Equilibrium analysis of the empirical crisis signaling game

Description

This method uses a formula and fixed data/parameters to allow for analysis of the crisis signaling game under specific settings. This function is very similar to predict.sigfit, but it is designed for analysis outside of conducting counterfactuals on a fitted model.

Usage

generate.eq(
  formulas,
  data,
  theta,
  type = c("actions", "outcomes"),
  na.action = na.omit,
  control = list(),
  parallel = FALSE
)

Arguments

Details

This function is used to consider comparative statics in the crisis signaling game, where the model of interest has pre-defined parameters. As such, it requires, at minimum, a seven-part formula and parameters. How this function behaves has to do with how data and theta are specified.

When the model is all constants (every part of the formula argument is either 0 or 1), then data is ignored. In these cases, equilibria are computed for every parameter vector, which are supplied as rows in a data frame to theta.

When there is one or more covariate in the model, then a data frame must be supplied to data. In these cases both data or theta must have at least one row. However, only one of these arguments can have multiple rows. In other words, only data or theta may vary, but not both.

For additional implementation details see predict.sigfit.

Value

An object of class sigProb containing three elements:

predicted

data frame of predicted probabilities. The first column of this data frame is called Row, which corresponds to the rows in either model or par. In the event of multiple equilibria, this column allows for mapping data and parameters to all computed equilibria.

model

data frame of covariates used to produce the predicted probabilities.

par

data frame of parameters used to produce the predicted probabilities.

See Also

plot.sigProb, predict.sigfit

Examples

## An example with one covariate
ftest1 <-  ~ 0 | #SA
             1 | #VA
             0 | #CB
             1 | #barWA
             x1 | #barWB
             1| #bara
             1 #VB
             
theta <- data.frame(VA = 1, barWA = -1.9, barWB = -2.9,
                    barWB1 = 0.1, bara = -1.2, VB = 1)
data <- data.frame(x1 = seq(from = -1,to = 2, length.out = 101))
test <- generate.eq(ftest1, data = data, theta = theta)
plot(test, prob = "pr")

## An example with all constants
ftest2 <-  ~ 0 | #SA
             1 | #VA
             0 | #CB
             1 | #barWA
             1 | #barWB  
             1 | #bara
             1 #VB
             
theta <- data.frame(VA = 1, barWA = -1.9, 
                    barWB = seq(-2.9, -2.2, length.out = 15),
                    bara = -1.2,
                    VB = 1)
test <- generate.eq(ftest2, theta = theta)
plot(test, prob = "pr")

Plot predicted probabilities from a sigProb object.

Description

This method takes a sigProb object produced by predict.sigfit and plots the comparative static(s) of interest.

Usage

## S3 method for class 'sigProb'
plot(x, prob, xvar, main = "", ylab, xlab, col = "blue", pch = 16, ...)

Arguments

See Also

predict.sigfit generate.eq plot par

Examples

data(sanctionsData)

f1 <- sq+cd+sf+bd ~ sqrt(senderecondep) + senderdemocracy + contig + ally -1|#SA
                    anticipatedsendercosts|#VA
                    sqrt(targetecondep) + anticipatedtargetcosts + contig + ally|#CB
                    sqrt(senderecondep) + senderdemocracy + lncaprat | #barWA
                    targetdemocracy + lncaprat| #barWB
                    senderdemocracy| #bara
                    -1#VB
## Outcome probabilities for first five using NPL probabilities
Phat <- list(PRhat=sanctionsData$PRnpl, PFhat=sanctionsData$PFnpl)
fit2 <- sigint(f1, data=sanctionsData, method="pl", phat=Phat)

## comparative static on \bar{a}, compute more precise equilibria with uniroot
new.theta <- data.frame(t(replicate(25, coef(fit2))))
new.theta[,19] <- seq(-6, 0, length=25)
pout <- predict(fit2, newdata=sanctionsData[93,], new.theta=new.theta, 
                control=list(gridsize=500))


plot(pout, prob="pc", ylab="Pr Challenge", xlab="Audience Costs")

Predicted probabilities and comparative statics for signaling games

Description

This method uses a fitted model of class sigfit to compute predicted probabilities or comparative statics. Users can provide either new data or a new parameters to generate counterfactuals of interest.

Usage

## S3 method for class 'sigfit'
predict(
  object,
  newdata,
  new.theta,
  type = c("actions", "outcomes"),
  na.action = na.pass,
  control = list(),
  parallel = FALSE,
  ...
)

Arguments

Details

This function is used to consider comparative statics in the crisis signaling game. The model of interest is fit using sigint. How this function behaves largely depends on how newdata and new.theta are specified.

When both newdata and new.theta are missing, all equilibria for every observation used to fit the model are computed. These equilibria are then used to calculated either choice probabilities (default, type = "action") the distribution over outcomes (type = "outcomes").

When only newdata is specified, then all equilibria are computed using the data frame in newdata and the coefficients from object. This produces standard comparative statics with respect to observed covariates.

When newdata is specified and new.theta is a one row data frame, then all equilibria are computed using the data frame in newdata and the coefficients from new.theta.

When newdata is a one row data frame and new.theta is specified, then all equilibria are computed using the data frame in newdata and the coefficients from new.theta. This is a comparative static on changing a structural parameter in the model.

If new.theta has more than one row, then newdata must be specified as a data frame with only one row. Anything else returns an error.

Equilibria are computed using a line search method. The control argument allows for user control over this process. Users can specify a list with the following named elements

gridsize

Integer. The number of points considered in the line search (default, 1e4). More points makes it more likely that all equilibria are discovered, but can slow down the search.

comp

Logical. Should an equilibrium be computed when discovered? When comp = FALSE (default), the mean of the grid points surrounding the equilibrium is used as an approximate solution. When comp = TRUE the function uniroot is called to find a more precise solution to the equilibrium constraint problem.

tol

Numeric. When comp = TRUE, this is the tolerance used by uniroot

.

When dealing with a larger problem, such as computing all equilibria for every observation, it can be helpful to parallelize process. If the user has the (suggested) parallel package, then the option parallel = TRUE will use the function parSapply is used.

Value

An object of class sigProb containing three elements:

predicted

data frame of predicted probabilities. The first column of this data frame is called Row, which corresponds to the rows in either model or par. In the event of multiple equilibria, this column allows for mapping data and parameters to all computed equilibria.

model

data frame of covariates used to produce the predicted probabilities.

par

data frame of parameters used to produce the predicted probabilities.

See Also

plot.sigProb, generate.eq

Examples

data(sanctionsData)

f1 <- sq+cd+sf+bd ~ sqrt(senderecondep) + senderdemocracy + contig + ally -1|#SA
                    anticipatedsendercosts|#VA
                    sqrt(targetecondep) + anticipatedtargetcosts + contig + ally|#CB
                    sqrt(senderecondep) + senderdemocracy + lncaprat | #barWA
                    targetdemocracy + lncaprat| #barWB
                    senderdemocracy| #bara
                    -1#VB
## Using Nested-Pseudo Likelihood  with default first stage    
## Not run:              
fit1 <- sigint(f1, data=sanctionsData, npl.trace=TRUE)
p.out <- predict(fit2, parallel=TRUE) #fitted choice probabilites for all observations

## End(Not run)

## Outcome probabilities for first five using PL method
Phat <- list(PRhat=sanctionsData$PRhat, PFhat=sanctionsData$PFhat)
fit2 <- sigint(f1, data=sanctionsData, method="pl", phat=Phat)
p1 <- predict(fit2, newdata=sanctionsData[1:5,], type="outcome")

## comparative static on \bar{a}, compute more precise equilibria with uniroot
new.theta <- data.frame(t(replicate(25, coef(fit2))))
new.theta[,19] <- seq(-6, 0, length=25)
p2 <- predict(fit2, newdata=sanctionsData[1,], new.theta=new.theta, control=list(comp=TRUE))

Print the summary table for a sigfit object.

Description

Prints the summary regression table for a model fitted with sigint.

Usage

## S3 method for class 'summary.sigfit'
print(x, ...)

Arguments

Details

Prints the standard regression results table from a fitted strategic model, along with the log-likelihood and number of games used in estimation.

Economic Sanctions Threats and Outcomes

Description

Dataset on economic sanctions threats and outcomes from 1970-2000

Usage

data(sanctionsData)

Details

These data were compiled using the Threat and Imposition of Sanctions (TIES), data project (Morgan, Bapat, and Kobayashi 2014), with additional data from the Correlates of War (COW), and Polity IV datasets. See Crisman-Cox and Gibilisco (2018) for more information. The unit of observation is the dyad-decade, and the variables are:

gameID

A dyad-decade identifier composed of COW country codes and the decade observed.

dyadID

A dyad identifier composed of COW country codes

tenyear

The observed decade

code1

Challenger's COW code

code2

Target's COW code

sq

The number of status quo observations in this dyad decade

cd

The number of times that the game ends with Challenge-Concede (Outcome CD)

sf

The number of times that the game ends with Challenge-Resist -Stand Firm (Outcome SF)

bd

The number of times that the game ends with Challenge-Resist-Back Down (Outcome BD)

senderecondep

Challenger's economic dependence (dyadic trade / Challenger's GDP per capita)

senderdemocracy

Challenger's Polity score

contig

Contiguity between states

ally

Are these state allied? (indicator)

anticipatedsendercosts

The Challenger's anticipated costs for enacting sanctions

anticipatedtargetcosts

The Target's anticipated costs for being sanctions

targetecondep

Target's economic dependence (dyadic trade / Target's GDP per capita)

lncaprat

Ratio of the Challenger's military capability to the Target's (logged)

targetdemocracy

Target's Polity score

PRhat

Estimated probability that the Target resists a challenge (fit using a random forest)

PFhat

Estimated probability that the Challenger stands firm given that it challenged (fit using a random forest)

PRnpl

Estimated probability that the Target resists a challenge (taken from the last stage of NPL iteration)

PFnpl

Estimated probability that the Challenger stands firm given that it challenged (taken from the last stage of NPL iteration)

References

Barbieri, Katherine, Omar M. G. Keshk, and Brian Pollins. 2009. "TRADING DATA: Evaluating our Assumptions and Coding Rules." Conflict Management and Peace Science. 26(5): 471-491.

Casey Crisman-Cox and Michael Gibilisco. 2018. "Estimating Signaling Games in International Relations: Problems and Solutions." Unpublished Manuscript.

Gibler, Douglas M. 2009. International military alliances, 1648-2008. CQ Press.

Marshall, Monty G., and Keith Jaggers. 2013. "Polity IV Project." http://www.systemicpeace.org/polity/polity4.htm.

Morgan, T. Clifton, Navin Bapat, and Yoshi Kobayashi. 2014. "The Threat and Imposition of Sanctions: Updating the TIES dataset." Conflict Management and Peace Science 31(5): 541-558.

Singer, J. David, Stuart Bremer, and John Stuckey. 1972. "Capability Distribution, Uncertainty, and Major Power War, 1820-1965." in Bruce Russett (ed) Peace, War, and Numbers, Beverly Hills: Sage, 19-48.

Stinnett, Douglas M., Jaroslav Tir, Philip Schafer, Paul F. Diehl, and Charles Gochman. 2002. "The Correlates of War Project Direct Contiguity Data, Version 3." Conflict Management and Peace Science 19(2):58-66.

See Also

sigint

Estimating the parameters of the canonical discrete crisis bargaining game.

Description

This function fits the Lewis and Schultz (2003) model to data using either the pseudo-likelihood (PL) or nested-pseudo likelihood (NPL) method from Crisman-Cox and Gibilisco (2018). Throughout, we refer to the data as containing D games, where each game is observed one or more times.

Usage

sigint(
  formulas,
  data,
  subset,
  na.action,
  fixed.par = list(),
  method = c("npl", "pl"),
  npl.maxit = 25,
  npl.tol = 1e-07,
  npl.trace = FALSE,
  start.beta,
  maxlik.method = "NR",
  phat,
  phat.formulas,
  pl.vcov = FALSE,
  phat.vcov,
  seed = 12345,
  maxlik.options = list()
)

Arguments

Details

The model corresponds to an extensive-form, discrete-crisis-bargaining game from Lewis and Schultz (2003):

 
  .       A 
  .      / \ 
  .     /   \ 
  .    /     \
  .   S_A     B 
  .    0     / \
  .         /   \
  .        /     \
  .      V_A      A 
  .      C_B     / \ 
  .             /   \
  .            /     \
  .     W_A + e_A    a + e_a 
  .     W_B + e_B    V_B

If A chooses not to challenge B, then the game ends at the leftmost node (SQ) and payoffs are S_A and 0 to players A and B, respectively. If A challenges B, B can concede or resist. If B concedes, the game ends at CD with payoffs V_A and C_B. However, if B resists, A decides to stand firm, which ends the game at SF with payoffs W_A + \epsilon_A and W_B + \epsilon_B. Finally, if A decides to back down in the face of B's resistance, then the game ends at the rightmost node BD, with payoffs a + \epsilon_a and V_B.

The seven right-hand formulas that are specified in the formula argument correspond to the regressors to be placed in S_A, V_A, C_B, W_A, W_B, a, and V_B, respectively. The model is unidentified if any regressor (including a constant term) is included in all the formulas for each player (Lewis and Schultz 2003). Often the easiest way to meet this requirement is set one formula per player to 0. When an identification problem is detected, an error is issued. For example, the syntax for the formula argument could be:

formulas = sq + cd + sf + bd ~ x1 + 0 | x2 | x2 | x1 + x2 | x1 | 1 | 0)

Where:

• sq + cd + sf + bd are the tallies of how many times each outcome is observed for each observation. When the game is only observed once, that observation will be a 1 and three 0s. When the game is observed multiple times, these variables should count the number of times each outcome is observed. They need to be in the order of SQ, CD, SF, BD.

• S_A is a function of the variable x1 and no constant term.

• V_A is a function of the variable x2 and a constant term.

• C_B is a function of the variable x2 and a constant term.

• W_A is a function of the variables x1, x2 and a constant term.

• W_B is a function of the variable x1 and a constant term.

• a is a constant term.

• V_B is fixed to 0 (or a non-zero value set by fixed.par.

Each row of the data frame should be a summary of the covariates and outcomes associated with that particular game. When each game is observed only once, then this will resemble an ordinary dyad-time data frame. However, if there are multiple observations per game, then each row should be a summary of all the data associated with that game. For example, if there are D games in the data, where each is observed T_d times, then the data frame should have D rows. The four columns making up the dependent variable will denote the frequencies of each outcome for game d, such that sq_d + cd_d + sf_d + bd_d = T_d. The covariates in row d should be summary statistics for the exogenous variables (e.g., mean, median, mode, first observation).

The model is first fit using a pseudo-likelihood estimator. This approach requires first stage estimation of the probability that B resists and the probability that A fights conditional on B choosing to resist. These first stage estimates should be flexible and we recommend that users fit a flexible semi-parametric or non-parametric model to produce them. If these estimates are produced by the analyst prior to using this function, then they can be provided by providing a list to the phat argument. This list should contain two named elements

• PRhat is the probability that B resists. This should be a vector of probabilities with one estimated probability for each observation.

• PFhat is the probability that A stands firm conditional on B resisting. This should be a vector of probabilities with one estimated probability for each observation.

If the user leaves the phat argument empty, then these first-stage estimates are produced internally using the randomForest function. Users wanting to use the random forest, can supply a formula for it using the argument phat.formulas. This argument can take a formula with nothing on the left-hand side and 1-2 right-hand sides. If two right-hand sides are provided then the first is used to generate PRhat, and the second is used for PFhat. If only one right-hand side is provided, it is used for both. Some examples:

• phat.formulas = ~ x1 + x2 predict PRhat and PFhat using x1 and x2.

• phat.formulas = ~ x1 + x2 | x1 + x2 predict PRhat and PFhat using x1 and x2

• phat.formulas = ~ x1 + x2 | x1 predict PRhat using x1 and x2, but predict PFhat using only x1.

If both phat and phat.formula are missing, then a random forest is fit using all the exogenous variables listed in the formulas argument

If method = "npl", then estimation continues. For each iteration of the NPL, the estimates of PRhat and PFhat are updated by one best-response iteration using the current parameter estimates. The model is then refit using these updated choice probabilities. This process continues until the maximum absolute change in parameters and choice probabilities is less than npl.tol (default, 1e-7), or the number of outer iterations exceeds npl.maxit (default, 25). In the latter case, a warning is produced.

If pseudo-likelihood (method="pl") is used, then pl.vcov is checked. There are four possibilities here:

• pl.vcov = FALSE (default), then no covariance matrix or standard errors are returned, only the point estimates.

• pl.vcov > 0 and phat.vcov is supplied, then phat.vcov is used to estimate the PL's covariance matrix.

• pl.vcov > 0, phat.vcov is missing, and phat is missing, then the random forest used to estimate PRhat and PFhat is bootstrapped (simple, nonparametric bootstrap) pl.vcov times.

• pl.vcov > 0, phat.vcov is missing, and phat is not missing, then an error is returned.

Value

An object of class sigfit, containing:

coefficients

A vector of estimated model parameters.

vcov

Estimated variance-covariance matrix. When pl.vcov = FALSE, this slot is omitted.

utilities

Each actor's utilities at the estimated values.

fixed.par

The fixed utilities if specified in the call.

logLik

Final log-likelihood value of the model.

gradient

First derivative values at the estimated parameters.

Phat

List of two elements

• PRhat The first stage estimates of the probability that B resists (method = "pl") or the final estimates that B resists (if method = "npl")

• PFhat The first stage estimates of the probability that A stands firms given that A challenged (method = "pl") or the final estimates that A stands firms given that A challenged (if method = "npl")

Note that PRhat will only be an equilibrium if method = "npl" and the NPL convergences

user.phat

Logical. Did the user provide phat?

start.beta

The vector of starting values used in the PL optimization.

call

The call used to produce the object.

model

The data frame used to fit the model.

method

The method ("pl" or "npl") used to fit the model.

maxlik.method

The optimization used by maxLik to fit the model.

maxlik.code

The convergence code returned by maxLik.

maxlik.message

The convergence message returned by maxLik.

Additionally, when method = "npl", the following are also included in the sigfit object.

npl.iter

Number of best response iterations used in fitting the NPL.

npl.eval

Maximum difference between the parameters at the last two NPL iterations. If the NPL method converged, this should be less than npl.tol specified in the function call.

eq.constraint

Maximum equilibrium constraint violation.

References

Casey Crisman-Cox and Michael Gibilisco. 2019. "Estimating Signaling Games in International Relations: Problems and Solutions." Political Science Research and Methods. Online First.

Jeffrey B. Lewis and Kenneth A. Schultz. 2003. "Revealing Preferences: Empirical Estimation of a Crisis Bargaining Game with Incomplete Information." Political Analysis 11:345–367.

Examples

data("sanctionsData")
f1 <- sq+cd+sf+bd ~ sqrt(senderecondep) + senderdemocracy + contig + ally -1|#SA
                    anticipatedsendercosts|#VA
                    sqrt(targetecondep) + anticipatedtargetcosts + contig + ally|#CB
                    sqrt(senderecondep) + senderdemocracy + lncaprat | #barWA
                    targetdemocracy + lncaprat| #barWB
                    senderdemocracy| #bara
                    -1#VB

## Using Nested-Pseudo Likelihood  with default first stage     
## Not run:             
fit1 <- sigint(f1, data=sanctionsData, npl.trace=TRUE)
summary(fit1)

## End(Not run)


## Using Pseudo Likelihood with user supplied first stage
Phat <- list(PRhat=sanctionsData$PRnpl, PFhat=sanctionsData$PFnpl)
fit2 <- sigint(f1, data=sanctionsData, method="pl", phat=Phat)
summary(fit2)

## Using Pseudo Likelihood with user made first stage and user covariance
## SIGMA is a bootstrapped first-stage covariance matrix (not provided)
## Not run: 
fit3 <- sigint(f1, data=sanctionsData, method="pl", phat=Phat, phat.vcov=SIGMA, pl.vcov=TRUE)
summary(fit3)

## End(Not run)

## Using Pseudo Likelihood with default first stage and 
## bootstrapped standard errors for the first stage covariance
## Not run: 
fit4 <- sigint(f1, data=sanctionsData, method="pl", pl.vcov=25) 
summary(fit4)

## End(Not run)

Summarize a fitted crisis signaling model

Description

The default method for summarizing a sigfit object.

Usage

## S3 method for class 'sigfit'
summary(object, vcov, ...)

Arguments

Details

Forms a block regression results table from fitted crisis signaling model.

Value

An object of class summary.sigfit. This object contains the information needed to print the summary

See Also

print.summary.sigfit.

Export a sigfit object into paper-ready LaTeX table

Description

This method converts one or more fitted models of class sigfit into a publication-ready LaTeX table. This conversion is performed by reformatting the models into a format that is fed to xtable, which generates the actual LaTeX code.

Usage

toLatexTable(
  ...,
  se.list,
  stars = c("default", "all", "none"),
  caption = "",
  label,
  align,
  digits = 2,
  se.note = "Standard errors in parenthesis",
  order,
  covariate.labels,
  model.names,
  dep.varnames,
  k = 1,
  print.xtable.options = list()
)

Arguments

Details

This function produces a ready-to-use LaTeX table for sigfit objects. Each column is its own model, along with model-based information. The generation of LaTeX code is done by xtable and so additional printing options can be passed via print.xtable.options. For a full list of these options see print.xtable.

See Also

xtable, print.xtable

Examples

data("sanctionsData")
f1 <- sq+cd+sf+bd ~ sqrt(senderecondep) + senderdemocracy + contig + ally -1|#SA
                    anticipatedsendercosts|#VA
                    sqrt(targetecondep) + anticipatedtargetcosts + contig + ally|#CB
                    sqrt(senderecondep) + senderdemocracy + lncaprat | #barWA
                    targetdemocracy + lncaprat| #barWB
                    senderdemocracy| #bara
                    -1#VB
## Using Nested-Pseudo Likelihood  with default first stage
## Not run: 
fit1 <- sigint(f1, data=sanctionsData, npl.trace=TRUE)

## End(Not run)

## Using Pseudo Likelihood with user made first stage
Phat <- list(PRhat=sanctionsData$PRhat, PFhat=sanctionsData$PFhat)
fit2 <- sigint(f1, data=sanctionsData, method="pl", phat=Phat)

## Using Pseudo Likelihood with default first stage and bootstrapped standard errors
## Not run: 
fit3 <- sigint(f1, data=sanctionsData, method="pl", pl.vcov=25) 

## End(Not run)

## Simple regression table
toLatexTable(fit2)

## More options: multiple models and user supplied standard errors
## Not run: 
toLatexTable(fit1, fit2, fit3,
        se.list=list(sqrt(diag(vcov(fit1))),
                     sqrt(diag(vcov(fit3))),
                     sqrt(diag(vcov(fit3)))),
        stars="all",
        caption = "Economic Sanctions",
        label = "tab:sanctions",
        model.names = c("NPL", "PL", "PL"))

## End(Not run)         
        
## Not run: 
## More options, from print.xtable including printing to a file
toLatexTable(fit1, fit2, fit3,
        caption = "Economic Sanctions",
        label = "tab:sanctions",
        model.names = c("NPL", "PL", "PL"),
        print.xtable.options=list(file="myTable.tex",
                                  booktabs=TRUE))

## End(Not run)