The odds algorithm (or Bruss 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.
Examples
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 1. (See
Compassionate use.)
Definitions
Consider a sequence of
independent events. Associate with this sequence another sequence
with values 1 or 0. Here
, called a success, stands for
the event that the kth observation is interesting (as defined by the decision maker), and
for non-interesting.
We observe independent random variables
sequentially and want to select the last success.
Let
be the probability that the kth event is interesting. Further let
and
. Note that
represents the
odds
Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.
Odds also have ...
of the kth event turning out to be interesting, explaining the name of the odds algorithm.
Algorithmic procedure
The odds algorithm sums up the odds in reverse order
:
until this sum reaches or exceeds the value 1 for the first time. If this happens at index ''s'', it saves ''s'' and the corresponding sum
:
If the sum of the odds does not reach 1, it sets ''s'' = 1. At the same time it computes
:
The output is
#
, the stopping threshold
#
, the win probability.
Odds strategy
The odds strategy is the rule to observe the events one after the other and to stop on the first interesting event from index ''s'' onwards (if any), where ''s'' is the stopping threshold of output a.
The importance of the odds strategy, and hence of the odds algorithm, lies in the following odds theorem.
Odds theorem
The odds theorem states that
# The odds strategy is ''optimal'', that is, it maximizes the probability of stopping on the last 1.
# The win probability of the odds strategy equals
# If
, the win probability
is always at least , and this lower bound is ''best possible''.
Features
The odds algorithm computes the optimal ''strategy'' and the optimal ''win probability'' at the same time. Also, the number of operations of the odds algorithm is (sub)linear in n. Hence no quicker algorithm can possibly
exist for all sequences, so that the odds algorithm is, at the same time, optimal as an algorithm.
Sources
devised the odds algorithm, and coined its name. It is also known as Bruss algorithm (strategy). Free implementations can be found on the web.
Applications
Applications reach from medical questions in
clinical trial
Clinical trials are prospective biomedical or behavioral research studies on human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel vaccines, drugs, diet ...
s over sales problems,
secretary problems,
portfolio selection, (one way) search strategies, trajectory problems and the
parking problem to problems in online maintenance and others.
There exists, in the same spirit, an Odds Theorem for continuous-time arrival processes with
independent increments such as the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
(). In some cases, the odds are not necessarily known in advance (as in Example 2 above) so that the application of the odds algorithm is not directly possible. In this case each step can use
sequential estimates of the odds. This is meaningful, if the number of unknown parameters is not large compared with the number n of observations. The question of optimality is then more complicated, however, and requires additional studies. Generalizations of the odds algorithm allow for different rewards for failing to stop
and wrong stops as well as replacing independence assumptions by weaker ones (Ferguson (2008)).
Variations
discussed a problem of selecting the last
successes.
proved a multiplicative odds theorem which deals with a problem of stopping at any of the last
successes.
A tight lower bound of win probability is obtained by .
discussed a problem of selecting
out of the last
successes and obtained a tight lower bound of win probability. When
the problem is equivalent to Bruss' odds problem. If
the problem is equivalent to that in . A problem discussed by is obtained by setting
Multiple choice problem
A player is allowed
choices, and he wins if any choice is the last success.
For classical secretary problem, discussed the cases
.
The odds problem with
is discussed by .
For further cases of odds problem, see .
An optimal strategy belongs to the class of strategies defined by a set of threshold numbers
, where