The model corresponds to the following extensive-form game, described in
Signorino (2003):
. 1
. /\
. / \
. / \ 2
. u11 /\
. / \
. / \
. u13 u14
. 0 u24If 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 left-hand side and the
other three are one-sided. For instance, suppose:
- u11 is a function of
x1,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 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 non-constant 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 with the BFGS method, using the function
maxBFGS from the maxLik package. For the sake of speed
and stability, the fitting procedure typically uses the gradient of the
log-likelihood. However, if fixedUtils or sdformula is
specified, the gradient is not used.