Imagine you are hired as a consultant of a company. This company wants you to decide how much each employee should get paid each month (in percentage terms of the month's profits).
However, they are really experimental. They want you to make the decision of every person's percentage of payout with virtually no information about them. You don't get to know their title or position, or anything.
You are allowed to ask the employees questions about how much they think everyone or anyone should get paid, but you can't ask anything else.
For example, you can ask, "Alice, should you make more or less than Bob?" but you can't ask, "Alice, does Bob get to work on time?"
So the question is - what do you ask them in order to make the most fair payout decisions on an ongoing basis?
To rephrase this, you're making a game that the employees are going to play. They're trying to maximize their pay over their coworkers. But we are taking it as our assumption that distributed amongst them is some kind of consensus that represents a more optimal distribution of pay than merely giving everyone equal pay. How do you extract that information from them without giving them the ability to manipulate the system individually or as a colluding sub-group?
In other words... How do you arrange the rules of a zero-sum betting game where all the players have somewhat hidden information such that the Nash equilibrium for all players is to just tell the truth?
Here's an example of what obviously won't work:
If you ask them, "how much should you make relative to each other person in the company?" and then take some kind of averaging of the answers then obviously those unscrupulous types that say "100% for me please, and everyone else should share 0%" will make the most money.
This is the opposite of what you want. This dumb, simple rule has incentivized a lie: they know they shouldn't make 100% of the profits, but they have to say that in order to maximize their pay. As more and more people just say 100% everyone's pay will approach the same number, and you will have mined no information out of them to inform your decision.
How do you incentivize them to reveal their hand and just tell the truth of what they really believe? How do you minimize their ability to manipulate the system for their own gain?
Here was my first idea, but I don't know if it would work:
You tell each person, every month, "This person made just more than you last month, and that person made just less than you. Do you think, for this coming month it would be more appropriate if they switched places?"
I think this still leaves the door open for collusion.
Here was my second idea, but again, I don't know if this would work:
At the beginning of each month, you ask every employee to provide for you a list of all employees ranked by who should make the most to who should make the least. You then rank the lists based on the following weighting: whoever's list last month was most accurate to what actually got paid out gets the most shares (weight), down to whoever's last month's list was furthest from the actual outcome of last month's payouts gets the least amount of shares (0 weight, perhaps) this month. Furthermore, if someone is 'most correct' multiple months in a row they get more weight that carries over into the future.
With this idea, I am trying to turn the ability to manipulate the system into a prediction of the what the group will decide the future payout the distribution should be. In other words, buy turning it into a prediction I attempt to minimize the outlandish manipulations of the system.
Does this incentivize groupthink such that it would undermine the optimal distribution?
I'm sure there are lots of other schemes, but I don't know what's best.
Can someone help me with this problem? I feel like I'm floundering without a framework, or a way to validate my theories, or even the proper education to make sense of it all.
Surely, someone has solved similar problems in the past?
Are you familiar with Mechanism Design? I think your question is well embedded in this topic.
Each of the employees has some kind of a true value for the company, which you don't know as the consultant. Think of this true value as a "type" of the employee. Your goal as a consultant is that each employee reports his "type" truthfully.
In mechanism design we have the so-called "Vickrey-Clarke-Groves (VCG) mechanisms", in which telling the truth is a dominant strategy. With the information that you provided, I cannot really say whether you can create a VCG mechanism here, but maybe you should have a look into this, I think it might be a good starting point.