Abstract
We provide the generalization of Ausubel’s 2004 ascending bid auction to public good environments. Like its private good counterpart, the public good Ausubel auction encourages truthful revelation of preferences, is privacy preserving, and yields an equilibrium allocation that is outcome equivalent to the public good Vickrey auction. Other properties are not ideal in a public good setting. We discuss two such issues and propose an alternative dynamic auction which solves these problems.
Keywords
Public goods Clarke tax Ausubel auctionJEL Classification
C72 D44 H411 Introduction
After almost 50 years of research on incentive design, it remains unclear whether a practical institution can be designed to overcome freeriding incentives in a public good environment. While this problem is easier if agents reveal their preferences to a decision maker, agents’ interests are not typically aligned with those of the decision maker. As a result, agents may misrepresent their preferences. In this paper, we look to recent contributions in dynamic private good auctions and experimental economics to give us insight into a new, and hopefully improved, way of overcoming the “preference revelation” problem in a public good setting.
Early work in this area came from Vickrey (1961) in a private good setting. He proposed an auction which efficiently allocates multiple units of a homogeneous good by encouraging revelation of preferences as a dominant strategy. His mechanism was later generalized by Clarke (1971) and Groves (1973) to accommodate public goods. Loeb (1977) shows the Vickrey auction can also be redefined to make efficient public good decisions. These mechanisms are known as VCG mechanisms and have generated a large literature.^{1} Despite the theoretical interest in VCG mechanisms, Rothkopf (2007), and others, have argued they are not practical due to lack of privacy preservation. Intuitively, an auction is privacy preserving for a bidder if, when the auction ends, the seller cannot construct a complete demand schedule for that bidder. A second price sealed bid auction, for example, is not privacy preserving since at the end of the auction a seller knows all the valuations. This feature encourages practices like ‘shill’ bidding and may dissuade bidders from participating. Privacy preservation is also a concern in public good settings, where consumers prefer the government not know their true valuations.
A potential solution to this critique is to use a dynamic auction to make allocation decisions. A Japanese auction, for example, is outcome equivalent to the second price sealed bid auction, but preserves the privacy of the winning bidder. In this auction, an auctioneer slowly raises the price and bidders signal each round whether they want to continue and drop out if the price becomes too high. The last bidder still in the auction wins and pays the price the second to last bidder dropped out at. However, unlike the second price auction, the winner never reveals his maximum willingness to pay. Thus, in a dynamic setting the auction can stop before bidders can reveal their whole valuation schedule. Ausubel (2004) introduced a dynamic auction that retains the nice revelation properties of the private good multiunit Vickrey auction, is outcome equivalent to the Vickrey auction, and preserves the privacy of some of the bidders.^{2} In addition, this auction has been more “behaviorally” successful than its sealed bid counterpart when tested in the laboratory. Kagel and Levin (2001) find that bidders in the private good Ausubel auction do, in general, bid truthfully when compared with bidders in uniform price sealed bid auctions. In a follow up paper, Kagel and Levin (2001) compare the private good, multiunit Vickrey auction against the private good Ausubel auction with different information feedback treatments finding the Ausubel auction outperforms the Vickrey auction.^{3} The authors credit these results to the relative transparency of Ausubel’s auction.
We define and study the public good Ausubel auction. The intuition for the transition is simple. In the private good Ausubel setting, all bidders take the auctioneer’s price as given and respond with (potentially) different quantity bids.^{4} In a public good setting, bidders face a uniform quantity of the public good, by definition, but have (potentially) different marginal valuations for each unit. Thus, it is natural to redefine the private good Ausubel auction by using an ascending “quantity” auction instead of an ascending “price” auction. In this new mechanism, the auctioneer starts by calling out a low quantity (instead of a price) and individuals respond by submitting value bids (instead of quantity bids). If the bids exceed the marginal cost (i.e., supply for that unit), the auction continues and the auctioneer increases the quantity. This process continues until the sum of the bids no longer exceeds the marginal cost—i.e., until there is no longer excess inverse demand for the public good. Truth telling is made incentive compatible by adopting a variation of Ausubel’s “clinching rule” to determine the individuals’ taxes.
The resulting auction is the ascending quantity Ausubel auction (AQAA) and it has a number of nice features. We show the AQAA is privacy preserving, that truthful revelation of bidders’ valuations is supported as an equilibrium in a variety of strategic environments, and that auction is outcome equivalent to the public good Vickrey auction. However, not all of the properties of the AQAA are ideal in a public good setting. We conclude the paper by discussing two shortcomings of the AQAA that do not appear in the private good Ausubel auction and introduce a dynamic auction with similar strategic properties that corrects these problems.
2 The public good economy

Bounded Type Assumption (BTA) No bidder type prefers to unilaterally finance production of the public good—i.e., \(0\le \theta _{i}^{1}\le c\epsilon \) for all \(i\).

Bounded Marginal Cost Assumption (BMCA) The marginal cost satisfies \(c\le (N1)(c\epsilon )\).
3 The public good Vickrey auction
Theorem 1
In a public good Vickrey auction with nondecreasing marginal cost, it is always a best response for each bidder to report his true type profile. Furthermore, if the BTA and BMCA are satisfied, then truth telling is a weakly dominant strategy for each bidder.
4 The public good Ausubel auction
4.1 Strategic analysis of the AQAA
Theorem 2
In the full bid information AQAA, truth telling by each bidder is an expost perfect Nash equilibrium. Moreover, the truth telling equilibrium allocation of the AQAA is outcome equivalent to the truth telling equilibrium of the public good Vickrey auction.
The following corollary follows almost directly.
Corollary 1
In both the aggregate bid information AQAA and the no bid information AQAA, truth telling by each bidder is an expost perfect Nash equilibrium.
Theorem 3
If bids are restricted to be weakly decreasing, then truthful revelation of bidding type in each round is always a best response to any bidding strategies of rival bidders in the “no bid information” AQAA. Furthermore, if the BTA and BMCA are satisfied, then truth telling is a weakly dominant strategy.
The robustness of the truth telling strategy profile as an equilibrium along with the privacy preservation property make the AQAA an appealing procedure. However, despite these features, the auction has two obvious shortcomings. First, it does not generate enough tax revenue to finance production—a major difficulty in application. Second, the AQAA protects the marginal valuations for later units in the auction (i.e., the low marginal valuations). This is unfortunate since it is presumably the high marginal valuations (i.e., valuations for earlier units) which the bidders would prefer that the government not know. In the next section, we provide an alternative auction which solves these two problems.
5 The descending Clarke–Ausubel auction
Theorem 4
In the full bid information DQCAA, truth telling by each bidder is an expost Nash equilibrium. Moreover, the equilibrium production level is efficient and the tax revenue generated by the auction covers the cost of production.
Since the auction is descending, the highest valuations are preserved. Truth telling is an equilibrium so the equilibrium production level of \(x\) is efficient by design. Finally, we are assured costs are covered when the auction ends. This follows from the fact that costs are covered when the auction begins, incentive taxes accrued during the auction are positive, and rebates are only given for units that are not produced. It is straight forward to verify that the outcome of this auction is equivalent to Clarke (1971).^{11} Other results such as the analogs to Corollary 1 and Theorem 3 can also be shown to hold for the DQCAA as well.
6 Conclusion
This paper has shown that the relationship between the dynamic private Ausubel auction and the static Vickrey auction can be extended to public good environments in a natural way by exploiting the dual nature of the private good/ public good problem. Once the description of the Ausubel auction has been augmented to fit the new public good environment and the appropriate assumptions are introduced many of Ausubel’s same results apply. The auction seems simple, transparent, scales well to increases in the number of bidders. Moreover, strategic incentives can be eliminated by restricting information during the auction. However, the AQAA does not protect the highest valuations of bidders or raise enough tax revenue to finance production. The DQCAA remedies both of these issues and has strategic properties that are similar to the AQAA. Whether one of these mechanisms is more desirable in practice is an empirical question and one that is well posed for future experimental research.
There are important differences between VCG mechanisms. To avoid confusion we refer to authors (i.e., Vickrey or Clarke) when refering to a specific VCG mechanism.
Bergemann and Välimäki (2010) is a dynamic treatment of the VCG mechanism in an infinite horizon environment.
Since this price uniformly applies to all bidders in a private good setting, it can be thought of as a public good.
As there may be multiple maximizing arguments, we shall assume the government chooses the largest \(x.\)
Interestingly, the AQAA is similar to a family of planning procedures initially studied by Malinvaud (1971) and Dreze and Pousin (1971). Often referred to as MDP processes, these mechanisms can be adopted to converge to any pareto optimal outcomes through different divisions of the social surplus. However, they are vulnerable to strategic manipulation.
See, for example, Cremer and McLean (1985) or Holmstrom and Myerson (1983). Ausubel uses a related concept called ex post perfect equilibrium.
See also, Tideman and Tullock (1976) for a discussion about the Clarke mechanism and its properties.
That the deviation has to come at this stage follows from the assumption of weakly decreasing marginal valuations and the fact that \(\tilde{b}_{i}\) deviates from \(b_{i}\) in only one stage.
Appendix
Proof of Theorem 2
Suppose the profile of truthful bidding strategies \(\beta \) is not an expost perfect Nash equilibrium. Then there exists \(\bar{\theta }\in \Theta \) such that the projection of strategy profile \(\beta \) to \(\Gamma ^{e}(\bar{ \theta })\), hereafter \(\delta \), is not subgame perfect. Since \(\bar{x}<\infty \) and there is “full bid information,” \(\Gamma ^{e}(\bar{\theta })\) is a finite horizon game with observed action. Since \(\delta \) is not subgame perfect it does not satisfy the “onestagedeviation principle” for finite horizon games.^{12} Thus, there is some bidder \( i\) and strategy \(\tilde{\delta }_{i}\) that agrees with \(\delta _{i}\) except at a single \(t\) and \(h^{t}\), where \(\tilde{\delta }_{i}\) is a better response to \(\delta _{i}\) than \(\delta _{i}\) conditional on history \(h^{t}\) being reached. Suppose conditional on stage \(t\) being reached the auction ends at stage \(L\ge t\) (i.e., \(x^{*}=L1\)) if all bidders report truthfully. Given the truthful reports of the other bidders, a 1 stage deviation in the AQAA at stage \(t\) can only result in three outcomes: Case 1, the deviation doesn’t change the outcome (i.e., the auction ends at \(L\)); Case 2, the auction ends earlier at \(t<L\); Case 3, the auction ends in round \(L+1\) (i.e., \(t=L\)). Case 1 is obviously not a profitable deviation since the auction ends at the same round and bidder \(i\)’s payment is independent of his own action. Suppose Case 2 is true, then the one stage deviation causes the auction to end earlier than \(L\ \)say round \(t=E\) (\(x^{*}=E1 \)). Bidder \(i\)’s payoff is \(v_{i}(E1,\theta _{i})\tau _{i}(E,b_{i})\). The payoff from truth telling is \(v_{i}(E1,\theta _{i})\tau _{i}(E,b_{i})+\sum _{k=E}^{L1}[\theta _{i}^{k}\tilde{s}_{i}(k,b_{i}^{k})]\). For each \(k\in \{E,...,L1\}\) it must be true that \(\sum _{j=1}^{N}\theta _{j}^{k}\ge c\), since otherwise the auction would have ended earlier when everyone was truthfully reporting. However, this implies that \(\theta _{i}^{k}>c\sum _{j\ne i}\theta _{j}^{k}\). Since \(\theta _{i}^{k}\ge 0\), by assumption, we also have \(\theta _{i}^{k}\ge \max \{0,c\sum _{j=1}^{N}\theta _{j}^{k}\}=\tilde{s}_{i}(k,b_{i}^{k})\). Thus, for each \(k\), we have \(\theta _{i}^{k}\ge \tilde{s}_{i}(k,b_{i}^{k})\) which in turn implies \(\sum _{k=E}^{L1}[\theta _{i}^{k}\tilde{s}_{i}(k,b_{i}^{k})]\ge 0\). Therefore ending the auction before \(L\) cannot lead to a profitable deviation. Suppose Case 3 is true, then the one stage deviation causes the auction to end at round \(L+1\) (i.e., \(x^{*}=L\) being produced).^{13} Bidder \(i\)’s payoff from this deviation is \(v_{i}(L1,\theta _{i})+\theta _{i}^{L}\tau ^{i}(L,b_{i}) \tilde{s}_{i}(L,b_{i}^{L})\). Bidder \(i\)’s payoff from truthful reporting is \(v_{i}(L1,\theta _{i})\tau _{i}(L,b_{i}).\) The single stage deviation is profitable only if and only if \(\theta _{i}^{L}\tilde{s} _{i}(L,b_{i}^{L})>0 \). However, when bidders were bidding truthfully the auction stopped at \(L\). From the continuation rule of the AQAA, it must be that \(\sum \nolimits _{i}\theta _{i}^{L}<c\) or \(0\le \theta _{i}^{L}<c\sum \nolimits _{j\ne i}\theta _{j}^{L}=s_{i}(L,b_{i}^{L})\), which contradicts claim truthful revelation was not subgame perfect in the realized game. Thus, \(\beta \) is expost perfect. It follows that equilibrium outcomes of the AQAA and Vickrey auctions under truth telling are the same. \(\square \)
Proof of Corollary 1
First, we look at the aggregate bid information AQAA. For each \(i\), consider the 2 bidder game between \(i\) and a representative agent for the other \(N1\) bidders whose value at each stage is the sum of the \(N1\) bidders’ values whom he represents. This is a 2 bidder AQAA with full bid information. From Theorem 2, we know that truth telling is an expost perfect Nash equilibrium. Therefore, for every realization of types, given aggregate bid information at each stage of the auction and truth telling of the \(N1\) other agents, truth telling is a best response. Since this is true for all players at each stage of the auction, it is true at each subgame. Thus, truth telling is an expost perfect Nash equilibrium. The no bid information case follows similarly. \(\square \)
Proof of Theorem 3
Let \(\hat{\beta }_{i}\) be \(i\)’s truth telling strategy. Since no bid information is being given, no bidder can distinguish between ‘auction continuing’ strategies of their rivals. So at any information set where the auction is continued, no bidder knows whether or not they were pivotal at any specific rounds. Suppose that given \((\hat{\beta }_{i},\beta _{i})\) the auction ends at round \(L\) yielding \(i\) a payoff of \(u_{i}(\hat{\beta } _{i},\beta _{i})\) and consider a deviation to \(\beta _{i}\ne \hat{\beta } _{i}\). If \(\beta _{i}\) ends the auction in round \(L\), then it gives \(i\) the same payoff since \(i\)’s tax at any given round is independent of \(i\)’s action. Since the other bidders can’t distinguish between \(\beta _{i}\) and \( \hat{\beta }_{i}\) for rounds \(1\) to \(L\), they cannot respond to the change. If \(\beta _{i}\) ends the auction in round \(E<L\). Bidder \(i\)’s surplus for the first \(E1\) rounds is exactly the same as when he was bidding truthfully so there are no gains for those units. Furthermore, since truth telling guarantees nonnegative surplus at each round, bidder \(i\) is potentially foregoing positive payoffs from round \(E\) to \(L1\). Last, if \(\beta _{i}\) ends the auction in round \(M>L\), then \(i\)’s surplus for the first \(L1\) rounds is exactly the same as under \(\hat{\beta }_{i}\). This is since, by the “No Bid Information” constraint, bidders other than \(i\) cannot respond to the change and the fact that \(i\)’s tax is independent of own action. Since the auction ended in round \(L\) when \(i\) was bidding truthfully \(\theta _{i}^{L}<c\sum _{j\ne i}b_{j}^{L}\). Therefore the tax for changing \(i\)’s bid in round \(L\) is bigger than the gains from having the auction continuing for that round. Gains in later rounds may offset the losses from this round. However, this cannot happen. Bids are decreasing so \(c\sum _{j\ne i}b_{j}^{t}\) is a weakly increasing function in \(t\). Marginal valuations are weakly decreasing. Thus, \(i\)’s payoff from continuing past round \(L\) is strictly decreasing– i.e., \(u_{i}(\beta _{i},\beta _{i})<u_{i}(\hat{\beta }_{i},\beta _{i})\). Thus, for all \(\beta _{i}\), \(u_{i}(\hat{\beta }_{i},\beta _{i})\ge u_{i}(\beta _{i},\beta _{i}) \). The second part of the proof follows Claim 2 in Van Essen (2010). \(\square \)
Proof of Theorem 4
Suppose \(x^{*}\) is the level of production under truth telling and consider a deviation by bidder \(i\). Because the other bidders are truth telling and since \(i\)’s bid must be increasing, the only changes in \(i\)’s payoff with a deviation occur at the changed units of \(x\) – i.e., if the deviation doesn’t change \(x^{*}\), there is no difference in payoff and deviation is not profitable. Suppose \(x^{*}\) is increased to \(\hat{x}\) from \(i\)’s deviation. Then \(i\) is not refunded for the \(\hat{x}x^{*}\) units and the tax for the first \(x^{*}\) units is the same. Two cases need to be considered. First, if \(s_{i}c\ge \) \(\theta _{i}^{k}\), the added benefit \(\sum _{i}\theta _{i}^{k}+(s_{i}1)c\) is smaller than the refund \( c+(s_{i}1)c=s_{i}c\) since \(\sum _{i}\theta _{i}^{k}\) \(<c\). Second, if \( s_{i}c\le \) \(\theta _{i}^{k}\), for each additional unit \(k\), then since, \( x^{*}<\hat{x}\), we have \(\theta _{i}^{k}\le c\sum _{j\ne i}\theta _{j}^{k}\). Thus, for each unit \(k\), \(\theta _{i}^{k}s_{i}c\le (1s_{i})c\sum _{j\ne i}\theta _{j}^{k}\) – i.e., any extra value the bidder gets from unit \(k\) is smaller than additional tax. So, neither of these cases results in a profitable deviation. Last, suppose the deviation causes \(x\) to be decreased to \(\check{x}\). Now the tax is the same for the first \(\check{x}\) units and \(i\) is refunded for the \(x^{*}\check{x}\) units. Again, two cases need to be considered. First, suppose \(s_{i}c\le \) \( \theta _{i}^{k}\) for each removed unit \(k\). Since, \(x^{*}>\check{x}\), we have \(\theta _{i}^{k}\ge c\sum _{j\ne i}\theta _{j}^{k}\) which implies the value lost \(\theta _{i}^{k}\) is larger than \(s_{i}c+c(1s_{i})\sum _{j\ne i}\theta _{j}^{k}=c\sum _{j\ne i}\theta _{j}^{k}\), or the fiscal savings for the removed units. Second, suppose \(s_{i}c\ge \) \(\theta _{i}^{k}\), for each removed unit \(k\). Since, \(x^{*}>\check{x}\), all the removed units \( k \) are such that \(\theta _{i}^{k}\ge c\sum _{j\ne i}\theta _{j}^{k}\). Thus, we have \((s_{i}1)c+\sum _{j\ne i}\theta _{j}^{k}\ge \) \(\theta _{i}^{k}\) \((c\sum _{j\ne i}\theta _{j}^{k})\ge \theta _{i}^{k}\) \(s_{i}c\ge 0\) –i.e., the new loss for unit \(k\) is larger than the original loss for unit \(k\). Therefore, there are no profitable deviations for \(i\). Since this argument hold for any \(\theta _{i}\), truth telling is an expost Nash equilibrium. \(\square \)