The Clarke-Groves Mechanism: How to Induce Honesty (notes by Simon Vicary)

In the module so far we have established conditions for the efficient provision of a public good. We have also found one mechanism, the Lindahl model, which in important respects replicates the fundamental theorems of welfare economics for an economy with a public good. However, this mechanism is almost certainly too costly to operate as a practical proposition, and is in any case vulnerable to individuals misrepresenting their preferences in an attempt to lower the taxes they pay whilst at the same time still enjoying the public good that is provided by the taxes paid by others. We have also found that majority voting may under some circumstances deliver an efficient quantity of a public good, but it cannot be relied upon as a general rule. All this work raises a deeper and rather more difficult question:

Does there exist a mechanism which will guarantee Pareto optimality in all circumstances, and which will induce people to be honest when asked to reveal their preferences?

The reason why the second part of the question is needed is that it is assumed that the government is going to provide the public good. In doing this it needs to know the MRSGx schedules for each individual (that is, individual preferences). Hence individuals have to give the government some information about their preferences. If this information is false, then clearly the provision of the public good will not be optimal. As the motivation for misrepresentation is to lower one’s tax bill this problem of getting individuals to reveal their preferences is part of what is rather loosely referred to as the free rider problem. Put this way a partial reformulation of the question could be:

Can we find a general solution to the free rider problem?

In stating the free rider problem way back in 1955 Samuelson thought the answer to this question was “no”. He did not analyse the issue in any depth however, and economists rather left the things at that for the next 15 years. Little thought was given to how one might try to solve the free rider problem, until a number of articles starting with Clarke and Groves separately in 1971, seemed to suggest that it was in fact possible to solve the free rider problem. This work is the focus for this lecture. We shall find that the solutions provided by Clarke and Groves are only partial. Indeed the conclusion of the literature seems to be that there is in fact no general way round the free rider problem. I would like to emphasise this point at the outset as the most accessible paper for you, and one you should read (Tideman and Tullock (1976)), seems to suggest the contrary.

The reasons for these pessimistic conclusions will be revealed in the fullness of time. It may, though, be useful to start by clarifying in our minds the exact nature of the free rider problem itself.


1. Interpretations of the free rider problem

McMillan (1979) outlines three strands of the free rider problem. We have encountered all of them already in one way or another.

No Government: Full information on Utility
This is the world of Nash equilibrium that we looked at early on. Technically, as a game theorist would tell you, the basic model assumes that all agents have full information about everybody’s preferences. (Even more technically any individual’s preferences are common knowledge among individuals.) The key point though is that individuals act in isolation, deciding on how much they should contribute or donate to a public good. As we saw, contributions are typically too low, and the provision of the public good is sub-optimal. Everyone would be better off if each person contributed a bit more to the public good, but it is in no one’s interest to do this unilaterally. For a summation public good, individuals find themselves in a prisoner’s dilemma.

Here people take, or attempt to take, a free ride on the contributions of others. Put more precisely, an increase in total contributions by all other individuals in the community results in any one given individual lowering their own contribution. The reason for sub-optimality is that people think only of themselves. That is they decide on their contributions on the basis of costs and benefits to themselves alone. They do not take into account the fact that their contributions also benefit other consumers of the public good.

Government: No Information on Preferences
In a world of complete information the problem of government provision would be trivial. Therefore, to capture what is likely to be a benevolent government’s problem we assume the government does not know individuals’ preferences. It has in some way to rely on individuals telling the government what their preferences are. This could come about directly (through such things as opinion polls), or indirectly through observing the way people behave (for example inferring the value people place on the environment by looking at the extra amount they are willing to pay for such things as organically produced vegetables, for houses in traffic free areas etc. Once preferences have been found, the Samuelson condition (in principle!) can be applied to deliver what might be an efficient quantity of the public good.

As we saw with the Lindahl model, however, it seems individuals do not in fact have an incentive to reveal honestly their true preferences. Again, free riding is to blame. In this case, it appears people will want to under-state their preference for the public good so as to lower their tax bill. The crux of the problem lies in what seems at first glance like a very good idea: that tax payments should be related in some way to the benefit one receives from public expenditure. This seems to provide the source of the gain from free riding in this context. Any attempt to get round this problem would therefore have to penalise individuals for deviating from their (unknown) true preferences. As any misrepresentation is made to inflict some cost on the rest of the community, it might seem reasonable that any charge people pay for misrepresentation reflects the cost they impose on others.

As we shall see, this is the key idea Clarke and Groves exploited. But before we tackle this problem head on we need to fill in some background.

Large Numbers
McMillan also mentions the large number problem of Olson, as a third variant on the free rider problem (it gets worse as community size increases). This is slightly different from the other two, and will not concern us over much in what follows.


2. Second price auctions and how to induce honesty

The question of inducing honesty is not unique to the public goods problem. In fact there are a whole host of variants on this theme in economics, coming under the general heading of asymmetric information. After pioneering papers by Akerlof, Spence and Stiglitz (for which they got a Noble Prize), this became a very active area of research in the 1980’s, and indeed continues to be so. The classic and original paper in this area was by Vickrey in 1961, and it will be useful to start with a simplified version of a key argument in that paper. It concerns the first seemingly unrelated area of auctions.

Suppose you have a single item to sell. Often for unique single items the sale is by auction. However, there are many different types of auction. The classic method, probably the one that first comes to your mind, is what is often called the “English Auction”. Here individuals make ascending bids for the item. This is done openly. As the highest price bid rises, potential buyers drop out, and the “winner” is the last person to stay in the auction/bidding. He or she gains the object and pays the last price they bid. A second possibility would be to require potential buyers to submit a sealed bid. In this case potential buyers submit a bid to the auctioneer. This is not observed by other bidders. The object is sold to the highest bidder, and they pay the price they bid. This is referred to as a first price sealed bid auction. There are quite a few other possibilities.

Now a moment’s thought should convince you that in these auctions people will not bid honestly. The best way to see this is to think about the first price sealed bid auction. Suppose each individual has a maximum price they are willing to pay for the object. Will they write this maximum price on their submission to the auctioneer? Obviously not, because the maximum price is such that one is indifferent between: (a) not having the object and (b) having the object but paying this maximum price. There is therefore no possible way in which a person who submits their maximum price can gain from participating in the auction. Lowering the bid must result in some non-zero probability of positive gain, and is therefore to be preferred. Suppose, however, that you wanted to find out what people’s maximum willingness to pay is. How would you design the auction to do this?

Vickrey showed that the following auction (now sometimes known as a Vickrey Auction) would do the trick. Technically it is a Second Price Sealed Bid Auction. This works in the same way as a first price sealed bid auction, except that the winner (the person who submits the highest bid) pays the second highest price bid. It is not too difficult to see why this elicits honesty. I won’t go into details. Can you gain by raising your bid? Not if you have the highest valuation, and you might lose if this is not the case. What if you lower your bid? If you are not the person with the highest valuation you cannot gain, and you can only lose if you are that person. In short by submitting anything other than your true valuation you can never gain, and in some circumstances you will lose. Faced with a second price sealed bid auction honesty really is the best policy.

There are a number of observations to make about the second price auction which will be useful to bear in mind as we go through the Clarke-Groves mechanism.

(a) It might be thought that a second price auction is unsatisfactory from the point of view of the seller. This is misleading at best, and generally false. It turns out that all the auctions mentioned and many others produce the same expected revenue for the seller. This was shown by Vickrey, and his result has been generalised subsequently. The error in the statement lies in assuming that bidding is invariant with respect to the type of auction faced. This is obviously not true. The amount you might have to pay if you bid highest must have some impact on what you are prepared to bid.
(b) Things differ if we think of the seller as a bidder. The point is that the seller may have a reservation price for the article, and not be prepared to sell for less than this price. Suppose the rules of the auction allow the seller to set a reservation price? Will the seller set the reservation price honestly? The answer is in general no. By raising the reservation price in the range between the highest and second highest bids, the seller can increase revenue. It is possible therefore that he/she will want to put down a higher reservation price than his/her true valuation.
(c) Putting these two points together we see that we can get honesty from individuals in a group when the money raised goes outside the group. However, if the money stays within the group (as happens when we add the seller), general incentives to be honest do not seem to exist.
(d) How do we interpret the price paid by the winner? One way to look at it is as a compensating sum within the set of bidders. Suppose James has the highest bid and would win the auction if he participated. Suppose too that Mary is the second highest bidder and places a value of £20 on the object. If James participates he deprives Mary of the object. Put quaintly, he deprives the rest of the community of an article which it values at £20 (the object is a private good). So when James participates he pays a sum of money that would compensate the rest of the community (the set of bidders in this case) for his participation.
(e) However, following on from the last point, and in a way linking up with point (c) it is vital that although James pays a compensating sum, the compensation is not paid to anyone within the group. A moment’s thought again should convince you that as soon as compensation is actually paid the incentive to be honest collapses. Some one will have some sort of incentive to raise their bid in order to get hold of some of the compensating payments.

This may seem a rather contrived set of points to make, but together they provide a link to the question of how we can devise a mechanism that induces an honest revelation of preferences in a public good economy.


3. The Clarke-Groves Mechanism: Discrete Case

To see how the above principles can apply, let us take the simpler example of a discrete public good. Suppose we have a good that is either provided in one unit (G = 1) or not at all (G = 0). Examples might be a bridge, the restoration of a church tower, the saving of Private Eye etc. Suppose it costs £100 to deliver the good, and suppose there are three agents whose value placed on G is given in the table:

Table 1
Individual Valuations of the Public Good
Individuals Valuation Assigned Tax Net Benefit
A 40 35 5
B 70 35 35
C 20 30 -10

A Clarke-Groves scheme for this problem would work as follows:

(a) Each person is asked to submit their valuation of the public good (The valuations in the table are known only to the individuals themselves).
(b) The good is provided only if the sum of declared (there is no way of knowing for sure whether submitted valuations are true or not) valuations is as least equal to the cost of provision £100.
(c) If the good is to be provided, there are two parts to each individual’s (individualised) payment: (i) an assigned share; (ii) the compensating sum.
(d) The first part of any tax payment is an assigned share. The table gives the assigned numbers for our example. These assigned taxes are imposed. Individuals have no control over them. A natural assumption would be to assign them equally across individuals, but this is not necessary and not what we do in the table. Willingness to pay net of assigned taxes appears in the 4th column of the table.
(e) The compensating sum for each individual refers back to points (d) and (e) above. It is worked out by summing the willingness to pay for all other individuals in the community, and subtracting from this the total amount of assigned taxes they paid if the good is provided. Given that the good is to be provided we have to work out whether our individual’s participation in the community would make any difference to the outcome. If not, then no compensating sum is paid. If so then the imposed taxes are increased by the compensating sum in question.
(f) Given point (c) made about the second price sealed bid auction it follows that the money raised by the compensating sum should be thrown away. Or at least it should go outside the community.

Now apply this scheme to the example in Table 1. Table 2 takes the story on

Table 2
Working out the Compensating Sum
Individuals Sum Net Benefit n-1 Decision Compensation Paid
A 25 Y 0
B -5 N 5
C 40 Y 0

Suppose that our three individuals declare their preferences honestly. As the total sum of benefits exceeds £100, or as total net benefits exceed zero the good will be provided. Look now at individual A. Suppose A did not participate in the “vote” (you can imagine that A’s taxes are available to B and C). In this case the sum of net benefits over B and C results in 25 (= 35 – 10). This is positive, so A’s participation makes no difference to the community decision, and hence she pays no compensating sum. By repeating this procedure we find that B’s participation alters the decision the community makes. In fact B’s participation imposes a cost of 5 (= – (5 – 10)) on the community. Hence B must pay £5 as a compensating sum. C, as can easily be checked pays no compensating sum, which is just as well as she is made worse off by the provision of the good.

The final question about the mechanism is simply this. Does it indeed elicit an honest revelation of preferences? It can again be checked that by declaring a valuation different from the true one, individual cannot gain, and might lose. Note first that the compensating sum paid is beyond the individual’s control. An individual can only alter tax payments by altering the provision of the good. So A, for example, could lower her declared valuation. If she does this she pays no tax, but she also loses the benefit of the public good. As with the good provided she enjoys a net benefit of £5, she can only lose by doing this. Raising her declared valuation does not alter the tax paid, and therefore will make no difference to her welfare.

Given that the good is provided individual B pays a compensating sum of £5. His net gain from having the public good is 30 = 70 – 35 – 5. Lowering the valuation either has no effect on provision (with a constant compensating sum) or causes the good not to be provided, in which case he loses £30. Raising his declared valuation makes no difference to anything.

Finally consider C. C loses from the provision of the public good. As C’s valuation makes no difference to the decision, lowering the declared valuation makes no difference to the outcome, and no difference to her tax bill. Neither does raising her declared valuation.

It appears then that there is no way for any agent to do better than to make a truthful declaration to the authorities. Doing otherwise either has no impact on utility or it makes the individual worse off than he/she would otherwise be. Is this a quirk of the example, or does this property hold generally? It is of course a general property of this mechanism.

To prove that this mechanism really does induce honesty, note two relevant factors:

(a) The public good may or may not be provided. The condition for provision is: , or , where A is the cost of provision, and ci is individual i’s assigned tax should the good be provided.
(b) To find out whether an individual is pivotal, the inequality in (a) must be compared with the inequality . If the signs of the two inequalities are the same then the individual is not pivotal, and pays no compensation tax. If they do differ then the individual is pivotal in the sense that he/she changes the outcome as opposed to what the rest of community would decide.

There are four cases to examine, depending on whether the individual is pivotal or not and also whether the good is provided or not.

Case 1: G = 1, Pivotal Individual
First take the case of the good being provided, and the individual being pivotal. This is the case of individual B in the example. The individual in this case must make a compensating payment . By construction:



(ci is the assigned tax.) In this case we have:

with

Hence it must follow that (substitute the tax identity into the first inequality):



The left hand side is the return if the individual is honest. The right hand side represents the return when the declared valuation is so low as to mean the public good is not provided (and our individual ceases to be pivotal). This being so, it would never be in the individual’s interest to lower her declared valuation below the true value. To do so incurs the danger of losing the benefit . As tax payments are fixed either exogenously (ci) or by the declaration of other agents ( ), there is no prospect of gain. Naturally there is no gain to be had by increasing one’s declaration. We conclude that if you are pivotal, and the good is provided, you can only lose by doing anything other than declaring the truth.

Case 2: G = 1: Non-pivotal Individual
Now suppose the individual is not pivotal, but that the good is provided (this is the case of A and C in the example). In this case no compensating taxes are paid. If the public good is not provided then the following inequalities hold:

with

Raising declared valuation makes no difference to the outcome (the good is still provided) and no difference to taxes paid (the individual is still non-pivotal). Thus our individual’s utility is:

vi – ci

It might be possible for i to lower declared preferences so that the good is not produced. This could apply to C in the example who loses as a result of the good being produced. However, were i to succeed in this (actually it’s not possible for C in the example, but she doesn’t know this) then he/she becomes pivotal and has to make a compensating payment:



The only thing that happens now to i is that the compensating payment is made. The good is not provided, and no assigned taxes are paid. To find out whether it is worth lowering declared preferences therefore i must compare:

vi – ci with



But



Hence



The left hand side of the inequality is the individual’s utility when he/she is honest and the good is provided. The right hand side represents the return when preferences for the public good are under-stated, and it is not provided. Hence it follows that if our individual succeeds in causing the public good not to be produced then he/she is worse off. Our conclusion now is that if the good is going to be delivered then there is no way any one can gain by misrepresenting preferences, and there is always the possibility of making oneself worse off.

Case 3: G = 0 Pivotal Individual
The method used to show that honesty is the best policy is the same for when the good is not going to be produced. If the individual is pivotal she “stops” the good being produced when the rest of the community would want this to happen. The analysis uses the following two inequalities:

with

Lowering declared preference for the public good makes no difference to the outcome, but raising declared preference might cause the good to be produced. If so, then no compensating taxes are paid.

Compare the utility (utility with honesty) with vi – ci¬ (utility if overstatement succeeds in getting the good produced). By subtraction you should be able to convince yourself that:

vi – ci <

Our individual can only lose by overstating preferences for the public good.

Case 4: G = 0 Non-pivotal Individual
As with the last case lowering declared preferences makes no difference to the outcome, but raising preference could cause the good to be produced. In this case we use:

with

However, if the good is produced our individual becomes pivotal and pays a tax equal to:



The comparison is between utility when honest (0) and when overstatement succeeds in getting the good delivered. That is, between:

0 and

Hence yet again honesty is the best policy. Our final conclusion is that no matter what the circumstances, individual dishonesty at best makes no difference. Otherwise, it results in our individual being made worse off. In this way the Clarke-Groves mechanism ensures an honest revelation of individual preferences and in this way offers a final solution for the free rider problem.


4. The Clarke-Groves Mechanism: Continuous Case

Before examining some of the limitations of the Clarke-Groves scheme, we shall just note how it gets extended to a continuous public good. Actually relatively little changes, and there is no point in going into details. See Cornes and Sandler if you are interested. It is simply a matter of applying the principles already learnt.

Consider Figure 1. It is assumed that each individual has a quasi-linear utility function Ui = xi + vi(G), and, to keep things simple we assume a constant MRTGx = 1. The procedure works as follows:

(a) The government asks each individual to state their preference for the public good. In this context this means to write out their MRS schedule.
(b) Having gained each MRS schedule the government applies the Samuelson criterion by choosing the output at which the sum of the MRS equals the MRT.
(c) In determining each individual’s tax bill there are again two components: a pre-determined tax share over which the individual has no control, and a compensating sum, determined by the impact our individual’s statement of preferences has on the outcome.
(d) Once the level of G is determined individuals pay the tax. The pre-determined tax revenue is spent on providing the public good, and the revenue from the compensating payments is thrown away.

Let us see how this works out on the diagram.  on the diagram represents our individual’s pre-determined tax share. Suppose people are honest.

(a) By equating the sum of declared MRS’s to the MRT the output of G is determined at 0Q.
(b) Now imagine that i does not participate in the process. The rest of the community would collectively, under this procedure decide on an output 0A, found by equating to the sum of their marginal tax rates (1 - MRTGx. This step corresponds to finding out whether the rest of the community would want a discrete public good given their (stated) preferences and costs.
(c) On the diagram our individual is pivotal in the sense that her participation raises provision from 0A to 0Q. As this is so, she must make a compensating payment.
(d) This compensating payment must equal, as before, the welfare loss suffer by the other n – 1 individuals in the community as a result of i’s participation.
(e) This aggregate loss is measured using standard welfare analysis (the area between the MC to the other individuals, 1 -  and their aggregate marginal benefit schedule , area JKL).
(f) The line SS (synthetic supply) is draw such that area WXY is the same as JKL. It is a mirror image of the aggregate marginal benefit schedule for the other n – 1 agents.
(g) Thus the area JKL = WXY gives the compensating payment i must make in this case.
(h) The output chosen will always be such that the Samuelson condition is met.

This scheme produces honesty in just the same way as we saw earlier. To see this, consult Figure 2 where we reproduce the relevant part of Figure 1. Suppose our individual decides to over-state his preference for the public good. The provision rises from OQ to OQ1. The benefit of this rise to our individual is QYNQ1 whilst the cost is QYRQ1. Our individual loses by the area YRN. This area will always be positive for any overstatement of the MRS schedule. A similar conclusion can be reached by assuming our individual under-states preferences for the public good. This is left as an exercise.

The conclusion is that under the assumed conditions the Clarke-Groves mechanism induces individuals to report honestly their preferences for the public good.

Before dealing with some of the weaknesses of the mechanism, one point is worth making. In running through the analysis we found that it was always in the interest of our selected individual i to tell the truth. This analysis of course applies to all other individuals in the economy. However, we did not however enquire too deeply into what these other individuals were doing. Were they telling the truth about their preferences, or were they disguising their preferences in some way? Actually, it doesn’t matter. While we sort of assumed other people were revealing their preferences, the conclusion goes through if they are being dishonest. The reasoning we followed really dealt simply with what these other people had declared. If we were to assume a false set of declarations by the other people our reasoning would proceed exactly as originally, and we would find it still in our individual i’s interest to reveal preference truthfully. We conclude:

Regardless of what other people have done it is in the interest of each individual separately to reveal honestly their preferences for the public good.

In the language of game theory we would say: Truth telling is a dominant strategy in the Clarke-Groves mechanism.

Another pretentious way to say this is that the Clarke-Groves mechanism is Strongly Individual Incentive Compatible.

A natural question to ask is this: Do there exist other mechanisms with this property? It can be proved that the answer to this question is “no” Green and Laffont (1979). If we are interested in mechanisms that induce individuals to reveal their preferences for the public good under all circumstances, then we can restrict ourselves to Clarke-Groves mechanisms. It is useful to bear this in mind in what follows.

If the mechanism is so wonderful, why is it not used more often? Let us find out why.


5. The Gibbard-Satterthwaite Theorem and its Implications

Our approach will be a little indirect, and requires us first to go back to the Arrow Theorem. Around about the time the Clarke-Groves mechanism was being developed (theoretically!) a remarkable theorem in the field of social choice was proved by Gibbard and Satterthwaite. It can be described as follows.

Suppose we have a community of n individuals, and suppose our community must choose between a number of alternatives S ={x1, x2,…….,xn}, where n ≥ 3. This problem is what Arrow originally had in mind. Suppose as with Arrow that the community has a mechanism for deciding which social state to choose given the declared preferences of the individuals. We can summarise this in the form of a function from the declared preference to the set of social states:

g: P→S x = g(P1, P2, ……,Pn)

where g is our function and Pi is the preference declared by individual i. g could be described as a social choice function. Voting, Lindahl and all the other procedures we have discussed so far would be classed as social choice functions.

We say a social choice function is manipulable at P = {P1, P2, ……,Pn} if there exists a false declaration of preference Pi* for some individual i such that:

g(P1, P2, …, Pi*,…,Pn) ≻i g(P1, P2, ……,Pn)

That is, individual i can make himself better off by making a false declaration of preferences. The work in our lecture up to now seems to suggest that the Clarke-Groves mechanism is non-manipulable, in that it is always in an individual’s interest to reveal preferences truthfully.

We would want any social choice function to have some desirable properties. Reflecting Arrow in some way, two can be stated:

Non-imposition
∀ x ∈ S, ∃ P such that x = g(P)

That is, take any social state. There must be some configuration of preferences that result in this social state being chosen. If you like, the constitution does not debar some alternatives. This, effectively, is Arrow’s unrestricted domain axiom.

Non-Dictatorship
∄ i such that x ∈ g(P) ⇔ x≻i y ∀ y ∈ S and for all preferences that could be stated by all individuals.

This is Arrow’s non-dictatorship axiom. There is no individual whose preference determines social choice regardless of the preferences of other people.

The Gibbard-Satterthwaite Theorem
If the number of social states is greater or equal to three it is impossible to construct a social choice function that has the properties of non-dictatorship and non-imposition.

Put another way, if a vote is to be taken over more than two alternatives, then no matter how the voting system is designed it will be possible in some circumstances for somebody to gain by misrepresenting their preferences. You cannot eliminate strategic voting for any “sensible” voting system.

As well as its interest for actual voting systems, the Gibbard-Satterthwaite Theorem poses a problem. The Clarke-Groves mechanism involves many alternatives (this is even true of the binary choice problem we examined in detail given that one aspect of the problem was the tax people had to incur), it is clearly non-dictatorial, and has the non-imposition property. Yet we found that it is non-manipulable in the sense defined. There is clearly something funny going on here. Our next section finds out what this is.


6. Limitations of Clarke-Groves Mechanisms

How do we reconcile the Clarke-Groves mechanism with the Gibbard-Satterthwaite Theorem? The answer lies in a half hidden assumption that was slipped in to the Clarke-Groves model.

No Income Effects
In explaining the Clarke-Groves mechanism for a continuous public good, as we did with Figure 1, we assumed utility to be quasi-linear. This ensures that the MRS functions never move around. It turns out that this assumption is quite vital to get the model to work. It is possible to extend the result so that truth telling is a dominant strategy with a wider class of utility functions. However, this cannot be done for general utility functions. (This is proved in Green and Laffont p 81). The reconciliation is therefore that the Clarke-Groves mechanism only “works” under a restrictive set of conditions. If you want to relate this point to earlier work, the assumption of quasi-linear utility violates the unrestricted domain assumption.

As an indication of the sort of theoretical trade offs involved, a parallel mechanism was devised in a heavily technical paper by Groves and Ledyard in 1977. This did work for general utility functions, but truth telling was only a Nash equilibrium strategy, not dominant strategy equilibrium. (How this would work in practice when people have to state preferences without knowing what other people’s preferences are is unclear). It seems therefore that some of the sharp properties of the Clarke-Groves model have to be sacrificed if it is to work for all utility functions. This general point is exactly what we would predict given the Gibbard-Satterthwaite Theorem.

There are, however, other limitations

Clarke-Groves Mechanisms do not work for all public goods
Examples of this would where income distribution is a public good (either as a matter of social policy or as a question of deciding on the finance of a given project). Here there are individualised benefits from the public good, and individuals have an incentive to misrepresent preferences in order to gain them.

The Budget Constraint Problem
Even if we were to put these problems (as well as the cost of operating the system) aside there is still a serious problem with the Clarke-Groves mechanism. It is simple to state and easy to see. The mechanism does not in general produce a Pareto Optimum. The reason for this lies in the compensating sum that in general we must expect to have paid. The revenue gained in this must be thrown away, otherwise individuals will have an incentive to misrepresent their preference in order to gain this revenue.

By revenue, as this is microeconomics, we mean real goods and services. But if real goods and services are being disposed of then obviously the economy has not got to a true Pareto optimum. In a general sense, the explanation for what is happening here is reasonably straightforward. Valuable information is being hidden from the planners. In order to extract it some cost has to be incurred, and one way to think of this “budget surplus” is as a cost of extracting the information. This failure to reach full Pareto optimality is in fact a standard property of models with information asymmetry.

As an aside, if we look at Figure 1 we can see that there is one happy case where full Pareto optimality would be achieved. This is where the imposed individualised price of the public good implied by the tax sharing rule () corresponds with the Lindahl price. In this case, there is no budget surplus problem.

Vulnerability to Coalitions
Just note. The Clarke-Groves mechanism is invulnerable to individuals misrepresenting their preferences. It is not invulnerable to groups of individuals misrepresenting their preferences.

To conclude…

Recall that Clarke-Groves mechanisms are the only ones that guarantee that truth telling is always in the interest of the individual agent. If there is a “solution” to the free rider problem it is here. However, we have found that they “work” only for a restricted set of utility functions, they do not produce Pareto optimality and they are vulnerable to manipulation by coalitions of individuals. If these problems are serious then the conclusion we have to come to is that except in special cases the free rider problem of revelation of preference is insoluble.