1. Model

States A and B compete over a good or a policy that is currently owned or controlled by B. At the beginning of the game, the states observe private information. State A observes (ɛ_A, ɛ_a), where ɛ_A and ɛ_a are additively separable payoff shocks to A's utility for war and backing down, respectively. Likewise, B observes ɛ_B which is an additively separable payoff shock to its war utility. All private information (ɛ_A, ɛ_a, and ɛ_B) is independently drawn from a standard normal distribution.

Interaction proceeds according to Figure 1. First, A decides whether or not to challenge B for control over the good, and if A does not challenge, then the game ends at node SQ with payoffs S _i for each state i. Second, after a challenge, B decides whether or not to resist A. If B does not resist, i.e., B concedes to A's demands, then the game ends at node CD, and payoffs are V _A and C _B for states A and B, respectively. Finally, if B does resist, then A must decide whether to fight or not. When A fights or stands firm, the states receive $\bar {W}_i + \varepsilon _i$ at node SF. Similarly, when A backs down and does not fight, the games end at node BD with A receiving $\bar {a}+\varepsilon _a$ and B receiving V _B.

Figure 1. The canonical crisis-signaling game.

Perfect Bayesian equilibria (equilibria, hereafter) for the game can be represented as choice probabilities. Let p _C and p _F denote the probability that A challenges and fights (conditional on challenging) B, respectively, and let p _R denote the probability that B resists. Let p = (p _C, p _R, p _F) denote a profile of choice probabilities. Furthermore, let θ denote the vector of payoffs, i.e., $\theta = ( \bar {a},\; C_B,\; ( S_i,\; V_i,\; \bar {W}_i) _{i=A, B})$. The following result is due to Jo (Reference Jo2011) and characterizes the equilibria of the game in terms of a system of nonlinear equations.

RESULT 1 (Jo, 2011): An equilibrium $\tilde {p}$ exists, and $\tilde {p}$ is an equilibrium if and only if it satisfies the following system of equations:

(1)$$\tilde{\,p}_C = 1 - \Phi\left({S_A - ( 1-\tilde{\,p}_R) V_A\over \tilde{\,p}_R} - \bar{W}_A\right)\Phi\left({S_A - ( 1-\tilde{\,p}_R) V_A\over \tilde{\,p}_R} - \bar{a}\right)\equiv g( \tilde{\,p}_R\semicolon\; \theta) ,\; $$

(2)$$\tilde{\,p}_F = \Phi_2\left({\bar{W}_A-\bar{a}\over \sqrt{2}},\; \bar{W}_A - {S_A - ( 1-\tilde{\,p}_R) V_A\over \tilde{\,p}_R},\; {1\over \sqrt{2}} \right)\left(g( \tilde{\,p}_R\semicolon\; \theta) \right)^{-1} \equiv h( \tilde{\,p}_R\semicolon\; \theta) ,\; $$

and

(3)$$\tilde{\,p}_R = \Phi \left({ h( \tilde{\,p}_R\semicolon\; \theta) \bar{W}_B + ( 1-h( \tilde{\,p}_R\semicolon\; \theta) ) V_B-C_B\over h( \tilde{\,p}_R\semicolon\; \theta) }\right)\equiv f \circ h( \tilde{\,p}_R\semicolon\; \theta) ,\; $$

where Φ is the CDF of the standard normal distribution and Φ₂( · , · , ρ) is the CDF of the standard bivariate normal distribution with correlation ρ.

In words, for a fixed θ, an equilibrium is completely pinned down by B's probability of resisting. In addition, the functions f, g, and h are essentially best-response functions. Specifically, the functions g and h compute how A best responds to B's probability of resisting p _R, and function f captures how B best responds to A. Furthermore, Jo (Reference Jo2011) illustrates that multiple equilibria exist in a nontrivial set of parameters, i.e., there exists several solutions to the equation f ○ h(p _R; θ) = p _R.

Before proceeding it is worth noting that there are different ways to specify how private information is introduced in the model. We discuss how the problems we consider appear under some of the most common information structures in Appendix A.

2. Estimation: problems and solutions

We consider D independent dyads or games. Each dyad is parameterized by covariates x _d and common payoff parameters β which determine the model's payoffs:

(4)$$\theta( x_d,\; \beta) =\left[\matrix{S_{dA}\cr S_{dB}\cr V_{dA}\cr C_{dB}\cr \bar{W}_{dA}\cr \bar{W}_{dB}\cr \bar{a}_d \cr V_{dB}\cr }\right]= \left[\matrix{x_{dS_A}\cdot\beta_{S_A}\cr \hskip-2.6pc 0\cr x_{dV_A}\cdot \beta_{V_A}\cr \hskip-2pt x_{dC_B}\cdot \beta_{C_B}\cr x_{d\bar{W}_A}\cdot \beta_{\bar{W}_A}\cr x_{d\bar{W}_B}\cdot \beta_{\bar{W}_B}\cr \hskip-1pc x_{d\bar{a}}\cdot \beta_{\bar{a}}\cr \hskip-3pt x_{dV_B}\cdot \beta_{V_B}\cr }\right].$$

Each x _d(·) vector above contains zero or more explanatory variables.Footnote ³ Hereafter, we are interested in the β parameters that are common across all games rather than θ(x _d, β).

Let $\beta ^\ast$ denote the parameters in the data generating process. Along with $\beta ^\ast$, the covariate vector x _d determines the equilibrium $p^\ast ( x_d,\; \beta ^\ast ) =( p^\ast _{dC},\; p^\ast _{dF},\; p^\ast _{dR})$ that generates T ≥ 1 outcomes $\{ y_{dt}\} _{t=1}^T$, where y _dt is a terminal node in {SQ, CD, SF, BD}. Thus, $p^\ast _d( x_d,\; \beta ^\ast )$ is a solution to the system of equations in Result 1, parametrized by payoffs $\theta ( x_d,\; \beta ^\ast )$. Additionally, the data are hierarchical: a complete observation is a dyad d with a single vector of exogenous traits x _d and a sequence of outcomes $\{ y_{dt}\} _{t=1}^T$.

There are two important assumptions implicit in our empirical setup. First, we assume that two states play from the same equilibrium conditional on x _d rather than allowing the equilibrium to vary over within-dyad observations. Substantively, this assumption reflects the international system which has several forces incentivizing states to focus on a single equilibrium over time, including persistent international norms/institutions (Keohane, Reference Keohane1984), a focal point specific to these two states (Schelling, Reference Schelling1960), or other factors that emerge from repeated interaction. Technically, this is a standard assumption that is required in the recent empirical literature on estimating games with incomplete information (Bajari et al., Reference Bajari, Benkard and Levin2007; Ellickson and Misra, Reference Ellickson and Misra2011). An alternative approach might assume that states play from the same equilibrium across dyads rather than within dyads. Such an assumption is more restrictive than ours, and it introduces an additional problem: because dyads are parameterized by different covariates, the number of equilibria may differ between two dyads even for a fixed θ, making it impossible to compare equilibria across observations.

Second, when dyads play T>1 rounds of play, we assume that a given equilibrium of the static game is played. Such an assumption can be justified if states play the game for a finite number of periods and private information is drawn independently over time. This assumption is a matter of convenience because our goal is to address the technical challenges that arise when estimating games with multiple equilibria even in the most straightforward environments. An added benefit of this simplicity is that we can easily enumerate the set of equilibria, allowing us to illustrate how equilibrium selection creates discontinuous likelihoods and the computational inefficiency of the tML in these situations. While we acknowledge the importance of considering dynamic interactions, this would require a different theoretical model, which is beyond the paper's scope.Footnote ⁴

Throughout we consider two numerical examples. Table 1 contains two sets of parameters that we use to demonstrate cases with a unique and with multiple equilibria. In both settings we include one regressor, x _d ~ U[0, 1], which enters B's war payoff.Footnote ⁵ There are a few additional things to note about the parameters. First, we normalize the status-quo payoffs S _i and B's concession payoff to zero, following standard identification assumptions (Lewis and Schultz, Reference Lewis and Schultz2003; Jo, Reference Jo2011). Second, the differences in the two columns are minor: by making small adjustments to only two parameters we can easily move into and out of situations where multiple equilibria exist. Third, these parameters reflect reasonable payoffs that satisfy the restrictions in Schultz and Lewis (Reference Schultz and Lewis2005). Both war and backing down from threats are worse than the status quo, and actors only receive positive payoffs when their opponent backs down.

Table 1. Parameters for Monte Carlo experiments

To illustrate the two settings, Figure 2 graphs the game's equilibrium correspondence with respect to x _d. In the left-hand panel of Figure 2, there are multiple equilibria for values of x _d between 0 and 1. Here, the gray triangles in the plots illustrate how we determine which equilibria generate the data in our Monte Carlo experiments. Specifically, when $x_d \in [ 0,\; {1\over 3})$, we use the smallest equilibrium probability of resisting p _R to generate the data for dyad d. When $x_d \in ( {2\over 3},\; 1]$, we use the largest. Finally, we use the moderate equilibrium in the remaining case. Notice that the equilibrium correspondence is smooth in the sense that it is upper hemicontinuous but selection creates discontinuities when modeling the probability of resistance p _dR as a function (not correspondence) of the covariate x _d. The right-hand side of Figure 2 graphs the equilibrium correspondence under parameters shown in the third column of Table 1, where there is a unique equilibrium for all values of x _d.

Figure 2. The equilibrium correspondences for numerical examples.

2.1. Problems with current practices

Current best practices closely follow the ML techniques discussed in Signorino (Reference Signorino1999). For every β, an equilibrium to game d is computed by solving the system of equations in Result 1; call this solution p(x _d, β). Note that this solution is not necessarily unique, and following standard practices, we do not search for all solutions.

Using p(x _d, β), we define the probability of reaching each of the terminal nodes as

(5)$$ \eqalign{\Pr[y_{dt}\mid p(x_d,\beta)] = \left\{\matrix{ (1-p_{dC})\hfill & if & y_{dt} = SQ \cr p_{dC}(1-p_{dR})\hfill & {\rm if }& y_{dt} = CD \cr p_{dC}p_{dR}(1-p_{dF})\hfill & {\rm if } & y_{dt} = BD \cr p_{dC}p_{dR}p_{dF} \hfill & {\rm if } & y_{dt} = SF.}\right.}$$

Under this setup, the log-likelihood takes the form

(6)$$L( \beta\mid Y) = \sum_{d=1}^D \sum_{t=1}^{T} \log \Pr[ y_{dt}\mid p( x_d,\; \beta) ] ,\; $$

and the tML estimates attempt to maximize this log-likelihood.

As described in Jo (Reference Jo2011), the current approach evaluates the likelihood function as if a unique equilibrium exists. That is, for each guess of the parameters, we compute an equilibrium, p(x _d, β), using a numeric equation solver. If there are multiple equilibria, then there is an indeterminacy in how analysts evaluate p(x _d, β). If the equation solver of choice selects the wrong equilibrium, i.e., not the one in the data generating process, then the likelihood is computed incorrectly, resulting in mistaken inferences. To better see this problem, suppose there are D dyads, and fixing parameters β, suppose each dyad admits n>1 equilibria. In this case, there are n ^D possible values of the log-likelihood for just this one guess at the parameter vector. Standard equation solvers return just one of the n ^D combinations. As D increases, it is increasingly implausible that the correct selection is made. An implication of this discussion is that two researchers can reach conflicting conclusions even when analyzing the same data if they implement the tML estimator with different equation solvers.

Before proceeding, we first consider potential fixes to the standard ML routine. We first ask: Can multiplicity in the crisis-signaling game be solved with traditional refinements? If so, tML techniques can be used so long as they are adjusted to always select the surviving equilibrium. Refinements based on off-the-path-of-play beliefs, such as the Intuitive Criterion or Divinity, are inconsequential here as all histories are reached with positive probability in every equilibrium. Because of this, an analyst may be tempted to use a refinement called regularity, which subsumes several other refinements such as perfection, essentialness, and strong stability (van Damme, Reference van Damme1996).Footnote ⁶ As we show in Appendix B, for almost all parameter values, all equilibria of the crisis-signaling game satisfy regularity.Footnote ⁷ Most importantly, the result demonstrates that multiplicity cannot be ‘refined away’ using standard criteria, and the predictive indeterminacy that plagues traditional maximum likelihood methods still persists.Footnote ⁸

With traditional refinements offering little headway, analysts may turn to ad hoc selection criteria such as selecting the equilibrium that maximizes a convex sum of A and B's payoffs. But determining the selection criterion forces an additional modeling choice onto the analyst. As we show in our empirical application, such a choice is consequential and can heavily influence the resulting estimates. Analysts could also consider empirical selection: for each dyad, select the equilibrium that maximizes the dyad's contribution to the likelihood. This would also remove the indeterminacy in p(x _d, β), but its implementation has several drawbacks. Researchers would need to reliably compute all equilibria for every dyad at every guess of the parameters, a computationally demanding task. In addition, imposing this (and other) selection criterion introduces discontinuities in the likelihood function as the number of equilibria and hence the solution to the criterion varies across different parameter values.Footnote ⁹ We return to this point in Appendix G.

2.2. Pseudo-likelihoods

Our first proposal involves a two-step estimator based on Hotz and Miller (Reference Hotz and Miller1993) that essentially removes the indeterminacy associated with multiple equilibria by using the observed data to select appropriate equilibrium beliefs. In the first step, we produce consistent (in T or D) estimates of the equilibrium choice probabilities $p_{dR}^\ast$ and $p_{dF}^\ast$, for d = 1, …, D. We label these estimates $\hat{\bf p}_{R} = ( \hat{p}_{1R},\; \ldots,\; \hat {p}_{DR})$ and $\hat{\bf p}_{F} =( \hat{p}_{1F},\; \ldots ,\; \hat{p}_{DF})$. While in theory we are agnostic about how an analyst obtains the first-stage estimates, in practice we have found that random forests tend to work very well across a variety of sample sizes and settings.

Next, consider how actors best respond to these first-stage estimates. By Result 1, the best responses take the form:

(7)$$\hat{\,p}( \hat{\,p}_{dR},\; \hat{\,p}_{dF}\semicolon\; x_d,\; \beta) = \left[\matrix{g( \hat{\,p}_{dR}\semicolon\; x_d,\; \beta) \cr h( \hat{\,p}_{dR}\semicolon\; x_d,\; \beta) \cr f( \hat{\,p}_{dF}\semicolon\; x_d,\; \beta) }\right].$$

In other words, if actors play the game as if they believed their opponents use strategies estimated in the first stage, ${\hat {p}_{dR}}$ and ${\hat {p}_{dF}}$, then $\hat {p}$ are their best responses. These best responses approach their true values as the first-stage estimates become more accurate. Using the first-stage estimates and the associated best responses, we build the pseudo-log-likelihood function as

(8)$$PL(\beta\mid \hat{\bf p}_{R},\hat{\bf p}_{F}, Y,X) = \sum_{d=1}^D \sum_{t=1}^{T} \log \Pr[y_{dt}\mid \hat{\,p}(\hat{\,p}_{dR},\hat{\,p}_{dF};x_d, \beta)].$$

What is the intuition behind the estimator? If we know the equilibrium choice probabilities, i.e., $\hat {p}_{dR} = {p}^\ast _{dR}$ and $\hat {p}_{dF} = {p}^\ast _{dF}$ for all dyads d, then the pseudo-likelihood is the likelihood in Equation 6 with the correct equilibrium selection. In addition, it is a continuous function of the parameters β. The equilibrium choice probabilities are unobserved variables, however. Thus, we estimate them from the data, which is possible given our assumptions that two states play from the same equilibrium conditional on x _d. For example, because the states in dyad d are playing from one equilibrium, when T is large, we can estimate $p_{dR}^\ast$ and $p_{dF}^\ast$ using frequency estimators:

$$ \eqalign{\hat{p}_{dR}^* = {\sum_{t=1}^{T} {\opf{I}}\left[y_{dt} \in \{SF,BD\}\right] \over \sum_{t=1}^{T} {\opf{I}}\left[y_{dt} \in \{SF,BD,CD\}\right]} \quad {\rm and} \quad \hat{p}_{dF}^* = {\sum_{t=1}^{T} {\opf{I}}\left[y_{dt} = SF \right] \over \sum_{t=1}^{T} {\opf{I}}\left[y_{dt} \in \{SF,BD\}\right]},}$$

where ${\opf I}$ is the indicator function. As we observe more draws from the same equilibrium, i.e., T goes to infinity, the frequency estimates converge to their true values because the equilibrium $p^\ast _d$ puts positive probability on all histories. Substituting the frequency estimates into Equation 8 demonstrates that the pseudo-likelihood converges to the true likelihood as T increases, and under standard regularity conditions the PL estimates converge to the true ML estimates. Thus, by estimating equilibrium beliefs from the data in a first-stage, we can select the appropriate equilibrium in a continuous manner when estimating payoff parameters during the second stage.

In finite samples, frequency estimators may be impractical. One alternative is to pool information across dyads and estimate the choice probabilities as functions of covariates x _d, albeit with highly flexible methods—hence our assumption that two observationally equivalent dyads play from the same equilibrium. As mentioned, we have found that random forests work particularly well in both our simulations and applications. Nonetheless, the PL estimator may perform poorly if the first stage is misspecified or imprecise. The two methods we discuss below attempt to overcome this issue.

2.2.1. Nested pseudo-likelihood

The NPL approach, proposed by Aguirregabiria and Mira (Reference Aguirregabiria and Mira2007), builds on the PL by using best responses to update the first-stage choice probabilities upon knowing the PL estimates. This process is repeated until convergence. More precisely, the NPL algorithm begins with the PL estimates,

$$ (\hat{\beta}^{NPL}_0, \hat{\bf p}_{R,0}, \hat{\bf p}_{F,0}) = (\hat{\beta}^{PL}, \hat{\bf p}_{R}, {\hat{\bf p}_{F}}),$$

and for the kth iteration, set

$$ \eqalign{\hat{p}_{dF,k} & = h( \hat{p}_{dR,k-1}; x_d, \beta_{k-1}) \cr \hat{p}_{dR,k} & = f ( \hat{p}_{dF,k-1}; x_d, \beta_{k-1}) \cr \hat{\beta}^{NPL}_k & = {\rm argmax}_{\beta} PL(\beta \mid \hat{\bf p}_{R,k},\hat{\bf p}_{F,k}, {\rm Y,X}).}$$

The algorithm is repeated until the parameters and choice probabilities cease changing. The intuition is to decrease the analyst's reliance on correct first-stage estimates by updating the choice probabilities with the new information captured in the estimated payoff parameters.

Without a particular stability condition on the data generating process, the NPL algorithm may fail to converge (Pesendorfer and Schmidt-Dengler, Reference Pesendorfer and Schmidt-Dengler2010). Specifically, if the data generating equilibrium is best-response stable, the above iteration will converge to the correct equilibrium as long as the starting value is not too far away. In contrast, if the data generating equilibrium is unstable, the above iteration may not converge to the true equilibrium. In Appendix C.3, we consider how sensitive the PL and NPL are to unstable equilibria. Overall, we find that both the PL and the NPL still outperform the tML even if best-response unstable equilibria dominate in the data.

2.3. Constrained MLE

An alternative approach is to use a full-information CMLE, as proposed by Su and Judd (Reference Su and Judd2012). Applied to this problem, we maximize the likelihood in Equation 6 subject to the equilibrium constraints in Result 1. Define

(9)$$ \bar{\,p}( p_{dR}\semicolon\; x_d,\; \beta) = \left[\matrix{g( p_{dR}\semicolon\; x_d,\; \beta) \cr h( p_{dR}\semicolon\; x_d,\; \beta) \cr p_{dR} }\right],\; $$

then the CMLE solves the following problem:

(10)$$\matrix{ \max\limits_{\beta, \; {\bf p_R}} & \quad \displaystyle\sum_{d=1}^D \displaystyle\sum_{t=1}^{T} \log \Pr[ y_{dt}\mid \bar{\,p}( p_{dR}\semicolon\; x_d,\; \beta) ] ,\; \cr {\rm s.t.} & \quad f\circ h( p_{dR}\semicolon\; x_d,\; \beta) = p_{dR},\; \, d=1,\; \ldots,\; D. }$$

Su and Judd (Reference Su and Judd2012) demonstrate that the CMLE is equivalent to the true MLE procedure in which equilibria are selected to maximize each dyad's contribution to the likelihood. Thus, the estimator is essentially using the data to select equilibria, which is similar to the PL procedure where data were used to estimate equilibrium beliefs. As mentioned above, modifying the tML to compute every equilibrium at every guess of the parameters and to select the ones that maximize the likelihood dramatically reduces its feasibility because it requires repeated equilibrium computations and introduces discontinuities.

The CMLE avoids these problems. By not requiring that p_R satisfy the equilibrium condition at every step in the constrained optimization, the CMLE avoids any equilibrium computation while ensuring that the objective function is well-behaved and continuous. As such, the true maximum likelihood estimates are discovered with a much lower computational burden than the tML with empirical selection discussed above. Additionally, the CMLE improves on the pseudo-likelihood procedures by eliminating the need to rely on first-stage estimates, resulting in both bias and efficiency gains.

Despite these improvements, the CMLE has two drawbacks. First, the full-information constrained optimization approach introduces D auxiliary parameters in the form of p_R; as such we need T >1 in order to use this estimator. In contrast, the pseudo-likelihood approaches cover the T=1 case. However, our Monte Carlo experiments demonstrate that the CMLE performs well even with a small number of within-game observations. Second, solving this constrained optimization problem requires specialized software; Appendix D contains complete implementation details.

3. Performance

We now evaluate the performance of the estimators in two settings: when there are multiple equilibria in the data generating process and when there is a unique equilibrium. We continue to use the parameter values from Table 1, where x _d is distributed standard uniform.Footnote ¹⁰ Throughout, we use the ordinary implementation of the tML as our baseline for comparison, which uses arbitrary equilibrium selection and Nelder–Mead's simplex method to find the estimates. These implementation choices match current practices as found in replication archives.Footnote ¹¹

To estimate equilibrium choice probabilities in the PL and NPL methods we use random forests. There are two models in the first-stage, where the dependent variables are the nonparametric frequency estimates of the probability that B resists (for ${\hat {\bf p}_{R}}$) and A fights (for $\hat {\bf p}_{F}$). We fit the former only with observations in which A challenges, and we fit the latter only with observations in which B resists. For predictors, we include the one regressor x _d.

We vary the number of dyads, D, between 25 and 200 and the number of within-game observation, T, between 5 and 200 to create simulated datasets of various sizes. For each combination of D and T, we draw x _d from the standard uniform distribution and then select the appropriate equilibrium that generates the data for the corresponding dyad as shown in Figure 2. Finally, we use the simulated data to estimate the game using all four estimators. Starting values for the PL and tML are drawn from a standard uniform distribution, and the same values are used within each Monte Carlo iteration. The CMLE and NPL use the PL estimates as starting values.Footnote ¹² We repeat this process 1000 times for each pair of D and T and for each of the parameter settings in Table 1.Footnote ¹³

The main results of the experiment are summarized in Figures 3 and 4, which compare the logged root-mean-square error (RMSE) of the estimators. The first thing we note is that the tML (dashed line) performs consistently bad and shows no improvement as the amount of data increases in either D or T. In many cases, its performance worsens as T increases.Footnote ¹⁴

Figure 3. RMSE in signaling estimators with multiple equilibria.

Figure 4. RMSE in signaling estimators with a unique equilibrium.

Contrast these results to those from the other estimators, which generally all improve with more data. The PL (solid line) tends to be best performing estimator when both T and D are small. Additional analysis in Appendix C shows that the estimator tends to have more bias than the others and that its strong performance is driven by low variance. The NPL (dot-dashed line) greatly improves the bias associated with the PL method without adding too much variance, and as a result, we see that it performs very well in most settings, particularly as the amount of data increases. Overall, the CMLE (dotted line) tends to be the best. However, this great performance often comes at the cost of decreased convergence rates and non-standard software choices.

Comparing Figures 3 and 4 reveals that the tML has uniformly poor performance regardless of the number of equilibria that exist in the signaling game generating the data.Footnote ¹⁵ What explains the poor performance of the tML in the unique equilibrium experiment? Even in this setting the tML's likelihood function is often evaluated at incorrect parameter values. For example, we pick starting values that are drawn uniformly over the interval [0, 1]. These are obviously incorrect, and the optimizer will need to search over the parameter space, evaluating the likelihood function at incorrect parameter values. In some instances, dyads parameterized (incorrectly) by these values will have multiple equilibria, and the objective function will need to select an equilibrium in an ad hoc manner. This selection will lead to discontinuities and creates the possibility that an incorrect equilibrium is selected, i.e., an equilibrium that has little relation to the one generating the data. These issues can lead even more robust optimizers astray.

We also find that the tML appears to face numerical challenges during the optimization process. Even in cases where we verify that the tML only considers candidate parameter vectors that are associated with a unique equilibrium, we find that the optimizer frequently converges to a wrong answer. These issues do not go away (and often get worse) when we consider alternative optimization routines. Additionally, with our empirical example we find that very small implementation differences, including simply changing software versions, result in wildly different tML estimates. Overall, this level of sensitivity indicates that the equilibrium computation in the tML's likelihood creates a highly nonlinear optimization problem that is difficult to solve. We do not observe these kinds of stability issues with the other methods.

With the above theoretic and numeric concerns in mind, it is worth considering how sensitive the tML's performance is to starting values; we investigate this in Appendix E. We find that the tML's performance improves if (a) there is a unique equilibrium at the true parameters and (b) the tML has starting values that are either the true parameters or the PL estimates. However, even when initialized with the PL estimates, the tML rarely improves much on, and sometimes worsens, the PL's performance, and it is almost always worse than the NPL or CMLE. Overall, relying on informed starting values and equilibrium uniqueness in the data generating process is perilous for applied researchers because neither can be verified before estimation. Furthermore, the PL, NPL, and CMLE perform at least as well as tML and oftentimes much better. Before turning our attention to economic sanctions, we report the following conclusions.

1. 1. The tML routine performs the worst in both multiple and unique settings.

2. 2. The NPL and PL methods consistently perform well, but the PL outperforms the NPL when the number of within-game observations is small, and vice versa when the number of within-game observations is large. In every experiment, the NPL is less biased than the PL.

3. 3. The CMLE is almost always the best, but it is the most difficult to implement.

4. Application to economic sanctions

Our application is motivated by Whang et al. (Reference Whang, McLean and Kuberski2013, WMK, hereafter) who use the empirical crisis-signaling game to study the implementation of economic sanctions. They test the hypotheses that greater economic dependence decreases the probability that state B resists and increases the amount of belief updating, finding substantial support for the former but not the latter. The game is reproduced in Figure 19 in Appendix F. The outcomes are status quo, concede to the threat, impose sanctions, and back down, which are denoted SQ, CD, SF, and BD, respectively.

An observation in WMK is a politically relevant directed dyad-decade. In their study, a directed dyad is politically relevant if there exists at least one sanction threat issued from State A to State B in the TIES dataset during the 1971–2000 period. Within each directed dyad, WMK aggregate the dependent variable to be the most extreme outcome within a directed dyad-decade, dividing the time frame into three groups 1971–80, 1981–90, and 1991–2000.

Like WMK we aggregate covariates x _d to the decade level, but unlike WMK, we dis-aggregate the outcomes y _dt to the monthly level (T=120). We treat the observed y _dt within each directed dyad-decade as if they are repeated draws from the same equilibrium.Footnote ¹⁶ Effectively this means that each game d consists of a decade-level covariate vector x _d that is thought to produce each directed dyad's monthly interaction over the course of the decade. In terms of our setup, each game is a politically relevant, directed dyad-decade, and we observe T=120 observations from each game.Footnote ¹⁷

For our purposes, this approach has two important advantages. First, the CMLE procedure requires within-game multiple observations for identification. Without this setup, we could not illustrate this estimator even though it performed quite well in the Monte Carlo experiments. Second, we do not ignore variation within each decade: a directed dyad with only one threat issued in a decade may be substantially different than one with several threats in the same period. Thus, our application does not replicate previous analyses but rather highlights the differences between tML routines and those that we propose.

Following WMK, we use the Final Outcome variable to record the dependent variable, which denotes how sanction-threat episodes end.Footnote ¹⁸ When there is no episode in a month, we record the status quo. When Final Outcome records either “acquiescence” by the target or a negotiated settlement, we record the outcome as B giving into A's threat (node CD). Likewise, whenever the Final Outcome variable notes that actual sanctions are imposed, we list A as standing firm on its threat (node SF). Finally, when Final Outcome denotes that A either “capitulates” or the situation is unresolved, we list A as backing down (node BD). After dropping irrelevant dyad-decades, i.e., those with no recorded threats or sanctions, we are left with 418 games, each with 120 within game observations that span one of the three time frames, 1971–80, 1981–90, and 1991–2000.Footnote ¹⁹

The independent variables, their sources, and how they enter the actors' payoffs are listed in Table 3 in Appendix F, following the specifications in WMK. All variables are measured on the dyad-decade level as discussed above.Footnote ²⁰

4.1. Point estimates

Table 2 displays our main results. Each column contains parameter estimates and standard errors using the different estimators. There are several notable patterns. First, the techniques derived from the dynamic games literature produce estimates that agree in direction, magnitude, and significance. Models 2–4 match signs for 14 out of 21 coefficients, and when we reject a null hypothesis using one estimator, we generally do the same for one of the others. Second, the tML returns estimates that diverge wildly from the other three. The problem appears particularly bad for coefficients that enter the target state's concession payoffs, C _B.

Table 2. Economic sanctions application

Notes: *p < 0.05. Asymptotic standard errors in parenthesis, see Appendix D.1 for details.

Not only does the tML routine return different point estimates, it also produces substantive implications that diverge from the other three estimations. For example, consider audience costs, i.e., the initiating state's payoff from backing down, $\bar {a}$. Notice that the relevant constant term is negative, significant, and large in magnitude in all three models that accommodate multiple equilibria. This suggests that states or leaders are indeed punished for backing down after issuing threats.Footnote ²¹ In fact, in Models 2–4, we reject the null hypothesis that $\bar {a} \geq 0$ at the p < 0.05 level in every observation. In contrast, we cannot reject the null hypothesis that $\bar {a}\geq 0$ at the p<0.1 level in the tML model for any observation. Our analysis suggests that researchers may underestimate audience or belligerence costs if estimation techniques do not accommodate the multiplicity of equilibria.

Overall, our results demonstrate that tML routines can produce point estimates and substantive implications that diverge from our proposed methods. To better illustrate that the differences are due to equilibrium selection and the computational problems addressed above, we conduct two additional analyses in Appendix G. First, we fit the sanctions model using a tML routine that is identical to what we use in Model 1 except for how it computes equilibria. Most surprisingly, the two tML results diverge in both sign (for 9/20 estimates) and significance (for 13/20 estimates). Second, we show the importance of starting values. Perhaps unsurprisingly, when we use the PL estimates as starting values, the tML improves, but is still worse than the NPL and CMLE in terms of log-likelihood. Thus, two researchers can reach substantively diverging conclusions with different software choices even when analyzing identical data sets.

4.2. Audience costs and substantive effects

How do audience costs affect the likelihood of leaders threatening sanctions? In the previous section, we demonstrated that tML routines can produce point estimates that diverge wildly from our solutions. In this section, we analyze the substantive effects of audience costs on the equilibrium probability of threatening sanctions, p _C, illustrating that the tML routines can fail to uncover important comparative statics. We focus on audience costs because of their importance to the economic sanctions literature (Martin, Reference Martin1993; Dorussen and Mo, Reference Dorussen and Mo2001; Drezner, Reference Drezner2003; Whang et al., Reference Whang, McLean and Kuberski2013). In addition, previous work has not connected audiences to the likelihood that leaders threaten sanctions.

We consider the directed dyad in which the US is the initiating state A and China is the target state B between 1991 and 2000, the most recent decade in the sample. We vary the US's audience cost, $\bar {a}$, from − 6 to 0 while fixing the remaining payoffs estimated using the tML and CMLE from Table 2. For every value of $\bar {a}$, we compute all equilibria using a line-search method. Then we plot the associated equilibrium probabilities of the US initiating a conflict, p _C, in Figure 5. For all values of $\bar {a}$ considered, there is a unique equilibrium, pictured with the black circles. The vertical line denotes the estimated value of US audience costs, around − 2.7 for the CMLE and − 0.6 for the tML. Throughout, we fix the other payoffs at their estimated values, thereby implicitly controlling for the other (belligerent) costs leaders face when choosing to start a crisis. Hence, our analysis allows us to isolate the effects of audience costs from belligerence costs, a traditionally difficult objective when using experiments or reduced-form analyses (Kertzer and Brutger, Reference Kertzer and Brutger2016).

Figure 5. Substantive effects of audience costs in US and China dyad, 1991–2000.

The figure illustrates three notable results. First, there is substantial difference between the substantive effects from the CMLE and tML. That is, even with the same theoretical model and data, the choice of estimation procedure matters. Second, given the CMLE, audience costs have a large substantive effect on the probability of threat initiation, covering the entire range between zero and one. These large effects are lost when using the tML estimates. Third, there is a U-shaped relationship between audience costs and threat initiation. Leaders only initiate threats when audience costs are very small or quite large. In the former case, leaders do not pay a cost for backing down and do so with impunity. In the latter case, their threats are quite credible, coercing rivals to concede with higher probability.Footnote ²² With intermediate audience costs, however, leaders almost never threaten rivals with sanctions, as their threats are not credible and backing down entails nontrivial costs.

Notice that if we were to increase the US's audience costs beginning from the value estimated in the data, then the model predicts an increase in sanction threats toward China. That is, the true value of audience costs tend to fall on the left-hand-side of the U-shaped curve, where larger (more negative) audience costs increase the likelihood of interstate threats. This pattern generalizes to other observations in the data. We compute the marginal effect of making audience costs, $\bar {a}$, more negative on the equilibrium probability of issuing threats. Conclusively, larger (more negative) audience costs increase the likelihood of states threatening their rivals with sanctions. This result holds in 97 percent of observations.