Opportunity Cost, Again, Rev 2

We have two jars. The left one is clear and contains $100 in $1 dollar bills. The right one is opaque and contains between 1 and 99 $1 bills with a uniform probability of 1 in 99 for each possible count.

A fair coin is flipped to choose between the left jar (heads) and the right jar (tails).

If the coin comes up tails, then the opportunity cost in dollars of the choice of the right hand jar is $100, the contents of the left hand jar.

If the coin comes up heads, then the opportunity cost in dollars of the choice of the left hand jar is between $1 and $99, the contents of the right hand jar.

Overall, the recipient of the money will be better off statistically with a larger numbers of dollars (in the range of 1-99) in the opaque right hand jar.

If this is true, and it is, why is an increased benefit to the recipient associated with an increased opportunity COST?

One way to make opportunity cost and benefit work in the same direction is to treat an opportunity cost not as the result of a particular choice, but rather as the sacrifice that must be made as a result of a requirement that a choice must actually be made.

If no choice is required to be made, then the recipient can keep both jars.

Then the actual sacrifice is always the jar that cannot be retained. This is always consistent with the normal formulation of opportunity cost.

addendum :

This may be more clear if we assume that both jars contain $100. There is then no possible cost associated with a particular choice, but the opportunity cost will always be $100, the sacrifice involved with having to choose.

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