We present an example of a one-stage three player game of incomplete information played on a sequence space {0, 1} Z such that the players' locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1} Z , each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g. all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is as much a philosophical issue as a mathematical one. We consider the double-shift Bernoulli probability space B Z 2 , where T 1 is the shift in the first coordinate, T 2 is the shift in the second coordinate, and x i,j is the i, j-coordinate of x ∈ B Z 2 . Let C be a compact and convex set with compact subsets)) in A x 0,0 for all x ∈ B Z 2 and that the inability of measurable functions to satisfy this property (in expectation) is bounded below by a positive constant dependent on the sets (A b | b ∈ B). We give an example of a one-stage zero-sum game played on B Z 2 that would not have a value (but would have equilibria!) if this conjecture were valid.

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We consider a stochastic differential equation that is controlled by means of an additive finite-variation process. A singular stochastic controller, who is a minimizer, determines this finite-variation process, while a discretionary stopper, who is a maximizer, chooses a stopping time at which the game terminates. We consider two closely related games that are differentiated by whether the controller or the stopper has a first-move advantage. The games' performance indices involve a running payoff as well as a terminal payoff and penalize control effort expenditure. We derive a set of variational inequalities that can fully characterize the games' value functions as well as yield Markovian optimal strategies. In particular, we derive the explicit solutions to two special cases and we show that, in general, the games' value functions fail to be C 1 . The nonuniqueness of the optimal strategy is an interesting feature of the game in which the controller has the first-move advantage.

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