So far, we have dealt with extensive form games, in which a given action profile by the players at a particular node always and unequivocally defines the next node to which this action profile leads. However, many strategic situations exist in which the development of the game does not depend solely on the actions of the players, and a certain randomness prevails over which the players have no control – either severally or jointly.

This randomness may be modeled with the aid of moves of nature at a chance node. This is a node on the game tree from which a number of branches divide; however, as distinct from nodes of the type we have dealt with hitherto, there are, at this node, no players who have to choose between the branches. Instead, there is a predefined probability at which each of the branches will be chosen. A node of this type may also be thought of as a node at which an imaginary player – “nature” – chooses how to act. Here, however, the probability of nature’s choice of each branch is given beforehand and is not a result of a conscious and mediated, intelligent choice. The “choice” of nature differs from the choices of the other players in that it is random and is not restricted to a definite choice of one of the branches.

When moves of nature are part of the game tree, the players’ strategies do not determine one unique path on the tree that leads to a particular leaf in a deterministic fashion. At every chance node, the game path splits into a number of possible continuations, in accordance with the probabilities dictated by the move of nature at that node; if the game has a number of chance nodes, the results of the lotteries at the various nodes are independent of one another. As a result, the players’ strategies determine a probability distribution over the leaves on the tree.

In the following sections we will analyze several extensive form games involving moves of nature.