  Bayesian game is a strategic decision-making model where each player's type
parameter characterizing its own objective is private information: each player
knows its own type but not its rivals' types, and Bayesian Nash equilibrium
(BNE) is an outcome of this game where each player makes a strategic optimal
decision according to its own type under the Nash conjecture. In this paper, we
advance the literature by considering a generalized Bayesian game where each
player's action space depends on its own type parameter and the rivals'
actions. This reflects the fact that in practical applications, a firm's
feasible action is often related to its own type (e.g. marginal cost) and the
rivals' actions (e.g. common resource constraints in a competitive market).
Under some moderate conditions, we demonstrate existence of continuous
generalized Bayesian Nash equilibria (GBNE) and uniqueness of such an
equilibrium when each player's action space is only dependent on its type. In
the case that each player's action space is also dependent on rivals' actions,
we give a simple example to show that uniqueness of GBNE is not guaranteed
under standard monotone conditions. To compute an approximate GBNE, we restrict
each player's response function to the space of polynomial functions of its
type parameter and consequently convert the GBNE problem to a stochastic
generalized Nash equilibrium problem (SGNE). To justify the approximation, we
discuss convergence of the approximation scheme. Some preliminary numerical
test results show that the approximation scheme works well.
