The Liberal Radicalism Mechanism for Producing Public Goods

The mechanism for producing public goods in Buterin, Hitzig, and Weyl’s, Liberal Radicalism is quite amazing and a quantum leap in public-goods mechanism-design not seen since the Vickrey-Clarke-Groves mechanism of the 1970s. In this post, I want to illustrate the mechanism using a very simple example. Let’s imagine that there are two individuals and a public good available in quantity, g. The two individuals value the public good according to U1(g)=70 g – (g^2)/2 and U2(g)=40 g – g^2. Those utility functions mean that the public good has diminishing utility for each individual as shown by the figure at right. The public good can be produced at MC=80.

Now let’s solve for the private and socially optimal public good provision in the ordinary way. For the private optimum each  individual will want to set the MB of contributing to g equal to the marginal cost. Taking the derivative of the utility functions we get MB1=70-g and MB2= 40 – 2g (users of Syverson, Levitt & Goolsbee may recognize this public good problem). Notice that for both individuals, MB<MC, so without coordination, private provision doesn’t even get off the ground.

What’s the socially optimal level of provision? Since g is a public good we sum the two marginal benefit curves and set the sum equal to the MC, namely 110 – 3 g = 80 which solves to g=10. The situation is illustrated in the figure at left.

We were able to compute the optimum level of the public good because we knew each individual’s utility function. In the real world each individual’s utility function is private information. Thus, to reach the social optimum we must solve two problems. The information problem and the free rider problem. The information problem is that no one knows the optimal quantity of the public good. The free rider problem is that no one is willing to pay for the public good. These two problems are related but they are not the same. My Dominant Assurance Contract, for example, works on solving the free rider problem assuming we know the optimal quantity of the public good (e.g. we can usually calculate how big a bridge or dam we need.) The LR mechanism in contrast solves the information problem but it requires that a third party such as the government or a private benefactor “tops up” private contributions in a special way.

The topping up function is the key to the LR mechanism. In this two person, one public good example the topping up function is:

Where c1 is the amount that individual one chooses to contribute to the public good and c2 is the amount that individual two chooses to contribute to the public good. In other words, the public benefactor says “you decide how much to contribute and I will top up to amount g” (it can be shown that (g>c1+c2)).

Now let’s solve for the private optimum using the mechanism. To do so return to the utility functions U1(g)=70 g – (g^2)/2 and U2(g)=40 g – g^2 but substitute for g with the topping up function and then take the derivative of U1 with respect to c1 and set equal to the marginal cost of the public good and similarly for U2. Notice that we are implicitly assuming that the government can use lump sum taxation to fund any difference between g and c1+c2 or that projects are fairly small with respect to total government funding so that it makes sense for individuals to ignore any effect of their choices on the benefactor’s purse–these assumptions seem fairly innocuous–Buterin, Hitzig, and Weyl discuss at greater length.

Notice that we are solving for the optimal contributions to the public good exactly as before–each individual is maximizing their own selfish utility–only now taking into account the top-up function. Taking the derivatives and setting equal to the MC produces two equations with two unknowns which we need to solve simultaneously:

These equations are solved at c1== 45/8 and c2== 5/8. Recall that the privately optimal contributions without the top-up function were 0 and 0 so we have certainly improved over that. But wait, there’s more! How much g is produced when the contribution levels are c1== 45/8 and c2== 5/8? Substituting these values for c1 and c2 into the top-up function we find that g=10, the socially optimal amount!

In equilibrium, individual 1 contributes 45/8 to the public good, individual 2 contributes 5/8 and the remainder,15/4, is contributed by the government. But recall that the government had no idea going in what the optimal amount of the public good was. The government used the contribution levels under the top-up mechanism as a signal to decide how much of the public good to produce and almost magically the top-up function is such that citizens will voluntarily contribute exactly the amount that correctly signals how much society as a whole values the public good. Amazing!

Naturally there are a few issues. The optimal solution is a Nash equilibrium which may not be easy to find as everyone must take into account everyone else’s actions to reach equilibrium (an iterative process may help). The mechanism is also potentially vulnerable to collusion. We need to test this mechanism in the lab and in the field. Nevertheless, this is a notable contribution to the theory of public goods and to applied mechanism design.

Hat tip: Discussion with Tyler, Robin, Andrew, Ank and Garett Jones who also has notes on the mechanism.


So that is it. We can make the food on the Titanic tastier.

Yes, Brazil is a titanic disaster, so tastier food is about the best consolation Brazilians can hope for, though not favela dwellers like yourself who are lucky to get pellets of dog food:

Here in the states, those are called fancy things like "Fruitloops" and "Wheaties".

No, they are pellets made out of expired food.

The corrupt elites of Brazil who prey on their fellow Brazilians can afford to purchase delicious, nutrient-fortified cereals made by American firms ("O grande sucesso entre as crianças americanas, certamente será muito popular em sua casa também!"):

It is a special industrial food. Brazilian experts created what they dubbed "human feed", a scisntifically designed meal to strenghten the children and save public money. The mayor who introduced this program is well liked and was elected with a record number of votes.

Free stuff, more free stuff! What could go wrong???

He introduced a rationalizarion plan to deliver public goods and protect the public purse. He defeated the populist incumbent (who now now runs a corrupt, pro-China political campaign) and introduced a scientific food for poor children.

Have you ever considered creating an e-book or guest authoring on other
sites? I have a blog centered on the same subjects you discuss
and would love to have you share some stories/information. I know my readers would appreciate your work.
If you are even remotely interested, feel free to send me
an e-mail.

Awesome! Its in fact amazing post, I have got much clear
idea regarding from this article.

Your alternative is workers sit idle and thus not only is their labor squandered, but their consumption as well, which reduces the total productivity of society even further, reducing the general welfare forever?

Time is an asset that is scarce and consumed relentlessly. Simplistically, the Puritan view requires devoting it to work or worship, both to further the building of tangible and intangible assets for the public and private good.

Puritans might be characterized as considering tastier food as hedonistic, thus worship and blender food would be the priority, but in practice, the work to produce tastier food was valued for the greater general welfare, as long as it seen as glorifying God and His bounty.

Speaking as a pk of those who quaked before God, and thus knowing the public good of tastier food from many church functions, for the glory of God.

The reference to the Titanic merely emphasizes the scarcity of time, but the Titanic is characteristic of management like Trump. Arrogant, deaf, blind, defying nature. Mother nature, God, will not be denied.

You are missing the point. It does not matter if we have shinning bridges and taste school lunches if Red China is allowed to destroy America's industry and enslave our allies. Brazil is fight for survival while we idle.

Brazilians enjoy life too much that's why China will forever eat their lunch. See Tyler's conversation with Bruno Macaes.

It is not true. Brazil is a peaceful country unlike aggressive Red China, with its wars and camps of concentration.

As N approaches infinity, I realize I'm a big cuck visiting a big cuck blog.

Yes, interesting. The question that fascinates me the most is the extent to which the top-up function depends on the shape/functional form/whatever of the individuals' utility functions. Because it's not much of a solution if you can get individuals to reveal their private utility functions only if you already know the shape of that function.

That's my main reaction as well (and Joel makes a similar comment). How did the authors come up with that top-up function? How well does it work with other utility functions, with n>2, with non-constant MC, with partially rivalrous (congestible) or partially excludable public goods, with uncertainty, etc.?

It would help to say upfront: *If public goods are funded by a mix of users and the government, how much government subsidy will encourage an optimal level of production?* This paper suggests a mechanism to determine how much the government should subsidize.
If the government promises in advance to follow that mechanism, then the rational decisions by individuals of how much to contribute will lead to producing the socially optimal amount.

Which make no sense to me at all. The government is endogenous and funded by the same people being asked to pay for the public good. How does the subsidy work if we don't believe lunches are free?

The lunch has a cost of 2 sqrt(c1 c2), paid out of taxes. The contributions c1 and c2 of the individuals plus this 2 sqrt (c1 c2) by the government together provide the funding that maximizes the total utility. If the government promises nothing, the individuals contribute nothing, and end up in a lower-utility state.

It makes sense, and there is no free lunch. Yes, the government gets whatever money it kicks in from C1 and C2. And realize there is no ethical claim here; C2 may feel the tax regime is oppressive; whatever.

The point is that the top-up mechanism uses the voluntary contributions of C1 and C2 to find the right amount to spend to maximize their collective utility. And we didn't need a political negotiation to do it.

So, is clean water a public good, and considering that problems like cholera tend to be very expensive in terms beyond the merely monetary, how would the math work?

The mechanism would certainly get the answers quite wrong in an absolute sense because it really is hard to quantify how bad a given increment in cholera risk is, but that's not a problem unique to the mechanism. How do you decide whether or not to put in the cost and effort to buy a personal water filter?

In this mechanism, people could think something like "it seems like if I contribute nothing this public water sanitation project would get $856070, if I contribute $50 it would get $864240, it intuitively feels like one percent more work on water sanitation is worth $50 for me so I'll do it". Alternatively people might think "I decided I'll spend $X of my income in the public goods market, and this represents 5% of what I care about recently, so I'll spend $X * 0.05 here".

The water itself is a private good. The reservoir, treatment, and distribution networks are either public goods or natural monopolies. There is a lot of literature on decomposing these networks into public and private goods and allocating resources accordingly.

One aspect I have not seen in public economics papers is an external threat, such as a foreign army, that adjusts its own level of public good according to your choice. That is, the optimal level of public good is decided in a larger game theoretical framework.

And if your foreign enemy is a dictatorship, then the game theory also must account for public choice.

This is part of why I've been trying to build bridges between "GMU econ blogger culture" and the blockchain space; many people in the blockchain space have spent years thinking about mechanism design in the context of (i) collusion, both horizontal (many participants working together), and vertical ("I bribe you $2 to spend $1 in the LR mechanism toward my fake charity and thereby redirect $N of other people's money to me"), and (ii) external actors who have a large incentive or budget to make the mechanism break. The presence of such actors *is the default security model*.

These harsher assumptions definitely make life harder for mechanism designers, but in some cases breakthroughs are possible.

'The water itself is a private good.'

Well, the water that comes out of tap. However, clean water can also be considered a public good in the sense that a river filled with industrial waste (think heavy metals or Kepone) has been ruined as a source of clean water, for a certain amount of time.

'There is a lot of literature on decomposing these networks into public and private goods and allocating resources accordingly.'

A cynical follower of the Virginia School just might wonder if a company like GM or Allied Signal was involved in funding any such work.

No. The definition of a public good is that it is nonrival in consumption and nonexcludable. You're not entitled to your opinion on what a public good is or isn't.

Water is a private good. Period.

"involved in funding"

That is called an "ad hominem." Your objection is overruled.

+2, one for each point.

You are certainly correct regarding the standard definition of a public good. However, given the context of the discussion and the analysis provided are we really talking about a public good? It's clear there must be some aspect of rivalry in consumption or we would not need output units.

I would find it helpful if you defined terms and variables. It seems like g is the amount spent, but MC = 80, so that can't be right.

g is the amount produced. In the first half of the article, g appears to be an independent variable (which is why Alex differentiated with respect to g to find marginal benefit). Then, later on, g is itself defined as a function of independent variables c1 and c2 (the voluntary contributions of each).

I think that, near the end, when Alex says "take the derivative of each", he means:

1. Take the partial derivative of B_1 with respect to c_1
2. Take the partial derivative of B_2 with respect to c_2

...which makes intuitive sense to me, since c_1 is the contribution that the first person controls, en route to personal benefit B_1. When I do that, I can match the two equations at the end, so I'm pretty sure my interpretation is correct.

Hope this helps.

"Where c1 is the amount that individual one chooses to contribute to the public good and c2 is the amount that individual two chooses to contribute to the public good. In other words, the public benefactor says “you decide how much to contribute and I will top up to amount g” (it can be shown that (g>c1+c2))."
c1 and c2 are how much each person contribute, and are compared to g. Did Alex mean c1 is how much of g that person 1 is "responsible" for?
MC = 80. Does that mean for 10 units of g the cost is 800?
I agree with Curious reader, an example (as promised in the 2nd sentence) would probably clear a lot of this up.


Glad I'm helping at least a bit.

Yes, c1 and c2 are the individual voluntary contributions of person 1 and person 2. I think that Alex means "Person 1 will voluntarily contribute quantity c1 of the public good", not "Person 1 will contribute c1 dollars". I was reading that part wrong at first, but you're getting it right in your comment.

g is then the quantity of the public good produced. And the proof that g > c1 + c2 is pretty trivial; just multiply it out and you see g = c1 + c2 + 2 * sqrt(c1) * sqrt(c2).

The MC is a flat 80, and I think your interpretation is correct there too. g = 10 implies total cost of $800.

So then MB and MC are dollars per unit of public good. MB is in dollars of private benefit; MC is in dollars of private cost.

My question in return to the audience (or Alex) is this: I plowed through this example to get the same result. I assume that, when more than two people are involved, the algebra gets worse. However, a famous result in abstract algebra is that not all polynomials of fifth order or higher can be solved directly. (Google "insolvability of the quintic" if you don't believe me.) Does that come up in more general cases?

Of course, the utility function of some folks has greater significance than the utility function of other folks. How is that? Investment in a public good is a political decision, and some folks have more influence over decisions made by politicians than others. And as Cowen reminds us in his second blog post this morning, people are greatly influenced by media, media that is used by some folks to influence public opinion. A drumbeat of media disparaging (for example) public transit (it's used by "those people") pretty much ends the possibility of public investment in transit, even though the combined utility function of "those people" is much higher than the combined utility function of the "right people" who oppose public goods because the utility function of the right people carries much greater weight.

You can't expect people suffering from false consciousness to make the right decisions. You need a vanguard of the woke.


Cardinal utility?

No consideration of opportunity cost in stating the socially optimal level?

I cannot help bu look at the graph and say person 2 is getting screws and person one enjoying a (partial) free ride.

Given that half of the population here is losing value how is this a social optimum?

Even with the Top-up function we're assuming that someone outside the system will insert the needed funds without having any impact on the two in the population. If government does that it's not clear how that would work. If some unstated private member in the population (that is it's not two but three) does it then we're simply assuming that the public good could have been produced in the first place by making some pareto optimal trade. That is a different story than what is told here it seem to me.

Student is automatically flunked for failure to label axes. Regardless.

There is a major problem here.

The government is not exogenous. It consists OF the two people and is funded BY the two people. Assuming away the public choice problem, at the very least the individuals must take as given in their U fx their share of the government top up through taxation.

This bothered me as well. Maybe the authors assume that taxes are unknown by the individuals 1 and 2? I think a further breakdown would be helpful. If individual one has a utility X for a public good y, wouldn't their MU curve be affected by their tax contribution? What of the tax rate and levels which feed directly into the government contribution?

What if the government provides and inferior good according to the top-up function? Wouldn't the utility curve by a wealthy individual be impacted by this? I suppose the tax obligation for each individual is public information for the government though....thoughts?

As N approaches infinity, an individual's effect on the total tax rate becomes negligible. At least, this is assuming the funding mechanism is a government that charges taxes. There are other possibilities, eg. charities, foundations, and potentially even companies, eg. Microsoft could use such a mechanism to fund public goods related their own developer ecosystem.

Why do you say " In this post, I want to illustrate the mechanism using a very simple example", and then, instead of using a relatable, concrete, specific example, you rephrase the mechanism using abstract and generic variables and equations?

So instead of complicated, abstract examples, how does this work in practice? If the local government thinks we need to subsidize public transportation they normally have some obscure process for setting ticket levels and additional funding. In this model, I guess the government would let a privately owned business provide service and set a ticket price with a guarantee to provide additional funding in some functional proportion to ticket revenue and let the privately owned business maximize profit? Is that how we imagine this working? I can certainly see a government funding backstop as a function of customer revenue working better than existing models, but hypothetical mathematical models that cannot be translated to realistic contracting and funding arrangements are not helpful.

I can imagine it applied to areas where governments allocate grants. Basic research, arts, charity, overseas aid. Areas where c1 and c2 are donations rather than prices.

What is interesting is that the paper treats preferences as fixed, and utility functions as constant.

What would be interesting would have been to add some behavioral economics into the paper.

In industrial organization there is a subcategory of behavioral industrial organization; public goods area looks right for subcategory of behavioral econ public goods theory.

Don't assume preferences are constant, when people look at what their neighbors choose; don't assume that framing doesn't affect choice, don't assume that gathering information is costless or equal across players, etc.

Here are some other specific limitations listed in Behavioral Law and Economics by Zamir et al on public choice theory:

1, Assumption that one maximizes one own utility without regard to the utility of others; some persons have prosocial motivitations; meta=analysis of decomposed games shows that 50 percent are decomposed games; 24 individualists; 13 percent as competitors; and 13% unclassified;
2. From behavioral studies, most people are reciprocators;
3. Inequality aversion; concern for overall social welfare of others;
4. Effect of threat of punishment in stabilizing repeated interactions.

1. instead of decomposed games ,should read: 50% classified as prosocial. Material was from met analysis of behavioral games.

What if c1, c2 or both is zero? g is not greater than c1+c2 in that case.

this type of system already exists with mass transit fares like subways. c1 or c2 is willing to pay the subway fare, and the government covers whatever the fares don't cover.

1. Does this theory make any assumptions about the stability of the different variables? I.e., what if MC changes in time? Or what if c_1 and c_2 change quickly -- people agreeing to contribute something then changing their mind at the last minute. Does the government have to continually change the "top-up" amount?

2. Can this mechanism work equally well for international organizations? E.g the individuals are countries, or groups of countries, and the government is the organization somehow?

That's interesting ! But where does this topping function, (square-root of c1 squared plus c2 squared), comes from. As I understand it (but perhaps I have missed something), this function is rather arbitrary (it is only important that it grows as a function of c1 and c2, and perhaps has some convexity conditions), chosen by the government or the benefactor, and publicly announced. Knowing this function, agents 1 and 2 will determine their contributions c1 and c2 to the public good, and these contributions plus the contribution of the benefactor will determine the total amount of public good produced. Good, but calling that amounts "socially optimal" seems somewhat of an overstatement, since it clearly depends of the choice of the topping function.

For instance just imagine you keep the same general form for the topping
function (square root of c1 squared plus c2 squares) but multiply it by a scaling factor lambda. If you choose, lambda close to 0, it is clear that the contribution of the topping will vanish and that we will tend toward the initial situation where c1=c2=0. If on the other hand lambda becomes bigger and bigger on the contrary, what will happen? The topping will crush the private contributions, which will just be chosen by the actor so as to maximize the utility of the public good, that is g=70 for the first agent and g=20 for the second. here will be some game theory to determine what happens, but probably the second agent will win, and the quantity of public good produced will be 20, not 10, almost-all paid for by the government.

So all thees quantities of public good produced (0,10,20) are what this model gives by changing somehow the topping function. It is clear that no one of them deserves to be called "socially optimum" without further arguments.

I don't have time to think about it right now, because I don't have the economics background or the game theory background to really get at the idea, but there's a branch of reinforcement learning that might be relevant - correlated q learning - which can solve optimal policies in multiagent games.

You could think of the development of pro-social norms reinforced by peer pressure as an example. But, this is inconsistent with the rational man model of individual utility.

A pedantic note, irrelevant for the point being made, is that the sum of the MB curves should jump to MB1 for g>20 (unless MB2 becomes negative for g>20.)

How is this different from a Shareholder Nation?

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