2. A war of attrition hazard estimator

In this section we propose an estimator that captures the main features of the model just described: a mixture over two types of contests, one of which has an endogenous time point by which all such contests have ended. Since the outcome of interest constitutes a duration we draw from methods developed to model time to failure from survival or duration analysis. Since strong–strong contests all fail at the same time and after all the other contests have ended, the predicted distribution of failure involves a positive density up through t*, followed by a density of exactly zero, and ending with a point mass for the end of the strong–strong contests. In empirical applications, however, we wouldn't know the values of t* or the end of time, T. Yet, maximum likelihood estimation (MLE) typically assumes that the estimated parameters do not determine the support of the distribution (see, e.g., King (Reference King1998: 75) for a discussion of this). If the support for the distribution of strong–strong contests did not include values of t immediately greater than t* (which it would not if it were a point mass strictly greater than t*), then our estimate of t* would in fact help set the support of the mixture distribution.

We therefore propose an estimator that deviates slightly from the theoretical prediction by allowing any non-negative duration value. We maintain the estimation of t* since that constitutes a critical feature of the war of attrition: the informational battle leads to an endogenous time horizon after which only one type of contest remains. We therefore allow failures for contests involving two strong players to fail at any point in time. We expect that in practice the hazard of such failures will be small relative to the hazard for contests involving weak players. To test comparative statics predictions from the theory, the estimator allows users to parameterize the duration process and the mixing probabilities.Footnote ³

2.1 The likelihood function

To build the likelihood for the theoretically-derived data-generating process, we start with the hazard for contests involving strong–strong pairs. Although the incomplete information model predicts a hazard of zero for these contests (until the end of the period, when they all end), we relax this for the reasons described above by assuming a non-zero hazard rate, which we model as a Weibull duration. If the structure of the theoretical model is correct, we expect this hazard to be small relative to the hazard for contests involving weak types. Although we have no particular sense of “smallness” here, to be consistent with the hazards depicted in Figure 1, the hazard for strong–strong contests should be near zero or at least almost entirely below the other hazard. Such a hazard could arise by averaging over the differing end points from each of multiple instances of the war of attrition, leading to a hazard when aggregated over those contests. Depending on how they are spread out, that distribution could be increasing, e.g., if they are uniform; flat, e.g., if they are exponential; or decreasing, e.g., if they are Weibull with negative duration dependence.Footnote ⁴

For strong–strong pairs the observed duration, $t_i^{SS}$, which we superscript by players' types for purposes of exposition, is generated according to a standard Weibull distribution:

$$t_i^{SS} = \exp ( {-\alpha } ) \times {\rm \epsilon }_i.$$

When ${\rm \epsilon }_i^p$ follows a standard exponential distribution, ε_i follows a Weibull distribution with shape parameter p _SS > 0. When the shape parameter is greater than one the data exhibit an increasing hazard over time and when it is less than one they exhibit a decreasing hazard. The Weibull has the following cumulative density function:

(16)$$F_{{\rm wbl}}( {\rm \epsilon }_i) = 1-\exp ( {-{\rm \epsilon }_i^{\,p_{SS}} } ) .$$

Since the rate of failure for pairs of strong players does not depend on any parameters in the theoretical model, we merely parameterize it with a constant hazard. Define λ ^SS = exp (α), where α captures the constant baseline hazard rate, which allows us to write ${\rm \epsilon }_i = \lambda ^{SS}t_i^{SS}$.Footnote ⁵ Inserting this into our c.d.f. and then taking the derivative with respect to t ^SS gives us the density of an observation as

(17)$$f_{{\rm wbl}}( t_i^{SS} \vert \lambda _i^{SS} ) = p_{SS}( {\lambda_i^{SS} } ) ^{\,p_{SS}}( {t_i^{SS} } ) ^{\,p_{SS}-1}\exp ( {-{( {\lambda_i^{SS} t_i^{SS} } ) }^{\,p_{SS}}} ) .$$

For contests involving at least one weak player, the war of attrition model predicts an increasing hazard and an endogenous end point, t*. Since a Weibull distribution allows an increasing hazard as a special case, we use it as the starting point for the distribution of their durations, which we denote $t_i^W$. To capture the end point we modify the associated Weibull distribution by truncating it at t*. This produces a distribution with support from zero to t* and whose cumulative distribution reaches one at t*. Thus, as in the theoretical model, all contests in this population will end by t*.

The truncated Weibull arises by dividing the cumulative distribution function of a standard Weibull, as already presented above, by the probability that $t_i^W \le t\ast$. Since we wish to parameterize the hazard of contests including a weak player we write $t_i^W = \exp ( {-X_i\beta } ) \times u_i$ and define $\lambda _i^W = \exp ( {X_i\beta } )$, where the superscript W indicates that this parameterizes the hazard for contests that involve at least one weak player. From this we derive the density and the associated hazard to use for interpretation after estimation:

(18)$$f_{t{\rm wbl}}( t_i^W \vert \lambda _i^W , \;t_i^W \le t^\ast ) = \displaystyle{{\,p_W{( {\lambda_i^W } ) }^{\,p_W}{( {t_i^W } ) }^{\,p_W-1}\exp ( {-{( {\lambda_i^W t_i^W } ) }^{\,p_W}} ) } \over {1-\exp ( {-{( {\lambda_i^W t^\ast } ) }^{\,p_W}} ) }}, \;$$

(19)$$f_{t{\rm wbl}}( t_i^W \vert \lambda _i^W , \;t_i^W > t^\ast ) = 0; \;$$

(20)$$h_{t{\rm wbl}}( t_i^W \vert \lambda _i^W , \;t_i^W < t^\ast ) = \displaystyle{{\,p_W{( {\lambda_i^W } ) }^{\,p_W}{( {t_i^W } ) }^{\,p_W-1}\exp ( {-{( {\lambda_i^W t_i^W } ) }^{\,p_W}} ) } \over {\exp ( {-{( {\lambda_i^W t_i^W } ) }^{\,p_W}} ) -\exp ( {-{( {\lambda_i^W t^\ast } ) }^{\,p_W}} ) }}.$$

The observed data represent a convex combination of these two duration processes depending on the types of players involved, which we do not observe. Let π denote the initial probability (i.e., after players decide whether or not to contest the issue but before the contest begins) that a contest involves two strong players and therefore corresponds to the exponential duration process. Combining these terms leads to the following likelihood function, in which we drop all superscripts on the outcome variable, t _i, since we cannot distinguish players' types for outcomes that end before t*:

(21)$$L( \beta , \;\alpha , \;\pi , \;p_{SS}, \;p_W, \;t^\ast \vert t, \;X) = \prod\limits_{i = 1}^n \pi f_{{\rm wbl}}( t_i\vert \lambda ^{SS}) + ( 1-\pi ) f_{t{\rm wbl}}( t_i\vert \lambda _i^W ) , \;$$

(22)$$\;\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = \prod\limits_{i = 1}^n \pi p_{SS}( {\lambda_i^{SS} } ) ^{\,p_{SS}}t_i^{\,p_{SS}-1} \exp ( {-{( {\lambda_i^{SS} t_i} ) }^{\,p_{SS}}} ) $$

$$\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + ( 1-\pi ) c_i\left[{\displaystyle{{\,p_W{( {\lambda_i^W } ) }^{\,p_W}t_i^{\,p_W-1} \exp ( {-{( {\lambda_i^W t_i} ) }^{\,p_W}} ) } \over {1-\exp ( {-{( {\lambda_i^W t^\ast } ) }^{\,p_W}} ) }}} \right], \;$$

where c _i ∈ {0, 1} indicates that an observations fails after t* to ensure that the density of the truncated Weibull contributes zero to the likelihood in such cases. Importantly, note that although t* determines the largest value of the truncated Weibull distribution, the possible values of t include all positive duration outcomes via the mixture with the standard Weibull for strong–strong pairs.

Lastly, we incorporate heterogeneity in the probability that a given contest involves two strong players by parameterizing π. This allows us to estimate variation across observations in the mixture probabilities for the possible types of contests, which improves prediction of the parameters of the duration for each type of contest. We model π with a logit equation, replacing it in the likelihood above with the observation-specific probability:

(23)$$\pi _i = \displaystyle{{\exp ( W_i\gamma ) } \over {1 + \exp ( W_i\gamma ) }}$$

One can then take the log and proceed to maximization. The likelihood is easily adapted to account for data involving right censoring, as we show in our Supplementary appendix.

3. Application 1: filibustering in the US Senate

In this section, we employ our new estimator to study the duration of filibusters in the United States Senate. This application is motivated by a long series of claims in the press (e.g., Will Reference Will1982) and academic literature (e.g., Binder and Smith Reference Binder and Smith1997) regarding a supposed trivialization of the filibuster over the last few decades starting with a change to the cloture rule in 1975. In particular, we assess the validity of claims by multiple experts consistent with Wawro and Schickler's (Reference Wawro and Schickler2006) conclusion that that “(t)he contemporary context of lawmaking in the Senate has essentially eliminated the informational benefits that used to accrue from these kind of battles.”Footnote ¹² Dion et al. (Reference Dion, Boehmke, MacMillan and Shipan2016) argue that the filibuster fits the war of attrition framework well with two sides—either weak or strong—clashing over some issue of value.

To assess these claims we utilize replication data from Dion et al. (Reference Dion, Boehmke, MacMillan and Shipan2016), which employs Beth’s (Reference Beth1994) measure of filibuster length.Footnote ¹³ We estimate the hazard of filibusters with variables accounting for the 1975 rule change and the importance of issues being filibustered. The former captures the shift in cloture from two-thirds to three-fifths of those present and voting (and revised to 60 votes in 1980). The latter codes the issues underlying filibusters as important, not important, and “unclear” using Binder's (Reference Binder1997) list of important issues. The war of attrition model predicts that issue importance decreases the hazard of weak filibusters and increases the probability of a strong–strong filibuster (Dion et al. Reference Dion, Boehmke, MacMillan and Shipan2016). We compare our results to those from a standard Weibull estimator.

To estimate the best overall value of the endogenous time horizon, t*, we fix it at a particular value and then estimate the other parameters. We repeat this procedure for all observed filibuster lengths. Figure 2 plots the final log-likelihood for every case that converged (those that did not are indicated by ticks inside the horizontal axis; for presentation we did not plot results for values greater than 50). The maximum final likelihood occurs when t* = 9. In line with our discussion of the estimation process the likelihood remains flat after t* = 18 when the Weibull c.d.f. underlying the truncated Weibull equals one; after this point the parameter estimates do not change.

Figure 2. Value of final log-likelihood for different values of t* (filibuster duration, no covariates).

Note: Plot represents the final value of the log-likelihood for the war of attrition duration estimator using all observed values of the outcome variable (filibuster length) in the data set. For presentation purposes, results for two values greater than fifty not included, but they continue the flat trend up through 50. The four upward ticks on the horizontal axis indicate values for which the estimator did not report regular convergence or exceeded 50 iterations.

Table 1 reports the results of this estimation along with a second model that include covariates along with estimates from the corresponding Weibull models. Consider first the results with no covariates. The first two equations report the truncated Weibull results for contests involving at least one weak player, the second two equations report the Weibull results for contests involving two strong players, and the final equation reports the logit model capturing the mixture of these two types of contests. From these results we observe that contests with a weak player exhibit positive duration dependence, as expected in a war of attrition. Contests involving two strong players do as well, but with a smaller value indicating a much flatter hazard. The logit model has a negative constant parameter, which indicates slightly less than half of all contests involve two strong players. The Weibull results reported in the first column attempt to capture these two type of contests with one equation and indicate even smaller positive duration dependence that either of the two equations in the war of attrition results. A likelihood ratio test for the Weibull against our estimator can be performed since the Weibull corresponds to the case in which t* → ∞ and the probability of a strong–strong contest is zero, though we make an adjustment since the test involves two boundary values. Shapiro (Reference Shapiro1988) shows that the appropriate statistic comes from the $\bar{\chi }^2$ which here represents the convex combination of three distributions, $\bar{\chi }^2 = ( 1/4) \chi _0^2 + ( 1/2) \chi _1^2 + ( 1/4) \chi _2^2$, and which has a critical value of 4.231.Footnote ¹⁴ The resulting test statistic exceeds this threshold.

Table 1. Weibull and partial cure hazard model results for duration of filibusters, 1919–1993 (hazard form)

Note: N = 274. Robust standard errors reported. See the text and Shapiro (Reference Shapiro1988) for information on the $\bar{\chi }^2$ test for model comparison between the Weibull and our estimator given the presence of two boundary values. The critical value with two such parameters is 4.23.

More helpfully, Figure 3 plots the predicted hazards from the war of attrition estimator with no covariates along with their 95 percent confidence intervals.Footnote ¹⁵ The resulting hazards correspond quite closely to the plots depicted in Figure 1, with the hazard for filibusters involving at least one weak player increasing from the beginning of a filibuster up through nine days, by which point all such contests will have all ended. The hazard for filibusters with two strong players stays quite low and increases slightly, as expected from the results. We have no expectation regarding the form of duration dependence for strong–strong contests, but do expect it to be small relative to the one for weak contests. Note that for presentational purposes we only plot the hazards up through 40 days, but the strong–strong hazard increases very slowly to the longest observed filibuster, which lasts 97 days. Finally, the logit equation estimates that 42 percent of the observed filibusters involve two strong players.

Figure 3. Estimated filibuster hazard rates from partial cure estimator.

Note: Estimates from Table 1 with no covariates (t* = 9). Black lines indicate the predicted hazard rate for each type of contest. Gray bars give a nonparametric 95 percent confidence interval estimated by sampling 10,000 draws of the parameters from their estimated distribution and calculating the corresponding value of the hazard at many values of filibuster length.

The other pair of models adds in covariates. As predicted, high importance and unclear importance issues have lower hazards than low importance issues, with both significant in the war of attrition estimator. This is what we expect since weak players contesting important issues are willing to fight longer based on their greater value. In contrast, the results indicate no difference in the hazard for weak contests after the 1975 cloture reform. These results reverse themselves in the equation for the mixture probability, with the post-1975 period containing fewer strong players but no evidence of variation by issue importance.

These results contrast with those from the standard Weibull model. Although both models indicate that high importance issues have lower hazards, the Weibull alone indicates a positive and significant effect of the 1975 cloture reforms, corresponding to a greater hazard and shorter filibusters. This result appears to occur in the standard Weibull due to its inability to account for the reduction in the probability of strong participants since 1975, which it instead misattributes to an increase in the hazard. Not surprisingly given these differences, the statistical comparison of the two models rejects the standard Weibull in favor of the war of attrition estimator. Taken together, these results run contrary to the common wisdom that the 1975 reforms “trivialized” the filibuster. The hazard for filibusters did not move much after 1975, rather our results indicate a sizeable drop in the proportion of strong types willing to fight for their issues. The structure of the filibuster game itself did not change—the commitment of the players playing it did.

Figure 4 summarizes these results by plotting the estimated hazard rates for strong–strong filibusters and for the different types of filibusters involving at least one weak player. We do not report confidence intervals to more clearly interpret the effects of the covariates on the hazards.Footnote ¹⁶ Again, the overall shape closely resembles that of our theoretical model. But we can also see how changes in various conditions affect the hazard for weak contests. The black line represents the baseline case, which consists of unimportant issues before the 1975 cloture reform. Increasing issue importance decreases the hazard by about one-half both for issues of unclear importance and important issues. The 1975 cloture reform has basically no effect, increasing the hazard ever so slightly relative to the baseline, which is consistent with the small estimated coefficient.

Figure 4. Estimated filibuster hazard rates from partial cure estimator with covariates.

Note: Estimates from Table 1 (t* = 15). The reported hazards for contests involving at least one weak player each vary just one of the indicator variables at a time from the baseline (which represents unimportant issues pre-1975). We omit confidence intervals on this figure since it would be too cluttered and hard to interpret. Adding them, however, confirms what we learn from the table, i.e., the hazards for important and unclear issues differ from the hazard for unimportant issues.

4. Application 2: the duration of international crises

We next apply our estimator to the duration of international crises. A significant part of the literature on crisis onset and escalation casts crises precisely as a “war of attrition” (Fearon Reference Fearon1994: 577). As Fearon (Reference Fearon1995) notes elsewhere, imperfect information is one of a handful of ways that we might observe war since parties could otherwise perfectly negotiate away their differences. This imperfect information is often cast in terms of types for the two states involved (e.g., Banks Reference Banks1990), with type capturing the level of cost states are willing to incur to win the prize (e.g., contested territory). Crisis escalation serves as a precursor to war, with both sides attempting to signal their types in the hopes that their opponent will back down.

We can therefore think of the crisis onset and bargaining stage in terms similar to the war of attrition model with two sides competing over a prize. Each state can be either weak or strong, which could correspond to their ability to suffer audience costs (Fearon Reference Fearon1994), the strength of the military forces (Morrow Reference Morrow1989), or their resolve to fight for the prize. They each choose whether to engage in the crisis and how long to escalate before backing down. The side that backs down loses the prize and bears the cost of the contest whereas the winner secures the prize. Strong states may gain from the crisis by enhancing their reputation or letting their leaders display their strength whereas weak states incur costs for fighting, such as audience costs. If both sides are sufficiently strong and neither state backs down by some endogenous time horizon, then they settle the contest through a war. The crisis therefore serves as an explicit mechanism for learning about the strength of ones' opponent (Morrow Reference Morrow1989; Fearon Reference Fearon1994).

Although many similarities abound, key differences in Fearon's (Reference Fearon1994) implementation lead to deviations from the predictions of the classic war of attrition model. He highlights two such differences: the option to escalate from a crisis to war rather than just continuing or backing down and the incurrence of audience costs only by the side that loses. In fact, his version of the war of attrition model for international crises predicts a negative expected hazard rate for crises up through the endogenous time horizon.

Despite these deviations from the classic war of attrition model, our estimator remains flexible enough to capture the salient features of these models of the duration of crisis bargaining in ways that previous research has not. The theoretical literature makes clear that crisis escalation serves as a signaling process through which weaker types attempt to appear strong, but often end up backing down. Stronger types, on the other hand, will remain committed and not back down. Finally, models of the process typically identify an endogenous time horizon after which neither party will exit (Fearon Reference Fearon1994). Our estimator captures these dynamics. Helpfully, it allows for both increasing and decreasing hazards among crises involving at least one weak player. Furthermore, as noted previously, it includes both the exponential and Weibull models as special cases. If the data do not mirror the implied process then we expect to obtain one of these special cases.

To guide our models, we draw on DeRouen and Goldfinch's (Reference DeRouen and Goldfinch2005) analysis of the duration of the length of all international crises from 1918 to 1994 using the International Crisis Behavior data (Brecher and Wilkenfeld Reference Brecher and Wilkenfeld2000).Footnote ¹⁷ The outcome variable indicates the number of days a state is in a crisis. DeRouen and Goldfinch's data include information on 710 actors from 220 distinct crises and capture information about both the target and the opponent. The remaining crises last anywhere from 1 to 1359 days. We replicate their analysis with the exception of employing a Weibull rather than a two-parameter gamma model, though they note little difference between the two. We employ the same eight independent variables.

Table 2 reports the results for the Weibull and war of attrition duration estimators. As with the filibuster analysis, the best way to interpret the results is by plotting the estimated hazard. Figure 5 reveals that such crises produce an increasing hazard rate through 44 days whereas crises involving two stronger states have a relatively small and decreasing hazard. Again, the plotted confidence intervals show clear separation between the two. The logit mixture equation estimates indicate that 78 percent of crises involve two strong players. Taken together, these results comport with those of the war of attrition model. A subtle difference from the filibuster hazards emerges with the truncated Weibull hazard for crises exhibiting an inflection point around 20 days. Its presence here reflects the smaller (though still positive) value of the duration dependence parameter interacting with the impending truncation point in the denominator of the hazard.

Figure 5. Estimated crisis hazard rates from partial cure estimator.

Note: Estimates from Table 2 with no covariates (t* = 44). Black lines indicate the predicted hazard rate for each type of contest. Light gray bars give a nonparametric 95 percent confidence interval estimated by sampling the parameters from their estimated distribution and calculating the corresponding value of the hazard at many values of crisis duration. The upper bound of the confidence interval for the Weibull hazard is not shown past 0.4 since it approaches 30 and including it renders the other results imperceptible.

Table 2. Weibull and partial cure hazard model results for duration of international crises, 1918–1994 (hazard form)

Note: Robust standard errors reported. See the text and Shapiro (Reference Shapiro1988) for information on the $\bar{\chi }^2$ test for model comparison between the Weibull and our estimator given the presence of two boundary values. The critical value with two such parameters is 4.23.

Contrast these results with those from the standard Weibull estimator, which indicate negative duration dependence, corresponding to a decreasing hazard rate. This finding emerges due to the downward pressure exerted by the high proportion of crises involving two strong players. Since their durations extend well beyond the endogenous truncation point and form a high proportion of all cases, the Weibull model for all durations accounts for this with an overall decreasing hazard. The decreasing rate comports with Fearon's (Reference Fearon1994) prediction. Although not inconsistent with this prediction, our war of attrition estimator adds some nuance to that finding by capturing the presence of two qualitatively different kinds of contests, one of which has an increasing hazard rate as predicted by the war of attrition model.

Figure 6 offers another way to compare the relative performance of the two estimators. Here we follow Maller and Zhou (Reference Maller and Zhou1995) by constructing a probability–probability plot that compares the predicted cumulative distributions from the Weibull and war of attrition estimators to the observed cumulative distribution. The closer the curves lie to the 45° line the better the performance of the estimator relative to the truth. Here we see that both underpredict failures in roughly the middle quintile. After about 0.35 the war of attrition estimator lies closer to 45° line, especially between 0.4 and 0.85. The correlations between the estimators and the observed data show the at Weibull at 0.91 and the war of attrition at 0.93.

Figure 6. Probability–probability plot for comparison of model performance.

Note: Estimates from Table 2 with no covariates (t* = 44). A probability–probability plot compares the observed cumulative distribution of the outcome variable with the cumulative distribution predicted by an estimator at the observed values. Perfect correspondence occurs when the plotted curve falls on the 45° line.

The second and fourth models add covariates and further demonstrate the value of the war of attrition estimator. These produce similar patterns of significance for the parameterized hazard components, with the level of violence and the presence of an ethnic dispute decreasing the hazard. The war of attrition estimator continues to indicate a sizeable proportion of crises involve two strong players, however, and the $\bar{\chi }^2$ test continues to reject the basic Weibull model. Adding covariates increases the estimate for t* from 44 to 309 days, though the overall shape of the hazards for both types of crises remains remarkably similar.Footnote ¹⁸

Our final model investigates whether the probability that both states are strong depends on some of the observed covariates. Since the literature argues that democracies are better able to generate audience costs (Fearon Reference Fearon1994) and that states with greater military resources ought to be more resolved (Morrow Reference Morrow1989), we might expect both variables to increase the chance of both players being strong. Although our results produce coefficients with signs as expected, only capabilities emerge as significant. Furthermore, the inclusion of these variables in the mixture equation alters the results in the duration equation for weak contests: joint democracy produces a larger effect and its coefficient becomes significant. These results indicate that although capabilities predict states' type and willingness to enter a crisis, they do not affect their willingness to escalate a crisis once it starts. In contrast, once we account for the role of capabilities and joint democracy in predicting players' types, we learn that crises involving two democracies have a smaller hazard and therefore tend to last longer.