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Statistical Analysis of Strategic Interactions with Unobservable Player Actions: Introducing a Censored Strategic Probit Mark David Nieman University of Iowa [email protected] Abstract Solution: Censored Strategic Probit Robustness: Split-sample DGP Empirical Application How to model strategic interactions when we can only observe the binary outcomes of joint actor choices? Existing strategic models require data for each of the actor’s actions. Unfortunately, this is not always available. Such data may not yet be collected or simply not exist. CSP probabilistically estimates each player’s actions when only aggregated outcome data are available. CSP explicitly accounts for strategic behavior on the part of the actors. Assuming πij is i.i.d. and normally distributed with mean 0 and variance 1, the likelihood takes the form: n Q yi 1−yi L= P (Yi = 1) P (Yi = 0) i=1 where P (Yi = 1) = pA pB , P (Yi = "0) = (1 − pA ) + pB (1 − pA ) = 1 −#pA pB , We can never be certain that our theory and estimator are correctly specified. What if, rather than being strategic and rational, Player 1 is drawn from two distinct populations? How does CSP perform under this alternative DGP? I apply CSP to Fearon and Laitin 2003 (F&L). F&L generate several hypotheses regarding the onset of civil war within an international state. I re-analyze their model, paying careful attention to the theorized mechanism (and utility) associated with each variable for insurgents and government actors. U1∗ (Con), U2∗ (Con) ∗ = Uij + πij and i is the player and j where Uij the payoff. The observable utility Uij represents a set of regressors while the unobservable component πij represents i’s private information. Problem: Unobserved Choices If actor choices are unobserved, it becomes difficult to separate the two types of “non-event”—SQ and Acq—as both are coded as “0” in the data. Problems with existing estimation techniques: Traditional Probit/Logit treats the strategic model as an additive function, ignoring the conditional nature of Player 1’s choices. Split-sample Probit and Logit assume two distinct “types” of Player 1—one who never engages with Player 2 (zero-inflated equation) and one who does (tradition probit/logit equation). The behavior of Player 1 is independent of Player 2. 0 15 30 Player 1 SQ Utility (Zero−inflated Equation) Results: CSP vs Traditional Probit −.5 0 1 −.5 0 .5 0 .5 Traditional Probit Censored Strategic Probit 1 1.5 1 1.5 Note: Dashed blue line represents the equation’s true coefficient. Results of 500 simulations with 5000 observations each. Root Mean Squared Error Probit 1.308 0.780 0.682 1.5 1 1.5 Player 2 Con Utility −.5 0 .5 Traditional Probit Censored Strategic Probit Variable U1 (SQ) U1 (Con) U2 (Con) Split−Sample Probit Coefficient Variable U1 (SQ) U1 (Con) U2 (Con) 1 Split−Sample Probit RMSE with Split-sample DGP Player 2 Con Utility −.5 .5 1.5 Player 1 Con Utility SSP 0.408 0.601 0.361 CSP 0.241 0.325 0.243 U1 = Insurgents, U2 = Government Traditional Probit Note: Dashed blue line represents the equation’s true coefficient. Results of 500 simulations with 5000 observations each. .5 0 1.5 Coefficient Player 1 SQ Utility −.5 1 Player 1 Con Utility Comparison of Estimated Coefficients 0 .5 0 15 30 Assuming the DGP is ∗ ∗ SQ if U (SQ) ≥ (U 1 1 (Acq) and U2 (Acq) ≥ U2 (Con) ∗ ∗ (SQ) ≥ (U or U 1 1 (Con) ∗ ∗ and U (Con) > U ∗ 2 2 (Acq) y = ∗ ∗ Acq if U1 (Acq) > U1 (SQ) ∗ ∗ (Acq) ≥ U and U 2 2 (Con) ∗ ∗ Con if U (Con) > U 1 1 (SQ) and U2∗ (Con) > U2∗ (Acq) and y = 1 if y ∗ = Con, and y = 0 otherwise. −.5 1 if s∗ > 0 0 if s∗ ≤ 0 0 15 30 U1∗ (Acq), U2∗ (Acq) B CSP allows the same variable to be an observable component of each actor’s utility calculation, meaning it can be included in multiple equations (this is the same as a traditional strategic probit). This allows for a direct test of whether GDP/capita 1) serves as an opportunity cost to potential insurgent recruits (U1 (SQ)), 2) acts as a deterrent by proxying the government’s military strength (U2 (Con)), 3) or both. and y ∗ = β1 U1 (Con) + β U (Con) + ǫ 2 2 1 if y ∗ > 0 where y = 0 if y ∗ ≤ 0 if s = 1, and 0 otherwise. Kernel Density ¬B s∗ = γ1 U1 (SQ) + ν where s = Comparison of Estimated Coefficients 0 15 30 U1∗ (SQ) Assuming the DGP is Monte Carlo Simulations 0 15 30 Many theories posit an interaction between two rational, utility-maximizing players. 1 ¬A A 2 , 0 15 30 Interaction between Two Players pB U1 (Con)+(1−pB )(U1 Acq)−U1 (SQ) q pA = Φ 2 2 p +(1−pB ) +1 B h i U2 (Con)−U2 (Acq) √ pB = Φ . 2 Kernel Density I derive a new estimator based on Signorino’s strategic probit that probabilistically estimates unobservable actor choices when only the interaction’s binary outcome is known. The estimator corresponds to the strategic logic underlying many political interactions and outperforms both traditional and split-sample binary choice models. MCs with Split-sample DGP Probit 1.436 0.670 0.657 SSP 0.218 0.237 0.234 CSP 0.362 0.312 0.250 CSP U1 (SQ): Prior War -0.39∗∗ (0.13) GDP/cap 0.20∗∗ (0.09) (0.03) Democracy -0.01 (0.11) GDP/cap -0.14∗∗ (0.03) U1 (Acq): Population 0.11∗∗ Mountain 0.09∗∗ (0.03) Constant 0.02 (3.38) (0.12) U1 (Con): Non-contig 0.18+ Oil 0.35∗∗ (0.12) Population 0.02 (0.25) New State 0.76∗∗ (0.16) Mountain 0.33+ (0.25) Instability 0.26∗∗ (0.10) Oil 3.70 (5.26) Ethnic Frac 0.34 (0.64) Democracy 0.01 (0.01) Ethnic Frac 0.09 (0.16) Relig Frac 0.38 (0.82) U2 (Con): Relig Frac 0.13 (0.21) Constant -3.22∗∗ (0.30) Prior War -0.61+ (0.43) GDP/Cap -0.01 (0.11) Population 0.17 (0.14) Non-contig 0.38 (0.32) Oil -0.62 (0.71) New State 1.24+ (0.84) Instability 0.43+ (0.32) Constant -2.20+ (1.56) ln(L) -481.42 -478.53 N 6327 6327 Note: **p <0.05, *p <0.1, two-tailed; + p <0.1, one-tailed. Conclusion The results highlight the importance of explicitly matching the underlying theoretical process to an appropriate estimation technique. The Monte Carlo simulations demonstrate that neither traditional nor split-sample binary choice estimators capture strategic processes. Moreover, CSP is relatively robust to an alternative DGP, especially compared to the split-sample estimator. CSP has many potential applications with theorized strategic interactions but limited data on actor choices, including intra-state conflict, inter-state conflict, intra-party discipline, effectiveness of EPA fines, etc.