Revised to include a recursive path to
a perspective of the result from a situation that did not occur.
. . . both Jack (J) and Kirby (K)
aren’t perfect logicians
and both had suboptimal strategies.
As such, . . . .
. . . both were destined to offer a swap with those \$ amounts
and, further, both correctly calculated
their probabilisticprobabilistically expected gains, . . .
. . . which is explained wellclear in
Bubbler’s solutionexplanation.
Scenario:
J and K are wellmutually-known perfect logicians
The followingA reduced example demonstrates how this would play out
with 10 twinned envelopes that allow for clean calculations.
Each player’s model of the other player includes uncertainty
that recursively bubbles all the way through to the two possibilities that
K has \$32 or \$64 and wouldn’t offer to swap.
Had both players assumed that each other would see how that works,
those two possibilities would have influenced
the possibility that did occur.
Suppose J has the minimal \$2 and considers two models of K,
one model where K has \$1 and the other where K has \$4.
J muses that the possible K who has \$4 would also consider
two models of a possible J where J has either \$2 or \$8.
Carrying this out produces a fractal-like recursive root system of models
that overlaps itself to form an endless lattice
whose right edge terminates at one of two models of
a possible K who has either \$32 or \$64 and will not offer to swap.
J has $2
/\
/ \
/ \
/ \
$2 J's model of K has $1 $2 J's model of K has $4
\ /\
\ / \
\ / \
$2 J's model of \ / \
$1/$4 K's model of J has $2 $2 J's model of $4 K's model of J has $8
/\ /\
/ \ / \
/ \ / \
/ \ / \
... model of K has $1 ..K has $4 $2 J's model of $4 K's model of
\ /\ /\ $8 J's model of
\ / \ / \ K has $16
\ / \ / \
\ / \ / \
... of K's model of J has $2 ..J has $8 $2 J's model of $4 K's model of
/\ /\ /\ $8 J's model of
/ \ / \ / \ $16 K's model of
/ \ / \ / \ J has $32
/ \ / \ / \
... model of K has $1 ..K has $4 ..K has $16 $2 J's model of $4 K's
\ /\ /\ model of $8 J's
\ / \ / \ model of $16 K's
\ / \ / \ model of $32 J's
\ / \ / \ model of K has $64
\ / \ / \ and won't swap
\ / \ / \
... of K's model of J has $2 ..J has $8 $2 J's model of $1/$4 K's model of
/\ /\ /\ $2/$8 J's model of
/ \ / \ / \ $4/$16 K's model of
/ \ / \ / \ $8/$32 J's model of
/ \ / \ / \ $16 K's model of
/ \ / \ / \ J has $32
/ \ / \ / \
... model of K has $1 ..K has $4 ..K has $16 $2 J's model of $1/$4 K's
\ /\ /\ model of $2/$8 J's
\ / \ / \ model of $4/$16 K's
model of $8/$32 J's
\ / \ / \ model of $16 K's
model of $32 J's
\ / \ model of K has $64
and won't swap
\
In the first rightmost branch J's model of \$4 K's model of \$8 J's model
of \$16 K's model of \$32 J's model of K has \$64 and certainly won't swap.
This lattice of a tree includes half of all possibilities
for the \$ amount that J might have
while all terminal branches have K not swapping for the same reason.
The other half of all possible J amounts
are included in a similar lattice spreading from a supposed J having \$4,
where all terminal branches consider
a model of K who has \$32 and also wouldn’t swap.
Thus J has no reason to offer a swap.
The reverse reasoning from here
was presented in this answer’s original posting and admittedly parallels
Florian F’s
previously-posted reasoning).
Here are the probabilistic gains and losses
from each player’s perspective
if the players swap at every time exceptchance other when K has \$32 or \$64.
Notice how the rowsThe rows for J’s \$16 and \$32
and the rows for K’s \$32 and \$64
reflect K’s not offering a swap at \$32 or \$64.
