The odds-algorithm is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the odds-strategy, and the importance of the odds-strategy lies in its optimality, as explained below.
The odds-algorithm applies to a class of problems called last-success-problems. Formally, the objective in these problems is to maximize the probability of identifying in a sequence of sequentially observed independent events the last event satisfying a specific criterion (a “specific event”). This identification must be done at the time of observation. No revisiting of preceding observations is permitted. Usually, a specific event is defined by the decision maker as an event that is of true interest in the view of “stopping” to take a well-defined action. Such problems are encountered in several situations.
Two different situations exemplify the interest in maximizing the probability to stop on a last specific event.
- Suppose a car is advertised for sale to the highest bidder (best “offer”). Let n potential buyers respond and ask to see the car. Each insists upon an immediate decision from the seller to accept the bid, or not. Define a bid as interesting, and coded 1 if it is better than all preceding bids, and coded 0 otherwise. The bids will form a random sequence of 0s and 1s. Only 1s interest the seller, who may fear that each successive 1 might be the last. It follows from the definition that the very last 1 is the highest bid. Maximizing the probability of selling on the last 1 therefore means maximizing the probability of selling best.
- A physician, using a special treatment, may use the code 1 for a successful treatment, 0 otherwise. The physician treats a sequence of n patients the same way, and wants to minimize any suffering, and to treat every responsive patient in the sequence. Stopping on the last 1 in such a random sequence of 0s and 1s would achieve this objective. Since the physician is no prophet, the objective is to maximize the probability of stopping on the last.