Two envelopes problem: Difference between revisions

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There have been many solutions proposed, and commonly one writer proposes a solution to the problem as stated, after which another writer shows that altering the problem slightly revives the paradox. Such sequences of discussions have produced a family of closely related formulations of the problem, resulting in voluminous literature on the subject.<ref>A complete list of published and unpublished sources in chronological order can be found in the [[Talk:Two envelopes problem/Literature|talk page]].</ref>

No proposed solution is widely accepted as definitive.<ref>{{cite journal |first=Ned |last= Markosian |title= A Simple Solution to the Two Envelope Problem |journal=Logos & Episteme |year=2011 |volume=II |issue= 3 |pages = 347–57|doi= 10.5840/logos-episteme20112318 |doi-access= free }}</ref> Despite this, it is common for authors to claim that the solution to the problem is easy, even elementary.<ref>{{cite journal |first1=Mark D |last1= McDonnell |first2= Alex J |last2= Grant |first3= Ingmar | last3 = Land |first4=Badri N |last4=Vellambi |first5=Derek |last5=Abbott |first6=Ken |last6=Lever |title= Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching |journal = [[Proceedings of the Royal Society A]] |year=2011 | doi= 10.1098/rspa.2010.0541 |volume=467 |issue= 2134 |pages=2825–2851|bibcode= 2011RSPSA.467.2825M |doi-access=free }}</ref> Upon investigating these elementary solutions, however, they often differ from one author to the next.

== Example resolution ==

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==Conditional switching==

As an extension to the problem, consider the case where the player is allowed to look in envelope A before deciding whether to switch. In this "conditional switching" problem, it is often possible to generate a gain over the "never switching" strategy", depending on the probability distribution of the envelopes.<ref name="rspa">{{cite journal |last1=McDonnell |first1=M. D. |last2=Abott |first2=D. |title=Randomized switching in the two-envelope problem |journal=[[Proceedings of the Royal Society A]] |volume=465 |issue=2111 |pages=3309–3322 |year=2009 |doi=10.1098/rspa.2009.0312 |bibcode=2009RSPSA.465.3309M |doi-access=free }}</ref>

== History of the paradox ==