A core-selecting auction for portfolio’s packages

Portfolio auctions are a type of auction where multiple assets are bundled and offered for sale as a single unit. These auctions are often employed to sell large and complex collections of assets, offering an efficient and low-risk approach compared to traditional auctions. Across finance, real estate, energy contracts, and art, portfolio auctions have become increasingly prevalent.

In the financial sector, portfolio auctions typically involve a basket of financial assets and have gained prominence due to the growth of ETFs and algorithmic trading. These auctions are designed as an automated execution protocol,²²2ICE Bonds, Tradeweb, and OneChronos are all examples of portfolio auctions for securities. These platforms allow asset managers and brokers to sell large and complex collections of securities transparently and efficiently. that facilitates the seamless auction and trading of a portfolio of securities, such as stocks and bonds. The execution can be done either by trading the entire portfolio at once or by “slicing” it into optimal packages using algorithmic trading strategies. The latter may achieve a better execution price than the single execution, yet it is possible to increase the execution time (Gsell, \APACyear2008; Konishi, \APACyear2002).

Portfolio auctions are also known as basket or program trading portfolio investment strategies. These strategies have been in the investment landscape for a while and have accounted for 50% to 60% of the total daily trading volume at the NYSE³³3In periods with high volatility, it can reach up to 90%. Since 2012, the NYSE no longer publishes weekly program trading reports., allowing broker-dealers to bid fast for a basket of trades executed as a single transaction. Before submitting bids, broker-dealers possess a clear understanding of each asset’s weight in the basket, expressed either as a dollar value or percentage (Giannikos \BBA Suen, \APACyear2007).

The brokerage plays an important role in basket trading strategy using two alternative strategies: principal and agency trading⁴⁴4In principal trading, the brokerage completes a customer’s trade with their inventory and can profit from the bid-ask spread, while in agency trading, the brokerage locates a counterparty to a customer’s trade willing to purchase or sell the security for the same price as the counterparty.. We focus on principal trade, in which a broker undertakes the risk price on behalf of the asset manager and executes the portfolio at an agreed price plus a commission fee. A key aspect of principal trading is the need for the broker to minimize any deficit caused by the price differential, i.e., the actual execution price versus the agreed price with the asset manager. The commission fee for the broker serves as compensation for undertaking the risk of executing the portfolio according to the agreed-upon terms and as a source of revenue for the services provided.

The impact of this investment strategy for asset managers is vast, especially in periods of high volatility where cost savings and speedy risk transfer are imperative. Even though asset managers always have the option to execute those trades at their discretion, using a portfolio auction trading tool enables them to have access to multiple liquidity providers simultaneously. In addition, this strategy reduces any information leakage and guarantees execution efficiency - an efficient way to deal with large and complex transactions. From the broker’s perspective, this strategy is an opportunity to access new order flows that might match their inventory or facilitate new business. The challenge is to bid low enough to win the auction and at the same time to cover the assumed price risk from the execution (Padilla \BBA Van Roy, \APACyear2012; Giannikos \BOthers., \APACyear2012).

Portfolio auctions typically employ single-round first-price sealed-bid format where the entire portfolio is allocated to a single broker. One of the caveats of this design is that it can cause weak demand since some brokers might have no preference for possessing the whole portfolio but have a higher valuation for a part of it. Thus, the asset manager might suffer an increased cost for the portfolio’s execution. Another issue is that it does not provide bidders with valuable information to avoid aggressive bidding, thereby the “winner’s curse”. Indeed, brokers may overestimate the portfolio’s value, potentially paying more than it is worth and causing a financial loss. Brokers may possess limited information or need more research capabilities to estimate the portfolio’s value accurately. Hence, the winner’s curse phenomenon is often manifested in portfolio auctions.

Motivated by this context and other real-world auctions, we examine here a package auction for portfolios. We assume that the auctioneer divides the portfolio into packages for a discrete number of brokers with different valuations. To avoid auction’s inefficiency and the “exposure problem”,⁵⁵5In environments with complementarities, simultaneous bidding for multiple packages exposes bidders with strong complementarities to the risk of winning by bidding more for a particular package than it is worth on its own (Rothkopf \BOthers., \APACyear1998; Roth, \APACyear2002) the packages are preset such that they minimize the portfolio’s execution cost and market impact losses. We characterize those who compete for the packages as “local” brokers and those who bid for the whole portfolio as “global” brokers. Each broker knows the individual securities in the portfolio and takes long positions in all securities. Our two-round design with information updates in the interim creates incentives for brokers to reduce their bidding further and increase the asset manager’s revenue.

We propose a new pricing rule, the D-NVCG, that aligns the bidder’s incentives with truthful bidding in both rounds and mitigates any free-riding opportunities that brokers might have if the auction occurred as a single round (Baranov, \APACyear2010; Roth, \APACyear2002). A local broker in the first round might get the desired package by bidding aggressively, with the expectation of receiving a higher fee in the second round if the coalition underbids the global broker. The rule considers the first-round bid as a reference point to adjust the final fees paid to the local brokers. We prove that this pricing rule is optimal for the coalition of brokers.

The auction occurs in two rounds. In the first round, the asset manager simultaneously performs sealed-bid auctions (a) for “local” brokers who compete for each package separately and (b) for “global” brokers who compete for the aggregate portfolio. The first round qualifies those brokers with the lowest fee (first price) - one global broker and one local broker for each package - to participate in the second round. The practice of qualifying one single global broker may puzzle some observers. It might seem to limit competition in the second round and harm the seller. However, the qualification stage promotes only those brokers with the lowest bid in the next round, i.e., the most competitive, and brokers’ qualification bids set an upper bound for the next round.

In the interim, the asset manager reveals information for the qualified bids of the first round. This attribute in the design provides a solution to brokers’ information asymmetries by revealing valuations and avoiding jump-bidding (Perry \BOthers., \APACyear2000). This private information update is valuable for the price discovery of brokers, as they can understand the likely price for each package and the whole portfolio. Thus, their valuation efforts are more productive, and the auction becomes more efficient and less susceptible to manipulations (Ausubel \BBA Baranov, \APACyear2017; P. Milgrom, \APACyear2021)

In the second round, the local brokers jointly compete in a sealed-bid auction for the whole portfolio against the global, who values packages as perfect complements.⁶⁶6The utility of the whole portfolio has a higher utility than the sum of the utilities for the individual packages (Cramton \BOthers., \APACyear2006). If the coalition of local brokers submits a bid lower than that of global, the coalition wins, and the fees are awarded to each local broker based on a pricing rule that ensures a core outcome.

In our setup, the core attributes the set of payoff vectors corresponding to the portfolio’s allocation, where no better outcome exists for the asset manager and the local brokers. Any payoff vector in the core is bidder-optimal if there is no other payoff vector that Pareto improves upon it, i.e., packages are allocated to bidders with the highest allocation. In other words, the coalition of local brokers is unblocked and feasible (R. Day \BBA Milgrom, \APACyear2008). Hence, our core-selecting auction provides a framework where the payoff of brokers is on the bidder-optimal-frontier (P\BPBIR. Milgrom, \APACyear2004), and may reduce the costs of the asset manager significantly due to the two-round setup (Perry \BOthers., \APACyear2000). In the existing literature, both the D-NVCG rule and the N-VCG rule lead to results in the bidder-optimal frontier. However, the D-NVCG includes a boundary related to the first-round bidding behavior that may further lower the costs for asset managers.

From a practical perspective, the D-NVCG can be applied in general package auctions such as spectrum auctions, energy contracts, or even real estate auctions. Its implementation as a pricing rule in discrete iterative auctions effectively incentivizes truthful bidding during the final two rounds. In iterative auctions, bidders reveal partial and indirect information about their valuation. The D-NVCG addresses one of the biggest challenges of iterative auctions: strategic bidding and economic efficiency (Parkes, \APACyear2006\APACexlab\BCnt1). The activity rule (P\BPBIR. Milgrom, \APACyear2004) that D-NVCG comprises in the pricing induces brokers to bid low from the first stage of the auction.

Most literature in package auctions (or combinatorial auctions) focuses on developing fast heuristics to solve the complex winner’s determination problem (Rothkopf \BOthers., \APACyear1998; De Vries \BBA Vohra, \APACyear2003; Bünz \BOthers., \APACyear2022), while economists focus on specific properties by using simplified theoretical models (R. Day \BBA Milgrom, \APACyear2008; P. Milgrom, \APACyear2007; Ausubel \BBA Baranov, \APACyear2020).

The underpinnings of package auctions can be traced to the seminal paper of Vickrey (\APACyear1961), in which each bidder is asked to pay an amount equal to the externalities he exerts on the competing bidders. Vickrey shows that this payment rule motivates bidders to submit a “bid” according to the actual demand schedules, regardless of the bids made by others. It is easy to think that a Vickrey auction could generate an efficient outcome for package auctions due to its appealing property of incentive compatibility. Nevertheless, in practice, the Vickrey auction is rarely used because this mechanism can lead to low payoffs for the auctioneer, even if bids are high enough (P. Milgrom, \APACyear2007; Ausubel \BBA Baranov, \APACyear2020). Also, the Vickrey pricing is determined by a non-monotonic function of the broker’s values in the sense that an increase in the number of brokers can reduce equilibrium revenues for the asset manager up to zero. Thus, brokers can use profitably “shill bidding” to increase competition in order to finally charge higher fees (Ausubel \BBA Milgrom, \APACyear2002, \APACyear2006).

The existing literature alleviates the aforementioned shortcomings by proposing alternative procedures. Ausubel \BBA Milgrom (\APACyear2002) developed a mechanism called the ascending proxy auction, while, Xia \BOthers. (\APACyear2004) reviewed several pricing schemes for incentive compatibility and ascertained that the revelation of the losing bids may reduce the value of the prices relative to winning bids. R. Day \BBA Milgrom (\APACyear2008) and R\BPBIW. Day \BBA Cramton (\APACyear2012) suggested a new cluster of payment rules for core-selecting auctions with respect to the reported values.

Recently, Ausubel \BBA Baranov (\APACyear2020) provided a theoretical justification for using core-selecting auctions. They propose an incomplete-information setting in which bidders’ values are correlated and analyze the equilibrium under a “local-global” approach. They found that in environments with positive correlations, core-selecting auctions can be significantly closer to the true core than the VCG outcome⁷⁷7Goeree \BBA Lien (\APACyear2016) show that a mechanism that guarantees a core outcome for true values does not always exist.. To our knowledge, Krishna \BBA Rosenthal (\APACyear1996) were the first to explore an independent private value setting⁸⁸8Rosenthal \BBA Wang (\APACyear1996) extended the setting with common values. for the simultaneous sale of multiple items in the “local-global” setting.

The problem that arises in the “local-global” setting is when a coalition wins and the VCG outcome is not in the core.⁹⁹9 If the VCG outcome is in the core, it is the unique bidder-optimal allocation.

Then, the closer the bidder-optimal frontier gets to VCG pricing, the fewer incentives for misreporting (R. Day \BBA Milgrom, \APACyear2008; Ausubel \BBA Milgrom, \APACyear2002). R\BPBIW. Day \BBA Raghavan (\APACyear2007) and R\BPBIW. Day \BBA Cramton (\APACyear2012) find alternative payment rules that minimize the bidder’s incentives for this strategic manipulation. This rule is called the Nearest-VCG (Quantratic Rule), a point in the bidder-optimal-frontier where the maximum deviation from VCG pricing is minimized. A critical assumption of the above research is that packages are equally weighted. Yet, equal sharing of the core outcome might be unfair for the small allocations, so another option is to share the core outcome with the weighted Nearest-VCG rule (Ausubel \BBA Baranov, \APACyear2017). The weighted Nearest-VCG rule is more relevant for portfolio auctions where each package has a different weight over the portfolio, and any pricing without encountering the weight would be unfair.

Erdil \BBA Klemperer (\APACyear2010) have proposed a new class of pricing rules for core-selecting package auctions focusing on the marginal incentives to deviate from “truthful bidding”. The idea is to select a point in the bidder-optimal frontier close to a reference point. Motivated by their suggestion, we construct a new payment rule, the Dynamic-Nearest-VCG, using an endogenous reference point suggested by the brokers’ strategic bidding behavior in the first round. The rule finds a point in the bidder-optimal frontier by calculating the distance from the first-round bid. By including the first-round bid in the final payments, the mechanism becomes robust since it provides incentives to bidders for “truthfull” reporting in both rounds.

Bünz \BOthers. (\APACyear2022) developed a computational search approach to design bidder-optimal-frontier rules, which are at the facet of the core polytope under a local-global setup. They find that some rules based on Sharpley value outperform the Nearest-VCG, yet their computation is complex. We investigate how to restore truthful incentives from the first round while they show that the best-performing rules are those that provide incentives to bidders with large values.

From a different perspective, it has been developed a vast literature on iterative combinatorial auctions¹⁰¹⁰10A multiple-round bidding process at which the auctioneer releases information regarding the provisional winners and the actual prices at the end of each round. Bidders obtain information regarding the bids of their rivals and can modify their bids in the following rounds (Parkes, \APACyear2006\APACexlab\BCnt2).. One of the merits of this approach is its ease of deployment. It allows bidders to learn about rivals’ valuations, and it is the most popular combinatorial auction format used in practice. For example, the FCC has used only multi-round formats for its auction design (Pekeč \BBA Rothkopf, \APACyear2003). Moreover, auction designs that allow the creation of synergies through bidder-determined combinations¹¹¹¹11In this context, bidders have the authority to decide what is biddable and which combinations hold economic significance to them. can yield economically impactful results, sidestepping potential computational complexities (Park \BBA Rothkopf, \APACyear2005).

The Combinatorial Clock Auction (CCA) in the spectrum auctions is relevant to our model and bears one similarity(Ausubel \BOthers., \APACyear2017; Cramton \BOthers., \APACyear2009; Bichler \BOthers., \APACyear2013). The similarity lies in the two-stage bidding process with information feedback in the interim. Yet, the whole design is different in both stages. In the qualifying round of our model, the auctioneer specifies the packages, while in the CCA design, bidders specify the desired packages at reserve prices. In the final round of our model the coalition of local brokers bids against the broker who is interested in the whole portfolio, while in the CCA bidders place multiple supplementary bids, which improve their clock-round bids and express values for other packages. (Ausubel \BBA Baranov, \APACyear2017).

Levin \BBA Skrzypacz (\APACyear2016) identify a weakness in the equilibrium outcome of CCA since bidders in the first round (clock phase) can exaggerate their bids and gain a surplus in the supplementary round through types’ revelation which might lead to a strategically increase in the rival’s costs (Janssen \BBA Kasberger, \APACyear2019). Thus, in the clock phase, bidders’ incentives might be distorted from the trade-off between efficiency and information revelation.

CCA even though it mitigates the communication complexity of the commonly used bidding¹²¹²12In XOR bidding language bidders specify values for all the possible combinations of items (Bichler \BOthers., \APACyear2023) language, it constrains the expressiveness of the bids and may cause bidders to win only a subset of the desired items at prices exceeding their valuation, known as “exposure problem” (Bichler \BOthers., \APACyear2023; Roth, \APACyear2002). The auction format that we adopt with the preset packages from the beginning allows bidders to state their preferences concretely and simplifies any assumption for computationally complexity¹³¹³13The number of possible allocations grows exponentially, and with many objects for sale, there is no guarantee the optimal allocation is found in a feasible time that would arise if bidders specified the packages (Roth, \APACyear2002; Rothkopf \BOthers., \APACyear1998; Goeree \BBA Holt, \APACyear2010; Scheffel \BOthers., \APACyear2012).

An interesting extension of our model would be to allow brokers to bid simultaneously for multiple packages in the first round and then participate in the second round by consolidating the packages’ weights from the qualifying round. While this approach presents computational challenges and necessitates efficient management of the portfolio’s execution time, it could yield a dual outcome. The asset manager might receive competitive bids from the first round, yet brokers might hedge on the packages that do not value, which likely would have a market impact on spot prices for the asset manager and the brokers (Padilla \BBA Van Roy, \APACyear2012).

An asset manager sells $m>2$ securities packaged in a divisible portfolio $\Theta\in\mathbb{R}_{+}^{m}$. A set of risk-neutral brokers, $\mathbb{N}=\{1,\dots,n\}$ are competing¹⁴¹⁴14We assume no entry costs for participating brokers. for a portfolio $\Theta$, with $n\geq 2$. Two types of brokers participate in the auction: a set of local brokers denoted by the generic element $\ell\in\mathbb{L}$, and a set of global brokers $\mathit{g}\in\mathbb{G}$, where $\mathbb{L},\mathbb{G}\subset\mathbb{N}$ are disjoint and $\mathbb{L}\cup\mathbb{G}=\mathbb{N}$.

We assume that $\Theta$ is divided in a finite set of $q$ packages $\theta_{j}$, with $j=\{1,\dots,q$}, such as $\theta_{j}\in\mathbb{R}_{+}^{m}$ and $\Theta=\sum\limits_{j=1}^{q}\theta_{j}$. The vector $p^{*}\in\mathbb{R}^{m}_{+}$ includes the agreed exercise price for $m$ securities, and the vector $p\in\mathbb{R}_{+}^{m}$ the anticipated price of $m$ securities when delivered. We denote by $\omega_{j}=\dfrac{p^{*}\cdot\theta_{j}}{p^{*}\cdot\Theta}$, the weight of $\theta_{j}$’s value over $\Theta$ with $\omega_{j}\in(0,1]$ and $\omega_{j}\leq\frac{1}{n-1}$. Even for the extreme cases, the definition of $\omega$ holds, and our results remain unaltered.

Each broker $i\in\mathbb{N}$ observes a private signal $s_{i}\in\mathbb{S}$ about the value of the $\Theta$ or $\theta_{j}$ and a public signal $\mathit{z}\in\mathbb{Z}$ for the aggregate characteristics of the portfolio. We denote by $s_{\ell}$ the private signal of local broker $\ell$, where $s_{-\ell}$ aggregates the private signal of all other local brokers and $s_{g}$ the signal of global broker accordingly. Information $(\mathbf{s},\mathit{z})$, where $\mathbf{s}=(s_{\ell},s_{-\ell},s_{g})$, $\forall\ell\in\mathbb{L}$ and $g\in\mathbb{G}$, is distributed according to a continuous i.i.d. function $F_{\ell}(\cdot)$, with the density function $f_{\ell}(\cdot)>0$ for local brokers, and $F_{\mathit{g}}(\cdot)$, with $f_{\mathit{g}}(\cdot)>0$ for global brokers respectively.

All brokers form expectations for the percentage change of the securities’ prices given by the vector,

where the vector $p^{*}\in\mathbb{R}^{m}_{+}$ includes the agreed exercise price for $m$ securities, and the vector $p\in\mathbb{R}_{+}^{m}$ the anticipated price of $m$ securities when delivered. Both random vectors, $p^{*}\in\mathbb{R}^{m}_{+}$ and $p\in\mathbb{R}_{+}^{m}$, are conditional on the signal received and the available public information. Evidently, when $\mathbb{E}[\frac{\Delta p}{p^{*}}|s_{i},z]>0$ brokers anticipate to incur a loss.

Example: Lets assume three securities, $m_{1}$,$m_{2}$,$m_{3}$, with $m_{1}=6$, $m_{2}=7$ and $m_{3}=14$ items respectively in the aggregate portfolio, such as $\Theta=(6,7,14)$.
The agreed execution price of each security is:

.
Also, we assume that the portfolio is divided into three packages¹⁵¹⁵15Suppose that package $\theta$ consists of only one security. In this case, $\theta$ can be represented as a sparse vector with only one non-zero element. For example, if the security is denoted as $m_{1}$, the sparse vector representation would be $\theta=(m_{1},0,0,\ldots,0)$. :

Then the weight of each package is with $\omega_{1}=\dfrac{p^{*}\cdot\theta_{1}}{p^{*}\cdot\Theta}\approx 0.2075$, $\omega_{2}=\dfrac{p^{*}\cdot\theta_{2}}{p^{*}\cdot\Theta}\approx 0.4049$ and $\omega_{3}=\dfrac{p^{*}\cdot\theta_{3}}{p^{*}\cdot\Theta}\approx 0.3876$.

Now lets assume that the anticipated prices of $m_{1}$,$m_{2}$,$m_{3}$ securities when delivered are:

Brokers form expectations for the percentage change of the securities’ prices:

.

We are assuming only long positions. Consequently, for security $m_{2}$, which is anticipated to trade at a price lower than the agreed-upon spot price, brokers are expecting to incur losses.

We start our analysis by characterizing our mechanism in the second round as pivotal. This means that the fee received by any local broker $i$ is equal to the loss imposed on other locals by adjusting $\phi_{\ell}^{2}$ to attribute $i$’s values. We define a bid $\phi_{i}^{2}$ submitted by broker $i$ as pivotal, if and only if $\phi_{g}^{2}=\sum\limits_{i\in Q}\omega_{i}\phi_{i}^{2}$ holds and if for any $\gamma>0$, a bid $\phi_{i}^{2}-\gamma$ attributes a non-empty package $\theta_{j}$, while $\phi_{i}^{2}+\gamma$ yields the null package.

A broker is pivotal if his report changes the auction outcome, in comparison to excluding the broker or attributing the null report to him, then the auction satisfies the pivotal pricing property (P\BPBIR. Milgrom, \APACyear2004; Ausubel \BBA Baranov, \APACyear2020). In a local-global setting, this property is satisfied for any core-selecting auction.

It is not hard to prove that the Dynamic-NVCG pricing rule results in allocations belonging to the bidder-optimal-frontier and minimizes misreporting incentives.

We do not disregard the Nearest-VCG; instead, we are using it as a touchstone to improve incentive compatibility further and reduce the degree of manipulation freedom in a two-stage framework (Parkes \BOthers., \APACyear2001). The following Proposition explains why the Dynamic-NVCG rule is optimal for distributing the commission of local brokers when some brokers have perverse incentives in the first round.

Additionally, Zarpala \BBA Voliotis (\APACyear2021) establish in Proposition 3 of their paper that, for all monotonically decreasing strategies $\phi$ in the first round, it is a symmetric equilibrium for each local broker to bid above the expected price differential. While the study focuses on a single portfolio in a two-stage design, the strategic behavior of brokers in the first round aligns with what is observed in our model.

Next, we provide some necessary assumptions.

Bosshard \BOthers. (\APACyear2017) have proved that the N-VCG rule does not always satisfy the non-decreasing condition, yet in our model, both rules are increasing in the bidder’s $i$. This means bidders are less prone to aggressive bidding. Lemma 3 confirms the revenue monotonicity by P\BPBIR. Milgrom \BBA Day (\APACyear2013) and R. Day \BBA Milgrom (\APACyear2008). We prove that the revenue monotonicity holds in the bidder-optimal-frontier even for more than two items for sale (Lamy, \APACyear2010).

In the VCG mechanism, brokers bid their valuation truthfully, and it is their weakly dominant strategy with no surplus. The following lemma suggests the global broker has a weakly dominant strategy in the second round.

Whereas for the local bidders, bidding lower than their expected losses is always worse off, $\alpha_{i}(s_{i},z^{\prime})>0$.

Next, following Ausubel \BBA Baranov (\APACyear2020) we derive the necessary optimality conditions. Suppose that all local brokers $j\neq i$ bid according to the profile $(\phi_{j}^{2*})_{j\neq i}$. Let $H_{i}\equiv\mathit{H_{i}}\big{[}\phi_{i}^{2},\alpha_{i}(s_{i},z^{\prime})\big{]}$ be the probability of winning for a local bidder $i$ who bids $\phi_{i}^{2}\in[\alpha_{i}(s_{i},z^{\prime}),\phi_{i}^{1}\big{]}$, and its marginal probability $h_{i}\equiv\mathit{h_{i}}\big{[}\phi_{i}^{2},\alpha_{i}(s_{i},z^{\prime})\big{]}$:

Also, we denote the expected commission fee of each local bidder $i$ with $c_{i}\equiv C_{i}\big{(}\phi_{i}^{2},\alpha_{i}(s_{i},z^{\prime})\big{)}$ and with $MC_{i}\equiv MC_{i}\big{(}\phi_{i}^{2},\alpha_{i}(s_{i},z^{\prime})\big{)}$ the expected marginal commission when each local broker $i$ bids $\phi_{i}^{2}\in\big{[}\alpha_{i}(s_{i},z^{\prime}),\phi_{i}^{1}\big{]}$.

The expected marginal commission expresses any change in the expected commission arising from the incremental increase in the bidding $\phi_{i}^{2}$. For instance, if brokers anticipate a loss in the expected prices, they will counterbalance their payoff by moving their bid upwardly.

Next, we define the first-order optimality conditions for the local broker’s maximization problem on the steps of Ausubel \BBA Baranov (\APACyear2020).

Intuitively, if $MC_{i}<0$ it means that $\alpha_{i}(s_{i},z^{\prime})<\phi_{i}^{2}$, and broker $i$ is not included among the winners. Otherwise, if $\phi_{i}^{2}<\alpha_{i}(s_{i},z^{\prime})$, broker $i$ will have to increase his bidding fee to reach the optimal payoff where $MC_{i}=0$.