### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

The focus of this study is on the A+B transportation procurement mechanism, which uses the proposed cost (A component) and the proposed time (B component) to score contractors’ bids. Empirical studies have shown that this mechanism shortens project durations. We use normative models to study the effect of certain discretionary parameters set by state transportation agencies on contractors’ equilibrium bidding strategies, winner selection, and actual completion times. We model the bidding environment in detail including multi-dimensional bids, contractors’ uncertainty about completion times, and reputation cost. The latter refers to a private penalty that accrues to tardy contractors from increased cost of posting bonds and reduced prospects of winning future projects. Our model explains why contractors may skew line-item bids and why winners frequently finish earlier than bid. It has several policy implications as well. For example, we recommend that agencies set the daily incentive, disincentive, and road user cost to be equal and not cap incentives. This is a departure from current practice, where incentives are often capped and weaker than penalties. Furthermore, we show that agencies may be justified in setting daily road user cost strictly smaller than the true cost of traffic disruption during construction.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

The A+B infrastructure procurement mechanism is a project-letting innovation used by state transportation agencies to provide incentives for faster completion. To date, A+B contracts have been quite successful: contractors offer short completion times and frequently complete projects on time or earlier than bid (Herbsman 1995). A+B is a first-price, sealed-bid mechanism that scores contractors on both cost (A component) and completion time (B component), awarding the contract to the bidder with the lowest total score. In issuing the request for bids (RFB), agency engineers specify quantities for each line item of the A component and three time-based parameters: a daily road user cost (RUC) to evaluate the time bid, an incentive rate for early completion, and a disincentive rate for tardiness. RUC is based on estimated cost of longer commute times and economic loss resulting from road closures. It may not equal the above-mentioned estimate, which we call the true RUC. Each contractor's multi-dimensional sealed bid consists of unit prices for line items included in the RFB and a time bid. When bids are opened, unit prices are multiplied by estimated quantities, whose sum yields the A component. The B component is the product of the RUC and the time bid.

The payment to the winner has two parts as well. The first part is the sum of the product of actual quantities consumed in project execution and winner's unit bids. Note that actual quantities may be different from estimated quantities that are used to select the winner. The second part is either an early-completion incentive or a lateness penalty, which is calculated relative to the winner's time bid. Agencies also announce minimum and maximum time bids permitted and the maximum number of incentive days. The latter caps the winner's incentive from early completion.

On the basis of the analysis of an agency's data and interactions with agency engineers and contractors who are experienced in preparing bids for A+B projects, we present a normative model of the A+B bidding environment. The need for such a model arises because some commonly held views in the practitioner literature are not supported by any existing model and because models presented in the auctions literature are not sufficiently detailed to be considered credible by practitioners. Moreover, results from auction theory neither explain nor can be extrapolated to explain several important observations from data, which our detailed model does. We elaborate on both these points after we summarize the key elements of our analytical approach. We decompose the contractor bid parameter selection problem into two parts. In the first part, each contractor picks optimal unit bids and time bid as functions of its private information, unresolved uncertainty about the effectiveness and cost of its efforts to expedite completion time, and total score. An important element of our analysis at this stage is the consideration of contractors’ reputation cost. In the second part, contractors pick equilibrium scores in a competitive bidding environment. Finally, we analyze the agency's problem of choosing time-based parameters.

Gransberg and Riemer (2009) argue that agency engineers’ estimates of required quantities are frequently inaccurate and biased upward. This provides a budgetary cushion for contingencies. The authors claim that this consistent bias in quantity estimates increases project costs and profits for the winning bidder. However, the authors do not present a model to support their claims, leaving open the question whether biased estimates indeed increase agency cost and contractor profit. State agencies often use historical unit bids to prepare budgets for future projects. However, data show significant variation in unit bids across bidders. Another important question for agencies is therefore how reliable are unit bids as estimates of line-item costs?

Agencies use contracts with time incentives to encourage contractors to exert greater effort on faster completion. Ideally, an agency wants the winning bidder to exert effort commensurate with the commuting public's daily cost of delay and avoid excess costs from bidders’ ability to manipulate multi-dimensional bids. The agency also wants to pick the overall lowest cost bidder. Finally, the agency wants the winner to complete the project on time, that is, as bid. Therefore, questions that naturally arise from practitioners' perspectives are as follows: Do contractors exert optimal effort upon winning the bid? Do contractors make excess profits, over and above what competition allows? Does the A+B mechanism pick the lowest cost contractor? Should the agency expect contractors to finish on time?

We develop a normative model that differs from previous attempts at modeling time incentives in construction projects in terms of the following features: (1) multi-attribute private information and multi-dimensional bids, (2) contractors’ uncertainty about the effectiveness of their efforts to expedite completion, and (3) reputation costs. Contractors’ private information consists of actual material quantities, cost and effectiveness of expediting effort, and reputation costs. We model the reputation cost as heterogeneous, private fixed cost, which may be substantial for some contractors. Reputation costs reflect increases in future costs due to completing the current contract later than bid. These stem from both increased costs in bonding and in lower scores on some future contracts.

The above model features together provide answers to the questions posed earlier and add to the auctions literature by revealing insights that cannot be obtained from an intuitive extrapolation of results presented in the literature. For example, previous models in the auctions literature (reviewed in section 'Literature Review') assume that contractors can accurately estimate project completion time and that they do not face reputation costs. Under these assumptions, we show that when the daily incentive rate is strictly lower than the disincentive rate or incentive days are capped, contractors should bid short completion times and finish late. Although this choice of parameters is common in practice, contractors predominantly finish earlier than bid. The auctions literature does not address why contractors finish early. Similarly, when daily incentive, disincentive and RUC are all set equal, previous studies conclude that the winner should exert optimal effort. We show that upon accounting for reputation cost and completion time uncertainty, the winner should exert greater than optimal effort when the previous conditions hold.

We show that the contractor that minimizes total social cost may not win when the incentive and disincentive rates are lower than RUC. Unit bids are not good estimates of contractor costs because, if a contractor's estimate is smaller (respectively, larger) than the engineers’ estimate for a line item, then it should offer a low (high) price for that item. However, which line-item bids a contractor should inflate or deflate also depend on time incentives, completion time uncertainty, and the target A+B score, making it difficult to calculate optimal bids intuitively.

Project engineers typically set the daily incentive rate strictly lower than the disincentive rate and cap total incentive pay. This practice is consistent with the view that incentive pay increases overall cost of the project and contractor profit. Furthermore, data show that contractors often complete earlier than bid even when incentives are relatively small or absent, which might serve to further strengthen engineers’ beliefs. Interestingly, the extant auctions literature on this topic suggests that setting incentive rates strictly lower than the disincentive rate does not affect contractor effort. Our results are significantly different. We show that both under- and over-investment in effort are possible under a variety of choices of incentive, disincentive, and daily RUCs. This insight is only possible because we consider both completion time uncertainty and reputation costs. We also address the question of truthful reporting of completion time estimate. In this sense, the presence of reputation cost makes contractors more likely to finish early or on time, but it also serves to increase time bids, which increases agency's total cost.

After characterizing contractors’ response to auction parameters, we address two questions. How should agencies choose auction parameters? Do strategic bids lead to excess profits for contractors? In response to the first question, we recommend setting all time incentive rates equal and not capping incentive days. This removes contractor incentive to bid short completion times and spend less than RUC on expediting effort in some cases. However, the contractor then exerts more daily effort than RUC in some cases. Therefore, the agency may set RUC strictly lower than the true RUC. This recommendation is consistent with previous empirical work (Lewis and Bajari 2014), but our result is based on a normative model. In response to the second question, we show that contractors’ profits depend only on the competitive environment, primarily the number of bidders. In particular, if agencies set daily rates different from our recommendation, then the lowest cost contractor may not win. Moreover, total cost (the sum of project and public inconvenience costs) may increase or completion may be delayed, but contractor profit is unaffected. Throughout this study, we utilize data from 38 A+B projects (188 bids), which we obtained from the Minnesota Department of Transportation (MnDOT), to motivate research questions and validate our findings. These data are summarized in Appendix A.

### 2 Literature Review

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

There are three bodies of literature related to A+B bidding. These are construction management and practitioner literature, economics literature, and operations research (OR) literature. Significant amount of practitioner literature can be found on the web in the reports published by the Federal Highway Administration (FHWA) and the American Association of State Highway and Transportation Officials (AASHTO) (http://ops.fhwa.dot.gov and www.transportation.org). These publications describe innovative contracting schemes and the experiences of early adopters.

Among construction management articles, Gransberg and Riemer (2009) study the effect of inflated engineers’ quantity estimates, due to the need for contingency budgets,1 on unbalanced unit bids. Unbalancing occurs when a contractor bids an amount that is significantly different from its true cost and markup. The authors argue that quantity inflation leads to higher bids. In contrast, it can be inferred from our model that payments resulting from obvious inaccuracies in quantity estimates would be factored into the bids with no effect on project costs.

Anderson and Damnjanovic (2008) report a survey of state transportation departments’ (DOTs’) employees to evaluate processes adopted by DOTs to select from a set of contracting methods and the effectiveness of different approaches to accelerate project completion. El-Rayes (2001) develops a model for optimum resource utilization in the A+B environment. Herbsman (1995) presents empirical evidence of early completion from the use of the A+B mechanism. In the vast majority of the projects studied there, the completion time was shorter than the time bid. The articles mentioned above ignore strategic bidding, and do not directly address the problems of choosing bid prices and agency parameters.

There is significant economics literature on procurement auctions. Excellent surveys on auctions, bids, and the use of incentives in procurement can be found in Engelbrecht-Wiggans et al. (1983), McAfee and McMillan (1987), Laffont and Tirole (1993), and Krishna (2002). Economics literature also contains studies that are focused specifically on construction procurement auctions (e.g., Hong and Shum 2002, Krasnokutskaya 2011). Therefore, we present a brief survey of these works and focus primarily on studies that analyze the A+B mechanism.

Among relevant studies from economics literature, Ewerhart and Fieseler (2003) model unit-price auctions, which allow for quantity adjustments, Maskin (1992) considers multi-attribute private information, and Asker and Cantillon (2008) and Che (1993) study multi-dimensional auctions with scoring rules. Construction auctions with time incentives are studied in Lewis and Bajari (2014, 2011). Our model has all the features of previous models. In addition, we consider the fact that bidders’ effort costs and actual completion time remain uncertain at the time of bidding and that bidders experience a reputation cost upon being tardy. No previous study has considered all these features simultaneously. Our model and its analysis provide insights, which we present as Observations 1–7 in sections 'Optimal Bid Parameters' and 'Equilibrium Bids'. When presenting these results, we also contrast them to those that follow from previous works.

The vast majority of previous studies in the auctions literature assume that for each bidder, all private information is known with certainty before it submits its bids. Esö and White (2004) and Lu and Perrigne (2008) extend auction models discussed above by explicitly incorporating ex-post uncertainty in deriving bidding strategies. This is also an important aspect of construction auctions, and we build on this research in our model. Studies in the auctions literature also typically assume independent private values (IPV) for reasons of tractability (Asker and Cantillon 2008, Che 1993, Ewerhart and Fieseler 2003, Lewis, and Bajari 2014, 2011). In contrast, we consider the more general affiliated private values (APV) framework. The APV assumption applies when contractors estimate their costs based on their signals, which are accurate reflections of true costs, but the signals themselves may be affiliated across contractors. For example, if there is ample supply of construction materials and labor for a project, then that reduces cost for all contractors. IPV arise as special cases of APV in which the affiliation across contractors’ signals are negligibly small. Evidence of IPV in pavement rehabilitation projects that are typically let using the A+B mechanism can be found in Hong and Shum (2002) and Krasnokutskaya (2011).

Among studies focusing on time incentives in construction contracts, Lewis and Bajari (2014) estimate the benefits to commuters from expediting, and compare these to the marginal cost curve they derive from data on A+B contracts in California. Based on their empirical analysis they recommend expanding the use of A+B to all projects, but scoring bids using a RUC much lower than commuters’ true cost. As a basis for estimating contractors’ cost, the authors also propose a relatively simple theoretical model for A+B bidding. Lewis and Bajari (2014) assume scalar private information that reveals to each contractor the relationship between its chosen actual completion time and project cost. They do not model unit bids, contractor effort, reputation cost, and possible discrepancy between time bids and actual completion times. Their focus is on evaluating when A+B contracts are *ex-ante* efficient and when they are ex-post efficient. For the latter, it is required that the RUC used for scoring the B component be the true RUC. In addition, to assure that contractors would expedite sufficiently to maximize social welfare, the disincentive rate should be no less than this RUC, although the incentive rate might be lower. For *ex-ante* efficiency, the authors find that awarding a contract based on a scoring rule, incorporating future incentives and penalties, is sufficient, assuring the contract is awarded to the bidder that maximizes social welfare.

The purpose of our study is to model the bidding environment in detail and to explain how contractors decide their time bids and how these may differ from actual completion time as well as from engineers’ estimates. In contrast, Lewis and Bajari (2014) are concerned only with the latter because they assume that contractors ex-ante choose actual completion time. This means that Lewis and Bajari's model cannot explain why contractors frequently finish earlier than bid, earning incentives.

Lewis and Bajari (2011) expand on the framework of Lewis and Bajari (2014) by incorporating uncertainty that is unresolved at the time of bidding. However, this study is confined to standard (A-only) contracts. Through an empirical analysis of standard projects, the authors model the impact of uncertainty on work rate to estimate contractors’ responses to uncertainty. They assume a quadratic cost-of-expediting function, and that capital and equipment are fixed for the duration of the project but labor investments may vary. In contrast, we model a general expediting cost function, potentially with capital as well as labor components, and evaluate the impact of incentives in A+B bidding environments.

Recently, several OR studies have investigated alternate mechanism design for multi-attribute procurement auctions to achieve specific objectives—see. e.g., Beil and Wein (2003), Parkes and Kalagnanam (2005), and Kostamin et al. (2009). However, the objectives and mechanisms described in these studies (e.g., multi-round auctions) are either not relevant for transportation construction auctions or are not permitted under FHWA and state regulations, which typically require the use of single-round, first-price, sealed-bid auctions for the award of such contracts.

### 3 Notation and Formulation

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

We present the sequence of events in Figure 1, and the notation used to formulate a model of A+B bidding in Table 1. All parameters in Table 1 are real valued (denoted by ℜ) unless stated otherwise. Analysis of the model requires additional notation, which is presented in Table 2. We shall describe all model variables in the ensuing discussion. We provide our variable naming scheme first to make it easier to read the paper: *s* for scores, *σ* for private signals, *q* for quantities, *c* for costs, *b* for bids, *t* for time parameters or decisions, and *ξ* for uncertainty. We use subscript *I* for incentive, *D* for disincentive, *U* for specified RUC, and *R* for reputation cost. The detailed model requires us to use sub- and superscripts, for example, *e* for engineers’ parameters, and over- and underlines to denote, respectively, the lower and upper bounds on the feasible values of either a parameter or a calculated threshold.

Table 1. Notation Used in Model FormulationAgency | | Contractor (index suppressed) |
---|

*General parameters* |

*n* = | Number of line items that are bid on a per-unit basis, *n* ≥ 1 & integer (*) | *ν* = | Number of bidders, *ν* > 1 & integer (*) |

| | *c*_{R} = | Contractor's private reputation cost from late completion, *c*_{R} ≥ 0, *c*_{R} ∈ ℜ |

*Time point 1: request for bids posted* |

Parameters |

*q*^{e} = | ( Engineers’ quantity estimates, and , , 1 ≤ *i* ≤ *n* (*) | *σ* = | (*q*, *c*, *c*_{R}, *t*(*γ*, *ξ*),*h*(*γ*, *ξ*)) = Contractor's private information, *σ* ∈ Σ |

*c*^{e} = | ( Agency's estimates of line-item costs, , , 0 ≤ *i* ≤ *n* | *ξ* = | Uncertain component of completion time and expediting cost, *ξ* ∈ Ξ |

| Acceptable ranges of unit prices, , (*) | *c* = | (*c*_{0},…,*c*_{n}) = Contractor's estimates of line-item costs, *c*_{i} > 0, *c*_{i} ∈ ℜ, 0 ≤ *i* ≤ *n* |

*t*^{e} = | Engineers’ estimate of project completion time, *t*^{e} > 0, *t*^{e} ∈ ℜ | *q* = | (*q*_{0},…,*q*_{n}) = Actual quantities used, *q*_{0} ≡ 1 and *q*_{i} > 0, *q*_{i} ∈ ℜ, 1 ≤ *i* ≤ *n* |

*c*_{T} = | True daily RUC, *c*_{T} ≥ 0 (*) | | |

Decisions |

*c*_{U} = | Announced daily RUC,0 < *c*_{U} ≤ *c*_{T}, *c*_{U} ∈ ℜ (*) | *γ*(*ξ*) = | Contractor's expediting effort, *γ* ≥ 0, *γ*(*ξ*) ∈ ℜ for each *ξ* |

*c*_{I} = | Announced daily early-completion reward, *c*_{I} ≥ 0, *c*_{I} ∈ ℜ (*) | *t*(*γ*, *ξ*) = | Completion time as a function of effort *γ* and uncertainty *ξ*,* t*(*γ*, *ξ*) ≥ 0, *t*(*γ*, *ξ*) ∈ ℜ |

*c*_{D} = | Announced daily lateness penalty, 0 < *c*_{D} ≤ *c*_{U}, *c*_{D} ∈ ℜ (*) | *h*(*γ*, *ξ*) = | Expediting cost as a function of effort *γ* and uncertainty *ξ*,* h*(*γ*, *ξ*) ≥ 0, *h*(*γ*, *ξ*) ∈ ℜ |

*t*_{I} = | Maximum number of incentive days, *t*_{I} ≥ 0, *t*_{I} ∈ ℜ (*) | | |

(*t*, | Announced minimum and maximum bid days, , (*) | |

*Time Point 2: sealed bids opened and winner selected* |

*s* = | Contractor's score (=*c*_{U}*t*^{B} + *b*·*q*^{e}) (*) | | |

Decisions |

Announce winner, the lowest score bidder (*) | *t*^{B} = | Contractor's time bid |

| *b* = | (*b*_{0},…,*b*_{n}) = Contractor's unit bids |

*Time Point 3: project executed* |

Agency realizes costs | *ξ* is realized |

Materials & Labor costs (*) | Contractor finishes in *t*^{*}(*ξ*) if *c*_{R} > 0, and in if *c*_{R} = 0 |

Time cost = *c*_{U}*t*^{*}(*ξ*)+*c*_{I}max{*t*_{I},(*t*^{B}−*t*^{*}(*ξ*))^{+}} − *c*_{D}(*t*^{*}(*ξ*)−*t*^{B})^{+} (*) | |

Table 2. Additional Notation Used in Model Analysis |

*t*_{1}(*ξ*) = | ; completion time when upon observing *ξ*, the contractor chooses effort rate *c*_{I} |

*t*_{2}(*ξ*) = | ; completion time when upon observing *ξ*, the contractor chooses effort rate *c*_{D} |

*t*(0, *ξ*) = | ; completion time when upon observing *ξ*, the contractor spends no extra effort |

| Given *ξ*, the smallest time bid at which the contractor will finish on time |

*η*(*τ*) = | ; the rate of change of expected profit as a function of *t*^{B} give *b*,* s*, and *σ* |

*λ*_{i} = | , the ratio of actual to estimated quantity for line-item *i* |

*x*(*σ*) = | Zero-profit bid score, *x* > 0, *x* ∈ ℜ |

*F*(·) = | Joint cumulative distribution of zero-profit scores observed by the *ν* contractors |

*f*(·) = | Density function of *F*(·) |

*Y*_{1} = | Lowest zero-profit score among other contractors indexed 2,· · ·, *ν* |

| Cumulative distribution of *Y*_{1} conditional upon contractor 1's signal being *x* |

| Density function of |

*p*(a|*x*) = | |

For ease of exposition, Table 1 divides notation into three time points, which correspond to three distinct events; namely, the posting of the RFB, the submission of sealed bids and the selection of the winner, and the realization of projects costs for the agency and either profit or loss for the contractor who won the contract. General parameters that are not necessarily related to a particular time point are presented first. Within each time point, we show agency and contractors’ parameters and decisions separately. Common-knowledge parameters and parameters that are revealed at each time point are denoted by an asterisk (*). Contractors are assumed to be risk neutral. Throughout this section and in sections 'Optimal Bid Parameters' and 'Equilibrium Bids', we refer to a tagged (arbitrary) contractor as “the contractor” and do not use contractor index.

The number of bidders *ν*, the line-item list of all pieces of work (*n*), and contractors’ reputation costs are considered general problem parameters. At Time Point 1, agency posts a RFB, which includes a detailed product design and engineers’ estimates of materials (*q*^{e}) required to complete each line item. There are two types of line items—those that are bid on a lump-sum basis (i.e., quantity equals 1) and those that are bid on a per-unit basis. Examples of lump-sum items are mobilization costs, contractors’ costs of setting up field offices, traffic control systems, and construction area signs. In this study, we put all lump-sum items into a single category () and call them mobilization costs. Mobilization costs are a large fraction of the A component. For items that are bid on a per-unit basis, agency engineers may be uncertain about the amount of materials that will be needed to complete each task at the time of issuing a RFB. However, agencies announce firm estimates because quantity estimates are needed to evaluate bids. In fact, agencies do not expect the project to consume precisely the estimated amount of materials for each line item, and routinely set aside contingency funds to account for discrepancies that arise.

Agency engineers also estimate parameters that are not posted along with the RFB. These are the lump-sum and per-unit costs (*c*^{e}) for all line items, the acceptable ranges of unit bids, and for item *i*, true daily RUC *c*_{T}, and engineers’ estimate of project completion time *t*^{e}. Agency decisions are the time-based parameters: daily incentive *c*_{I}, daily tardiness penalty *c*_{D}, daily RUC *c*_{U}, the maximum number of incentive days *t*_{I}, and bounds on the time bid, denoted by and . If a contractor's time bid *t*^{B} is not within these bounds, then its bid is disqualified. In a typical A+B project, the announced RUC (*c*_{U}) is less than or equal to the true RUC (*c*_{T}) to ensure that no legal challenges arise as a result of choosing unsupported values of daily RUC.2 Also, the incentive rate (*c*_{I}) is typically no larger than the disincentive rate.

Acceptable ranges of unit bids are used to determine which bids are unbalanced. If a contractor bids very high unit prices on some items (*b*_{i} greater than for item *i*) and very low unit prices on others (*b*_{i} lower than for item *i*), then the bid is declared unbalanced. Such bids are disqualified. Increasingly, bids are submitted via the Internet, which allows contractors convenient access to the range of prices bid by contractors on similar items in earlier projects. Therefore, unbalanced bids are rare—there were no unbalanced bids in our sample of 38 projects, which contained a total of 188 bids. We use these facts to justify our assumption in section 'Notation and Formulation' that contractors know the ranges of values of unit bids that would be considered acceptable. In this study, each contractor's estimates of quantities needed (*q*) and unit costs (*c*) are assumed to be private and affiliated. Note that this includes as a special case the independent private values assumption commonly encountered in the literature—(see Hong and Shum 2002, and Krasnokutskaya 2011).

Contractors can affect project completion date by engaging in activities such as working longer hours or double shifts, working through weekends and holidays, and using extra crews, prefabricated components, and equipment capable of producing higher daily output. Their time bids take into account competition and both positive and negative consequences of bidding an amount different from their initial estimate. Shorter time bids make proposals more competitive, but upon finishing later than bid, contractors pay a penalty to the agency, and also experience a private tardiness penalty, which we refer to as the reputation cost. Therefore, they also need to factor cost of expediting, number of concurrent projects, availability of working capital, and uncertainty about subcontractors’ completion times in their decisions. We assume that functions that relate a contractor's expediting effort (*γ*) and uncertainty (*ξ* ∈ Ξ) to cost (*h*(*γ*, *ξ*)) and actual completion time (*t*(*γ*, *ξ*)) are private information and independent of such estimates by other contractors. The contractor learns its private information at Time Point 1 in the form of signal *σ*. Consistent with practitioner literature, we assume that *t*(*γ*, *ξ*) is decreasing convex in *γ* (i.e., *t*^{′}(*γ*, *ξ*) < 0,*t*^{′′}(*γ*, *ξ*) ≥ 0) and *h*(*γ*, *ξ*) is increasing convex (i.e., *h*^{′}(*γ*, *ξ*) > 0, *h*^{”}(*γ*, *ξ*) ≥ 0), for each fixed *ξ* (see, e.g., Pyeon and Park 2010, Shr and Chen 2004, Sillars and Riedl 2007).

At Time Point 2, sealed bids are submitted and the winner is selected. Contractors typically submit bids just before the due date. We assume, therefore, that bid submission and winner selection coincide. The contractor's decision consists of bid parameters: that is, time *t*^{B} and unit prices *b* = (*b*_{0},…,*b*_{n}). Each bid is evaluated on the basis of its score *s*(*b*, *t*^{B}) = *c*_{U}*t*^{B} + *b*·*q*^{e}, and the lowest score bidder wins. Finally, at Time Point 3, uncertainty about *ξ* is resolved, the winner chooses actual completion time, and the agency realizes project cost.

Because reputation cost *c*_{R} plays an important role in our model, we describe its origin next. Before project award, winners are required to post bonds equal to the bid prices to assure completion. Nearly all contractors use approved third-party sureties to post bonds. Sureties use contractors’ performance history including all instances of having paid delay costs (also called liquidated damages [LD]) to agencies to assess their credit worthiness and set bond rates3. Lower credit worthiness lowers the bonding offered by the surety, and also increases the bonding cost. Contractors estimate that frequent project delays could increase their bonding costs by between 0.75% and 1.5% of their entire project portfolio. It is this cost that we model as a heterogeneous and private fixed cost. In addition, previous instances of liquidated damages are included in some bid evaluations by agencies. Past occurrences of project delays penalize contractors when bidding in Design-Build projects.

Although uncertainty about effort cost and how effort affects completion time may unfold continuously over the course of the project, for the purpose of modeling, we represent it as a realization of *ξ* at a single time point.4 That is, the pair (*γ*, *ξ*) uniquely determines the completion time *t*(*γ*, *ξ*). Note that for each realization of *ξ*, the choice of effort *γ* is equivalent to the contractor choosing a completion time *t*. In a slight abuse of notation, we now perform a variable transform such that the argument of *h*(*γ*, *ξ*) is changed to *h*(*t*, *ξ*). Because the original function *h* is increasing convex in *γ* and *t*(*γ*, *ξ*) is decreasing convex in *γ*, the transformed function *h*(*t*, *ξ*) is decreasing convex in *t* for each fixed *ξ*. We denote optimal completion time by *t*^{*}(*ξ*).

After the winner completes the work, it receives a payment of *b*·*q* for labor and materials and time-based incentive or disincentive of *c*_{I}min{*t*_{I},(*t*^{B} − *t*)^{+}} − *c*_{D}[(*t* − *t*^{B})^{+}], where *t* is the actual completion time. The contractor's conditional profit function, , which is the amount that the contractor with signal *σ* will earn if it wins the project upon bidding *s* and *ξ* realizes, can be written as follows:

- (1)

where *b* and *t*^{B} are related according to the equation *b*·*q*^{e} + *c*_{U}*t*^{B} = *s*. The term is the contractor's private reputation cost if it completes later than *t*^{B}. The term is the incentive payment, which is capped at *c*_{I}*t*_{I}, whereas is the disincentive payment. Both sets of terms cannot be simultaneously non-zero. We refer to these terms as time-related windfall. Similarly, *b*·*q* − *b*·*q*^{e}, the difference between the actual payment for material and labor and the A component is the windfall due to unit bids, which may be positive or negative.

Note that if daily incentive and penalty are equal (*c*_{I} = *c*_{D}), reputation cost *c*_{R} = 0, and the maximum number of incentive days *t*_{I} is not capped, then is a smooth and differentiable function of *t*, the actual completion time. However, *c*_{I} ≠ *c*_{D} causes there to be a kink in , and *c*_{R} > 0 causes a discontinuity at *t* = *t*^{B}. In absence of *c*_{R}, if *c*_{I} = *c*_{D} = *c*_{U}, the contractor's conditional profit function is independent of *t*^{B} and it should choose to exert effort at a daily rate of *c*_{U}. Similarly, if *c*_{I} < *c*_{D} ≤ *c*_{U}, it can be shown that the contractor will bid and finish late. However, the presence of *c*_{R} serves to make the calculation of optimal completion time more complex. We present this analysis next.

### 5 Equilibrium Bids

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

Upon substituting the value of (*b*^{*}, *t*^{B*}) from Theorem 1 and optimal completion time from Proposition 1 into Equation (10), we obtain the contractor's optimal expected profit if it chooses bid score *s* upon observing *σ* and wins. We write this expression below.

- (18)

From Equation (18), we obtain certain properties of the contractor's expected profit as a function of bid score *s*. First, the conditional expected profit is monotone increasing in *s*. This follows from the fact that if *s*_{1} < *s*_{2}, then the optimal bid parameters under *s*_{1} are feasible under *s*_{2} after the contractor increases *b*_{0} by *s*_{2} − *s*_{1}. Each extra dollar budgeted in *b*_{0} increases the contractor's profit by the same amount. Second, we observe that the contractor's expected profit may increase more than linearly when *s* is small. This happens because for a small bid score, the contractor does not have sufficient budget to benefit fully from discrepancies in material quantities and time incentives. However, when *s* is sufficiently large, the contractor's profit increases linearly in *s* because larger *s* causes to increase, while , *i* ≥ 1 and *t*^{B*} remain invariant. Also, in such cases, *b*_{0} is strictly greater than 0.

Upon examining the MnDOT data, we found that *b*_{0} is never zero in all bids tendered. This suggests that *c*_{0} is sufficiently large that discrepancies in material quantities and time incentives do not cover contractors’ mobilization costs. Therefore, they choose *s* in a region where *b*_{0} > 0 after taking into account discrepancies in estimates of material quantities and time incentives. In this range of values of *s*, the contractor's profit is linear in *s*. That *π*(*s*|*σ*) is linearly increasing in *s* is implicitly assumed in all previous transportation procurement auction studies although this assumption is not explicitly stated because previous studies do not model multi-attribute private information and multi-dimensional bids. Our analysis helps clarify that although such an assumption is justified in practice, it is tantamount to assuming that contractors face large mobilization costs. We also make this assumption for tractability.

Using Equation (18) and upon solving the equation *π*(*s*|*σ*) = 0, we obtain a zero-profit score for an arbitrary contractor. We denote this quantity by *x*(*σ*). That is, for each signal that the contractor draws there is a corresponding zero-profit score *x*(*σ*). Note that no rational bidder will bid less than *x*. For the purpose of determining equilibrium bid *s*, we can interpret the draw of the signal *σ* as a draw of the zero-profit score *x*, where the latter is obtained after applying the results of Theorem 1. Although a particular *x*(*σ*) may be obtained by many different combinations of bid parameters, it is sufficient for the contractor to know its zero-profit score. In effect, we demonstrate how multi-attribute private values are converted into an equivalent scalar parameter under optimal bidding, and the requisite conditions.

We use *F*(*x*_{1},*x*_{2},…,*x*_{ν}) to denote the joint cumulative distribution function (CDF) of signals received by the *ν* contractors, with support , and density function (PDF) *f*(·). We assume that *x*_{i}'s may be affiliated, which means that a smaller realized zero-profit score for one contractor makes lower zero-profit scores for other contractors more likely (see, Milgrom and Weber 1982). After the contractor receives its private signal *x*, its conditional profit function for a bid *s*, if it were to win, can be written as Π(*s*, *x*) = *s* − *x* + *ϵ* and its conditional expected profit as *π*(*s*, *x*) = *s* − *x*. Put differently,

- (19)

has five terms, where each term accounts for the difference between an expected and a realized value of either a cost term or a windfall term. We call *ϵ* the random component of a contractor's conditional profit function and use *D*(*ϵ*) to denote its distribution function. Note that *E*(*ϵ*) = 0. The random component of the conditional profit stems from the uncertainty about *ξ*, which affects *t*_{1}, *t*_{2}, and *t*^{*}. However, the effect of these parameters is additive to the conditional expected profit because *ξ* is independent of bid parameters *b*^{*} and *t*^{B*}, and , *i* ≥ 1, and *t*^{B*} remain invariant in *s* ≥ *x*.

With risk-neutral contractors an additive random component is incorporated in a straightforward manner into the bid, as we show below via informal arguments (see Milgrom and Weber 1982 for complete details). Also consistent with the auctions literature, we restrict attention to symmetric, monotone bidding strategies, characterized by the bidding function (or strategy) *s*(*x*). In terms of our earlier discussion, *s*(*x*) is the function that returns the bid score that a bidder will target given that its zero-profit score is *x*. This also means that a bidder with signal *x* bids *s*(*x*) and upon winning earns a conditional expected profit equal to *s*(*x*) − *x*. From Milgrom and Weber (1982) there exists a unique Nash equilibrium strategy—(see Esö and White (2004) and Lu and Perrigne (2008) for the case of stochastic outcomes).

Suppose the tagged contractor is indexed 1. Denoted by *Y*_{1} the lowest zero-profit score among the (*ν*−1) contractors indexed 2,⋯,*ν*. Suppose that each of these bidders submits a bid of *s*(*y*) upon observing its private cost signal to be *y*. Because *Y*_{1} and *x* are affiliated, we use and to denote the CDF and the PDF of *Y*_{1}, conditional upon contractor 1's signal being equal to *x*. We also define

- (20)

because it helps explain how contractors choose their A+B scores in Proposition 2.

Proposition 2. An equilibrium bidding strategy, given contractor 1's signal equals *x*, can be obtained as follows:

- (21)

A proof of Proposition 2 is included in the online Appendix S2. The first term in Equation (21) is the contractor's break-even bid score and is the maximum zero-profit score among all bidders. The second term shows how much the contractor *shades* its bid, given auction parameters and the level of competition from other bidders, which depends on both the number of bidders and the affiliation among their signals. The term *p*(*a* | *x*) captures the effect of signal affiliation and competition. If the cost signals were not affiliated, then the second term above would simplify to the conditional expected zero-profit score of the bidder with the second lowest score minus *x* given that the tagged contractor's score *x* is the lowest score. In that case Equation (21) would give the optimal bidding strategy in the IPV setting. Affiliation among contractor signals causes each bidder to submit less aggressive bids (higher values of *s*) because when *x* is high, the tagged contractor would assume higher values of zero-profit scores of other contractors as well.

Observation 7. Equilibrium bidding strategy identified in Proposition 2 implies that competitive pressures, not time incentives, determine contractor profit. Anticipated windfall is incorporated into bids and *x*(*σ*). For example, setting *c*_{I} < *c*_{D} may impact expediting, sorting of contractors, and agency cost, but it does not substantially impact profit for the winning contractor. The latter are determined primarily by the number of bidders. Similarly, inflated estimates by engineers to provide budget for contingencies generate anticipated windfall for contractors, but do not increase project cost.

### 6 Policy Implications

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

This section summarizes key policy implications of results obtained in sections 'Optimal Bid Parameters' and 'Equilibrium Bids', and compares A+B mechanism with some alternate ways in which agencies may provide time incentives. We begin with policy implications.

- Multi-dimensional bids allow contractors to choose unit bids strategically. Our analysis suggests that line-item bids are not reliable estimates of true cost. Outside estimates of line-item costs are justified. This statement follows from Observation 4.
- Our key recommendation to agencies is setting
*c*_{I} = *c*_{D} = *c*_{U} and eliminating the cap on incentive days. The common practice of setting *c*_{I} < *c*_{D} ≤ *c*_{U} < *c*_{T} and capping incentives frequently leads to under-investment in expediting, relative to socially optimal benchmark (see Observation 2). This has important implications for bid sorting. A contractor who may have the lowest score under the socially optimal benchmark, may realize a higher score when time-based incentives are weak. This, in turn, may lead to awarding the project to an inefficient contractor. In this regard, the A+B mechanism shifts profit across contractors. Our recommendation eliminates that possibility. - Uncertainty in the relationship between effort and completion time can lead to either under- or over-investment in effort, relative to the socially optimal benchmark. By setting
*c*_{I} = *c*_{D} = *c*_{U}, the agency eliminates incentives for the contractors to under invest. However, in some cases contractors invest more than *c*_{U} to avoid reputation cost *c*_{R}. This increases agencies’ costs. The under- and over-investment consequences under different values of time-based parameters follow from Observations 2, 5, and 6. - Reputation costs increase time bids. By setting
*c*_{I} = *c*_{D} = *c*_{U}, agencies reduce contractor incentives to bid short completion times and complete late. As a result, contractors are more likely to complete as bid. This statement follows from Observation 6. - Bid shading by contractors, when placing their bids, is not due to the structure of the A+B mechanism. Profit, conditional on winning the auction, is determined by the level of competition. This statement comes from Proposition 2 and Observation 7.

The second statement above suggests that agencies may be justified in choosing *c*_{U} < *c*_{T}. Setting *c*_{U} < *c*_{T} is a common practice, which is justified by arguing that the inconvenience cost to the public does not come out of the agency's budget, but increasing incentives increases expediting effort and direct project costs. Our model provides a different explanation that applies even in those situations where the true cost of inconvenience to traveling public could be monetized.

The difficulty of finding an optimal *c*_{U} arises because agencies need to estimate the amount by which reputation costs affect *t*^{B} and project costs. Since reputation costs vary by contractor, the optimal incentive level is contractor-specific. By choosing *c*_{I} = *c*_{D} = *c*_{U} = *c*_{T}, the agency gives a competitive advantage to contractors with lower expediting costs, but may pay more than *c*_{T} per day for expediting effort. If the agency sets *c*_{I} = *c*_{D} = *c*_{U} < *c*_{T}, then contractors expedite less with the following implications. First, completion may be delayed relative to the socially optimal benchmark. Second, a contractor with higher construction cost and lower expediting costs is less competitive when facing *c*_{I} = *c*_{D} = *c*_{U} < *c*_{T}, potentially changing the sorting of contractors relative to daily rates of *c*_{T}. This may cause the lowest cost contractor (under *c*_{T}) to lose the contract, when facing rates equal to *c*_{U}. As implied by Proposition 2 misspecification of incentive rates does not increase profit for the winning contractor. Unfortunately, actual expediting effort remains random and agency may need to resort to simulation to find an optimal value of *c*_{U}.

The analysis of line-item bids in Theorem 1 indicates that when contractors’ quantity estimates differ from engineers’ estimates, line-item bids are extreme. This raises the possibility of modifying the A component of bids to lump-sum and not allowing quantity adjustments based on actual usage. While this change eliminates windfall payments associated with biased bids, differences between engineers’ estimates and actual usage remain, with contractors bearing this risk. To limit possible losses, contractors would raise initial bids for the A component. Moreover, substantial changes in line-item quantity estimates would lead to frequent renegotiation, lengthening projects. For these reasons, both agency engineers and contractors favor unit bids over lump-sum.

Next, we discuss another contracting mechanism, referred to as the *Lane Rental* mechanism. In that case, the agency chooses the lowest cost bidder based on materials and labor costs, but charges the winning bidder a fee of *c*_{U} = *c*_{T} per day that it takes to complete the project. Contractors are not required to propose a time bid. Reputation costs do not play a role anymore because there is no fixed benchmark, and consequently contractors exert an optimal amount of effort. Lane rental contracts are also commonly used. Although agencies avoid the under- or over-investment problem by choosing the Lane Rental mechanism, they do not receive any signal about what completion time to anticipate. Depending on the need to schedule contingent follow up activities (e.g., next phase of construction), this may make lane rentals less desirable than A+B contracts. That is, although it is difficult to guarantee an optimal amount of expediting effort in A+B contracts, they do create incentives for contractors to bid competitively and to complete either on time or early.

### 7 Concluding Remarks

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

The need for substantial investments in transportation infrastructure enhancements and commuters’ inconvenience from road closures necessitate the use of innovative mechanisms to affect faster rehabilitation. This study analyzes in depth the A+B mechanism, which provides time-based incentives and has been demonstrated to shorten project duration. This is the first study to comprehensively analyze A+B bidding strategies in a normative model with multi-attribute private information and multi-dimensional bids, residual uncertainty at the time of bidding, and reputation costs. In particular, our analysis models reputation cost explicitly, which allows us to establish that agencies cannot avoid one of two negative consequences. If they set *c*_{I} = *c*_{D} = *c*_{U} < *c*_{T}, they ensure that in every realization of residual uncertainty, the winning bidder incurs effort cost of at least *c*_{U} per day. However, the A+B mechanism may not pick the lowest cost bidder and contractor may exert effort less than *c*_{T} delaying completion relative to the socially optimal benchmark. Alternatively, if agencies set *c*_{I} = *c*_{D} = *c*_{U} = *c*_{T}, then because of the presence of reputation costs, the actual effort cost may exceed *c*_{T}, project completion may be shortened more than the optimal, and direct project costs may be higher. Given that inconvenience cost to the commuting public is difficult to monetize, these observations lend support to the practice by agencies to choose *c*_{U} < *c*_{T}.

Our analysis of contractors’ response to auction parameters helps inform agencies such as MnDOT about their current practices. First, setting *c*_{I} < *c*_{D} does not guarantee optimal effort in every realization of residual uncertainty and creates an incentive to bid short completion times. This practice is not recommended. Second, if *c*_{I} = *c*_{D} = *c*_{U} but *c*_{U} < *c*_{T}, it is possible that the overall low-cost contractor is not awarded the project. Although agencies such as MnDOT should set *c*_{I} = *c*_{D} = *c*_{U}, they may be justified in setting *c*_{U} smaller than *c*_{T} because the presence of reputation costs may lead to greater effort on reducing completion time than what the agency intended. Cost of such efforts is passed on to the agency. Reputation costs are, to some extent, dependent on environmental factors and agencies may change policies to reduce the effect of forces exerted by reputation costs. For example, if A+B were to become a more prevalent contracting mechanism, agencies may not treat completion delays as liquidated damages, affecting how bonding companies weigh tardiness in A+B projects when assessing contractors’ credit worthiness. Third, agencies should expect extreme bids on at least some line items and limit the use of historical bids as estimates of reasonable line-item costs for future bids.

How may an agency design a contract that provides the appropriate incentive for expediting, picks the lowest cost bidder, and gives the agency a reasonably accurate signal about the anticipated completion time? In section 'Policy Implications', we argue that a lane rental contract will create appropriate expediting incentives and pick the lowest cost contractor. However, it does not provide any signal to the agency about the contractors’ estimate of completion time, which depend on its private cost of expediting. In that sense, it may not be appropriate for projects where contingent activities need to be scheduled in advance. In addition to the mechanisms discussed in the study, we examined several alternate contract designs in the context of single-round sealed bids. Such analyses are not reported in the study because we found that all of the agency's objectives cannot be met. The analysis of more complicated bidding environment, for example, those involving multiple rounds of bidding is beyond the scope of this study.

### C. Proof of Theorem 1

- Top of page
- Abstract
- 1 Introduction
- 2 Literature Review
- 3 Notation and Formulation
- 4 Optimal Bid Parameters
- 5 Equilibrium Bids
- 6 Policy Implications
- 7 Concluding Remarks
- Acknowledgments
- Appendix: A: Summary of MnDOT A+B Bids’ Data
- B: Parameters for the Example in Figure
- C. Proof of Theorem 1
- References
- Supporting Information

If , then there must exist a *λ* ⩾ 1 such that . From (24) we find that for all *i* = 1,⋯,*n*. This implies that: (1) If , then because *μ*_{i,1} − *μ*_{i,2} > 0, *μ*_{i,j} are non-negative and only one of the *μ*_{i,j}'s can be strictly positive. (2) And, for reasons similar to those mentioned above, if , then .

In addition to requiring *λ* ⩾ 1, we observe that value of changes only when *λ* belongs to the set {*λ*_{1},⋯,*λ*_{m}}, where *m* ⩽ *n* is the number of items for which . That is, for the purpose of choosing optimal bid parameters, we need to consider at most *n* possible values of *λ*. Suppose we pick . If for all *i*, then set *m*_{0} = 0. Then, we claim that *m*_{0} must be such that for all *i* < *m*_{0}, for all *i* > *m*_{0} and

- (31)

If *m*_{0} = 0, then (31) gives the value of . Otherwise, if the value of calculated in (31) is such that , then . If , then , where the latter can be calculated from the fact that . Finally, if , then the current choice of *t*^{B*} is infeasible. We are now ready to present an algorithm that can be used to find an optimal set of bid parameters for each fixed *s*. The algorithm searches over possible values of *m*_{0}.

If *t* is a potential solution, then the above algorithm will return a solution. Otherwise, we conclude that *t*^{B*} cannot equal *t*. Note that this occurs when either *λ*_{1} ≥ 1 and , or *λ*_{1} < 1 and .

In this case *γ*_{1} = *γ*_{2} = 0, *λ* ⩾ 1 and *t*^{B*} is determined from the equality *η*(*t*^{B*}) = *λc*_{U}.

Case (a): *λ* ∈ {*λ*_{1},⋯,*λ*_{m}}. For each value of *λ* ⩾ 1, there is a corresponding value of *t*^{B*}, which comes from the equality *η*(*t*^{B*}) = *λc*_{U}. Using (24), we have that if , then for all *i* < *m*_{0}, for all *i* > *m*_{0}. [If *λ*_{1} < 1, then the value of *t*^{B*} is determined from *η*(*t*^{B*}) = *c*_{U} and in this case *m*_{0} = 0. We do not discuss this case in detail because its analysis is similar to the treatment presented in Case 1.] Therefore for all *i* < *m*_{0}, for all *i* > *m*_{0} and

- (32)

If the value of calculated in (32) is such that , then . If , then and the latter can be calculated from the fact that . Finally, if , then the current choice of *t*^{B*} is infeasible.

Solving for potential bid parameters in Case 2(a) is an iterative process. Beginning with *λ* = *λ*_{m}, we iteratively choose the next higher value of *λ* ∈ {*λ*_{1},⋯,*λ*_{m}} and identify all solutions that satisfy the first-order optimality equations. At the end of this process, multiple optimal values of *t*^{B*} are possible and corresponding to each such value, there will be a different set of optimal *b*^{*} values.

Case (b): *λ* ∉ {*λ*_{1},⋯,*λ*_{m}}. In this case, Using (24), we have that if , then for all *i* ⩽ *m*_{0}, for all *i* > *m*_{0}. Therefore for all *i* ⩽ *m*_{0}, for all *i* > *m*_{0}. In addition, by (23) we also have that *b*_{0} = 0. Therefore, *t*^{B} is calculated by . If , *t*^{B*} is not feasible. If and , then *t*^{B*} is feasible and (22) is satisfied. The case *m*_{0} = 1 is the same.

Solving for the potential bid parameters in Case 2(b) is similar to the procedure in Case 1, except that we choose *t*^{B*} after choosing all 's.