<P> The unexpected hanging paradox is a paradox related to backward induction . Suppose a prisoner is told that she will be hanged sometime between Monday and Friday of next week . However, the exact day will be a surprise (i.e. she will not know the night before that she will be executed the next day). The prisoner, interested in outsmarting her executioner, attempts to determine which day the execution will occur . </P> <P> She reasons that it cannot occur on Friday, since if it had not occurred by the end of Thursday, she would know the execution would be on Friday . Therefore, she can eliminate Friday as a possibility . With Friday eliminated, she decides that it cannot occur on Thursday, since if it had not occurred on Wednesday, she would know that it had to be on Thursday . Therefore, she can eliminate Thursday . This reasoning proceeds until she has eliminated all possibilities . She concludes that she will not be hanged next week . </P> <P> To her surprise, she is hanged on Wednesday . She made the mistake of assuming that she knew definitively whether the unknown future factor that would cause her execution was one that she could reason about . </P> <P> Here the prisoner reasons by backward induction, but seems to come to a false conclusion . Note, however, that the description of the problem assumes it is possible to surprise someone who is performing backward induction . The mathematical theory of backward induction does not make this assumption, so the paradox does not call into question the results of this theory . Nonetheless, this paradox has received some substantial discussion by philosophers . </P>

When can backward induction be used to arrive at the equilibrium for a​ game in the case​ of