is expressed as a negative number.

Axiom 2 is essentially a statement about entropy and the ease with which states of disorder can be reached, as compared to the amount of effort required to create or impose order.

Axiom 3: Effect of unused power

Power that is not used constructively or destructively, but that is instead retained by an agent as a stock, either depreciates or has a lower growth rate than power used constructively. Without this axiom, the incentive for agents to cooperate with each other would be undermined. The amount of growth/depreciation is governed by the parameter λ, such that λ < β.

Principles of Edge Change

The remaining principles govern how agent relationships in a power structure change over time.

Axiom 4: Pursuit of absolute and relative power

Agents prefer other agents to be relatively weak. They are preoccupied with how much power other agents have, wanting to be relatively powerful while keeping others relatively powerless. Given the choice, agents would prefer that their competition be weak and divided, rather than strong and united. For example, an agent would generally prefer five competitors with one unit of power each, to one competitor with five units of power. At the same time, agents want to become more powerful in absolute terms, meaning that they want the amount of power they have to increase.

Later we define a utility function that quantifies these preferences, taking into account the absolute and relative sizes of the agents. Within this function, the parameter α controls the importance of absolute versus relative power.

Axiom 5: Social inertia

It cannot be the case that agents are free to choose whatever tactics they like. In the real world, the past, or more specifically the present, binds their options. For example, a country could not one day suddenly alter its entire foreign policy. Processes like trade agreements, peace talks, mergers and acquisitions, and divorces all take time, because social relationships have a kind of stickiness that resists rapid change. This phenomenon is called social inertia, and it makes it less likely that agents will be able to effect dramatic tactical changes. The parameter ρ expresses the degree of social inertia present in a power structure.

Axiom 6: Ongoing interaction

Agents engage in ongoing interaction. They do not interact merely one time, nor do they necessarily know how long the interaction will continue. However, they expect to continue interacting with each other in the future. There is no terminal state or end game; agents just continue the struggle for power forever. How far ahead agents look in time is influenced by the parameter δ, or the discount rate of intertemporal utility, a concept which is explained later.

Heuristic 1: Reciprocity

Though agents can behave asymmetrically towards each other, with one engaging in a constructive action and the other engaging in a destructive one, such asymmetrical actions are likely to be short-lived. A constructive action will likely only be continued if it is reciprocated, and an agent who is attacked will tend to attack back (Axelrod 1984). So relationships are presumed to be reciprocal. This means that agent actions towards each other typically have the same polarity, even if each agent does not necessarily allocate the same amount of power. To resolve cases in which agents cannot agree on a relationship, this axiom assumes that one agent can start a conflict unilaterally, whereas cooperation requires mutual consent.

There is no parameter associated with the reciprocity heuristic. This may suggest that it is not actually fundamental and that it can instead be derived from the other axioms.


To recap, the principles and their associated parameters are:

  • A1. Effect of constructive action (β > 1)
  • A2. Effect of destructive action (μ > β)
  • A3. Dissipation of unused power (λ < β)
  • A4. Pursuit of absolute and relative power (2 ≤ α ≤ 3)
  • A5. Social inertia (0 < σ < 1)
  • A6. Ongoing interaction (0 < δ < 1)
  • H1. Reciprocity

The Axioms and Political Realism

Though on the surface it might appear that these starting assumptions are a radical departure from realism as traditionally conceived, they can be interpreted as a generalization of traditional realism. The traditional assumptions that states have military capacity and interact under anarchy are, here, abstracted into the simple idea that agents can use their power to render each other powerless, and possibly even nonexistent. And in both paradigms, agents interact repeatedly and are assumed to take self-help action to promote their own survival in light of the current distribution of power. What this new formulation adds are the following: First, the conception of power is more general. It's not solely about destructive capacity and it's not zero sum: power can be created as a result of cooperation. Second, agents make tactical decisions based upon their interrelationships and not solely upon the distribution of power, as in traditional realism. Finally, agents' use of power has quantitative effects on other agents' power levels, causing the distribution of power to change in potentially measurable ways. Overall, this generalization allows quantifiable consequences to follow directly from the starting assumptions.

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