Bidding with Allowances: Moral Hazard in Package Auctions Per Paulsen and Martin Bichler Decision Sciences & Systems, Technische Universität München

1

Introduction

Game-theoretical models of auctions are traditionally based on the assumption of profitmaximizing economic agents. This assumption allows for the implementation of the truthful and efficient Vickrey-Clarke-Groves (VCG) mechanism (Clarke, 1971; Groves, 1973; Vickrey, 1961). Overall, efficiency of auctions is often used to advocate their use as allocation mechanisms over beauty contests or other forms of allocation (Peter Cramton, 1998; McMillan, 1995). Binmore and Klemperer (2002, p. 2) write “Most importantly, a well-designed auction is the method most likely to allocate resources to those who can use them most valuably.” In many markets, this utility model might not be adequate to describe bidders. As a consequence equilibrium strategies can be quite different from the ones assuming quasi-linearity. The IS literature has always been concerned with models about bidder behavior on markets. Many IS contributions have analyzed behavior in online or multi-item auctions that deviate from traditional game-theoretical assumptions (Adomavicius & Gupta, 2005; Bapna, Goes, & Gupta, 2000, 2003). The design of markets and the development of adequate market models has become a significant IS research stream contributing to the interdisciplinary field of market design (Roth, 2008). We focus on multi-unit package auctions where bidders are firms and there is a principalagent relationship within the firm (Hölmstrom, 1979). The board of directors, representing the shareholders, can be considered the principal, whereas the management, representing the bidding

team, assumes the role of the agent. The principal does not participate in the auction, but provides his agent with allowances to bid up to certain amounts of money to win in the auction. All remaining money from the bidding belongs to the principal, who also pays the price for the package won. We treat the auction as a one-shot game without repeated interaction between principal and agent within the firm, and between all bidding firms. Furthermore, the agent has no effort of bidding in the auction and therefore does not receive compensation in form of wage payments. It is also well-known by practitioners that principal-agent relationships within bidding firms are wide spread. Shapiro, Holtz-Eakin, and Bazelon (2013) describe principal-agent relationships within firms in spectrum auctions. Here, firms have preferences over different packages of spectrum licenses. Each of these packages can be assigned a business case with a specific net present value. Preferences are then defined by the net present values of business cases associated with the corresponding license packages. The principal sets allowances less or equal to these valuations, which determine how high the agent can bid in order to win a package. Shapiro et al. (2013) write that such pre-determined budgets are implied by capital rationing and constitute a sufficient tool to direct the agent. Our principal-agent model is characterized by a moral hazard problem. The principal has a quasilinear utility function and wants to maximize expected profit. In case of winning, he receives the entire net present package value and fully bears the burden of the package price. Although the agent faces the same net present package values as the principal, he is not profit maximizing. The agent is not a shareholder of the company and does neither benefit directly from the package value nor internalize its price in his objective function. All remaining money from the allowance belongs to the principal. The agent wants to win the packages with the highest expected values, because he indirectly benefits from the respective large business cases. The

expected value-maximizing motive directs the agent’s attention towards more valuable packages irrespective of their profitability. “Empire building” motives are a wide-spread reason for such behavior of agents in the principal-agent literature (Jensen, 1986). Engelbrecht-Wiggans (1987, p. 116) argue that similar agent motives within the principal-agent relationship can also be observed in sales of mineral leases, defense systems, and construction contracts. “For example, in bidding for mineral leases, a firm may wish to maximize expected profits while its bidder feels it should maximize the firm's proven reserves.'' Engelbrecht-Wiggans (1987) discuss pre-defined budgets as a possibility for the principal to limit the agent’s spending in the auction. We argue that even if both parties agree on the net present package values, they do not necessarily follow the same objectives. Their preferences over allocations differ with prices, because the principal internalizes the package prize, whereas the agent does not. The differences in the utility functions lead to substantially different strategies. Strategic demand reduction has served as an explanation for bidding behavior in some auctions in the field (Lawrence M Ausubel & Cramton, 2002). However, there are also many examples of auctions where bidders did not reduce demand and the auction ended with very high prices, which are hard to explain by the marginal value of a larger package. We will refer to this phenomenon as price wars. Examples include the British and German spectrum auctions in 2000 (Jehiel & Moldovanu, 2001; Klemperer, 2002), as well as the German spectrum auction in 2010 (Peter Cramton & Ockenfels, 2014), each of which ended in very high prices.1

1

In the German spectrum auction in 2000 six bidders could have reduced their demand to two blocks after the

seventh bidder dropped out such that the auction would have ended at a price of EUR ~2.5 bn per block, but eventually two bidders even continued to fight for the third block until the price reached EUR ~4.2 bn per block. There have been different explanations (Grimm, Riedel, & Wolfstetter, 2001). Jehiel and Moldovanu (2001, p. 16) describe it as “bizarre,” and question that the outcome is in equilibrium. (Klemperer, 2001) compares the aborted

We define the moral hazard problem by the use of a hidden action model. The principal cannot observe his agent’s bids in the auction and consequently cannot write a contract conditioning on ex-post observable actions. The complexity of bidding strategies and the environment in high-stakes auctions is typically such that it is almost impossible for the principal to control every bid of an agent. Furthermore, such a contract might be very time consuming and too costly to set up. Therefore, our assumption of a hidden action model is an appropriate abstraction of the real-world principal-agent relationship. We do not use hidden information to model the present moral hazard problem. A hidden information model would imply the agent to have better information about net present package values than the principal. Although the principal might be uncertain about the net present value of each package, in spectrum auctions, for example, the net present values of packages are defined via the corresponding business cases. The bidding team is unlikely to have more detailed knowledge about the various business projects than the board of directors. In contrast to a hidden information model, the hidden action model also allows us to define an optimal strategy of the principal based on the standard assumption of independent private values, which can then be compared to the strategies of the agents. We assume risk averse agents such that the principal cannot use a profit-sharing contract to align the agent’s incentives with his own (Wolfstetter, 1999). Risk-aversion is a reasonable assumption in high-stakes markets such as spectrum auctions. The bidding team might simply

attempt to acquire large licenses as irrational behavior. Also, in the German spectrum auction in 2010, there was little disagreement in the early rounds. 234 rounds later the allocation was similar to the initial allocation but EUR 4.3 bn. more expensive. These auctions were simultaneous multi-round auctions, which did not allow for package bids. However, bidders also try to win packages of licenses. Large package bids can also be observed in the combinatorial clock auction, however, there are also additional explanations for such behavior in this auction format (Kroemer, Bichler, & Goetzendorff, 2014).

disagree to be made liable for the amount of money involved. Moreover, the owners of the company might not want to share the profit with the bidding team either. Again, setting up such a contract can be very time consuming and costly. Also in high-stakes spectrum auctions profit sharing contracts between the principal and the agent are typically not implemented. The development and future profits of a telecom cannot easily be traced back to a particular allocation or even bid in an auction years later. Engelbrecht-Wiggans (1987, p. 116) notes that “the precise amount of expenditures or receipts may remain uncertain until long after the auction itself”. Finally, the principal will not transfer the entire budget to the agent to provide incentives for profit maximization. If budget in height of the allowance belongs to the agent, the latter cares about remaining money, i.e., profit. This profit can be substantial and the transfer of money might then not be profitable in relation to alternative investment opportunities of the principal. The principal’s opportunity costs of leaving the remainder of the allowance with the agent might simply be too high. We assume that the agent has limited liability and could not afford to win a package in the auction without the budget provided by the principal. In addition, neither the auctioneer, nor another third party would subsidize the agent to achieve efficiency. In summary, the principal can solely determine allowances for his agent’s bidding to align the latter’s incentives with his own. In our model, we analyze a limit case: the net present values of all packages are known with certainty by principal and agent, the agents have no utility for profit and their bids cannot be observed by the principal. This limit case provides a foundation for the understanding of situations where agents might have some residual utility for profit. We argue that this model adequately describes the strategic situation in many high-stakes auctions with bidders being firms participating. This limit case provides a foundation for the understanding of more complex situations where agents might have some residual utility for profit, face different signals about

package values than their principals and their actions are observable. Our model gives managerial advice on how to incentive align the agent by means of simple allowances in multi-unit package auctions. We focus on multi-unit package auctions for several reasons. First, in package auctions bidders can express complementary valuations for multiple units of a good. This is impossible in traditional multi-unit auctions. The exposure problem, is a well-known strategic challenge for bidders in traditional multi-unit or multi-item auctions where bidders have complementary valuations (Rothkopf, Pekeč, & Harstad, 1998). Second, package auctions or (multi-item) combinatorial auctions have found numerous applications in spectrum auctions world-wide (P. Cramton, 2013), for the procurement of bus services for bus routes in London (Cantillon & Pesendorfer, 2006), of school meals in Chile (Olivares, Weintraub, Epstein, & Yung, 2012), and raw materials at Mars Inc. (Bichler, Davenport, Hohner, & Kalagnanam, 2006). Our results for the multi-unit package auctions can easily be extended to multi-item combinatorial auctions for heterogeneous goods. The former simply constitutes a special case of the latter. We focus on ascending package auctions, simple clock auctions, because this format is practically very relevant. Actually, almost all spectrum auctions worldwide are being organized as ascending auctions. We demonstrate a welfare loss in comparison to simple profit-maximizing bidders, if principals do not implement incentive aligning allowance schemes for their agents.

2

The Model and Contributions

In our model we consider

ex-ante symmetric firms , ,

object combinatorial auction for this setting as

×

units ∈

= {1, … ,

∈ = {1, … , } competing in a multi} of a homogeneous good. We denote

package auction. Let us first outline the payoff environment in our model

before we discuss the auctions.

2.1 The Payoff Environment Each firm

is the net present value of firm =[

∈

faces cardinal ex-interim net present package values

1 ,…,

⊆ ℝ ∀ ∈ , where

for the bundle of

units. The vector

] ∈ ℝ contains the net present value draws for all

packages of any firm

. Let us define the valuations vectors of all agents other than i as

. We assume net present
1 bidders. These lemmate are independent of the

allowance scheme with which the agent is provided. Lemma 1: It is a weakly dominant action for any agent to remain active for the package of two units until its price reaches his respective allowance in the 2 ×

ascending package auction:

+ 2 =/ 2 . Proof: For opponents’ fixed bids + , suppose agent quits from bidding on the bundle of 2 units at a price strictly lower than his allowance. Remaining active until the price reaches his

corresponding allowance, + 2 = / 2 , instead, does not reduce his payoff; he still obtains a utility of 3

2

when winning. Moreover, this action strictly raises the probability of winning

the respective bundle. This comes at the opportunity cost of proportionately lowering the chances to win the one-unit package. However, valuations are strictly increasing in the number of units, 2 >

1 , and agent ’s utility function 3 ∙ is strictly increasing net present package value

. Thus, the smaller package offers less utility than the larger package. Submitting a bid in excess of his allowance on the two-unit package cannot be optimal for any agent as it is an unacceptable outcome. QED. Lemma 2: Any agent ’s optimal action for the single-unit package in the 2 ×

ascending

package auction is either to not start the clock at all, or to remain active until its price reaches his corresponding allowance: + 1 = / 1 . Proof: Any single-unit bid of agent , + 1 , from the range of (0, / 1 ] does not affect payoff when winning the one-unit bundle. He receives utility of 3

1 . For his opponents’ fixed bids

+ , staying active on one unit until the price reaches his respective allowance, + 1 = / 1 , however, maximizes the probability of coordinating on the small bundle. It thereby minimizes the chances of winning the two-unit package. Moreover, immediately dropping out from bidding on one unit, + 1 = 0, cancels the possibility to win this bundle, and maximizes the probability of winning the two-unit package. If the small package is preferred, a bid of + 1 = / 1 is a weakly dominant action. Otherwise + 1 = 0 weakly dominates any other bid on the single unit. Any bid from within the range of 0, / 1

neither maximizes nor minimizes the probability of

obtaining the one-unit package and therefore cannot be optimal. Again, any bid greater than the allowance is an unacceptable outcome. QED.

Be aware that these Lemmata do not imply agent

to in fact pay his respective allowances.

Remember, any package clock stops when the second last bidder exits and the package belongs to the winning allocation. With these important findings we can now derive the agents’ equilibrium strategies in the second stage of the principal-agent 2 × 2 ascending package auction. Let us assume the principal provides the agent with package-dependent allowances / = [/ 1 , / 2 ] of the form / 1 ≤ / 2 . Note that this allowance scheme contains the overall allowance as a special case and helps to better understand the underlying moral hazard problem. We show that agents have a strong incentive to shade their allowance on the smaller package, but truthfully reveal their allowance for the large package. Let us first introduce an adapted definition of straightforward bidding for agents in the second stage of the principal-agent 2 × 2 ascending package auction model. Definition 1: A straightforward bidding strategy for agents in the second stage of the principalagent 2 × 2 ascending package auction model is defined as follows: An agent begins to bid on the largest package and remains active until he is overbid. As long as he is winning, he does not bid for a smaller package. If he is overbid, he starts his clock for the next smaller package and again remains active until he is overbid. This process continues iteratively for all next smaller packages. An agent prefers a larger package to a smaller one independent of its price, as long as the price is weakly lower than his allowance. Theorem 1: In the second stage of the principal-agent 2 × 2 ascending package auction model, in which any agent

faces package-dependent allowances / = [/ 1 , / 2 ] of the form

/ 1 ≤ / 2 , straightforward bidding constitutes an ex-post equilibrium. In this equilibrium

the agent with the highest allowance for two units does not get active on one unit. He receives the large package at a price equal to the second highest allowance. Proof: Lemma 1 implies both agents to start their clocks on the two-unit package immediately. Each of them remains active until his allowance for two units is reached. According to Lemma 2, each bidder has to decide whether to get active on one unit or not. If a bidder is active, he stays active until his allowance for one unit is reached. Without loss of generality assume agent is the last remaining active bidder for the package of two units. Suppose he decides to start his clock for the single unit. Let also opponent

be active on this unit. Then the sum of both agents’ single-

unit prices might eventually exceed bidder ’s allowance for two units. This cannot happen if agent

does not bid on one unit. In this case, it is a weakly dominant action for bidder

start his single-unit clock. Remember, agent

not to

strictly prefers two units to one unit. Given this

behavior, opponent is in fact indifferent between becoming active and not starting the clock on the single-unit package, as he cannot win anyway. Even knowing the opponent’s values and allowances, no agent can benefit by deviating from his equilibrium strategy. Thus, straightforward bidding constitutes an ex-post equilibrium in the second stage of the principal-agent 2 × 2 ascending package auction model with agents facing allowances / = [/ 1 , / 2 ] of the form / 1 ≤ / 2 . QED. Theorem 1 describes an ex-post equilibrium that is robust against any form of risk-aversion. Moral hazard occurs, because agents have a strong incentive to shade their demand for one unit instead of reducing their demand to the single package. In the 2 × 2 ascending package auction the strongest agent for the double-unit package has an ultimate “veto power” on the dual winner award under straightforward bidding.

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