Applied Mathematics in Political Science: A Complement to Tradition

Political Science


Cornell government majors are bound by a slew of requirements: introductory classes on International Relations (IR), American politics and political philosophy; intermediate courses on comparative government, political theory and American government; advanced seminars on topics ranging from normative issues in IR to the philosophies of Kant and Adorno. These classes train us to philosophize, to discuss, to consider issues from a deontological perspective. They do not, however, necessarily teach us how to create testable models or to systematically challenge existing theories. They do not, as the name “political science” suggests, teach us how to be scientists. But game theory does.

Some professors of political science will insist that the real-world explanatory power of mathematical methodologies and paradigms is disappointingly limited. Others will tell you that research produced in the field of political science is insubstantial without the incorporation of mathematical models as they are necessary for the discovery of external truths. Hopefully reasonable people can agree, however, that “science” allows us to orderly arrange knowledge and, on a more theoretical level, to combine elements of traditionalism (common sense), rationalism (reason) and empiricism (observation and testing).

There is a longstanding tradition in political science of purposefully disregarding biological, psychological and metaphysical theories of the “Will” (instincts and drives are, after all, excessively difficult to measure and valuate) and to instead view individuals as rationalist thinkers committed to achieving simple, identifiable objectives. The principles behind game theory merge seamlessly with this practice since game theory is a means of analyzing social outcomes in terms of interaction between participants, each of whom are seeking to accomplish a self-identified goal. Through game theory, we are presented with opportunities to organize investigation into rational choice and to term our results as testable hypotheses. Increasingly, political scientists are using game theory to analyze strategic interactions across many different settings. Each of the sub-fields, to differing degrees, has seen game theoretic concepts enter its vocabulary and students of political science must be prepared to accept this direction of their field.

By way of summarization, a game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players’ interests, but does not specify the actions that the players do take. A solution is a systematic description of the outcomes that may emerge in a family of games. Game theory suggests reasonable solutions for classes of games and examines their properties. A game theorist tries to design rules that capture the constraints faced by decision-makers and asks, “What actions would be chosen by players facing these rules?” If the predicted actions are similar to the behavior of real-life decision-makers, then the model helps us “understand” that behavior.[1]

While game theory does present individuals as goal-oriented actors, it does not deny the importance of free choice or humanistic values, unlike popular deterministic assumptions that transform humans into stubborn machines without agency. Students of political science, unable to satisfactorily evaluate the impact of free will on processes of decision-making, shy away from irregular, un-generalizable behavior. A tempting approach, yes but impractical. Psychology simply cannot be detached from political phenomena.

Fortunately, game theory combines exercises of generalization and free choice by suggesting that participants, aware of their own personal preferences, gauge the degree to which alternative strategies might satisfy these preferences in the face of similar calculations by opponents. Rather than insisting on a universe governed by inflexible causal laws, game theory allows for choice and interaction. At the same time that it promotes generalization, game theory also compensates for a common error: many observers to a conflict or situation tend to attribute their own goals to others or to attribute what they predict their goals would be were they to find themselves in another person’s position.

Theories of post-Westphalian political realism, which argue that power is the decisive category in international politics, are criticized because of their apparent inability to explain recent developments in global integration and differentiation in norms and laws. Likewise, state-centric theories are challenged because of their inherent contradictions and detachment from the role of major organizations such as the United Nations and European Union as well as ideas of democracy, public/private dynamics, local/global linkages, poverty, etc.. Underlying the state-centric paradigm and at the root of most of its criticisms are the ideological facts of nationalism. This point of view offers an image of the political reality that interprets what happens in the rest of the world through the lens of what occurs inside our nation. The separation of the study of domestic politics from that of international politics is a theoretical error since this approach prevents a proper understanding of the mutual relations between these two spheres of politics and, therefore, a comprehensive study of politics.[2] But there is an alternative to found in game theory.

Let us first more closely examine realism, at the center of which is uncertainty about distributions of power and the reliability of inter-state commitments (for more on this philosophy of international politics, refer to the works of Kenneth Waltz and Hans Morgenthau). To summarize, realism holds that a balance of power fosters peace since states are secure in the knowledge that their survival is not endangered. An imbalance, however, “naturally” culminates in war. Although a total dismissal of long-standing, traditional balance-of-power hypotheses would be excessive, it is worth noting that in practice, game theoretic analyses do not display a straightforward, monotonic relationship between distributions of power and the probability of international conflict. Given our definition of balance-of-power arguments, it may seem intuitive that they would lead to generalizations insufficient and inadequate to explain the course of history; nonetheless, this very point has been empirically proven (see Niou and Ordeshook[3] or Niou, Ordeshook and Rose[4]). These studies demonstrate that under neorealist assumptions, any state that can transform a losing coalition into a winning coalition (an “essential state”) will survive whereas states lacking this capacity will not. But what about the USSR in 1992? Despite functioning as an essential state, it ceased to exist! The same applies to a defeated Austria-Hungary in 1918. In the opposite direction, the United States, a nonentity in 1787, gradually flourished to an incredible degree.

In other studies, Powell[5] applies noncooperative game theory and assumes that states are rational actors to demonstrate that central precepts of neo-realist thought (such as the one that anarchy is inevitable in international politics and another that the threat of war is inescapable) cannot be logically sustained. Other researchers, Kim and Morrow[6] among them, have combined the forces of statistics and game theory to show that neither power balance nor power preponderance between rivals is enough to motivate war. More specifically, Bueno de Mesquita and Lalman[7] have concluded that within the confines of neorealist assumptions, a .5 probability of victory is necessary but not sufficient for war; they then proceed to demonstrate how this figure contradicts centuries’ worth of evidence.

Wittman[8], moreover, shows that establishment of a military power advantage does not, in and of itself, facilitate conflict resolution since demands are contingent upon changing expectations regarding the probability of victory or defeat. In other words, the price at which rivals will settle depends on their changing performances. It should make sense that as prospects for victory improve, the expected victor’s “settling price” will escalate in the face of updated expectations. Adding to this, Downs and Rock[9] point out that a high-probability of defeat might embolden political leaders to take self-destructive risks, a choice that would not be at all consistent with state-centric, balance-of-power, noncooperative models. As if not enough critiques of neorealism have already been introduced, it is also worth mentioning that game-theoretic analyses/political economy perspectives have convincingly challenged the central notion that what goes on inside states is of little consequence for international politics. For instance, the political cost of defeat in war is higher in societies that function under accountable democratic leaders (as opposed to autocrats or monarchs) and “democrats” therefore only engage in war when the probability of success is high. Following this logic, it makes sense that rival democrats will opt not to engage each other in war since each probably has a record of success in war, a concept aptly demonstrated by political economy models.

Having expounded on the merits of game theory, it is also important to bear in mind its limitations. Game theory implies the use of mathematical models, which are, by nature, limited. They can only approximate reality and the correlations they demonstrate may not necessarily imply causation. Causal analyses demand control of numerous uncontrollable, sometimes subjective variables. (Imagine trying to mathematically measure the extent to which Saddam Hussein’s advisors influenced his policies, for example.) Finally, the field of political science is restrained by time and place, which is to say that political scientists face and cannot really overcome “replication problems” that prevent them from participating in truly comparative research. According to some, modern game theory has become so extremely mathematical, notation heavy and removed from everyday life that it can no longer live up to its promises.

In the words of John Lott, author of Are Predatory Commitments Credible?, “If economics isn’t testible, you don’t have a science. Having a certain richness is nice, but there are simply too many game theory models that end up making similar predictions. When you can’t even differentiate monopoly behavior from perfect competition in predation what good is it? Indeed the goal frequently seems to be how many different models can be generated.”[10] Nonetheless, the limitations of mathematical approaches in political science cannot remove them as a primary requirement in that arena. Despite their shortcomings, mathematical methods have the power to transform political science into a true science, one that allows us to explore the realm of social engineering.

[1]Martin J. Osborne, “Bach or Stravinsky? Human behavior as game-playing.”
[2] Lucio Levi, “Globalization and the decline of the state-centric paradigm,” Cosmopolis: Revue de cosmopolitique (2007).
[3] Emerson Niou and Peter Ordeshook, “Stability in Anarchic International Systems,” American Political Science Review (1989): 1207-34.
[4] Emerson Niou, Peter Ordeshook, and Gregory Rose, “The Balance of Power” (1989).
[5] Robert Powell, “Guns, Butter, and Anarchy,” American Political Science Review (1993): 115-32.
[6] Woosang Kim and James D. Morrow, “When Do Power Shifts Lead to Power Shifts?,” American Journal of Political Science 36 (1992): 896-922.
[7] Bruce Bueno de Mesquita and David Lalman, “War and Reason: Domestic and International Imperatives” (1992).
[8] Donald Wittman, “How a War Ends: A Rational Model Approach,” Journal of Conflict Resolution 23 (1979): 743-63.
[9] George W. Downs and David M. Rocke, “Conflict, Agency, and Gambling for Resurrection,” American Journal of Political Science 38 (1994): 362-80.
[10] John Lott, Are Predatory Commitments Credible?: Who Should the Courts Believe? (1999).