CHAPTER 3: ENDOGENOUS MOBILIZATION THEORY – WHY ELECTORAL
D: The ‘Formal’ Model – Endogenous Mobilization Theory: The Marginal Effects of Elections on COIN
Table 3-1: Semi-Formal Model: Overview Summary
-Players • I – Incumbent Regime (Government)
• C – Challenger (Insurgency)
• CP – Civilian Pivot (Reservation Price Selected by Nature)
-Possible
Moves/Values25
• I: |0-1| • C: |0-1| - c
• CP: π (High and Low)
-Outcome State Utilities26
• Victory: πH-Premium • Settlement: πL-Premium
• Defeat: 0
The empirical observation that interstate conflicts have overwhelmingly been
terminated by some form of negotiated settlement, and not by the outright disarmament of one of the principal combatant parties by the other, has served as a principal undergirding facet supporting the application of economic bargaining models to the study of war (Goemans 2000; Wagner 2000). “Nearly all wars end not because the states that are fighting are incapable of further fighting but because they agree to stop”(Wagner 2000, 469). However, the simple descriptive reality of how intrastate conflicts have been predominantly concluded provides a stark contrast. In the nineteenth century incumbent states were able to outright defeat their
25 ‘C’ here refers to any surplus value transfer which would be required to cover the transaction cost
associated with undertaking revisionist violence. The principal implication is simply that for incumbents this would be zero as a perpetuation of the status quo would not require any such cost. And by extension insurgents would need to do so by some non-zero amount so as to be able to match offers made by incumbents.
26 ‘Premium’ means simply any disutility which results from overbidding. Any amount offered in excess of
the unknown minimum threshold value which was required to meet either the high or low reservation price.
insurgent challengers over 80% of the time; and in the modern era insurgencies have been terminated through some form of settlement only one third of the time.
Rather, in a majority of contemporary conflicts an insurgency has ended either by the outright overthrow of the incumbent regime or the wholesale elimination of the insurgent organization. From this perspective, insurgency in the modern era seems far more akin to Clausewitz’s understanding of absolute war, than to his characterization of real wars which comprise the predominant basis of interstate conflicts. “War [absolute] is nothing more than a duel on a larger scale…Each tries through physical force to compel the other to do his will; his
immediate aim is to throw his opponent in order to make him incapable of further resistance” (Wagner 2000, 472: Original in Clausewitz 1976, 75). In light of this distinction, the
appropriateness of a rigid application of a ‘bargaining model of war’ framework to the case of insurgency appears to be at least partially misguided.
The extensive form representation (below) presents a stylized ‘moment’ in any counterinsurgency campaign. An ongoing war is always at risk of experiencing one of four possible transition states: victory for the incumbent regime, victory for the insurgents,
achievement of terms constituting a negotiated settlement between them, or a continuation of the war which is then replayed according to the same structure. The war’s termination status depends upon the relative magnitude of the concessions offered by both incumbents and insurgents, and the extent to which they meet or exceed the reservation values of a sufficient subset of the population being contested, herein represented by π. The payoffs then for each player (and most importantly for the incumbent regime) are determined by both the extent to which the value of a successful bid is associated with meeting the minimum threshold of the pivot’s reservation price; minus the surplus concession offered to pay off subsets of the
population whose allegiance was not necessary to do so. In this simple depiction, offers are characterized as being sufficient to meet the high reservation price (SH), sufficient to meet the low (SL), or not sufficient to satisfy either (NS).
Table 3-2: Semi-Formal Model: Outcomes and Payoffs27
1: Incumbent Government Victory I |0-1| >= πH
2: Negotiated Settlement I |0-1| >= πL & C |0-1| >= πL
3: War Continuation (Repeat Game) I |0-1| >= πL & C |0-1| < πL
4: Challenger (Insurgent) Victory I |0-1| < πL & C |0-1| >= πH
5: War Continuation (Repeat Game) I |0-1| < πL & C |0-1| >= πL
6: War Continuation (Repeat Game) I |0-1| < πL & C |0-1| < πL
Figure 3-4: Endogenous Mobilization Theory: Spatial Voting Inspired Representation
27 While it is always possible for an incumbent to offer and opening bid that is ‘sufficiently high’ (SH) two
related principal impediments preclude the simple adoption of such a move. The first is the unknown value of the civilian pivot’s reservation price, and the second is the disutility suffered by incumbents associated with compromising their own ideal policy position. The interplay of these factors is more fully examined in the supporting calculations.
A useful analogy for explicating the strategic dynamic of the contest is that of a silent auction.28 The two strategic players, Incumbent Regimes and Insurgent Challengers, begin at
opposing positions along a univariate policy spectrum, with the Civilian Pivot located between them. Nature moves first and determines the value of the Civilian Pivot’s reservation price. While this exact amount is unknown to either the Government or the Insurgents, the
distribution of its expected value is public information. Given that the ‘standard’ logit normal is a symmetric distribution centered at y=1/2, the starting locations of the two strategic players is immaterial. As such Incumbents are assigned an initial ideal position of 0 and Insurgents a preference of 1.29
While both players move simultaneously, the outcome of their strategic interaction is depicted solely as a function of the Incumbent’s choice set.30 The potential for overlapping ‘ties’
between the moves of both players is precluded owing to the transaction cost associated with revisionist violence. Insurgents are effectively constrained to plays within the policy spectrum not dominated by the choices of Incumbents, given that a utility maximizing Civilian Pivot would demand a premium to compensate for the transaction cost of overthrowing the status quo regime.31 To be sure, the argument here is not that insurgents are not strategic players in an
absolute sense. The primary intention of the formalization presented herein is to reconcile the
28 In contrast to the format of an auction however the civilian pivot does not derive any surplus positive
utility from receiving a bid in excess of its own reservation price. Overbidding in this present context means simply that the loyalty of some additional marginal subset of the populace is secured above and beyond the required threshold.
29 The expected value of the civilian pivot’s reservation price serves a crucial role in the model as it
provides the basis of constructing the benefit function. A formal derivation and defense of its functional form can be found in the following section, along with the other supporting calculations.
30 In the proper sense then the model may be more accurately characterized as decision theoretic. 31 An alternative justification might appeal to prospect theory – the distinction here between a
government’s ability to shift status quo policy and an insurgents considerably more limited ability to only offer promises of future policy alterations post conflict.
conflicting theoretical expectations generated by the Institutionalization & Collective Action research traditions, as they apply to the issue of COIN war outcomes. Rather, it is in this limited context the argument is made that insurgents enter into the model as a relatively non-strategic player.
States have discretion in terms of how much incumbency value they are willing to trade off in exchange for increasing their survival prospects, via the conduit of differing electoral policies. Two ideal types of incumbents are evaluated.32 Risk averse types attempt to limit their
exposure to prospective defeat, and do so by continuing to expand their selectorates until the marginal cost of doing so begins to exceed the estimated marginal benefit. Risk acceptant types by contrast are willing to incur costs only up until the point of maximal return, defined by the maximum value obtained by the cumulative return to scale function. The ‘standard’ logit normal distribution defines the functional form of benefits for both types; though the probability density function (PDF) is utilized for the case of risk averse incumbents and the cumulative distribution function (CDF) in the risk acceptant case.33 For both player types the equilibria are
identified based on the marginal implications of competing strategy profiles, with risk averse types incurring cost up until the threshold of negative returns to scale and risk acceptant types until the limit of increasing marginal returns. After solving for the equilibria, the comparative statics for the games are computed using the values of y associated with the marginal results as the upper bound of integration. The resulting area under the density curve represents the
32 These ideal types are roughly approximated by two of the primary independent variables tested in the
cross sectional statistical chapter. Electoral States correspond to those who opt for pursuing a higher variance strategy profile, while Competitive Electoral States more closely approximate risk averse players. Detailed definitions for the coding of these variables can be found in the following chapter.
33 This distinction in the construction of the benefit functions is predicated on the competing interests and
objectives of the two ideal types of incumbents under evaluation. The PDF measures the change in magnitude at each point along the distribution while the CDF provides a cumulative measure, or running total.
probability of capturing the Civilian Pivot under equilibrium play, and the final comparative static prediction of the game.
Results: Risk Averse Incumbents34
Figure 3-5: Cost and Benefit Functions
Blue=Benefits, Red=Constant Cost, Orange=Continuous Growth Costs
The figure above contrasts the distribution of marginal benefits (PDF) with those of both marginal costs functions. The game is played at the margins, with the intersection of the
functions representing the limit of rational play associated with a risk averse incumbent. A potentially useful analogy is that of the Riemann Sum definition of integration35, where the
relative difference between the heights of the functions and the x axis at any given value of Y represent the sign and magnitude of the marginal return of the strategy profile at that exact point. The plot below represents the marginal return to scale functions directly, measured as
34 Details for the calculations of all values are contained in the accompanying appendix.
35 Given that a density function of a random variable has no value at any point, only between any two, the
Riemann Sum definition visualizes the area between any two points under a curve as the sum of the areas of an infinite number of rectangles, each with infinitely small widths. Accordingly, one may think of the change in the sign of the marginal return to scale as the difference between the heights of said rectangles. An alternative conceptualization would be vectors which range from the x axis to the point of intersection with each respective function, and the corresponding deltas between their magnitudes.
0.2 0.4 0.6 0.8 1.0
0.5 1.0 1.5
the differences between the benefit and cost functions at every point within the domain of the game [0, 1], with the green curve depicting constant costs and the purple continuous growth.
Figure 3-6: Marginal Returns to Scale (MRS) Green=Constant Costs, Purple=Continuous Growth Costs
The game is evaluated by calculating the absolute value of each MRS function, and then solving for the global minimum.36 The difference between the assumptions motivating the two
alternative conceptions of cost are not overly significant for the game’s results. The approximate intersection under the assumption of unit costs is at y=0.825 as y approaches from the left; and at y=0.856 as y approaches from the left under the assumption of continuously accelerating costs.37 While the game is played at the level of marginal costs and benefits, its resulting statics
and implications are calculated at the absolute level. The values of y, resulting from the marginal game, are used as the upper bounds of integration to compute the comparative statics, with the result being the probability of capturing the expected value of the ‘Civilian Pivot’. When the
36 The rationale being that the difference between marginal benefits and costs will be at parity when
equal to zero. In the context of the original function’s absolute value, this lowest point will be that which is equal to zero.
37 This function is defined as y^e. The motivation is that each additional marginal unit of policy comprise is
experienced as a greater cost than the prior, the rate of which grows continuously as one moves further away from their ideal preference. A more detailed explanation can be found in the following section of supporting calculations. 0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.5 1.0 1.5
game is played under the assumption of unit costs, the estimated result is a probability of 94%; and under the assumption of continuously growing cost the predicted probability is 96%.
Results: Risk Acceptant Incumbents
Figure 3-7: Cost and Benefit Functions
Blue=Benefits, Red=Constant Cost, Orange=Continuous Growth Costs
By contrast, risk acceptant incumbents attempt to maximize their return on investment; not to minimize their exposure to risk. Accordingly, the value function (Blue) is represented by the cumulative distribution function (CDF), with both costs functions represented the same. As before, the second plot directly depicts the return to scale functions (cumulative), with the green curve representing constant costs and the purple continuous growth.
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
Figure 3-8: Cumulative Returns to Scale (CRS) Green=Constant Costs, Purple=Continuous Growth Costs
In this setup of the game incumbents attempt to maximize their return on investment, and accordingly the equilibria are identified by a shift in the rate of cumulative returns to scale.38 To solve for the policy positions associated with these two points (values of y), one can
take the first derivative of each function and calculate the points at which it is equal to 0.39
Under the assumption of constant costs the equilibria play is estimated at y=0.7899, with an associated probability of capturing the ‘Civilian Pivot’ of 90%. Under the alternative continuous cost assumption, the approximate value of y is 0.6764, and yields an associated prediction of capturing the ‘Civilian Pivot’ of 77%.
38 In contrast to the risk averse type scenario, where a change in the sign of returns to scale was defined
as a limiting threshold of play, risk acceptant incumbent types will cease further investment after achieving a zenith in the value of the cumulative return to scale function.
39 The first derivative measures the instantaneous rate of change at every defined value of the function.
Negative slope values would indicate a region of diminishing returns, and positive slopes those of increasing returns. A slope of 0 indicates a point of inflection where the sign of the rate of cumulative returns to scale inverts.
0.2 0.4 0.6 0.8 1.0 0.1 0.1 0.2 0.3 0.4
Table 3-3: Summary Implications of the Games’ Results:
Risk Averse Types Limit of Rational Play
Probability of
Capturing the ‘Civilian Pivot’
Odds Log Odds
Constant Costs y=0.825 94% 15.6 2.74 Continuous Growth Costs y=0.856 96% 24 3.17 Risk Acceptant Types Limit of Rational Play Probability of
Capturing the ‘Civilian Pivot’
Odds Log Odds
Constant Costs y=0.7899 90% 9 2.20
Continuous Growth Costs
y=0.6764 77% 2.33 0.84
The tables listed above summarize the equilibria values and corresponding comparative statics associated with both ideal types of incumbents; along with the predicted probability of capturing the civilian pivot expressed in the forms of a percent, an odds, and as the natural log of the odds. Overall, the results definitively support an expectation that incumbents who expand the selectorates of their regimes will enjoy a considerable reduction in the prospect of being defeated by an insurgency. Nevertheless, equilibrium play is still associated with some positive probability of defeat, owing to the excessive opportunity costs associated with trying to capture extreme potential values of the civilian pivot random variable. The model’s results suggest that while both the institutional cooptation and collective action mechanisms are plausibly operative when governments opt to hold elections, their net effect empirically on the outcomes of counterinsurgency wars is anticipated to be overwhelmingly favorable towards the interests of incumbents. This resultant implication is all the more tenable when one considers
that incumbents possess complete agency in the decision itself, of whether or not to engage in electoral politics during a counterinsurgency war, and would therefore plausibly elect to only if doing so carried a positive expected utility.