Two Envelopes Paradox Calculator: Expected Value and the Switching Fallacy

Your details

The dollar amount you found when you opened your chosen envelope. Must be positive.
Choose whether to work with a specific dollar amount or the general algebraic form of the paradox.
If you know the smaller of the two envelopes must be at least this much, enter it here. Leave at 0 if unknown. This affects the threshold strategy analysis.
Naive switching EV (the fallacy)Classic paradox
125$

The flawed expected value of the other envelope: 0.5 * 2X + 0.5 * X/2 = 1.25X. This argument is wrong.

Correct EV if you stay100$
Correct EV if you switch125$
Illusory gain from switching25$
If your envelope has the larger amount (2x)50$
If your envelope has the smaller amount (x)200$
Implied minimum envelope amount75$
Threshold strategy verdictWithout a known minimum, neither staying nor switching has an expected-value advantage. The paradox: the naive argument says switch, but the correct analysis says it does not matter.
Stay (your certain value)100
Naive switch EV (the fallacy)125
If yours is the larger envelope50
If yours is the smaller envelope200

You see $100.00. The naive argument says switching nets you $25.00 more - but this is wrong.

  • The flawed reasoning: the other envelope is either $200.00 (if yours is smaller) or $50.00 (if yours is larger). Averaging those gives $125.00, which beats $100.00.
  • The error: the variable X is used inconsistently. In one branch it represents the smaller amount; in the other, the larger. They cannot both equal your observed $100.00 simultaneously.
  • The correct view: label the envelopes m and 2m. E(yours) = E(other) = 1.5m regardless of which you hold. Switching does not improve your position in expectation.
  • Your envelope contains $100.00 with certainty. The other envelope is either $50.00 or $200.00, each with 50% probability.

Next stepIf you can set a threshold T before looking: switch only when your amount is below T. This threshold strategy provably achieves better than 50% accuracy at picking the larger envelope, even without knowing the amounts in advance.

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