Random Stopping Times in Stopping Problems

A mixed strategy is a probability distribution over the set of pure strategies. When a player implements a mixed strategy, she chooses a pure strategy at the outset of the game, and follows that pure strategy all through the game.

A behavior strategy is a function that assigns a mixed action to each of the player’s information sets. When a player implements a behavior strategy, whenever the play reaches an information set the player chooses an action according to the mixed action that corresponds to that information set.

Thus, if the play crosses twice the same information set, a mixed strategy will choose the same action in both visits, while a behavior strategy will choose each time the action to play independently of past play.

The well known Kuhn’s Theorem states that in extensive-form games with perfect recall, the notions of mixed strategies and behavior strategies are the same: it is irrelevant whether the player makes her choices when the play visits the information set, or whether she makes these choices at the outset of the game, because the condition of perfect recall implies in particular that the play can cross each information set only once.

Kuhn’s Theorem holds for finite games: the depth of the game is finite, and there are finitely many actions in each decision nodes. The theorem can be extended to the case of infinite trees (countably many nodes), provided the number of actions in each decision nodes is finite.

In stopping problems the number of nodes is of the cardinality of the continuum, and therefore Kuhn’s Theorem does not apply. Stopping problems are a model that is used in mathematical finance: at every stage an agent receives payoff-relevant information, and she has to choose when to stop (when to sell stocks, or when to exercise options). The payoff is given by some stochastic process. Yuri Kifer asked me a question that boiled down to the following: in the model of stopping problems, are mixed strategies and behavior strategies equivalent? My first response was “sure”. Yuri wanted to see a proof. When I tried to write down the proof, a technical issue emerged. It required some time and the energy of three people to be able to provide a definite answer: “sure”.