Download Statistical Analysis of Strategic Interactions with Unobservable Player Actions:

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.