# Making efficient public good decisions using an augmented Ausubel auction

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## 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 auction### JEL Classification

C72 D44 H41## 1 Introduction

After almost 50 years of research on incentive design, it remains unclear whether a practical institution can be designed to overcome free-riding 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 re-defined 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 multi-unit 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, multi-unit 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 re-define 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 (AQ-AA) and it has a number of nice features. We show the AQ-AA 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 AQ-AA are ideal in a public good setting. We conclude the paper by discussing two shortcomings of the AQ-AA 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

*some*of our results:

*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 \(c\le (N-1)(c-\epsilon )\).

## 3 The public good Vickrey auction

^{5}Specifically, each bidder \(i\) sends a bid \(b_{i}\in \Theta _{i}\) to the government. Let \(b\in \Theta \) be an arbitrary bid profile and \(b_{-i}\in \times _{j\ne i}\Theta _{j}\) be an arbitrary bid profile by bidders other than \(i\). The government takes the bids as proxies for bidders’ valuation schedules and constructs a reported valuation function \(\tilde{v}_{i}\), for each \(i\), where

^{6}

**Theorem 1**

In a public good Vickrey auction with non-decreasing 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.

^{7}

## 4 The public good Ausubel auction

^{8}If \( \sum \nolimits _{i}b_{i}^{t}\ge c\), the auction continues to the next round (\(x=t+1\)) and it stops otherwise. Denote the round the auction stops by \(L\) so the quantity produced is \(x^{*}=L-1\). Bidder \(i\)’s residual supply function at round \(t\) is a function of the other players’ bids (i.e., \( b_{-i}^{t}\in B_{-i}^{t})\) and is defined by \(\tilde{s}_{i}:X\times B_{-i}^{t}\rightarrow \mathbb R \), where \(\tilde{s}_{i}(t,b_{-i}^{t})=\max \{0,c-\sum \nolimits _{j\ne i}b_{j}^{t}\}\). At each round \(t\) where \(b_{i}^{t}\ge \tilde{s} _{i}(t,b_{-i}^{t})\), bidder \(i\) accrues a tax equal to \(\tilde{s} _{i}(t,b_{-i}^{t})\). His total payment at the end of the auction is

^{9}

### 4.1 Strategic analysis of the AQ-AA

*pure strategy*\(\beta _{i}\) for bidder \(i\) in the game \(\Gamma ^{e} \) is a collection of functions (one for each stage), where each function maps a bidder’s history and type into a feasible action.

*ex-post pure strategy.*For any realization of types \(\hat{\theta }\), a strategy profile \(\beta \) in \(\Gamma ^{e}\) can be

*projected*into an ex-post pure strategy profile of the realized game \(\Gamma ^{e}(\hat{\theta })\) by setting \(\delta _{i}^{t}(h_{i}^{t})=\beta _{i}^{t}(h_{i}^{t},\hat{\theta }_{i})\) for each \(t\). A strategy profile \(\beta \) is an

*ex-post (perfect) Nash equilibrium*of \(\Gamma ^{e}\) if, for each \(\theta \in \Theta \), the ex-post pure strategy profile \(\delta \) is a (subgame perfect) Nash equilibrium of the game \(\Gamma ^{e}(\theta )\), where \(\delta _{i}^{t}(h_{i}^{t})=\beta _{i}^{t}(h_{i}^{t},\theta _{i})\) for each \(i\), \(t\), and \(h_{i}^{t}\).

^{10}Our first result is that truthful bidding behavior is an ex-post perfect Nash equilibrium of the AQ-AA in the full bid information setting—all of the proofs are in the Appendix.

**Theorem 2**

In the full bid information AQ-AA, truth telling by each bidder is an ex-post perfect Nash equilibrium. Moreover, the truth telling equilibrium allocation of the AQ-AA 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* AQ-AA and the no bid information AQ-AA, truth telling by each bidder is an ex-post perfect Nash equilibrium.

*not*a dominant strategy. To illustrate, consider a two bidder case where bidders \(A\) and \(B\) have the degenerate type profiles \(\theta _{A}=(5,4)\), \(\theta _{B}=(4,3)\) respectively, can bid any positive integer, and marginal cost is \(c=6\). Consider the strategy profile \(\beta =(\beta _{A},\beta _{B})\) defined by:

**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” AQ-AA. 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 AQ-AA 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 AQ-AA 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

*decreasing*quantity auction with strategic properties similar to the AQ-AA, but which protects the highest marginal valuations for the bidders and covers the cost of production. This mechanism, dubbed the decreasing quantity Clarke-Ausubel auction (DQ-CAA), is motivated by the familiar “Clarke tax.” The auction begins at stage \(1\) where the public good is initially set at \(x= \bar{x}\). The cost of the allocation \(\bar{x}\) is \(C(\bar{x})\). The government imposes an tentative cost sharing rule\(\ (s_{1},...,s_{N})\), where \(i\)’s tentative fixed cost share is \(s_{i}C(\bar{x})\) for \(s_{i}\ge 0\) and \(\sum _{i}s_{i}=1\). At each round \(t=1,...,\bar{x}+1\), each \(i\) submits \(b_{i}^{t}\in B_{i}^{t}\) to the government who uses the following continuation rule: If

*there exist*a bidder \(i\) such that \( \sum \nolimits _{j\ne i}b_{j}^{t}\le (1-s_{i})c\), then the auction continues to round \(t+1\), otherwise the auction stops. Bids are constrained to be

*weakly increasing*. The quantity \(x\) produced is determined by the

*first*round \(t\) where \(\sum _{i}b_{i}^{t}\ge c\) – i.e., \(x=\bar{x} -(t-1)\). Finally, at each stage \(t\), bidders receive rebates of their initial cost shares determined by the function

*ex-post Nash equilibrium*.

**Theorem 4**

In the full bid information DQ-CAA, truth telling by each bidder is an ex-post 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 DQ-CAA 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 AQ-AA does not protect the highest valuations of bidders or raise enough tax revenue to finance production. The DQ-CAA remedies both of these issues and has strategic properties that are similar to the AQ-AA. Whether one of these mechanisms is more desirable in practice is an empirical question and one that is well posed for future experimental research.

## Footnotes

- 1.
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.

- 2.
Bergemann and Välimäki (2010) is a dynamic treatment of the VCG mechanism in an infinite horizon environment.

- 3.
See also Kagel et al. (2007).

- 4.
Since this price uniformly applies to all bidders in a private good setting, it can be thought of as a public good.

- 5.
As there may be multiple maximizing arguments, we shall assume the government chooses the largest \(x.\)

- 6.
- 7.
The revenue generated in the public good Vickrey auction always runs a deficit—see Loeb (1977).

- 8.
We define this set more formally in the next section.

- 9.
Interestingly, the AQ-AA 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.

- 10.
- 11.
See also, Tideman and Tullock (1976) for a discussion about the Clarke mechanism and its properties.

- 12.
See, for instance, Fudenberg and Tirole (1991) p. 109.

- 13.
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.

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