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The Hold-Up Problem in Mud World

DISCLAIMER: This article is NOT an attempt to practice serious science in the mudly world. The author does not wish to make a fool out of himself by the mere suggestion of such.

Instead, this is a small attempt to look at muds in an abstract way and draw some funny conclusions. It takes a bit of math, however I hope that won't deter you. I regulary try to comment for the sake of understanding. The math is just a tool, nothing more.

It is all about an abstract 'Mud World', in which 'The Mud' is one of many muds. Let's see what this all looks like:

* 'The Mud' knows but one good, called 'Fun'.
* Two production factors (inputs) are being used to produce Fun:
  - IM, or 'Immortal'
  - P, or 'Player'
* The Mud is characterized by a division in two sectors:
  - Sector 1 is the 'Autarky Sector', in which only IM is used as an input.
  - Sector 2 is the 'Joint Production Sector', in which one unit of Immortal and one unit of Player are combined to form one Joint Production unit (JPU) at production level: Y.
* Supply of Immortal is given at IM = 1 (simplification).
* Player can be invested in 'The Mud' or instead be used to invest in 'All Other Muds'. In the latter case, Player will have a fixed return of Wp = 1, the wage of a player.
* Player in a new Joint Production Unit is totally specific, or tied, whatever you prefer to call it. The result is that second-hand Player is worthless.

Interpretation: I hear you think, what the hell has this to do with muds. Let's try and place the model in mudly reality. We are facing a mud here, in which two groups are active, being Immortals and Players. Both are trying to have fun: trying to produce 'Fun'. But they are facing a decision. You see, an Immortal can choose to work on his/her own, or together with one Player. Players of course are facing a decision as well. That is a Player can work along with an Immortal, or leave the mud for another mud. However, as soon as a player is working together with an Imm, breaking up the relationship means the player is so frustrated he/she will find it hard to start at any other mud and can be considered wasted for the mud world.

There were introduced a couple of simplifications, which may seem confusing. But for the analysis it doesn't matter if we take supply of Immortals to be 1 or 10000. Neither does it matter if we assume one Immortal combined with one Player, or one Immortal combined with a 100 Players. But the first thing is much easier to work with.

What is interesting now, is what decisions Immortals and Players will make, and if we can actually get some insight from the model, instead of only a hard time.

Sector 1:
production function:
f(IM1) = 2/(1 + 1/ETA)[1 -(1 - IM1)^(1+1/ETA)]
      ETA --> just a parameter, don't worry over it
      IM1 --> supply of Immortals in sector 1
first order condition:
f'(IM1) = Wim
    Wim = 2(1 - IM1)^(1/ETA)
    Wim = 2(IM2)^(1/ETA) (for IM = 1 = IM1 + IM2)
      Wim --> wage of an Immortal
      IM2 = ((Y-1)/2)^ETA
      IM2 --> supply of Immortals in sector 2

The latter means the marginal productivity, that is the extra production of 'Fun' you get from adding one extra Immortal to sector 1, should be equal to the wage of an Immortal. This is logical if you think of it. It means Immortals should enter sector 1 as long as the gains are larger than the costs.

Note that if IM1 rises, that is the use of Immortals in sector 1 rises, Wim falls. This shows there are diminishing returns. With every extra Immortal that starts to work in sector 1, the productivity goes down, and so do the wages. Watch carefully, it also means that if Immortals leave sector 1, productivity will go up (and wages with it)!

Interpretation: Sector 1 is not that important, so if you can't follow it all, don't worry. In mud terms, it says that working alone on the mud can be productive, but as with most activities, there's an optimal number of Immortals to work alone. Too many of these means Immortals will start to get in each others way.

By the way, wages of Immortals are just the share of 'Fun' they want for putting time in The Mud. Just like players demand their share Wp.

Sector 2:
There is an incentive for setting up more JPUs if:
Y > Wim + 1

So the production Y should be larger than the sum of:
- What that one Immortal could have earned in sector 1, Wim, and
- What that one Player could have earned in another mud, Wp = 1.

Once the JPU starts running, it keeps running as long as:
Y > Wim
Remember: A player receives nothing if the JPU is broken up!

Surplus S of the JPU:
S = Y - Wim > 0

Suppose Players receive 50% of this surplus, Immortals claim the other half. This means Players will ex-ante invest if:
S/2 > 1
That is, the Player invests only if his/her share in the surplus is expected to be larger than 1, being what he/she can earn for sure at another mud.

Under totally free entry, while facing this surplus to reap,
Players will start JPUs with Immortals until:
S/2 = 1 --> S = 2
At which:
Wim = Y - S
    = Y - 2
IM2 = (Wim/2)^ETA = ((Y - 2)/2)^ETA

The latter formula is NOT equal to the one we found under Sector 1 for IM2. Under Sector 1 was listed the EFFICIENT OUTCOME. However, that is utopia. As you can see, sector 2 is now much smaller than under the efficient outcome. Compare:
IM2 = ((Y-1)/2)^ETA
IM2 = (Wim/2)^ETA = ((Y - 2)/2)^ETA

Why is this? Well, Players ex-ante realize their investment in the JPU to be specific. That is, when the JPU starts running, they can no longer get any return elsewhere if the JPU were to break up. This gives the Immortals a strong bargaining position. The Immortals will as soon as the JPU start running, seize a share of the surplus, on top of their wage. The Players of course realize this ex-ante, and therefore do not invest as much as they would have if a perfect contract between Players and Immortals would have been possible.

Players in this case are suffering from the HOLD-UP PROBLEM. Their specific investments make them prone to the increased negotiation power of their partners, the Immortals.

Argh Fletchez. Give me some clue as to how this relates to mud. Well... Immortals in The Mud are 'mobile'. That is they have a suitable alternative for working along with players. They can always start working alone to have their 'Fun', without working along with Players. Player on the other hand, can only endure one mud. They need to make a decision and stick to it. Is that true? Hmm, yes I think. A Player needs to learn the rules of a mud, its commands, tactics, co-operating with the Immortal, and so on. Quite a specific investment. And when you get fed up with the Immortal and leave the mud over it, you can imagine no longer thinking muds to be fun.

So there will be Players deciding not to play The Mud, because they know the Immortals will not keep their promise of giving them their desired share of 'Fun'. Instead the Imms will take it themselves! The insight is, that IF it would have been possible to give Players that share, The Mud would have had more players. Sadly enough this isn't possible in this model. *Grins.*


A funny thing to conclude with.
Immortals in sector 1 make: Wim
Immortals in sector 2 make: Wim + S/2
so Wim2 > Wim1

This is strange. After all we would expect the wages of Immortals to be equal in both sectors. After all, Immortals from sector 1 can now go to sector 2 and earn more, right?


To be able to work in sector 2, an Immortal needs a Player to start a JPU... And guess what? Because of the hold-up problem, Players are not willing to join. In other words, Immortal jobs in sector 2 are rationed...

Sounds plausible. If you don't have enough players, you can't co-operate with them. So you are forced to work alone, even though you may prefer co-operation.

That's it, hope you can all appreciate some abstracting. However, maths are just a tool, though a useful one. And you can draw funny conclusions.

Further Readings: The model described in this article is a derivative of the model used by Caballero and Hammour. It is certainly interesting material, if you happen to be interested in economics. Note that the Caballero and Hammour model is easy to generalize, allowing social security for Labour (Immortals) as well as guaranteed returns for Capital (Players). Under those conditions, it can be shown that the Capital market can become segmented as well. Which is interesting if linked to the South-East Asian crisis, in which capital guarantees played a major role. And if linked to the above model, it would mean the Immortals not being willing to invest in a JPU, because they know the Players won't keep their promises. How very close to reality models sometimes seem to be.

Caballero, R.J. & Hammour, M.L., 'The 'Fundamental Transformation' in Macroeconomics', American Economic Review May 1996, p.181-186.

About the Author:

Fletchez (J.W.R. Lambregts), is one of the Admins of 'Ad Hesperia', where he, among other things, is responsible for the supervision of area creation and the general development of new concepts. In real life, he hopes to receive his Master's Degree in General Economics this month at Tilburg University (The Netherlands), for a thesis on 'Price Dispersion, Strategic Pricing and Electronic Markets: An Empirical Analysis of the Electronic Books Market'. Fletchez can be contacted at, or at 'Ad Hesperia', to be found at or telnet to 7000