The office is having a penny war fundraiser.
The obvious way to win such a contest is to only contribute pennies, and only to one’s own pool, since any defensive actions, other coins to other pools, must necessarily be divided among the other pools.
If there are five pools, the expected contribution to one’s own pool is +100 points.
But a dollar in any other pool, or a quarter in each other pool, has only the effect of +25 points for one’s team.
Now, if we observe other coins in any pools, and we assume all players are rational, then we have a puzzle to explain.
I suggest two related reasons we might observe other coins:
1. Pennies are cumbersome.
2. People give what they’ve got.
Now, transforming non-pennies into pennies takes time and energy and can be a hassle.
So: value of transforming non-pennies into pennies = points from pennies – transactions costs of transformation.
V(~F) = F-c
Where F=face value of penny or non-penny, and c = transaction cost.
V(~F) = F(1/(n-1))
Where n= # of pools.
Substituting and rearranging:
We should then be able to extract the transactions costs of transforming non-pennies into pennies.
Let’s assume there are five pools.
So, if I have a dollar in my pocket, and I’m trying to decide whether to plunk it into the opponent’s bucket or to transform it into pennies, I’m really asking if it is worth 75 cents to make the transformation. If going to a bank to get pennies requires more than 75 cents of effort, people will just drop the dollar.
If I have $100, and I want to contribute it, the trip to the bank will have to be more costly than $75.
Guessing, I’d say it is roughly worth $10 to go to a bank. Therefore, I do not expect to see an bills greater than $10/.75, or $13.33
The nearest bill is a $20.
I expect to find no $20, and yet an occasional $10.