Monday, December 24, 2012

A Talmudic Shipping Problem

We don't do Christmas but I do encourage my kids to take advantage of Christmas specials. So when ThinkGeek offered a 20% discount on orders over $100 my kids knew they had an opportunity. 

It turns out that working out what you might want is quite tricky. This is because ThinkGeek has clearance items like Giant Inflatable Robot Fists that apparently no one other than my son wanted. But they eventually did it and their cart looked like this.

They came to me to enter credit card information.

"OK but I need to know what I am taking out of each of your bank accounts?"

"Well, just what each of us paid for what we wanted."

"Yes, but what about shipping? That's $42.84 to Canada. What are each of you paying for that?"

"Well, we will divide it by three."

"Then your 8 year old sister, who is ordering one Hair Bow and a few Guitar picks will end up paying $14.28 for that alone when her stuff only cost $3.98. You need to come up with something fairer."

And then the problem ensued. What was fair? They would allocate the shipping costs using some other dimension. 

Child No.1 argued that they divide it on the basis of the number of items ordered. Child No.2 argued that they divide it on the basis of the price paid. So they did the maths. This is what they came back with.

Well, Child No.1 was shrewd. While dividing on items had her pay much more than Child No.2 for shipping it was the better deal. It was also better for Child No.2. However, the amount for Child No.3 was still above what I would have regarded as fair. So they proposed to use the price-based cost allocation.

"I'm still not sure that is fair. Your sister is order very light weight things. Surely they don't contribute as much to shipping?"

"Well, how will we tell? We don't know what ThinkGeek is doing."

"You'll have to work that out."

So they went back and decided to look at what it would cost to ship each of these separately. For Child No.1 it was $29, for Child No.2 it was $31 and for Child No.3 it was $6.95. It turns out that weight was a consideration but there was some fixed component for each order -- perhaps for the box. The two eldest then tried to assert that the original $1.67 allocation to their sister was fair.

"But is she really causing that amount of cost? What happens to shipping if you just leave off her order?"

With a sigh, they went back and did that run. It turned out that it made no difference to shipping cost.

"So shouldn't she pay zero then?"

"No, she is still getting a benefit of sharing our box. What is more, she is getting a discount. That wouldn't happen if not for us."

"Then how much should she pay?"

"She should pay an amount equal to the discount she is getting. She should pay $1."

"But that means you two get her discount. Why should you get all of that?"

I argued, on Child No.3's behalf, that she should only have to pay 50 cents and she should share half her discount with her siblings. Child No.1 dug her heels in and argued with Child No.2 who want to accept the deal. 

That took a little while but eventually they came back and argued Child No.3 should share 50 cents of her discount and contribute 50 cents towards 'the box.' While it was the same as the previous accepted deal, it was better argued so I accepted that.

Many will recognise this as a classic cost allocation problem. Eventually, with prompting, the kids ended up with a solution that was in the 'core.' This is something that Talmudic scholars had discovered centuries ago

It was then time to finally order the goods. Unfortunately, this negotiation (and maths exercise) had taken too long. By the time it was resolved an hour or so ago, ThinkGeek had ended their promotion. The discount and an important basis for the whole exercise was gone. It turns out that this was one of those time sensitive negotiations but we didn't quite know it. Another lesson to accompany the maths and social choice of the day.

I persuaded the kids that they might be better waiting until December 26. There was argument over that. I fear this issue will last all week.