The merchant Harout has just acquired a certain number of very valuable gold coins. He wants to send them from Yerevan to Zurich. Doing so by land requires passing through many unfriendly countries. Doing so by insured carrier is safe but requires paying 50 percent in insurance and shipping.

He discusses all this over tea with a rather untrustworthy but very capable smuggler who calls himself Jack. Jack says, "No matter how many coins you send with me in a shipment, I'll take a commission of only one gold coin for that shipment."

Harout smiles then breathes deeply. "Both of us know that you may just steal the whole shipment from me. I promise you one thing: if you ever steal from me, I'll never do business with you again."

"True, I have a bad reputation," Jack nods. "But then again, it's either me or the insurance premiums."

"Right," Harout says. "Let's start with the assumption that you know how many coins I have. Given your spy network, I must assume that much. You also know that I'm unlikely to get any more given the recent government crackdown. If I have one coin left, your commission will consume it entirely. Also, if I send my last two coins with you, you'll steal those for sure. You have no incentive to do otherwise and I know you to be entirely rational."

Warm-up: What if Harout has four coins? Should he send any with Jack?

Solution to warm-up: If Harout sends all four, Jack will steal them. If he sends three, Jack will steal them because Jack stands to gain more by stealing them than by being honest. If Harout sends two at first, then Jack, knowing that Harout will use the insured carrier for the last two, might as well steal those first two.

Question 1. How many coins must Harout have for it to be worthwhile to use Jack at all?

Harout and Jack worked out the above question together. After seeing what would happen if they both played without any trust, Harout makes a proposal, "Look Jack, you know I'm an honest man. I suggest the following protocol. I will divide up my coins into a series of lots whose sizes I will tell you from the beginning. If you are honest for the first k lots, for k >= 0, I will send you the next lot. But, if you steal even once, I will use the insured carrier from then on. Note that this means that I will send even the last lot with you if you've been honest with me up to that point. Who knows? Maybe you have changed your ways."

Jack laughed. "I didn't want to claim to have become as honorable a man as you, but now that you mention it, my children have been pressing me to live by the Zoroastrian virtues. That may be too much to ask, but if I can profit as much from being honest as being dishonest, I will be honest. Further, I know you have earned your reputation for honor."

Suppose that Harout will live up to the protocol he promises and Jack knows this. Suppose that Jack will prefer honesty to dishonesty if his profit remains the same. If not, he will choose the most profit every time.

Question 2. For 10 coins, what would be the lot sizes that would allow Harout to get as many coins to Zurich as possible? If done that way, how many will Harout get to Zurich, how many will Jack get by payment or theft, and how many will go via the insured carrier?

Question 3. What about 20 coins?

Question 4. What about 50 coins?

Question 5. How would you answer questions 2, 3, and 4 if Jack were less influenced by his children than by a long-lived insult he feels he has suffered at the hands of Harout? So, given equal profits, he would steal from Harout rather than give Harout the satisfaction of greater deliveries. Assume Harout knows this.