Wednesday, July 27, 2011

Dividing the Loot: Sealed Bids

So, I'm currently taking Math 101 in the Summer Post-Session at East Stroudsburg University.  It's called "Excursions in Mathmatics," and it's every bit as easy for a 36-year old geek as you might think it is.  Unlike a lot of fluff classes, however, I'm actually learning a lot from a math perspective.  Already this week, we talked about voting methods and measurements of voting power.  That part was a lot of game theory basics that might actually be useful in an "end-game" scenario where there are a finite and known power blocs.  Having a relative understanding of who controls what degree of power is interesting, but relies heavily on intelligence (in the military/political sense, not the cognitive) and in our particular paradigm, the individual system used.  That's not why we're here today.  By looking at what we will be talking about tomorrow in class, I've determined we're here to talk about something much more concrete to the every-day adventurers: dividing the loot.  Here's a fable:
The Lion, the Fox, and the Ass entered into an aggreement to assist each other in the hunt.  Having secured a large booty, the Lion on their return to the forest asked the Ass to allot its due portion to each of the three partners in the treaty.  The Ass carefully divided the spoil into three equal shares and modestly requested the two others to make the first choice.  The Lion, bursting out into a great rage, devoured the Ass.  Then he requested the Fox to do him the favor to make the division.  The Fox accumulated all that they had killed into one large heap and left himself the smallest possible morsel.  The Lion said, "Who has taught you, my very excellent fellow, the art of division?  You are perfect to the fraction," to which the Fox replied, "I learned it from the Ass, by witnessing his fate.
 You can thank Aesop for that one.   "Happy is the man who learns from the misfortunes of others," he says.  Dividing treasure can sometimes be a contentious issue, particularly when it involves things like gems or magic items which are not as mathematically divisible as coinage.  My math textbook, Excursions in Modern Mathematics, gives many options for fair-division games.  The one I'd like to examine is the method of sealed bids, where players bid on objects secretly, and everyone either gets the objects they desire or a sum of money that is fair.

After a trip into the Church of Environmental Ambivalence, our intrepid adventurers (Ace, Bob, Sue, and Zed) returned with naught but boots of the elvenkind, a rope of climbing, and a wand of fire.  Three objects, four people.  Someone is getting left out, right?  Not with the sealed-bid method.  Our quartet has previously agreed to use this method and have Ostler Gundigoot (of the Inn of the Welcome Wench fame) be the arbiter.

First, each player makes a bid.  Ace, a fighter, doesn't have a whole lot of use for each item, but could handle selling them for some bread.  He bids 22,500gp for the wand, 4,000gp for the boots, and 8,000gp for the rope.  Bob is a thief, so he could use both the rope and the boots, and while he can use the wand, it's not AS valuable to him.  13,000gp for the wand, 7,500gp for the boots, and 15,000gp for the rope.  Sue, a magic-user, wants the wand and has little use for anything else.  She bids 30,000gp for the wand, 2,500gp for the boots, and 5,000 or the rope.  Zed, a fighter, thinks he could use the rope, but doesn't put much stock in the other items.  He bids 10,000gp for the rope, the boots 3,000gp, and the wand 20,000gp.  These bids are made on a sheet of paper and handed to Ostler (the DM).

Now the items are divided.  Sue gets the wand of fire, while Bob gets both the boots of the elvenkind and the rope of climbing, as the highest bidder.  Are the other members of the party going to get screwed?  No.  Here's why.

The next step is the first settlement.  Each adventurer will either owe the group or be owed by the group some money.  This comes from each character's fair-dollar share  of the loot.  Basically, what they claimed the loot was worth, divided by the number of adventurers.  Ace thought everything was worth a combined 34,500gp, Bob 35,500, Sue 37,500, and Zed 33,000.  So each character ends up with a fair-dollar share of 8,625, 8,875, 9,375, and 8,250 gold pieces, respectively.  If the character's items are worth more than the fair-dollar share they bid, they owe the group the difference.  If they received items of lesser value (or no items at all), they receive the balance of the share.  So Ace and Zed can expect a respective amount of 8,625gp and 8,250gp, while Bob owes 3125gp and Sue owes 20,625gp (daaaang!).

We're not quite done, though.  Because everyone placed different values on the items, the money in doesn't match the money out.  There is a surplus of 6,875gp!  Divide that surplus by the number of adventurers and either tack it on to their refund or deduct it from what they owe the group.  In this case, they get the credit of 1,719gp (I rounded).

In the final account, Ace gets 10,344gp, Bob gets boots of the elvenkind and rope of climbing, but owes the group 1,406gp, Sue gets her wand of fire but is on the hook to the sum of 18,906gp, and Zed gets 9,969gp.

This is probably more complex work than most players are willing to undertake, but it is the most equitable if people evaluate the objects honestly.  Let's say that Zed and Sue, for whatever reasons, swapped bids.  Now this would result in  Zed the fighter with a magic item he can't use and enough debt to bankrupt a small town.  Well, assuming Zed can get face value for the wand.  He'd sell it for 25,000gp, profiting 6094gp.  Not as much profit as he made low-balling the rest of the party.

If you look at it in terms of net profit/loss using the gold piece value of the magic items, Ace profited 10,344gp, Bob profited 13,594gp, Sue profits 6,094gp, and Zed 9,969gp.  Is it fair to Zed that Ace gets 500gp more for no reason?  Actually, yes.  By having a lower value attached to each of the items, he increased his chances of a payout, but at the cost of a reduced payout.  Is it fair that Bob profited twice as much as Sue?  Yes, again.  Sue really wanted that wand, so by overbidding its value, she was able to get it at a much higher cost, but she was able to get it.

Any thoughts?