Formal Theories of Conflict: Quantitative Approaches

Most theories of conflict rely on words: motivations, grievances, identities, and ideologies described in rich detail. Formal theories take a different route. They treat conflict as something that can be measured, modelled, and even predicted using numbers and equations. By translating arms races, hostility, and competing interests into mathematical relationships, these approaches aim to uncover the hidden logic behind why states fight, why they cooperate, and why some situations escalate while others stabilise. This article explains the key figures, models, and concepts that built this quantitative tradition.

What are formal theories of conflict? 🔗

Formal theories analyse conflict through quantitative methods, expressing the relationships between actors as mathematical equations rather than as narrative descriptions. The core assumption is simple but powerful: if hostility, fear, and self-interest follow patterns, those patterns can be captured numerically and studied with the same rigour scientists apply to physics or economics.

This approach offers two main advantages. First, it forces precision. A vague claim like “fear drives arms buildups” becomes a testable equation with measurable variables. Second, it allows prediction. Once you know how variables relate, you can estimate what happens when one of them changes. The tradition draws heavily on three pillars: Lewis Richardson’s mathematical models of arms races, Von Neumann and Morgenstern’s utility theory, and the game theory that grew out of it.

Lewis Richardson and the mathematics of arms races 🔗

The British physicist and meteorologist Lewis Fry Richardson was the first scholar to design mathematical models for the quantitative description of armament processes, which is why such models still carry his name. Trained as a physicist and already a pioneer in weather forecasting, Richardson believed that the buildup of weapons between rival nations followed laws that could be expressed with equations.

How the Richardson model works 🔗

Richardson’s study of conflict modelling was inspired by his meteorological work. Just as he tried to predict weather by finding general laws common to all conditions, he tried to predict war by finding laws common to all nations. After studying World War I, he derived a set of differential equations to describe the arms buildup between major European powers in the 1930s.

His model rests on a clear logic. Each country increases its own armament level in proportion to the weapons held by its opponent, weighted by a defence coefficient that measures how threatened it feels. At the same time, each country reduces its armament in proportion to its own existing stockpile, weighted by a fatigue coefficient that reflects the economic strain of maintaining weapons. A third element, the grievance term, captures underlying hostility or ambition independent of the other side’s behaviour.

When the armament levels of both sides stop changing, the system reaches a point of stability. This equilibrium corresponds to what political scientists call the balance of power. If the defence coefficients dominate, the model predicts a runaway arms race; if fatigue dominates, the buildup slows and stabilises. Richardson’s work appeared in his books Arms and Insecurity and Statistics of Deadly Quarrels, and it was later brought to wider attention through Anatol Rapoport’s influential 1957 review.

Why Richardson still matters 🔗

Richardson’s contribution was not just a single equation. He demonstrated that mathematics could contribute to peace and conflict research, using ideas of system dynamics and stability borrowed from the natural sciences. His pioneering work opened an entire field. In modern textbooks, formal modelling of arms races is now treated as one of the major scientific approaches to studying international relations, and prizes in his name exist in both meteorology and peace research.

Von Neumann, Morgenstern, and the measurement of preferences 🔗

Richardson showed that hostility could be modelled. The next breakthrough was showing that human preferences themselves could be measured numerically. This came from mathematician John von Neumann and economist Oskar Morgenstern, whose 1944 book Theory of Games and Economic Behavior is recognised as the first book on game theory.

What utility theory does 🔗

The central problem they tackled was this: how do you compare different outcomes when each actor values them differently? Their answer was utility theory, which seeks to describe how people evaluate and compare outcomes using mathematical tools. Instead of saying one outcome is simply “better,” utility theory assigns each outcome a number on an interval scale, allowing the strength of a preference to be measured, not just its direction.

An interval scale is important here. It means the gaps between values carry meaning. The difference between a utility of 10 and 20 is the same size as the difference between 50 and 60. This lets analysts say not only that an actor prefers peace to war, but by how much, and crucially how that actor weighs risky gambles against certain outcomes.

The role of the axioms 🔗

To guarantee that preferences could be represented by such numbers, von Neumann and Morgenstern set out a small set of reasonable assumptions, or axioms, including completeness, transitivity, and continuity. The IISc Bangalore notes on the theory explain that transitivity is a natural requirement because violating it leads to a “money pump” situation, where an actor with inconsistent preferences could be drained of resources through a series of trades.

The result is now known as the von Neumann-Morgenstern utility theorem, which demonstrates that rational choice under uncertainty takes the form of maximising the expected value of a cardinal utility function. In plain terms, a rational actor facing risk should pick the option with the highest average payoff, weighted by probability. This idea forms the foundation of expected utility theory and, with it, the entire structure of game theory.

Game theory: analysing strategy mathematically 🔗

Once preferences could be measured, conflicts of interest could be analysed as strategic interactions. This is the domain of game theory, a mathematical framework for studying the strategic decisions of rational actors in situations where each one’s outcome depends on the choices of others. In international relations, this framework has become a central concern for literature on diplomacy, regional integration, and conflict resolution.

A “game” in this sense is any situation with players, strategies, and payoffs. A game matrix represents the situation by laying out the choices that must be made and the resulting payoffs for each combination. Game theory then asks which strategies rational players are likely to choose, and what outcome the interaction will settle into.

Zero-sum games 🔗

The simplest category is the zero-sum game. Here one player’s gain exactly equals the other’s loss, so the payoffs always add up to zero. These are strictly competitive situations with no room for mutual benefit. As one analysis puts it, some political games, including arms races or cross-border conflicts, are treated as zero-sum, where players fight over a fixed and finite amount of territory, wealth, or influence. In a zero-sum world, cooperation offers nothing, because anything one side wins must come directly out of the other side’s pocket.

Non-zero-sum games 🔗

Reality, however, is rarely so stark. In a non-zero-sum game, the total payoff is not fixed, which means it is possible for both sides to gain or for both to lose. These are sometimes called variable-sum games. International trade agreements are a prime example, where countries cooperate to reduce tariffs and increase trade in ways that benefit all parties. Climate negotiations work the same way: coordinated action can leave everyone better off, while collective failure harms everyone.

This distinction matters enormously for conflict resolution. Non-zero-sum games offer scope for cooperation to reach outcomes that are best in aggregate, opening the door to negotiated settlements that pure competition would rule out. The challenge is that even when cooperation would help everyone, individual incentives can still push players toward conflict.

The Prisoner’s Dilemma 🔗

The most famous illustration of this tension is the Prisoner’s Dilemma, a classic non-zero-sum game. Two suspects are interrogated separately and each must choose to confess or stay silent without knowing the other’s choice. The trap is that confessing is the dominant strategy for both prisoners, meaning it gives a better or equal payoff regardless of what the other does. Yet when both follow this individually rational logic, they reach an outcome worse than if both had stayed silent.

This captures a deep problem in international politics: the clash between individual rationality and collective benefit. During the Cold War, the superpower arms race followed exactly this pattern. As the University of Notre Dame’s teaching materials describe, both sides poured money into weapons, and their levels of security dropped in a negative-sum game where the growing arsenals made everyone less safe rather than more.

Nash equilibrium and stable outcomes 🔗

A key concept for understanding where games settle is the Nash equilibrium, a state in which no player can improve their payoff by unilaterally changing strategy. In the Prisoner’s Dilemma, mutual confession is the Nash equilibrium even though it is suboptimal for both. The concept is widely used in political science to analyse strategy. Diplomatic negotiations, including arms control treaties, often rely on the idea of equilibrium to ensure no country finds it beneficial to deviate from agreed terms, which is what makes an agreement self-enforcing and durable.

From competition to cooperation: insights for peace building 🔗

The real value of formal theories for conflict resolution lies in how they expose the difference between competitive and cooperative dynamics. Recognising whether a conflict is genuinely zero-sum or merely perceived as such can transform how it is managed. Many disputes feel like a fixed pie that must be divided, when in fact creative bargaining could expand the pie for everyone.

Game theory also illuminates how cooperation can emerge even among self-interested actors. When the same game is played repeatedly, reputation and reciprocity come into play, and strategies based on conditional cooperation can outperform pure defection over time. Researchers studying repeated interactions have shown that this is how trust gradually builds between rivals, whether they are businesses, communities, or nations. Mediators, too, draw on these ideas: understanding the equilibrium point helps a mediator guide parties toward a resolution that is stable even if it is not perfect for either side.

Strengths and limitations of the formal approach 🔗

The formal tradition brought scientific discipline to a field once dominated by intuition. Its strengths are clarity, testability, and the ability to model complex strategic interactions that words alone cannot capture. By reducing a tangled situation to its essential incentives, these models reveal patterns that might otherwise stay hidden.

Yet the approach has real limits. Critics point out that human behaviour is rarely perfectly rational, that emotions, misperceptions, and culture shape decisions in ways equations struggle to capture, and that the rich phenomenon of genuine cooperation resists being squeezed into a few neat assumptions. Some scholars note that formal modelling has developed competition and conflict far more thoroughly than it has captured the complexity of cooperation. The simplifying assumptions that make a model tractable can also make it unrealistic. Used wisely, formal theories are best treated as one lens among several, sharpening our questions rather than dictating final answers.

What do you think? If many international conflicts are actually non-zero-sum games where cooperation could benefit everyone, why do states so often behave as if they are locked in a zero-sum struggle? And can a conflict driven by deep emotion and identity ever be fully understood through mathematical models of rational choice?

References

1. https://link.springer.com/chapter/10.1007/978-1-4613-2805-6_7
2. https://link.springer.com/chapter/10.1007/978-3-030-31589-4_8
3. https://www.researchgate.net/publication/337954743_The_Influence_of_the_Richardson_Arms_Race_Model
4. https://en.wikipedia.org/wiki/Oskar_Morgenstern
5. https://gtl.csa.iisc.ac.in/gametheory/ln/web-ncp7-utility.pdf
6. https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem
7. https://www.sciencedirect.com/topics/earth-and-planetary-sciences/zero-sum-games
8. https://www.numberanalytics.com/blog/mastering-non-zero-sum-games
9. https://www.numberanalytics.com/blog/prisoners-dilemma-in-international-relations-theory
10. https://www3.nd.edu/~dlindley/handouts/gametheory.html
11. https://www.numberanalytics.com/blog/ultimate-guide-nash-equilibrium-game-theory
12. https://mediators.substack.com/p/game-theory-post-4-the-nash-equilibrium