  The paper proposes a natural measure space of zero-sum perfect information
games with upper semicontinuous payoffs. Each game is specified by the game
tree, and by the assignment of the active player and of the capacity to each
node of the tree. The payoff in a game is defined as the infimum of the
capacity over the nodes that have been visited during the play. The active
player, the number of children, and the capacity are drawn from a given joint
distribution independently across the nodes. We characterize the cumulative
distribution function of the value $v$ using the fixed points of the so-called
value generating function. The characterization leads to a necessary and
sufficient condition for the event $v \geq k$ to occur with positive
probability. We also study probabilistic properties of the set of Player I's
$k$-optimal strategies and the corresponding plays.
