Formal Modeling Of Politics Research Paper

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Positive theories of politics provide explanations for why political phenomena occur in the manner that they do. Examples of such phenomena include which parties or candidates are elected at certain times, what policies legislative bodies adopt, and when and how wars are fought between countries. Formal models complement these explanations by using the language and analysis of mathematics to derive the phenomena as necessary consequences of certain sets of underlying factors. As with mathematical models in other disciplines, the main advantage of such an approach is twofold: to give an unambiguous representation of the relevant moving parts of a theory (the clarity of the argument) and to enforce a logical coherence on a theory (the consistency of the argument). The main drawback is found in the required simplification of the phenomena down to a manageable number of moving parts, and the subsequent possibility of simplifying away the very essence of the problem.

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1. The Basic Structure

Most formal models of politics fall in the category of rational choice (e.g., see Rational Choice in Politics), where the presumption is that observed outcomes are the result of decisions made by a certain relevant set of individuals. These could be voters and candidates in the first example above, elected representatives and appointed ministers in the second, or heads of state in the third. The general behavioral postulate is that individuals attempt to influence the outcome in a manner reflecting their preferences (tastes, values, etc.) over the set of possible outcomes. For instance, in selecting campaign platforms, a candidate who only cares about winning the current election might look to adopt policies appealing to the greatest number of voters; conversely, a candidate who cares intrinsically about the policies to be implemented might shy away from politically popular but personally unpalatable alternatives. Whatever their specific form, individuals’ preferences are taken to be primitives of the model, and so are held fixed throughout the analytic determination of an outcome. However, different specifications of preferences can give rise to different outcomes. Indeed, to the extent the predicted outcome varies with changes in the underlying individual preferences, one has a potential explanation for any variation in observed outcomes as being due to variation in these preferences. The extent to which these preferences can themselves be modeled as functions of observable parameters, such as wealth or legislative seniority, permits one to generate a theoretical relationship between these parameters and observable political outcomes as well.

The influence of preferences on outcomes occurs through the choices the individuals make. The most common assumption is that each individual’s choice is optimal with respect to their preferences, for example, in the above scenario the candidate selects a vote-maximizing policy. This optimality assumption may be far from a realistic representation of individual decision-making, implying as it does that individuals can effortlessly solve what are at times relatively difficult mathematical problems. However, assuming otherwise typically requires the presence of auxiliary factors (for instance, decision costs) to rationalize nonoptimizing behavior, factors which, while reasonable, tend to create additional complications in the model. Also the optimality assumption allows one to import existing results from optimization theory to aid in the analysis.

As seen in the examples above, most formal models of politics involve more than one individual, and so this individual level of optimization often takes place within the confines of non-cooperative game theory. Such models explicitly describe the interaction among the individuals that ultimately produces a collective decision. This description begins with a ‘game form’ consisting of (a) the set of strategies available to each individual, and (b) the specification of which outcome occurs when a given profile of strategies is chosen (with each profile consisting of one strategy per individual). Continuing with the electoral setting, one example of a game form is where the set of strategies for a candidate is equal to the set of available policies, a voter’s strategy describes which candidate to vote for based on the candidates’ chosen policies, and an outcome, consisting of who wins the election and with what policy, is determined by the candidate receiving the most votes. The remaining piece of the puzzle then is the derivation of the individuals’ strategies, with the fundamental concept being that of the ‘Nash equilibrium.’ Note that, even assuming all individuals have well-defined preferences over the set of outcomes, the notion of optimality in this multiperson environment is necessarily conditional, as an individual’s best choice of strategy may depend nontrivially on the strategy choices by the other individuals. For instance, in the electoral game above, the location of a candidate’s optimal policy may depend on the voters’ behavior as well as the policies endorsed by rivals. A Nash equilibrium is a profile of strategies wherein each individual’s strategy is optimal, given the strategies of the others, that is, conditional on the choices by the other individuals, no one individual has an incentive to alter her choice. A Nash equilibrium outcome is thus a collective decision consistent with individually optimal behavior and constitutes the basic object of analysis in any noncooperative model. (In certain game forms, some Nash equilibria are inherently more plausible than others and so refinements of Nash equilibria, such as subgame perfection, are used.)

As with the choice of preferences, the choice of game form is in the hands of the modeler. This choice may be influenced by observable features of the political phenomenon in question, as in the above electoral example. In such situations, there will invariably be a trade-off between making the game form descriptively accurate and keeping the game form analytically tractable. Thus while the electoral rule may easily be incorporated, simplifying the campaign stage of the process to a single choice of policy, thereby avoiding the complexities found in actual elections, may be necessary to guarantee the existence of and to solve for equilibrium outcomes. At other times, there may not exist any such observable features to leverage, an example of which would be when two heads of state are bargaining over land and other resources in the shadow of war. Here one would be hard-pressed to assert that the unstructured negotiations taking place can be definitively captured in any one particular game form. Hence descriptive accuracy is not an issue, and the modeler is relatively unconstrained in the search for a ‘good’ game form, that is, a game form capable of generating new and valuable insights into the political phenomenon in question.

Yet to the extent that the game form incorporates observable features of the political process and that the equilibrium outcomes vary with these features, one has a second source of variation in outcomes above and beyond changes in preferences, namely, changes in the political process. For instance, the equilibrium outcomes under the above plurality electoral rule may differ from those associated with a run-off rule (where, if no candidate receives a majority of the votes, the two candidates receiving the highest number of votes are paired in a second election, in which majority rule determines the winner) even if the candidates’ and voters’ preferences are held fixed. This allows one to make statements about the degree to which differences in observed collective choices are due to differences in preferences, or to differences in political institutions. More ambitiously, this type of comparative institutional analysis can be used to predict the effect changes in existing political institutions will have on collective outcomes.

2. An Alternative Approach

While noncooperative game theory has tended to dominate the modeling landscape in recent years, much of the early formal work employed the techniques of cooperative game theory. Rather than explicitly modeling outcomes as the aggregation of individual decisions, under this approach one looks for outcomes that are stable in some sense, with this ‘sense’ implicitly characterizing the (unmodeled) underlying strategic interaction. Given the normative appeal and frequent use of majoritarian-based political procedures, the focus was primarily on identifying outcomes which are majority-preferred to all others in pair-wise comparisons. Such an outcome would then be immune to changes proposed by any majority coalition, whereas all other outcomes would not.

It turns out that a majority-stable outcome necessarily exists in certain environments, the most compelling being when the collective choice concerns the selection of a single number. For example, suppose the issue before a legislature is the size of the nation’s military budget, the set of possible outcomes can be taken to be the set of nonnegative real numbers. Suppose further that the legislators have ‘singlepeaked’ pBibliography: each has some best number, and the value to them of the outcome falls as one moves away from this best number. Then a stable outcome will exist and, if the number of individuals is odd, will be unique and equal to the median of these best numbers. This ‘median voter theorem’ has been at the core of numerous and varied empirical studies of collective choice, from local school budgets in the United States to European Union decision-making. Its comparative advantage as a formal model is obviously its parsimony, requiring as it does only a description of how individual preferences over outcomes are related to any underlying parameters of interest (e.g., household income in the local school budget case). Further, by imposing additional assumptions on preferences, one finds that the median voter’s preferences coincide with those of the majority over the entire set of outcomes, implying that the median voter is in a literal sense representative of the collective as a whole. The benefits from this result accrue principally to the noncooperative modeler: suddenly a one-dimensional model of the interaction between the US President and the 435 members of the US House of Representatives over the size of the military budget, with majority rule dictating the decision-making among the latter, can be simplified to a two-person interaction between the President and the median member of the House.

The main shortcoming of the median voter theorem of course is its restriction on the set of outcomes to a single dimension. In certain situations this is benign, as in the above example when the policy choice in question consists of selecting the military budget. However, a policy choice consisting of the distribution of this budget across more than two groups is inherently a multidimensional problem and the conclusions about majority rule for multidimensional problems, relative to their single-dimensional cousins, could hardly be more dramatic. In particular, for most descriptions of preferences, no majority-stable outcome exists (where ‘most’ has a mathematically precise definition in this setting). As an example of such nonexistence, suppose in the above scenario that each legislator cares only about how much their favorite group receives, with one-third of the legislators associated with each of three groups. Then for any distribution of the budget, the legislators for two groups (which together constitute a majority) prefer to take away any amount going to the third group and redistribute it amongst themselves, thereby guaranteeing that no distribution is majority-stable. Furthermore, when no stable outcome exists, not only is there no representative voter but in fact one can construct majority preference cycles throughout the policy space (i.e., policy a is majority-preferred to policy b, which is majority preferred to policy c, which … which is majority-preferred to policy z, which is majority preferred to policy a). Thus when majority rule breaks down, it breaks down completely. Subsequent research has shown that these negative conclusions continue to hold, to a greater or lesser degree, for most nonmajoritarian rules as well.

As with the median voter theorem, it is difficult to overstate the impact these ‘chaos theorems’ have had on the formal modeling of politics. Clearly, in a multidimensional world, the modeler has lost the analytic shortcut of representing a collective’s decision-making as being governed by a single individual (or, at least, the logical rationale for doing so). More importantly, the theorems gave rise to a new-found appreciation for political institutions, such as committee systems, restrictive voting rules, and the like as key elements in the determination of collective outcomes. Indeed, such institutions are often depicted as necessary ingredients in any multidimensional model of politics, in that, but for such institutions, collective decision-making would fall prey to the inevitable onslaught of cycling and chaos. This interpretation, while quite popular, appears inconsistent with the central tenet of formal modeling, that of equilibrium analysis, as the chaos theorems are properly seen as nonequilibrium results. Put differently, these theorems do not predict that (as commonly phrased) ‘anything can happen,’ for as nonequilibrium results they do not predict anything at all. Rather, they demonstrate the impossibility of any general theory of collective behavior based solely on the (majority) aggregation of preferences, and hence the necessity of being more explicit about the decision-making process so as to provide the analytical traction required to formulate a well-posed model. From this perspective, the lesson to be learned from the chaos theorems is one for the modeler of collective decision-making, rather than as one about collective decision-making itself.


  1. Austen-Smith D, Banks J S 1999 Positive Political Theory I: Collective Preference. University of Michigan Press, Ann Arbor, MI
  2. Morrow J D 1994 Game Theory for Political Scientists. Princeton University Press, Princeton, NJ
  3. Ordeshook P C 1986 Game Theory and Political Theory: An Introduction. Cambridge University Press, Cambridge, UK
  4. Shepsle K A, Bonchek M S 1997 Analyzing Politics: Rationality, Behavior, and Institutions. Norton, New York


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