A Theory of Networked Alliances

This is Lieutenant Peter Devine's companion paper to our event, "Applied Economics in Foreign Policy." You can watch the full discussion here.

Motivation

Why do wars occur? There are as many answers to this question as conflicts. Nevertheless, are there some common threads that will help us prepare and possibly influence the next one?

Such a noble pursuit as preventing future wars has attracted scholars from every discipline. However, time and time again, scholars have learned to temper their expectations for any pacifying effect academic analysis can have. In "The Arrogance of Power," William Fulbright claims that wars are started by emotion rather than reason. If so, then using economic analysis to study, predict, and affect decisions to fight might not be effective. However, even the most 'irrational' states and dictators still operate rationally within their incentive matrix. Take North Korea; when it threatens nuclear war, it is not on a whim. North Korea needs to maintain power over its population and deter any foreign state from intervening while generating economic concessions. In this context, their behavior starts to look very much like a rational strategy.

Carl Von Clausewitz asserted two essential concepts in his inscrutable work, "On War": first, wars cannot be made obsolete through a mathematical solution. Wars will always be fought. No amount of out-maneuvering will guarantee a world of peace and an elimination of conflict. Nevertheless, his second insight is critical to establishing a tractable study of conflict: war is an extension of politics by other means. That means that wars are executed and fought based on rational responses to incentives, deterrence, and expected gains. This insight, caveated by an understanding of human behavior, provides a firm enough assumption about the nature of war to unpack, analyze, and generate results.

This paper outlines a new model of conflict prediction through network analysis and private information. Most models of alliances assume all information about all states is known to everyone (perfect information). Assuming perfect information allows rational actors to directly evaluate their rivals' relative strengths and weaknesses and choose to attack or not based on that assessment. When all information is known, equilibria are determined based on whatever parameters about the nature of conflict the author assumes. E.g., is conflict rich rewarding or poor rewarding? Is it a strategic complement or substitute? There is still valuable insight from this approach, but a real-world application can be limited. The models tend to predict corner solutions, either one country achieves world domination, or no wars occur.

Empirically, states invest significant resources into intelligence, counter-intelligence, and misinformation, suggesting that accounting for private information is a critically important factor in understanding alliances and conflict. This paper outlines a method that incorporates private information into the network model, not about military strength directly, but about the strength of the alliance; essentially, how relevant is a potential ally when facing a decision to fight or join an alliance?

This paper first introduces some stylized facts drawn from military history and geopolitical literature. Second, it defines a model that attempts to explain these facts and draw new conclusions. Third, it interprets current actions in the Western Pacific and offers some recommendations. Finally, it identifies some paths of further research.

Stylized Facts

1- Weaker countries tend to join an alliance in the face of impending conflict.

2- These 'defensive' alliances can be with the defending national bloc or the attacking national bloc.

3- Nations that remain neutral believe they are either insulated from consequences or stand to gain from war regardless of the outcome.

4- The closer a war gets to a nation's network border, i.e., fact (1) is increasingly dominating fact (3), the more likely they are to join a network.

5- Alliances are joined based on the lower cost of conflict or the higher reward of conquest rather than political ideology.

6- Nations have long collective memories and entrenched beliefs about their rivals' intentions, capabilities, and types.

7- (5) and (6) jointly imply nations tend to make choices based on cultural and ethnic factors rather than ideological ones.

8- Nations are rational actors, although miscalculations of incomplete information often start wars.

The Model

This model uses a simplified network consisting of an alliance of 3 member nodes {1, 2, 3} opposed to a single Leviathan node {L} with no allies.

Order of Play

In essence, this is a repeating stage game.

1. All states have private information about their strength and commitment and signal their draw to the other states.

2. The Leviathan picks a single alliance member and enters a Hawk Dove game. If both states choose Hawk, a conflict starts.

3. Once a conflict begins, the Leviathan engages with one of the alliance members. The two unengaged alliance members enter a contribution game that determines if they will honor their commitment to the alliance and fight with the engaged member.

4. After payouts are determined, states update their beliefs for the next round.

Solution Concepts

We can solve this by starting at the end of the game and bringing the expected results back up the decision tree ("looking forward and folding "backward"). The crux of this system is what happens in the contribution game, i.e., what determines if an allied state will honor its commitment or not?

The contribution game assumes that states gain some value, vi, from winning a conflict (where vi = 0 if the conflict is lost), but fighting is painful, so they incur a cost, c. Critically, these values are idiosyncratic to each state and are known only to that state. Each state now has a specific type defined by the continuous variable vi, but a finite decision to either honor the alliance and contribute forces, or not: {C, NC}.

The alliance members have a probability of winning the engagement equal to all the members who choose to contribute proportionally to the total military strength.

I make a simplifying assumption that investment in military strength xi is a monotonic function of vi. Those transformations will drop out algebraically, and we can reduce the number of variables to just vi without losing any dimensionality.

When each state has a continuous type { vi }, but finite action set {C, NC}, an equilibrium solution can be a cutoff strategy such that if vi > 𝒗 , then that alliance member will contribute. I assume that vi is uniformly distributed on the interval [0,1], so the probability that a state does not contribute {NC} is the cdf of v: F(v), which is common knowledge to all states.

Using this solution, we can work back to the Hawk Dove game and recognize that a mixed strategy exists in which the Leviathan will choose {Hawk} with probability α > 0 and the alliance member will choose {Hawk} with probability 𝛽 > 0 guaranteeing that with some positive probability a conflict will occur initiating a contribution game.

Finally, while all states do not know the specific strengths of those around them, they know the probability distributions F(vi) for each and calculate expected results. Before a state chooses to risk starting a conflict (choosing an action from the set {Hawk, Dove}), they will try to signal to gain an advantage.

Results and Implications

While the results are still being built and checked for robustness, there are some clear implications to the stylized facts established earlier. The Leviathan wants to face as weak an alliance as possible, so choosing which member to engage is determined based on expected counter strength.

The member states of the alliance, however, have a much more complicated picture. The most likely to be engaged state wants a strong alliance and will try to prepare the ties through credible commitments. The least vulnerable state prefers a weaker commitment. This is because more substantial commitments could encourage the engaged member to choose {Hawk} with higher probability and therefore impose the costs of conflict more frequently. This opens up serious questions about the asymmetry between alliance members in strength, centrality within the network, and degrees to the Leviathan.

A Brief Example in the Pacific

This model has an immediate application to the Indo-Pacific, where alliances are present, if sometimes informal, and a rising power affects the strengths of multilateral alliances. Take, for example, the Republic of China (Taiwan). Taiwan is surprisingly important to the Indo-Pacific strategic alliance networks and the Peoples Republic of China (PRC).

The Importance of Taiwan to Each Network

According to the PRC, Taiwan is a renegade province that the party must establish control over. Additionally, the longer Taiwan remains independent, the more difficult reintegration will be. For President Xi, this presents a political opportunity to elevate his status, underline the PRC's hegemonic rise, and send a powerful signal to every third-party nation in the region that a defensive alliance to counter the PRC will be counterproductive.

For the loose democratic alliance, the stakes may be higher. Arguably, the United States and allies have been the underwriters of security in the Indo-Pacific since World War II, which is especially true for Taiwan, where America has made political promises (but only limited legal ones) to support Taiwan's defense. If the PRC could bring Taiwan under party control, it would be an undeniable signal that the network of alliance members is weakening and cause all thirdparty states to update their beliefs and recalculate their geopolitical and geoeconomic strategies.

For many years, the PRC's strategy towards Taiwan has been to bring the island state into the fold through stronger cultural, governmental, and chiefly economic ties. One country two systems, the same model as Hong Kong, was a mollifying promise to the Taiwanese who fretted over losing democratic governance. However, the PRC's actions in Hong Kong over the last year have exposed that offer as unlikely and raised the value (vi) of winning a conflict for both sides. All things being equal, this is likely increasing the probability of a conflict across the Strait.

Predicted Results

According to our model, as soon as the PRC calculates it can win, it will engage Taiwan in a Hawk Dove game. The question then is how does the PRC make that determination? It's not the power of the U.S. 7th fleet that will directly determine this, it's a question of how strong Taiwan's network is. The PRCs strength, the strength of the opposing states, and the probability that the alliance members will honor their commitments all factor into the calculation.

Both the PRC and the alliance members know this and have formed beliefs about each other's private information: if a conflict occurs, how likely will allies contribute to Taiwan's defense? So before entering a Hawk Dove game, both sides will try to influence the other's decision by indicating their relative strength and the possible costs:

"Some people may think that by talking to other countries before meeting with China, they can give out their voice and show their strength. This is unnecessary and may not work. It is just like when people walk alone at night, they may sing to dispel their fears, which may not be useful"

– Chinese Ambassador to the United States Cui Tiankai, March 17th

In line with the model prediction, the PRC is signaling that the democratic allies will not come Taiwan's (or anyone else's) aid. Conversely, the United States is trying to signal the strength of the alliance, recently affirming the specifics of the "Six Assurances" and passing the Taiwan Allies International Protection and Enhancement Initiative. However, the United States, which isthe stronger ally, is concerned about increasing the likelihood of Taiwan taking a provocative action (choosing Hawk) and leaves strategic ambiguity in its signal of commitment. Calibrating the model will help find the optimal military, political, and economic signaling strategy for Taiwan and other allies.

Further Research

While this model is a potential gain for strategic planning in a network environment, further research is needed to understand its implications and robustness. So far, this game has been treated as static, however, adding multiple rounds with Bayesian updating needs to be explored to understand its real-world applications better. Allowing the alliance member states to choose to contribute sequentially rather than simultaneously should also be checked and allow the Leviathan to enter its own networked alliance.

Furthermore, an exciting question emerges from prior beliefs about private information: are democratically governed states at a disadvantage in limited networks? Democracies, by definition, have to discuss their decisions publicly, so the probability distribution of their strength and commitment may be less variable than an authoritarian state. This could be a disadvantage. However, that might change in a more extensive network. More research is needed to see if large democratic neighborhoods that incorporate reputations create a stronger, more war-stable, network turning a disadvantage into an advantage.

Lieutenant Peter A. Devine, United States Navy currently serves as an Instructor of Economics at the United States Naval Academy and an STC at the World Bank Group focusing on the Blue Economy. From 2017-2019 he served as the Speechwriter to the Superintendent at USNA. He is a qualified F/A-18 pilot and the distinguished graduate of his training class, his most recent operational assignment was to VFA-102, forward deployed in the Pacific Theater. He completed four operational Indo-Pacific patrols from 2014-17. He earned a B.A. from Notre Dame in 2009 an M.A. from Duke in 2011 and is currently completing a PhD all in Economics. His research focuses on game theory and network architecture applied to alliances, conflict, and political economy.