CAPITULO 1 : MARCO TEÓRICO
1.5 METODOLOGÍA DE DESARROLLO
Table 3-4: Summary of the Games Key Elements
Structure • Single Shot Simultaneous Move Strategic Players • I – Incumbent Regime
• C – Insurgent Challenger
Information • Both players know the density function describing the ‘Civilian Pivot’ player’s location. Neither knows its exact reservation price.
Cost • Compromise of player’s ideal policy position
• Univariate political spectrum ranging from 0 to 1 • Value of cost y =
o y^1 – constant cost to scale
o y^e – continuous growth in cost to scale
Benefit • MRS (Marginal Return to Scale)
o PDF of the ‘Standard’ Logit Normal Distribution o Marginal change in probability of preferable war
outcome
• CRS (Cumulative Return to Scale)
o CDF of the ‘Standard’ Logit Normal Distribution o Cumulative probability of preferable war outcome
Non-Strategic Players • The Civilian Pivot – The last marginal person within the population endogenous to the conflict each strategic player needs to capture to determine the game’s outcome
• Nature – Chooses the value of the Civilian Pivot’s
reservation price by drawing samples from a ‘standard’ logit-normal density
Strategy Profiles • Risk Averse Player – PDF
o Minimize Loss (Potential): Equality of marginal
cost and benefit
• Risk Acceptant Player – CDF
o Maximize Gain (Expected Value): Optimal ROI
Solution Concept • Bayesian Nash Equilibrium Comparative Static
Result
• Prospect of capturing the Civilian Pivot under equilibrium
play
• Resolution of the net impact between the competing
institutional expansion and collective action mechanisms
Setting Up the Game
The center of gravity in counterinsurgency wars may be conceptualized as the interaction of two latent variables. The first is the ‘popular support function’, a univariate probability density function (pdf) which represents the distribution of political preferences amongst the civilian population, arranged along a linear policy spectrum with the two opposing endpoints representing the ideal positions of the incumbent regime and the insurgent challenger,
respectively. The second, ‘war difficulty’, is an index measure which aggregately reflects the net sum of all salient structural features of a conflict that either promote or impede the efforts of the strategic players. More casually, one may think of these two latent factors as representing the amount of popular support which an aspiring insurgent movement requires if it is to successfully overthrow an incumbent government, and the extent to which that requisite support is forthcoming. And by extension that same value equally defines the threshold of popular acquiescence to status quo political institutions a regime must maintain to preserve its incumbency. The result of these two interactive factors summarizes the proportion of the populace in any given conflict whose allegiance and support will dictate the outcome of the war; hereafter referenced as the ‘Civilian Pivot’.
Proposition 1 (The Civilian Pivot): Let the interactive effect of the ‘popular support function’ and ‘war difficulty’ be represented by a single value, henceforth referred to as the ‘Civilian Pivot’; a random variable with continuous univariate support over the bounded unit interval (0 to 1), which defines the proportion of the population whose support is required by either strategic player (the incumbent government or insurgency) to determine the outcome of a counterinsurgency war.
The ‘Civilian Pivot’ random variable can be usefully represented by a probability density function, representing the range of possible values it has taken across all cases of insurgency
wars. The plot of such a distribution is likely skewed in favor of the insurgents on account of their agency in initiating the onset of a conflict.40 However, while this underlying distribution is
not fully known to either strategic player in any specific conflict, the distribution of its expected value can nevertheless be effectively represented pursuant to the properties of the central limit theorem and the law of large numbers. The resulting distribution of sample means can be further standardized according to a simple linear transformation of its values and be represented as standard normal.
Proposition 2 (CLT & LLN): The underlying probability density function of the ‘Civilian Pivot’ random variable is unknown to both incumbents and insurgents. By invoking the asymptotic properties of the central limit theorem and the law of large numbers, one can represent the ‘Civilian Pivot’s’ distribution of sample expected values as normal; and as the size of the sample mean and number of samples taken approaches ∞, the esamated mean and variance of the sample distribution will converge to a normal distribution with mean µ and standard deviation . A z transformation will yield a standard normal distribution, N~[0, 1], represented by the probability density function:
StandardNormalPDF = √2 A Standard Normal PDF
40 This is not to suggest that in observed instances of insurgency war the insurgents possess a net
advantage over incumbent governments, they are of course significantly weaker. ‘Skew’ here refers to the more favorable conditions likely present in the subset of observed wars that took place, relative to country-years that did not experience an insurgent threat at all.
4 2 2 4
0.1 0.2 0.3 0.4
Accordingly, the density function of the resulting standard normal distribution can be used to represent the probability that the expected value of the ‘Civilian Pivot’ random variable is more or less favorable to either insurgents or incumbents in any given conflict. The strategic dynamic of the game is centered on the probability of either player capturing the ‘Civilian Pivot’, and the corresponding cost associated with doing so. Electoral political institutions here
represent the mechanism by which incumbents implement their offer of concession. Furthermore, the relative net value of any given policy position can be represented by the marginal cost of compromise (corresponding to its location) and the marginal benefit, associated with the additional probability of having met the civilian pivot’s reservation price. Constant costs represent the assumption that the foregone value of policy compromise is invariant across the political spectrum. An accelerating cost function is alternatively entertained based on the assumption that the opportunity cost of political compromise increases with respect to the magnitude of its extent.
Proposition 3 (Constant Cost): Let Y1 define the opportunity cost of policy compromise
between a players actual and ideal policy positions, with y1 taking values between 0 and 1.
Proposition 4 (Continuous Growth Cost): Let Y2 define the opportunity cost of policy
compromise between a players actual and ideal policy positions, with ye taking values
between 0 and 1.
Unfortunately, given that the standard normal distribution supports all real values ranging from −∞ to +∞, it is not possible to directly map a discrete cost function to its range41. A
truncated normal distribution permits one potential resolution to this issue by allowing for
41 At least not without fundamentally altering the functional form of the variable and according relation to
uniform reapportioning of the density function to lie within a discrete interval, where a and b
represent the lower and upper bounds of truncation.
Truncated Normal DistributionPDF √2 1 2 $ Erf' ( √2) 1 2 Erf'√2* )
Though, relying on the truncated standard normal would unfortunately introduce a considerable arbitrary assumption into the model. While the ‘excess’ probability values are appropriately redistributed over the new truncated discrete range, the resulting relationship between the distribution of value and that of cost would admittedly be driven to a considerable extent by the somewhat arbitrary choice of the interval [a,b]. To illustrate, the unbounded standard normal distribution is represented in beige and the truncated in blue, with truncation at the interval [-3, 3] standard deviations depicted on the left plot and [-1, 1] on the right.
A preferable solution then is the use of the logistic function to map the values of the standard normal distribution back to the original range of possibility associated with the
theorized underlying data generating process. The resulting “standard” logit-normal distribution serves to provide an appropriate representation of the ‘Civilian Pivot’ random variable’s density of expected value, within the bounded interval from 0 to 1, and can be transformed from and to
a standard normal distribution via application of the logistic function and its inverse (logit), respectively.
LogisticTransform =1 + .LogitTransform = Log[1 − -]-
There are several methodologies available for transforming the density function of one random variable into that of another. For univariate cases, where each of the original underlying values of the initial distribution are mapped to distinct values in the subsequent, a one-to-one transformation can be performed.42 This approach involves an application of the inverse
transformation function and the absolute value of the Jacobian of the transformation, with respect to the original density.
Proposition 5 (Distribution of Benefits): Let the distribution of benefits, the expected value of the ‘Civilian Pivot’ random variable, be represented by a logistic transformation of the standard normal distribution and its resulting probability density function defined by the “standard” logit-normal. Accordingly, both the scale and value of the cost and benefit functions range from 0 to 1.
-Let X be a random variable with density f(x)
-Let Y be a random variable with density u(X), where u(X) denotes the Logistic Transformation of X
-Therefore let the pdf of Y, g(y) = f(u-1(y)) |J|
-Where x= u-1(y) is the inverse function of y=u(x) and J=du-1(y)/dy denotes the Jacobian
/~1'0,1] =
√2
42 The standard normal distribution is univariate. The logistic transformation does not map any distinct
values of the standard normal distribution to the same point in the resulting logit-normal. An example of a link function that would violate this requirement is x2. Both negative and positive values in the original
- 4^ 1(7) = Log[1 − 7]7
9 :'Log[1 − 7], 7] =7 7 − 71
;(7) Log[1 − 7]|7 7 − 71 |
=>Standard> Logit − Normal = ?@AB[ C ?C]
√2(7 − 7)
A ‘Standard’ Logit-Normal PDF
The game models the relative returns associated with an expansion of the selectorate; the differential marginal costs and benefits of doing so across the range of the distribution. Playing one’s ideal position carries no cost and yields no benefit, while playing that of your opponent carries both a cost and benefit of 1. Given that the distribution is unimodal and symmetric about the mean (1/2), the marginal value of strategy profiles towards the center will yield a greater relative return than those towards either of the tails.
Two different functions associated with the ‘standard’ logit normal distribution are used in constructing the return to scale functions for two different ideal types of incumbent (detailed further in the preceding section). The marginal returns to scale measure is constructed using the PDF while the cumulative returns to scale assesses value according to the CDF. Benefits and
0.2 0.4 0.6 0.8 1.0
0.5 1.0 1.5
costs are each evaluated at common values of y (ranging from 0 to 1), with their difference defining the density of each return to scale function. These alternative metrics for the function of benefits are used to estimate the differences in equilibrium play between more and less risk averse incumbent types, with risk averse players judging value at the margins and risk acceptant ones cumulatively.4344
Proposition 6 (Returns to Scale): The functions ‘MRS’ and ‘CRS’, characterize the relative return of all strategy profiles associated with values of y ranging from 0 to 1, and are defined by the difference between the corresponding benefits and costs associated with each possible policy position. Based on the PDF, the MRS function evaluates the marginal returns to scale; whereas the CRS, based on the CDF, measures the cumulative returns to scale. For the MRS, let the positive return to scale strategy profiles be defined over the range of Y where the marginal increase in the probability of capturing the expected value of the ‘Civilian Pivot’ is greater than the marginal cost in policy compromise associated with doing so. For the CRS, let the positive return to scale strategy profiles be defined over the range of Y where the slope of the function, as evaluated by its first derivative, is positive. The range of negative return to scale strategy profiles is accordingly defined inversely.
MRS(PDF) = ?@ABF C ?CG √2(7 − 7)− 7?,H CRS(CDF) = 0.5 +1 2 Erf[ Log[ 71 − 7] √2 ] − 7?,H
43 These two positions are roughly approximated by the Competitive Electoral State and Electoral State
variables, respectively, in the cross sectional empirical analysis. They are explained in greater detail in the following chapter.
44 Disutility, is measured by the remaining area under the curve that would not be captured by an
incumbent’s concessions. The area of this sub-section represents the probability that the reservation price of the civilian pivot would not be met, and provides a value estimate for the risk of an incumbent