Implications of PrinceRank

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In this section, we explore the implications of the PrinceRank metric. Most of the examples use ternary tactics, although PrinceRank also works on continuous and asymmetric power structures.

Structural Ideals

An Agent's Fantasy

What would an agent's ideal world would be like according to PrinceRank? That is, if an agent could structure the relations of a system of equally-sized agents however it wished, what would that power structure be? When we sample continuous tactic matrices and select the one that gives the focal agent the highest PrinceRank, we get something like this:

IPR ideal.png

The focal agent's ideal arrangement is for every agent in the system to be giving it constructive power, and for it to be reciprocating to some lesser degree. This pattern is reminiscent of imperial systems and their tributary states, and of hierarchical relationships generally. Historically, weak nations often paid tribute to a strong one, which in turn protected the weak from attack. This type of power relationship has been known to exist since the dawn of recorded history and even though it doesn't exist in the same formal sense in modern international relations, it is still the case that less powerful political entities often provide a disproportionate volume of benefits to more powerful ones, who seek to prevent the disruption of that flow of benefits. These tributary links form the backbone of hierarchical power structures. If this kind of hub-and-spoke structure above is every agent's fantasy, we would expect that when they have the ability to shape networks, they will generally try to impose this topology upon it.

What is considered optimal in an agent's ideal power structure depends upon the specific parameter values used, particularly δ and α. When an agent is shortsighted (low δ) or not interested in absolute growth (low α), then its ideal structure is just to receive constructive power from every other agent and not reciprocate anything in return. In contrast, forward-thinking agents reciprocate in order to create a positive feedback loop of mutual growth, albeit one based upon unequal exchange that allows them to grow at a faster rate than their minions.

Pairwise Preferences

Preferences in Dyads

When there are only two actors in a power structure, what relationships do they prefer as their relative sizes vary? When the two agents are approximately equal in strength and ternary tactics are used, PrinceRank shows that they have an incentive to cooperate with each other. As indicated by the three possible relationships below — positive, neutral, and negative — both agents' PrinceRanks are maximized when they cooperate:

IPR pairwise 1.png

On the other hand, when there is a slight size difference between the two agents, as shown below, the stronger one has an incentive to attack as a way to keep its rival at bay, dominating it while it still can. Here one cannot help but be reminded of Machiavelli's repeated injunctions that power politics requires the destruction of rivals before they present a danger.

IPR pairwise 2.png

When there is a significant size difference between the two entities and the weaker one is no longer considered a threat, conflict is not worth the expense to the stronger agent, and neutrality appears to be the preferred option:

IPR pairwise 3.png

This last example gives the impression that a large agent and a much smaller one will not engage with each other, a conclusion that would appear to be in tension with the incentives noted above regarding hierarchy formation. The reason for the apparent inconsistency is that the scenario above uses ternary tactics, which are not sensitive enough to contemplate small allocations of power when there are only two agents. With more agents, the large agent can subdivide its outgoing allocation among numerous recipients. With only two agents, however, the large one's allocation would increase the size of the smaller one disproportionately. In a more nuanced, non-ternary set up, the stronger party could allocate a smaller amount of power, in which case a cooperative protectorate-tributary relationship would then be in both agents' interests:

IPR pairwise 4.png

As shown above, when continuous tactics are used, exchange is the preferred option, not neutrality as was the case with ternary tactics. So the incentive structure of PrinceRank does support hierarchy formation, even if it's not always apparent when ternary tactics are used.

Alignment with Realist Theory

We can crudely summarize the bilateral preferences just described as: agents generally engage in positive exchange, but cut down potential rivals. They allow quantitative realism to model the phenomenon that some international relations scholars call the Thucydides Trap (Allison 2017). This is the observation that a rising power presents a threat to an established power, which feels pressure to attack, as was the case at the outset of the Peloponnesian War. As famously noted by Thucydides, who chronicled and interpreted that war, "What made war inevitable was the growth of Athenian power and the fear which this caused in Sparta." We can portray a Thucydides Trap as a sequence like this:

IPR Thucydides Trap.png

As the rising power (left) starts to grow (t=1,2,3), the established power's PrinceRank turns green, reflecting its mounting displeasure. At some point (t=4), the rising power is considered a genuine threat, and the established power attacks it while it can still prevail. Both parties are reduced by the conflict (t=5), until the established power regains a comfortable size difference, at which point it ceases the attack. A tributary relationship may very well form in the aftermath of this conflict (not shown). The Peloponnesian War was obviously much more complex than this, involving numerous parties, many years of combat, and a variety of plot twists that made the result far from inevitable. Nonetheless, in the end, the upstart Athenian Empire was destroyed.

These bilateral incentives also align with certain interpretations of political realism, such as that of Henry Kissinger. In Diplomacy, Kissinger asserts that two types of international systems are possible: a universal empire with a single dominant state, and a balance of power system in which states form coalitions to counterbalance against potentially dominant or aggressive states. PrinceRank reveals incentives for both types of systems. First, it shows that states with large power differences have incentives to form tributary relations and, by extension, imperial structures. Second, it posits that relatively equal entities have disincentives to fight with each other and that those with slight size differences have incentives to fight. Each of these entities can be a coalition of allied states, meaning that balanced coalitions have disincentives to fight — the essential claim of balance of power theory. So quantitative realism is not only in accord with Kissinger-style realism, but provides a theoretical foundation for it.

Triadic Structures

How would an agent order three-agent, or triadic, power structures according to PrinceRank? Let's consider all triadic, ternary structures in which all agents have the same size. There are 18 such scenarios that are unique from a focal agent's perspective. When we sort these by PrinceRank, there is a fair amount of variation in the ordering based on whatever parameters are chosen. However, the most preferred and least preferred structures tend to remain in the same general place in the list, regardless of the parameter choices. Here are four of the most preferred structures:

IPR triadic best.png

The first scenario is the schadenfreude structure, in which the focal agent takes numerical pleasure in the suffering that the other agents are inflicting upon each other. Their conflict is causing the focal agent's PrinceRank to increase. Some realists call this pattern bloodletting, which is when a state happily watches two rivals weaken each other. The second structure we've seen before: the hub-and-spoke pattern is every agent's fantasy. The third scenario is a composite: the focal agent has allies who are helping it grow, but the allies are weakening each other. And in the fourth, the focal agent has an ally, resulting in growth.

Now let's look at the triadic structures that tend to be among the least preferred, using the same parameters:

IPR triadic worst.png

What these power structures have in common is that the focal agent is being attacked, sometimes by two agents and sometimes when the other agents have formed an alliance. These examples align fairly well with intuition. In the most preferred scenarios, the focal agent is not being attacked, there are no rivals in a position to surpass it, and it is poised to grow in absolute terms. In the least preferred structures, the focal agent is the object of violence, including by hostile alliances.

Here are all 18 triadic, ternary structures (with symmetries eliminated), ordered by the focal agent's PrinceRank, along with the parameters used:

IPR triadic all 1.png

The ordering is sensitive to changes in α. For example, if we raise that parameter, we see the focal agent's preferences shift slightly in favor of structures that will give it real growth:

IPR triadic all 2.png

The agents in the power structures above all have the same size. Here's what PrinceRank has to say about triadic, ternary structures with random relationships and agent sizes:

IPR triadic random.png


PrinceRank reflects agent preferences for being at the epicenter of hierarchical power structures. This section examines the basic incentives that lead to the establishment, maintenance, and overthrow of hierarchies.

Establishing Hierarchy

As is evident from an agent's ideal power structure and from the triadic preferences shown above, agents have strong affinities for hierarchy — specifically, hierarchies in which they are the hub. Why do hegemon-tributary relations form? When one agent is significantly more powerful than another and the flow of mutual power balanced such that the stronger agent gets an even or disproportionate benefit out of the relationship, the stronger agent need not be threatened by the growth of the weaker one. From the perspective of the weaker agent, it's better to pay tribute than to be disconnected, particularly when resistance to the hegemon is futile. Additionally, the weaker agent can still grow in an absolute sense, even under conditions of unequal exchange. Under such circumstances, a cooperative hegemon-tributary relationship is preferable for both parties.

For example, the power structures below show two possible outcomes for the focal agent. It can join the hierarchy as a tributary state (left) or it can remain unaffiliated (right):

IPR hierarchy tributary.png

It is obviously in the hegemon's interest for the focal agent to join. It is in the focal agent's interest as well, as indicated by the slight increase in its PrinceRank. This is typical of the mutual incentives that lead to the creation of hierarchies.

Notably, a hierarchy maximizes the total PrinceRank in a system. Consider the following power structures, which all have the same size vector:

IPR hierarchy total juice.png

The hierarchy (far left) has the highest total PrinceRank out of the examples given. Presumably, this is a universal rule, due to the fact that hub agents are rewarded with extreme levels of PrinceRank. If PrinceRank is equated with well-being, then a hierarchical structure is the topology that maximizes the population's well-being — setting aside the fact that that well-being is not distributed equitably. This structural factor tends to push systems towards hierarchies.

Maintaining Hierarchy

Since at least Machiavelli, one strand of realist thought has concerned itself with the maintenance of hierarchy. The idea is that a hegemonic agent must be vigilant about maintaining its position against rivals and must discipline coalition members in order to maintain its supremacy. PrinceRank provides a glimpse into what motivates this.

Mutual Defense

First, PrinceRank indicates that hub agents typically don't like when their tributary states are attacked by third parties. As shown below, the hegemon's PrinceRank decreases in the scenario on the right, in which one of its sources of power is being diminished.

IPR hierarchy third party.png

PrinceRank also reflects incentives for tributary states to join in the hegemon's campaigns against third parties. For example, in the sequence below, at t=1, agents #1 and #2 are allies but they have inconsistent policies towards agent #3. At the next time step, agent #2 is faced with a choice: continue its alliance with #3 and lose support from #1 (t=2a), or harmonize its foreign policy with agent #1 (t=2b). PrinceRank shows that agent #1's threat of playing time branch 2a has the potential to coerce agent #2 into time branch 2b.

IPR hierarchy help hegemon.png

These preferences can create a kind of mutual defense arrangement in which a strong state defends a weak one to protect its supply of power, and the weak acts as a mercenary to the strong — a point we develop in more detail later on.

Infighting and Collusion

Given a hub-and-spoke structure dominated by a powerful agent, two general challenges present themselves to the leader: coalition members fighting with each other and coalition members colluding with each other.

IPR hierarchy collusion.png

Infighting is a problem because it drains the system of constructive power that would otherwise enrich the hegemon, and collusion is a problem because it threatens the hegemon with the rise of an alternate power center. PrinceRank shows that such behavior is generally displeasing to hegemonic agents. The hegemon facing infighting or collusion will have to apply carrots and sticks to correct these situations, such as by temporarily withholding support from the offending parties or by inflicting punishment. On the other hand, a hegemon may not mind infighting if it serves to weaken potential rivals, as shown here:

IPR hierarchy infighting.png

The focal agent's PrinceRank is higher in the structure on the right, where the infighting is occurring.


The desire to rebel against a power structure, and the converse desire of powerful agents to resist disturbance, forms a theme that permeates all of history. The representation of those desires in the context of quantitative realism is straightforward, and PrinceRank reflects the kinds of incentives that one would naturally expect to see.

Consider the sequence below. The canonical starting point is a hub dominated by a powerful agent (t=1). Rebellions are only successful when a critical mass of agents act in concert to overthrow the powerful, and the subsequent time steps show how the preferences of the various parties change as more agents participate in the revolt. The PrinceRank of the focal (hub) agent plummets as more agents join the fray.

IPR rebellion 1.png

PrinceRank also reflects the preference of agents to free ride on another's aggression. For example, the PrinceRank of those who don't rebel at t=3 is higher than it would be if no one had rebelled at all (t=1), though this is not distinguishable in the diagrams above. The non-rebelling agents reap the benefits of the fight without incurring any of its costs. Moreover, the PrinceRank of those who do rebel is generally lower than it would have been had they not rebelled at all, reflecting the reality that there's a cost to rebelling. Part of the calculus of revolution is for agents to determine whether those short-term costs are worth taking in light of the probability of overall success.

The PrinceRank of a rebel goes up when more agents rebel along with it, because the hegemon then has to fight on multiple fronts and weakens more quickly. In the example above, the largest rebel benefits the most when everyone rebels. Not only is its PrinceRank higher than that of all the other agents (above, at t=4), including the soon-to-be former hegemon, it is also higher than it was in the original, hierarchical power structure. This "beta" agent is the only one who would ultimately prefer a full rebellion.

PrinceRank shows that rebels sometimes have an incentive to cooperate with each other. For example, even though one can't discern it in the image below, the smaller agents have incentives to support the beta agent as a counterbalance to the hegemon:

IPR rebellion 2.png

Notice also how the focal agent is unhappy about this support, for two reasons: it bolsters a rival and it diverts power that would otherwise be allocated to the hegemon. These arrangements are likely to be short-lived if the hegemon has the ability to quell the uprising and reimpose order, and it seems natural that a threatened hegemon would want to fight back. However, the hegemon could also increase the amount power allocated to the disgruntled agents, essentially buying their loyalty.

The final stage of a successful rebellion or revolution is when a new political order is established. Suppose we play out the initial scenario, with all of the agents rebelling simultaneously (t=4,5). After the hegemon is defeated (t=5), the victors form a new hierarchy (t=6). In this particular example, all of the rebels end up with a higher PrinceRank than they did when the revolution began, despite the fact they they have all been diminished in absolute size. This is because they have eliminated agent the original hegemon, a giant that was dominating them.

IPR rebellion 3.png

PrinceRank can account for the basic incentive patterns of revolution: agents willing to absorb the short-term pain of fighting in order to achieve the long-term benefits of a newly balanced order, and established powers struggling to maintain the status quo.

Divide and Rule

Divide and rule, also known as divide and conquer, is a classic strategy of political control. It entails breaking up larger concentrations of power by sparking rivalries and preventing smaller agents from linking up. In quantitative realism, an agent can't interfere directly with the relationship of two other parties, and it can't create dissension by spreading lies, as there are simply no mechanisms in the model to accomplish that. However, agents can nonetheless choose tactics that use rewards and punishments to break up rival coalitions.

What kind of opposing coalition would an agent would prefer to deal with? As the figure below illustrates, agents would generally prefer a group of rivals to be divided by infighting rather than unified against it, all else being equal:

IPR DnR divided.png

Assuming that an agent finds itself in a situation like the one on the left, how might it attempt to turn it into one more like that on the right? One possibility is to introduce discord into the coalition by attacking some members and co-opting others. This strategy can imbalance relationships, triggering waves of strife and punishment within the coalition, because the co-opted agents have an incentive to turn on each other.

Let's examine the simplest possible case of divide and rule. Suppose that agents #2 and #3 have an alliance, and that agent #1 is stronger and excluded from it (t=1). Agent #1 commences divide and rule operations by co-opting agent #2 and attacking #3 (t=2). Agent #2 then has to choose between two outcomes: remain with #3 and be attacked by #2 (t=3a), or betray #2 and align with #1 (t=3b). As agent #2's PrinceRank indicates, the latter option is preferred, enabling #1 to accomplish its objective of fracturing its opposition.

IPR DnR simple.png

In the example shown, the agent carrying out divide and rule (#1) used a combination of positive and negative tactics. However, divide and rule can also be more subtle, entailing any kind of differential treatment among similarly situated agents, as well as flip-flopping between temporary favorites. Agents can also use divide and rule to exploit existing divisions within an opposing coalition, causing some members to get sucked into the fray and diverting destructive power away from the instigating agent.

Even though this glosses over some of the finer points — for example, the sizes of the agents here do not change at the various time steps — it should be evident that PrinceRank helps illuminate the essential logic underlying divide and rule: both the preference structure that motivates it and the mechanism of its attainment.

Competing Alliances

Most of the examples above involved hierarchical power structures, even if some of them were undergoing processes of degeneracy. But multiple hierarchies often coexist together in a single system, forming alliances that face off in potential competition. This section explores PrinceRank in that context.


Each of the three power structures below depicts multiple alliances in what are arguably bipolar worlds. PrinceRank nevertheless manages to reasonably reflect each agent's level of satisfaction within these structures.

IPR alliances polarity.png

Exclusivity of Tribute

The first observation is that hegemonic agents do not like when their tributary states have positive relationships with other powerful agents. They prefer that tribute be given exclusively to them. For example, in the diagram below, the focal agent's PrinceRank declines when one of its tributaries also pays allegiance to a rival power.

IPR alliances exclusive tribute.png

Given this hegemonic preference for exclusivity of tribute, weaker members of a multipolar system are often faced with a choice of which major power to align with. In the scenarios below, the focal agent can choose between joining either of two alliances:

IPR alliances choice.png

The variations in PrinceRank show that each hegemon would prefer that the focal agent join its gang. For them, every alternate alignment is a double loss: less power for themselves, and more power for a competitor. In the example given, it would be better for the focal agent to align with the alliance on the left. However, the choice might be different if the major power on the right were significantly stronger.

Balancing, Bandwagoning, and Buck-Passing

The international relations literature often refers to idiomatic behavior such as balancing, bandwagoning, and buck-passing. Balancing is a broad term that seems to refer to any conduct that attempts to reestablish a balance of power. It could involve shifting alliances to oppose an aggressor state, direct confrontation with an aggressor, internal strengthening, or some sort of signaling. The term bandwagoning is used to describe situations where a weaker state joins with an aggressor, rather than helping to balance against it. And buck-passing is when a state gets another state to do the balancing, either by providing it with support or, more commonly, by doing nothing, i.e. "passing the buck."

Balancing, bandwagoning, and buck-passing can for the most part be represented in the context of abstract power structures. Consider a scenario in which there is one hierarchy, led by a focal agent, and a separate, rival structure:

IPR alliance setup.png

The question is what options the focal agent has for containing its large rival in the southeast. Balancing here can be construed in two different ways: as the initiation of aggression to cut down a rival or as the formation of an alliance that counterbalances the rival by building strength:

IPR balancing 1.png

Bandwagoning and buck-passing are depicted below:

IPR balancing 2.png

Even without a full game tree to help us reason through this situation, PrinceRank exposes the preferences of the relevant parties to help us understand what outcomes are possible and mutually desirable.

Latent Tension

Some power structures have intrinsic stress due to the incentives of certain agents to initiate conflict. We can explore this latent tension by looking at power structures in which there is no active conflict, but where PrinceRank suggests that a party has an incentive to start one. Consider the following power structure, in which the great powers that have a slight size difference:

IPR tension 1.png

At the next time step, the great powers can form one of the three possible ternary relationships:

IPR tension 2.png

PrinceRank suggests that the focal agent can improve its position at t=2 by instigating a fight with the other great power, which is slightly weaker. It doesn't matter that the other agent's PrinceRank is lowest in this scenario, because of Heuristic 1: if one agent wants to fight, a fight will occur. Accordingly, the power structure at t=1 has a latent tendency for great power conflict — an instability inherent within it that is not immediately obvious on the surface.

Compare the situation above with a variation in which the two most powerful agents are roughly equal in size:

IPR tension 3.png

Here the incentives of the great powers are different, and PrinceRank indicates that it is in their mutual interest to cooperate. Accordingly, there is less latent potential for great power conflict.

IPR tension 4.png

The same latent tensions can exist between any two agents in a system, and not just between great powers. It doesn't mean that two agents will necessarily fight merely because some structural tension is present. Agents have to consider all of their other possible relationships and any counter-reactions to their actions. So the discussion above is not meant to suggest that conflict is inevitable under any given set of circumstances.

Power structures with latent tension illustrate the fear and insecurity that agents experience in an anarchic setting, and PrinceRank highlights the fact that structures that appear to be in equilibrium may actually be fraught with destructive potential.

Unity under Threat

Another theme in power politics is that agents tend to unify in the face of a common threat. To see how PrinceRank accounts for this incentive, consider the four power structures below:

IPR unity 1.png

In diagrams (a) and (b), in which all agents are the same size, when two agents unify it increases their PrinceRank by a factor of 2.6 (=.37/.12). In contrast, in diagrams (c) and (d), in which there is a more powerful agent in the system, cooperating increases PrinceRank by a factor of 4.3 (=.26/.05). The mere presence of a dominant agent intensifies the incentives of smaller agents to unite with each other. This sort of four-way comparison is necessary because agents generally like cooperation and prefer to be the hub of the network, and so would be inclined to cooperate regardless. However, the presence of a more powerful agent has the causal effect of amplifying that desire.

The same incentive exists when the powerful agent is not merely present but also behaving aggressively towards weaker ones:

IPR unity 2.png

Here the focal agent's PrinceRank increases by a factor of 1.8 (=.23/.13) when transitioning from (a) to (b), and a factor of 2.3 (=.12/.05) when going from (c) to (d). Note that this assumes that the power that the smaller agents allocate to cooperation does not detract from the power needed to defend themselves against their attacker.

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