Simple resolution Two envelopes problem

the total amount in both envelopes constant



c
=
3
x


{\displaystyle c=3x}

,



x


{\displaystyle x}

in 1 envelope ,



2
x


{\displaystyle 2x}

in other.

if select envelope



x


{\displaystyle x}

first gain amount



x


{\displaystyle x}

swapping. if select envelope



2
x


{\displaystyle 2x}

first lose amount



x


{\displaystyle x}

swapping. gain on average



g
=


1
2


(
x
)
+


1
2


(
−
x
)
=


1
2


(
x
−
x
)
=
0


{\displaystyle g={1 \over 2}(x)+{1 \over 2}(-x)={1 \over 2}(x-x)=0}

swapping.

swapping not better keeping. expected value



e
=


1
2


2
x
+


1
2


x
=


3
2


x


{\displaystyle \operatorname {e} ={\frac {1}{2}}2x+{\frac {1}{2}}x={\frac {3}{2}}x}

same both envelopes. no contradiction exists.