Description Usage Arguments Details Value Author(s) References See Also Examples
Fits a strategic model with two players and three terminal nodes, as in the game illustrated below in "Details".
1 2 3 4 
formulas 
a list of four formulas, or a 
data 
a data frame containing the variables in the model. 
subset 
optional logical expression specifying which observations from

na.action 
how to deal with 
link 
whether to use a probit (default) or logit link structure, 
type 
whether to use an agenterror ("agent", default) or privateinformation ("private") stochastic structure. 
startvals 
whether to calculate starting values for the optimization using statistical backwards induction ("sbi", default), draw them from a uniform distribution ("unif"), or to set them all to 0 ("zero") 
fixedUtils 
numeric vector of values to fix for u11, u13, u14, and u24
respectively. 
sdformula 
an optional list of formulas or a 
sdByPlayer 
logical: if scale parameters are being estimated (i.e.,

boot 
integer: number of bootstrap iterations to perform (if any). 
bootreport 
logical: whether to print status bar when performing bootstrap iterations. 
profile 
output from running 
method 
character string specifying which optimization routine to use
(see 
... 
other arguments to pass to the fitting function (see

The model corresponds to the following extensiveform game, described in Signorino (2003):
1 2 3 4 5 6 7 8 9  . 1
. /\
. / \
. / \ 2
. u11 /\
. / \
. / \
. u13 u14
. 0 u24

If Player 1 chooses L, the game ends and Player 1 receives payoffs of u11. (Player 2's utilities in this case cannot be identified in a statistical model.) If Player 1 chooses L, then Player 2 can choose L, resulting in payoffs of u13 for Player 1 and 0 for Player 2, or R, with payoffs of u14 for 1 and u24 for 2.
The four equations specified in the function's formulas
argument
correspond to the regressors to be placed in u11, u13, u14, and u24
respectively. If there is any regressor (including the constant) placed in
all of u11, u13, and u14, egame12
will stop and issue an error
message, because the model is then unidentified (see Lewis and Schultz
2003). There are two equivalent ways to express the formulas passed to this
argument. One is to use a list of four formulas, where the first contains
the response variable(s) (discussed below) on the lefthand side and the
other three are onesided. For instance, suppose:
u11 is a function of x1
, x2
, and a constant
u13 is set to 0
u14 is a function of x3
and a constant
u24 is a function of z
and a constant.
The list notation would be formulas = list(y ~ x1 + x2, ~ 0, ~ x3, ~
z)
. The other method is to use the Formula
syntax, with one
lefthand side and four righthand sides (separated by vertical bars). This
notation would be formulas = y ~ x1 + x2  0  x3  z
.
To fix a utility at 0, just use 0
as its equation, as in the example
just given. To estimate only a constant for a particular utility, use
1
as its equation.
There are three equivalent ways to specify the outcome in formulas
.
One is to use a numeric vector with three unique values, with their values
(from lowest to highest) corresponding with the terminal nodes of the game
tree illustrated above (from left to right). The second is to use a factor,
with the levels (in order as given by levels(y)
) corresponding to the
terminal nodes. The final way is to use two indicator variables, with the
first standing for whether Player 1 moves L (0) or R (1), the second
standing for Player 2's choice if Player 1 moves R. (The values of the
second when Player 1 moves L should be set to 0 or 1, not
NA
, in order to ensure that observations are not dropped from the
data when na.action = na.omit
.) The way to specify formulas
when using indicator variables is, for example, y1 + y2 ~ x1 + x2  0
 x3  z
.
If fixedUtils
or sdformula
is specified, the estimated
parameters will include terms labeled log(sigma)
(for probit links)
or log(lambda)
. These are the scale parameters of the stochastic
components of the players' utility. If sdByPlayer
is FALSE
,
then the variance of error terms (or the equation describing it, if
sdformula
contains nonconstant regressors) is assumed to be common
across all players. If sdByPlayer
is TRUE
, then two variances
(or equations) are estimated: one for each player. For more on the
interpretation of the scale parameters in these models and how it differs
between the agent error and private information models, see Signorino
(2003).
The model is fit using maxLik
, using the BFGS optimization
method by default (see maxBFGS
). Use the method
argument to specify an alternative from among those supplied by
maxLik
.
An object of class c("game", "egame12")
. A
game
object is a list containing:
coefficients
estimated parameters of the model.
vcov
estimated variancecovariance matrix. Cells referring to
a fixed parameter (e.g., a utility when fixedUtils
is specified) will
contain NA
s.
log.likelihood
vector of individual log likelihoods (left unsummed for use with nonnested model tests).
call
the call used to produce the model.
convergence
a list containing the optimization method used
(see argument method
), the number of iterations to convergence, the
convergence code and message returned by maxLik
, and an
indicator for whether the (analytic) gradient was used in fitting.
formulas
the final Formula
object passed to
model.frame
(including anything specified for the scale parameters).
link
the specified link function.
type
the specified stochastic structure (i.e., agent error or private information).
model
the model frame containing all variables used in fitting.
xlevels
a record of the levels of any factor regressors.
y
the dependent variable, represented as a factor.
equations
names of each separate equation (e.g., "u1(sq)", "u1(cap)", etc.).
fixed
logical vector specifying which parameter values, if any, were fixed in the estimation procedure.
boot.matrix
if boot
was nonzero, a matrix of bootstrap
parameter estimates (otherwise NULL
).
localID
an indicator for whether the Hessian matrix is negative definite, a sufficient condition for local identification of the model parameters.
The second class of the returned object, egame12
, is for use in
generation of predicted probabilities.
Brenton Kenkel (brenton.kenkel@gmail.com) and Curtis S. Signorino
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.
Curtis S. Signorino. 2003. "Structure and Uncertainty in Discrete Choice Models." Political Analysis 11:316–344.
summary.game
and predProbs
for
postestimation analysis; makeFormulas
for formula
specification.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  data("war1800")
## Model formula:
f1 < esc + war ~ s_wt_re1 + revis1  0  regime1  balanc + regime2
## ^^^^^^^^^ ^^^^^^^^^^^^^^^^^ ^ ^^^^^^^ ^^^^^^^^^^^^^^^^
## y u11 u13 u14 u24
m1 < egame12(f1, data = war1800)
summary(m1)
m2 < egame12(f1, data = war1800, link = "logit")
summary(m2)
m3 < egame12(f1, data = war1800, subset = year >= 1850)
summary(m3)
m4 < egame12(f1, data = war1800, boot = 10)
summary(m4)
summary(m4, useboot = FALSE)
## Estimating scale parameters under fixed utilities
utils < c(1, 0, 1.4, 0.1)
m5 < egame12(esc + war ~ 1, data = war1800, fixedUtils = utils)
summary(m5)
m6 < egame12(esc + war ~ 1, data = war1800, fixedUtils = utils, sdByPlayer = TRUE)
summary(m6)
## Estimating scale parameters with regressors
m7 < egame12(f1, data = war1800, sdformula = ~ balanc  1)
summary(m7)
## Using a factor outcome
y < ifelse(war1800$esc == 1, ifelse(war1800$war == 1, "war", "cap"), "sq")
war1800$y < factor(y, levels = c("sq", "cap", "war"))
f2 < update(Formula(f1), y ~ .)
m8 < egame12(f2, data = war1800)
summary(m8)

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