# Tragedy of the Commons: More In Depth

Posted by dguenther

Tragedy of the Commons: More In Depth

Here I plan to go more in depth into a Tragedy of the Commons (TotC). Let’s go back to goats.

There are 100 families in our community. The all live equidistantly from the common pool resource of the field. The field can optimally support 1,000 goats- which more precisely means that it can keep 1,000 goats sufficiently fat and healthy such that they will fetch the best price at market and will have a rate of grass consumption such that with near certainty, the same amount of grass will return next season.

So, the field has 1,000 consumable “grass units”.

Recap: 1 goat needs 1 grass unit for 1 season to fetch $1 at market. Our field has the “utility potential” to be worth $1,000 for the whole community. Here, I will stipulate by blog fiat that nothing you can do can make that field worth more than $1,000.

**World 1: “Utopia”**

Let’s say in a perfect utopian and egalitarian world “Utopia”, every family is keeping 10 goats on the field.

100 families with 10 goats = 1000 goats. Since our carrying capacity (I hope I am using the term correctly here) is 1,000 goats, everybody is making $1 per goat or $10 a family.

So, in Utopia, every Family unit makes $10.

Now after a couple of years in Utopia, a particular Family_{β }starts to wonder if they couldn’t possible work the system. So, Family_{β} sneaks in an 11^{th} goat. Good thing all those goats are nearly identical. What do our numbers look at like now?

**World 2: “One Family Defect”**

We will call this world “One Family Defect”.

We have 1,001 goats but only 1,000 consumable grass units. Good news is that a goat won’t die if it eats a little bit less. But it will be a little less fat and a little less healthy. In the case of 1 extra goat, almost negligibly so. But, the addition of Family_{β}’s goat **necessarily** devalues the worth of every goat in our scenario.

{Why? Well every goat has to be a little skinnier and a little less healthy. Add in the increased chance of the goat dying, feeling uncomfortable with so many other goats, fighting with another goat, etc. I’m not even adding in the environmental impact yet (I will at the end). There are even inflation deflation and market responses to consider, but I don’t want to. The goat store may change prices based on the number of goats in the commons. Having an additional goat probably also adds to the unpredictability of the rate of future goat birth, which is a consideration too. Maybe you don’t want to have 1,000 goats if you might have a exceptional birth period that doesn’t keep up with goat death. If you have 1,200 goats because of too many goats born, and not even because people tried to graze an extra goat, you still have a tragedy. In that case you might just eat the extra goat. But, for our purposes, 1,000 goats will be 1,000 goats next season.}

How much does it devalue it? Well, I thought about it long and hard and this is what I think.

Each goat is now worth 1000/1001 of what it was before. Before each goat was 1000/1000 or $1. Now each goat is worth 0.999000999000999.

So every non-defecting Family_{α} has 10 goats at the 0.999000999000999 price, or a new net goat worth of $9.99000999000999.

The defecting family Family_{β} has 11 goats at the same 0.999000999000999 price or a new net goat worth of $10.98901098901099.

So in our “One Family Defect” world, 99 Family_{α}s are making $9.99 a season and Family_{β} is making $10.99.

**World 3 “One Family Defects A Lot!”**

The alpha families probably wouldn’t care about their lost penny. They might not even notice it. Let’s make them notice it.

Same land details, but our defecting family beta liked the extra 99 cents. What if they try to raise 10 extra goats or a total of 20 goats?

That leaves us with 1,020 goats or a goat price of 1000/1020 or $0.9803921568627451.

Family_{α}s are making 10*0.9803921568627451 or $9.803921568627451

Family_{β} is making 20*0.9803921568627451 or $19.6078431372549

The alpha families are starting to notice at this point which leads us to another scenario.

**World 4 “One Family Defects a Lot, but So Do Others!”**

80 Family_{α}s keep raising their 10 goats.

1 Family_{β}, the original defector, keeps grazing 20.

19 Family_{σ}s now try to outdo and preempt the beta family, so they graze 25 goats each.

So now we have a total of 800+20+475=1295!

That makes our goat value drop drastically to 1000/1295 or 0.7722007722007722.

Broken down:

Alpha families are making 10*0.7722007722007722 or $7.72200772200772 per family.

The beta family is making 20*0.7722007722007722 or $15.44401544401544.

The sigmas are making 25*0.7722007722007722 or $19.30501930501931 per family.

**World 5 “The final act of the tragedy”**

Now even though the per goat price has been declining as each goat got skinnier due to less grass for all goats, people raising more goats were making more money. I’ll now stipulate that at any less than $.50 per goat price (representing both its health and worth) the goat dies and is worth nothing.

Now, even the 80 alpha group families have gotten upset and started raising 20 goats.

The 1 beta family, preempting the sigmas, raises 100 goats.

The 19 sigmas, realizing the “error of their ways” drop down to 20 goats each.

We now have (80*20) + (1*75) + (19*20)=2,055 goats

Our per goat price is now 1000/2055 or 0.48661800486618.

Each alpha family has 20 dead goats worth $0 each.

The beta family has 100 dead goats worth $0 each.

The sigmas each have 20 dead goats worth $0 each.

Furthermore, the goats still ate all of the grass such that new seeds were not able to germinate and therefore there will be no grass next season, not that there would have been any goats to graze their anyways.

—

Concluding thoughts:

I’ve tried to throw some math in here. My most important premise is that our field is one big cake that can be divided many ways, but the total value cannot be made higher. One extra goat got that person an extra 99 cents and cost everybody else only a penny. At 2,040 goats, every goat died and the field was ruined.

What are we to do? On to Ostrom!!

Posted on October 1, 2011, in Uncategorized and tagged common pool resource, commons, CPR, game theory, Ostrom, prisoners' dilemma, selfish, social dilemma, tragedy, tragedy of the commons. Bookmark the permalink. Leave a comment.

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