The Marriage ProblemFinding an Optimal Stopping Procedure Examples of Optimal Stopping ProceduresThere are many legitimate real life applications where we may want to find an optimal stopping procedureSelling a house !i"en a series of offers# how many should you consider before accepting an offer$Computer Chess %ow many positions should the computer consider before ma&ing a mo"e$Finance %ow long do you wait to exercise a financial option between the time of purchase and the expiration date' Examples of Optimal Stopping Procedures(hat you are about to see is not really a legitimate application of an optimal stopping procedure)*t+s ,ust simpler to state this problem# it+s &ind of fun in a sort of per"erse way# its mathematically interesting# and it has a simple and surprising answer' Statement of the Problem -ou will meet many people in your life and suppose you are loo&ing for the .best/ mate' 0ssume you &now you will meet n potential mates and could actually ran& them after you ha"e met them all# on a scale from 1 to n# where 1 is the best' This is an extreme o"ersimplification 2ob"iously not "ery realistic 2 please do not try this at home 2 but it does lead to an interesting mathematical 3uestion %ow do you pic& the best one$ Statement of the ProblemSuppose that you can+t wait to see them all'Suppose that you ha"e to ma&e a decision at some point# not &nowing whether the ones you see in the future may get a higher rating than those you ha"e seen so far' Example with 4 potential mates For example# suppose you will meet 4 potential mates in your life' -ou don+t &now what order they will enter your life# suppose any order is e3ually li&ely' Further# assume you will meet them one at a time# and if you let them go# you can not call them bac&' There are 4) 5 46761 5 8 different possible permutations174147714741417471Suppose your selection rule is to pic& the first person you meet'This rule would wor& on the first 7 possible permutations# meaning your probability of success would be 798 5 194 5 44: Example with 4 potential mates0nd if your rule was simply to pic& the second person# you would ha"e the same probability of selecting the best one# which is 194'0gain# your probability of selecting the best person would be 194 if you used the rule .pic& the third person you meet/'%ow about a new rule to impro"e your chances$174147714741417471Consider this rule:Let the first one go and pick the first one you see better than the one you let go. (hat are your chances now$*f there are 8 different possible permutations# in which of those 8 possibilities does this rule lead you to pic& the best mate$*t doesn+t wor& in the first two permutations listed abo"e# but it would wor& in the cases 714# 741# and 417'(hy won+t it wor& in the last case 471$ Example with 4 potential mates174147714741417471The rule ;called a stopping rule in decision theory