Strategic model with 2 players, 3 terminal nodes
Fits a strategic model with two players and three terminal nodes, as in the game illustrated below in "Details".
egame12(formulas, data, subset, na.action, link = c("probit", "logit"), type = c("agent", "private"), startvals = c("sbi", "unif", "zero"), fixedUtils = NULL, sdformula = NULL, sdByPlayer = FALSE, boot = 0, bootreport = TRUE, profile, method = "BFGS", ...)
- a list of four formulas, or a
Formulaobject with four right-hand sides. See "Details" and the examples below.
- a data frame containing the variables in the model.
- optional logical expression specifying which observations from
datato use in fitting.
- how to deal with
data. Defaults to the
- whether to use a probit (default) or logit link structure,
- whether to use an agent-error ("agent", default) or private-information ("private") stochastic structure.
- 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")
- numeric vector of values to fix for u11, u13, u14, and u24
NULL(the default) indicates that these should be estimated with regressors rather than fixed.
- an optional list of formulas or a
Formulacontaining a regression equation for the scale parameter. The formula(s) should have nothing on the left-hand side; the right-hand side should have one equation if
FALSEand two equations if
TRUE. See the examples below for how to specify
- logical: if scale parameters are being estimated (i.e.,
NULL), should a separate one be estimated for each player? This option is ignored unless
- integer: number of bootstrap iterations to perform (if any).
- logical: whether to print status bar when performing bootstrap iterations.
- output from running
profile.gameon a previous fit of the model, used to generate starting values for refitting when an earlier fit converged to a non-global maximum.
- character string specifying which optimization routine to use
- other arguments to pass to the fitting function (see
The model corresponds to the following extensive-form game, described in Signorino (2003):
. 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
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 left-hand side and the
other three are one-sided. For instance, suppose:
- u11 is a function of
x2, and a constant
- u13 is set to 0
- u14 is a function of
x3and a constant
- u24 is a function of
zand 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
left-hand side and four right-hand 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
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
na.action = na.omit.) The way to specify
when using indicator variables is, for example,
y1 + y2 ~ x1 + x2 | 0
| x3 | z.
sdformula is specified, the estimated
parameters will include terms labeled
log(sigma) (for probit links)
log(lambda). These are the scale parameters of the stochastic
components of the players' utility. If
then the variance of error terms (or the equation describing it, if
sdformula contains non-constant regressors) is assumed to be common
across all players. If
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
An object of class
- estimated parameters of the model.
- estimated variance-covariance matrix. Cells referring to
a fixed parameter (e.g., a utility when
fixedUtilsis specified) will contain
- vector of individual log likelihoods (left unsummed for use with non-nested model tests).
- the call used to produce the model.
- a list containing the optimization method used
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.
- the final
Formulaobject passed to
model.frame(including anything specified for the scale parameters).
- the specified link function.
- the specified stochastic structure (i.e., agent error or private information).
- the model frame containing all variables used in fitting.
- a record of the levels of any factor regressors.
- the dependent variable, represented as a factor.
- names of each separate equation (e.g., "u1(sq)", "u1(cap)", etc.).
- logical vector specifying which parameter values, if any, were fixed in the estimation procedure.
bootwas non-zero, a matrix of bootstrap parameter estimates (otherwise
- an indicator for whether the Hessian matrix is negative definite, a sufficient condition for local identification of the model parameters.
c("game", "egame12"). A
gameobject is a list containing:
egame12, is for use in generation of predicted probabilities.
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.
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)