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Applications: Add secretary problem illustration

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{{short description|Method of computing optimal strategies for last-success problems}}

In [[decision theory]], 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 <math>n</math> 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 <math>n</math> 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 <math>n</math> [[independent events]]. Associate with this sequence another sequence of independent events <math> I_1,\, I_2,\, \dots ,\, I_n</math> with values 1 or 0. Here <math> \,I_k =1</math>, called a success, stands for
the event that the kth observation is interesting (as defined by the decision maker), and <math>\, I_k=0</math> for non-interesting.
These random variables <math> I_1,\, I_2,\, \dots ,\, I_n </math> are observed sequentially and the goal is to correctly select the last success when it is observed.

Let <math> \,p_k = P( \,I_k\,=1)</math> be the probability that the kth event is interesting. Further let
<math> \,q_k = \,1- p_k </math> and <math> \,r_k = p_k/q_k</math>. Note that <math> \,r_k</math> represents the [[odds]] 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

:<math> r_n + r_{n-1} + r_{n-2}\, +\cdots, \, </math>

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

:<math> R_s = \,r_n + r_{n-1} + r_{n-2} + \cdots + r_s. \, </math>

If the sum of the odds does not reach 1, it sets ''s'' = 1. At the same time it computes

:<math> Q_{s}=q_n q_{n-1}\cdots q_s.\,</math>

The output is

# <math>\,s</math>, the stopping threshold
# <math>\,w = Q_s R_s</math>, 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 <math>w= Q_s R_s</math>
# If <math>R_s \ge 1</math>, the win probability <math>w</math> is always at least {{math|1=1/[[e (mathematical constant)|''e'']] = 0.367879...}}, 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==
{{harvnb|Bruss|2000}} 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==
{{secretary_problem.svg}}
Applications reach from medical questions in [[clinical trial]]s over sales problems, [[secretary problems]], [[portfolio (finance)|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]] ({{harvnb|Bruss|2000}}). 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 estimate]]s 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 {{harv|Ferguson|2008}}.

== Variations ==
{{harvnb|Bruss|Paindaveine|2000}} discussed a problem of selecting the last <math> k </math> successes.

{{harvnb|Tamaki|2010}} proved a multiplicative odds theorem which deals with a problem of stopping at any of the last <math> \ell </math> successes.
A tight lower bound of win probability is obtained by {{harvnb|Matsui|Ano|2014}}.

{{harvnb|Matsui|Ano|2017}} discussed a problem of selecting <math>k </math> out of the last <math> \ell </math> successes and obtained a tight lower bound of win probability. When <math> \ell= k = 1,</math> the problem is equivalent to Bruss' odds problem. If <math>\ell= k \geq 1,</math> the problem is equivalent to that in {{harvnb|Bruss|Paindaveine|2000}}. A problem discussed by {{harvnb|Tamaki|2010}} is obtained by setting <math> \ell \geq k=1. </math>

=== Multiple choice problem ===
A player is allowed <math>r</math> choices, and he wins if any choice is the last success.
For classical secretary problem, {{harvnb|Gilbert|Mosteller|1966}} discussed the cases <math>r=2,3,4</math>.
The odds problem with <math> r=2, 3 </math> is discussed by {{harvnb|Ano|Kakinuma|Miyoshi|2010}}.
For further cases of odds problem, see {{harvnb|Matsui|Ano|2016}}.

An optimal strategy for this problem belongs to the class of strategies defined by a set of threshold numbers <math>(a_1, a_2, ..., a_r)</math>, where <math>a_1 > a_2 > \cdots > a_r</math>.

Specifically, imagine that you have <math>r</math> letters of acceptance labelled from <math>1</math> to <math>r</math>. You would have <math>r</math> application officers, each holding one letter. You keep interviewing the candidates and rank them on a chart that every application officer can see. Now officer <math>i</math> would send their letter of acceptance to the first candidate that is better than all candidates <math>1</math> to <math>a_i</math>. (Unsent letters of acceptance are by default given to the last applicants, the same as in the standard secretary problem.)

When <math>r=2 </math>, {{harvnb|Ano|Kakinuma|Miyoshi|2010}} showed that the tight lower bound of win probability is equal to <math> e^{-1}+ e^{-\frac{3}{2}}. </math>
For general positive integer <math>r</math>, {{harvnb|Matsui|Ano|2016}} proved that the tight lower bound of win probability is the win probability of the [[Secretary problem#Pick the best, using multiple tries|secretary problem variant where one must pick the top-k candidates using just k attempts]].

When <math> r=3,4,5 </math>, tight lower bounds of win probabilities are equal to <math> e^{-1}+ e^{-\frac{3}{2}}+e^{-\frac{47}{24}} </math>, <math> e^{-1}+e^{-\frac{3}{2}}+e^{-\frac{47}{24}}+e^{-\frac{2761}{1152}} </math> and <math> e^{-1}+e^{-\frac{3}{2}}+e^{-\frac{47}{24}}+e^{-\frac{2761}{1152}}+e^{-\frac{4162637}{1474560}}, </math> respectively.

For further numerical cases for <math>r=6,...,10</math>, and an algorithm for general cases, see {{harvnb|Matsui|Ano|2016}}.

== See also ==
* [[Odds]]
* [[Clinical trial]]
* [[Expanded access]]
* [[Secretary problem]]

== References ==
*{{cite journal |last1=Ano |first1=K.|first2=H. |last2=Kakinuma |first3=N. |last3=Miyoshi |title=Odds theorem with multiple selection chances |journal=Journal of Applied Probability |volume=47 |year=2010 |issue=4|pages=1093–1104 |doi=10.1239/jap/1294170522|s2cid=17598431|doi-access=free }}
* {{cite journal | last=Bruss | first=F. Thomas | authorlink=Franz Thomas Bruss | title=Sum the odds to one and stop | journal=The Annals of Probability | publisher=Institute of Mathematical Statistics | volume=28 | issue=3 | year=2000 | issn=0091-1798 | doi=10.1214/aop/1019160340 | pages=1384–1391 | doi-access=free }}
** "[https://www.cambridge.org/core/journals/journal-of-applied-probability/article/note-on-a-lower-bound-for-the-multiplicative-odds-theorem-of-optimal-stopping/B759B6E4A9D83DB84D6EE1B4C827785B A note on Bounds for the Odds Theorem of Optimal Stopping]", ''[[Annals of Probability]]'' Vol. 31, 1859–1862, (2003).
** "[https://web.archive.org/web/20230409100437/https://www.ems-ph.org/journals/newsletter/pdf/2006-12-62.pdf The art of a right decision]", ''Newsletter of the [[European Mathematical Society]]'', Issue 62, 14–20, (2005).
*{{SfnRef inline|Ferguson|2008}}[[Thomas S. Ferguson|T. S. Ferguson]]: (2008, unpublished){{sfn whitelist|CITEREFFerguson2008}}
*{{cite journal |first1=F. T. |last1=Bruss |first2=D. |last2=Paindaveine |title=Selecting a sequence of last successes in independent trials |journal=Journal of Applied Probability |volume=37 |pages=389–399 |year=2000 |issue=2 |doi=10.1239/jap/1014842544 |url=https://mpra.ub.uni-muenchen.de/21166/1/MPRA_paper_21166.pdf}}
*{{Cite journal|last1=Gilbert |first1=J |last2=Mosteller |first2=F |title= Recognizing the Maximum of a Sequence |journal=Journal of the American Statistical Association |volume=61 |pages=35–73 |year=1966 |issue=313 |doi=10.2307/2283044 |jstor=2283044 }}
*{{cite journal |last1=Matsui |first1=T |last2=Ano | first2=K |title=A note on a lower bound for the multiplicative odds theorem of optimal stopping | journal=Journal of Applied Probability |volume=51 |pages=885–889 |year=2014 |issue=3 |doi=10.1239/jap/1409932681 |doi-access=free }}
*{{cite journal |last1=Matsui |first1=T |last2=Ano | first2=K |title=Lower bounds for Bruss' odds problem with multiple stoppings |journal=[[Mathematics of Operations Research]] |volume=41 |pages=700–714 |year=2016 |issue=2 |doi=10.1287/moor.2015.0748 |arxiv=1204.5537 |s2cid=31778896 }}
*{{cite journal |last1=Matsui |first1=T |last2=Ano | first2=K |title=Compare the ratio of symmetric polynomials of odds to one and stop |journal=Journal of Applied Probability |volume=54 |pages=12–22 |year=2017 |doi=10.1017/jpr.2016.83 |s2cid=41639968 |url=http://t2r2.star.titech.ac.jp/cgi-bin/publicationinfo.cgi?q_publication_content_number=CTT100773751 }}
* Shoo-Ren Hsiao and Jiing-Ru. Yang: "[https://www.cambridge.org/core/journals/journal-of-applied-probability/article/selecting-the-last-success-in-markovdependent-trials/09D192C389B4BA5D4E2CA678E11B31CC Selecting the Last Success in Markov-Dependent Trials]", ''[[Journal of Applied Probability]]'', Vol. 93, 271–281, (2002).
*{{cite journal |last1=Tamaki |first1=M | title=Sum the multiplicative odds to one and stop |journal=Journal of Applied Probability |volume=47 |pages=761–777 |year=2010 |issue=3 |doi=10.1239/jap/1285335408 |s2cid=32236265 |doi-access=free }}
* Mitsushi Tamaki: "[https://www.cambridge.org/core/journals/journal-of-applied-probability/article/optimal-stopping-on-trajectories-and-the-ballot-problem/4DEDE2A52A75420648C0DF39797E4A64 Optimal Stopping on Trajectories and the Ballot Problem]", ''[[Journal of Applied Probability]]'' Vol. 38, 946–959 (2001).
* E. Thomas, E. Levrat, B. Iung: "[https://hal.archives-ouvertes.fr/hal-00149994/document L'algorithme de Bruss comme contribution à une maintenance préventive]", ''[[Sciences et Technologies de l'automation]]'', Vol. 4, 13-18 (2007).
{{reflist}}

==External links==
* Bruss Algorithmus http://www.p-roesler.de/odds.html

{{DEFAULTSORT:Odds Algorithm}}
[[Category:Optimization algorithms and methods]]
[[Category:Statistical algorithms]]
[[Category:Optimal decisions]]

Odds algorithm - Revision history

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