First of all, suppose the villagers have a strong sense of shame if they are caught overusing the commons. They figure that all they have to do is is to publicly identify anyone found cheating the miscreant would pay up the fine or face the wrath of the whole village. But they do need someone to act as a monitor and patrol the common looking for overgrazing goats.
Suppose one of the farmers becomes a full-time monitor. He is no longer able to work on his farm, but gets an income from the other farmers.
Suppose the income of a farmer under the group plan is qg.
The income of a farmer in the old system was qi < qg.
The total village product under the plan is Qg = sum qg.
Now farmers redistribute some of their income to pay the monitor. They are egalitarians, and both farmers and monitor end up with an income qf where qi < qf < qg.
Suppose that monitoring is 100% effective. That is, if you overgraze you will always be caught. If you are caught you end up with a penalty income d < qf.
The individual farmers decision is now: go along with the group plan and get qf, or cheat and get d. Being rational, farmers respect the group plan.
The farmers decision is simpler than before, in that now they dont need to consider the actions of the other farmers. Your outcome is not dependent on the actions of the group as a whole, but just of yourself and the monitors.
But what about the monitor? Assuming monitors arent of some higher moral level than the other villagers, they also face a choice about whether or not to cheat.
Monitoring consumes time and energy. If you could still get your monitors pay qf, you would rather stay at home with your feet up, or work on your own farm to get extra income. And unless there is someone else monitoring the monitor, there will be opportunities to slack.
Perhaps a farmer catches the monitor slacking. He could report the monitor to the village meeting. Or he could take advantage of the monitors absence and get in some extra grazing.
But if the monitor is clever, he won't slack too much. The more you slack, the more chance of being reported, and the more villagers will start taking advantage and returning to overgrazing. If this goes on too long, we are back to the old state of affairs again.
And the clever monitor has a better way of supplementing his income. For monitoring to work, it must be the case that the monitor is given disproportionate say when it comes to identifying cheats. If someone is accused of cheating there may be some process of trial or appeal. But the village has to have some degree of trust in the evidence of the monitor.
Where there is power, there is the possibility of corruption. The monitor could take bribes from farmers to let them graze for longer. If you let a farmer graze all day the farmer gets a high cheats income qc instead of the normal qf. The monitor can set any bribe less than (qc qf) and both cheating farmer and monitor will be better off.
Again, the wise monitors should limit his corruption to make sure that the system does not breakdown completely.
In fact the best strategy for a corrupt monitor is blackmail rather than bribery. The monitor wants the group plan to work he wants the maximum village income Qg. Ideally you make farmers observe the plan but threaten them with false reporting as cheats unless they pay a bribe. The bribe from each farmer can be anything up to b = qf d.
In an extreme case, if a monitor could get away with extorting the limit from all other (N-1) farmers, he ends up with a total income of (N-1) (qf d) + qe, and everyone else is right down to the penalty income d (or just above it). The villages total product is still Qg, an improvement on the old state of affairs. But the farmers themselves could be even worse off than before if d < qi.
If that was to happen, the farmers would be better off getting rid of the monitor and going back to the old ways. But so long as farmers are better off under a monitor than with rampant overgrazing, there is an argument for keeping the system even though it is creating corruption and inequality. Perhaps these are the price we have to pay for a better standard of living for all equity-efficiency trade-off.
The monitor has no power over the other farmers except for the weight that is given to his reporting. If the villagers become too disgruntled with the regime, they can decide to withdraw their trust in a monitor. But if the villagers are better off under the monitor than with overgrazing, they will not sack a monitor to go back to the old ways. They will only get rid of one monitor to replace him with another. The extent of corruption will be limited to a large extent by how easy it is for the village to change monitors. Maybe some tolerance level of corruption would emerge where villagers will come to expect that anyone they get as a monitor will cream a certain amount.
Note - what if there is more than one monitor?
Even if everyone takes turns at monitoring, the same incentives to corruption exist when you are on your turn.
If there are a number of monitors, there is a collective action problem amongst monitors of how much to cream. Unless they can organise themselves, they will take too much and end up wrecking the system.
There is scope for corruption because monitors have something to offer farmers which they were prepared to pay for. Individual farmers got better outcomes if they pay the monitors by bribes or blackmail - than if they dont pay.
Secondly, the group as a whole gets a better outcome by having a monitor, even though they pay for an accepted level of corruption.
This is because monitoring creates an improvement or surplus over the uncoordinated outcome. The monitors take a cut of this surplus, but there is still some left for the farmers as well.
In general maybe exploitation means an individual or sub-group of group members acts so as to improve the outcomes for the group as a whole, but they benefit more than other group members.
A key part of this concept is that there is an improvement for the whole group. That is we can think of two possible states of affairs one with and one without the action that creates the improvement or surplus.
This is an interesting discussion. What strikes me is the way the goat-police reconfigure the problem. Although there are still GameTheory elements, the payoffs are now drastically parameterized by a seperate informational component : how easy it is for monitors to catch cheats and how easy it is to monitor the monitors. A new InformationTechnology or principle for organizing the monitors can totally upset the existing equilibria and dynamics of the GameSpace.
In GameTheory terms I suppose you build all these things as extra modifyers in the PayoffMatrix. From a CyberNetics perspective, I suppose you could see these as two CoupledSystems : the farmer / goat system and the monitor / monitor-checking system.
I'm not sure how important the 'informational' issue is here.
First, if monitors don't always catch cheats, all this does is modify the expected payoff to farmers from cheating.
Eg if there is a 50% chance of getting caught expected payoffs are:
cheat - (qc + d) / 2
conform - qf
and village can set d low enough that cheating is discouraged.
As for monitoring the monitors, I think the issue is who rather than how.
In the real world, there are technologies to oversee police and reduce corruption - eg all police interviews in UK are now recorded, and there is a whole science around detecting whether these tapes are doctored (this has been in the news here lately around the PC Blakelock / Broadwater Farm case).
But the issue always comes down to human decision-making - human beings must ultimately judge whether a citizen - or a monitor - has cheated. Just as one technology is introduced to catch cheats, other technologies are developed for evasion. (It used to be that police interviews were recorded in writing, and cases revolved around handwriting analysis and impressions left on notepads.) Ultimately all a technology does, however advanced, is provide evidence which can be used to support alternative decisions.
The key to the goat monitors story is the issue of power. The monitors' power consists of the fact that their word counts for more than that of an ordinary citizen not in authority. This is a key factor in policing systems irrespective of technologies.
By the same token, you could imagine a super-monitor is appointed to investigate cheating by monitors. But the situation in the monitors-supermonitors game is now analogous to the old farmers-monitors game before. The supermonitor has the choice of taking his basic income, or creaming off the monitors. etc etc
The issue of organising amongst the monitors is the interesting one, I think.
There are different ways you could think of the interaction within group of monitors (a game within a game). The basic decision for each one is what level of surplus to cream.
eg. suppose there is an optimum level of creaming that the village as a whole can take. The best group plan for the monitors would be to jointly cream this amount, each one getting a share. But then we have a new Commons problem between the monitors, with each having an incentive to cheat and cream more until it brings the whole system crashing down. (Much like Cournot-type oligopoly but against the interests of 'consumers'.)
On the other hand, a basic insight in many collective action problems is that it is easier to coordinate small groups. It may be the goat village as a whole is too big to 'coordinate' other than by introducing an authority sub-group, but the authority sub-group is small enough to coordinate itself without introducing further hierarchy.