The actual problem in question is as follows (and perhaps I should have just said as much earlier, but I liked keeping things general) –
There’s a scientific apparatus that needs someone watching it 24/7 in case something goes wrong, but nothing ever does. The person is basically stuck in a room with nothing to do, but if something did manage to go wrong, it would really suck if no one was there. So, people need to watch it. The workforce is a group of ~100 graduate students, post doctoral scholars, and professors of various rank. The whole team decides that everyone is created equal when it comes to this unwanted task (no seniority) which will continue for the lifetime of the experiment (a few years).
To see who isn’t pulling their weight, you could just give out one point for any shift taken, and sort the list of people by points accumulated (perhaps with a decay or occasional reset on their points). But, someone who takes a dirty night shift ought to get more than one point. The question arises: How much should each shift be worth?
Actual money can’t get involved, obviously. The role of a (points-based) market would be to have people self-value the shifts on-the-fly.
I think the auction idea gets the system going, by establishing the value of the commodities in the first place. A system of trade still needs to be specified, though. Also, the auction would have to start out at a nonzero price for far-future shifts, since (I think) you wouldn’t want a pressure motivating people to procrastinate on shift sign-ups. So I could see empty shifts being worth something like a+f(t), where t is the time until the shift starts (relative to when you claim the shift), f(t)–>\infty as t–>0, and f(t) is monotonic. This is unattractive since the form of f(t) and the baseline “a” have to be specified a priori, rather than just falling out naturally from a market. But even allowing that feature, once shifts are claimed, there still ought to be a mechanism for exchange. Would exchange only be allowed at the purchase value? Or, could it be given away for more than it was worth at purchase? The seller shouldn’t really have the right to increase the value of shifts arbitrarily, lest inflation run rampant.
It would be a nice feature is the system didn’t allowed meanness. For example, some professors are limited by class schedules and travel constraints, so you wouldn’t want a savvy grad student signing up for all the Tuesday and Thursday shifts, and then selling them for mucho points to professors who can’t show up on Mon/Wed/Fri. But it may be the case that such game-ability is inevitable in a market scenario.
Apologies for the vague statement of the problem throughout. Thanks for the thoughts thus far. (This is just an extension of a rambling dinner conversation, by the way – no one is actually going to get subjected to any of this complexity.)
Sure. Shift selection can be done by both advance planning and seniority.
In a nutshell, they pick shifts for next month. Assuming everyone has to work 5 8 hour shifts a week x 100 employees there are 500 shifts to be filled per week.
Every week an employee accumulates 10 “duty dollars” +1 dollar for every 3 months of seniority.
When it comes time to pick shifts, they bid for the priveledge of filling in their 5 shifts first. Senior employees will generally win but a newb could save up to make sure he gets certain days off that are important to him. You don’t care about when you work, take your chances on drawing names for the ties and don’t cares.
what do you do with the excess duty dollars? Buy some prizes and have an auction every 6 months or so… Let some of the prizes be a paid day off, a small raise, or the parking spot by the door for 6 months, movie tix, restarunt gift cards, whatever.
This will create an environment where those with seniority get to have their choice of work, those without still bid for priveledges in their experience/duty dollar peers. Those who take the less preferred shifts and save their duty dollars compete for some nice prizes. A senior guy who saves his duty dollars would clean house at the auction…guess who I was
I just figured out how to say something I was trying to say earlier:
Each person’s utility has to stay within the system. Outside value can’t enter, and inside value can’t leave. That is, the currency/points/etc. that they value in the problem has to be an entirely internal concept, lest the (externally) wealthy value shifts less than the (externally) poor.
This sounds like a variant of the Marriage problem. Basically, each employee can submit a ranked ordering of which shifts they prefer and there is a deterministic algorithm that can figure out the optimal matching. The matching is such that no two employees can switch shifts such that they will both be better off.
I like drachillix’s method, which reminds me a little bit of how World of Warcraft raids divide up scarce loot: a market of sorts called Dragon Kill Points. Seniority in the guild earns you some nominal salary of DKP each week, and participation in successful raids earns you points (typically based on the amount of treasure that raid earned for the guild). Points may be docked for things like showing up late to a raid.
Once the raids for a week are done, the guild has an auction where each member may bid on the treasure that dropped that week. If Al, Bob, and Carol (and 17 other guild members) all helped kill Gigantor (who is worth just-call-it-200-points), then each of them gets 10 points. Al has 15 left over from last week because none of the gear he wanted was available. Bob is relatively senior and gets a 3-point bonus to any raid he attends, so he has 13 total. The two items that Gigantor dropped are both very desirable to Bob; Al wants one very badly, and Carol wants the other very badly. Al and Bob each commit to purchasing the first item. Bob can only bid 13 points, so Al has to pay 14 points to get the item. Similarly, Bob has to pay 11 points to beat Carol in the second auction.
In this way, people with seniority or lots of rough service get first choice of shifts, but they have to decide if narrowly winning a particular shift is worth going to the back of the line.
If you award DKP for seniority, weekend shifts, and the undesirable overnight shifts, then the most desirable shifts will become expensive.
To meet the restriction of no outside payments, I’ve removed the middle part of Chronos’ post. This is just an Edgeworth box story. There is some initial allocation. People offer to make trades if they will. Each trade is Pareto superior to the former allocation. With largish numbers the core shrinks to the Pareto optimum. If shift length is fixed tangencies won’t quite be achieved.
If a person can’t trade away an undesirable shift they must be the person who dislikes it least - or they would be able to make a trade. The process of a competitive market with fixed preferences and a given initial allocation of endowments reveals preferences.
There may be some mathematical approach, but I doubt it. I have spent my entire career in shift work (Emergency Department) and observed with some amusement the various strategies. I’m a Doc and not an academician, so I can’t help with the math, but here are my observations from real life that you might want to try mathemetizing:
I assume there is a total pool of money–a budget–to fill every shift. There is therefore an average wage/shift. Let’s call it $1,000/shift to make it easy. (I’m an ED Doc, remember…) So if there are 100 shifts/month, your total budget is $100k/month.
The simplest practical approach is to pay a differential which entices people to trade the nuisance of working more hours for the nuisance of working less desirable shifts (assuming everyone is targeting the same total yearly salary). The worker preferences decide this, and not mathematics. On average, weekday day shifts are preferred; for the sake of simplicity let’s say 40% of shifts (Grp 1) are great times, 40% are less-preferred (Grp2) and 20% are least preferred (Grp3). The first strategy is to assign shift differentials that smooth out the natural tendency to work only the most desirable shifts. Let’s pay Grp 1 shifts $700, Grp 2 $900 and Grp 3 $1200. We’ll keep back $12,000/month for bonus pay…(you will need to check all my arithmetic).
The manager’s problem is to get all the shifts covered. Generally the two approaches are to either assign them all equally and let the employees trade among themselves, or let the employees assign themselves. The latter method requires that some sort of structure be put in place. This can be a group meeting or a seniority preference or whatever, but there must be a fairly complicated structure since those who choose first constrain the choices of those who choose next.
At the end of the first round of assignments, there will be odd shifts left over. Use the $12,000 to assign bonuses for working those shifts. In my opinion bidding for shifts (which has become popular at my Hospital) works well for covering the odd shifts, but you can’t really use it to generate the entire schedule. You have to have a fairly automatic mechanism for that.
There are so many complexities around this it would take me all day to elaborate on them. In the end, you need a system which assigns most of the shifts in a fairly orderly fashion and pays extra for the least desirable shifts. Note that in the typical real world, each category of shift usually has its own set of workers. Most ordinary employees work only days or only evenings, or whatever. Docs are unusual in rotating through all shifts. Your OP seems to suggest that all employees can be assigned to any shift. Although this was true with my group it is most definitely NOT true with any other of the groups who work in the ED (RNs, Techs, Admins…)
It is not that case that you will be able to find some sort of mathematical formula to fill a blank schedule. There are many shift-assignment programs which are on the market. I do feel that the more flexibility a manager is given in wage assignments, the easier it is to get all the shifts assigned. I used to tell my Director that he would have no problem covering Christmas if he paid that shift an entire year’s wages. Somewhere between $1,000 and $250,000 is the right amount to pay an ED Doc to work Christmas day if you never want to have problems covering the shift. (I think we ended up paying double to cover the most popular Holidays…)
