The Combinatorial Multi-Round Ascending Auction (CMRA): proposal for a new auction format

Combinatorial auctions have gained popularity in recent years for the award of radio spectrum licences. This is because combinatorial auction formats are particularly suitable for allocating spectrum in small blocks and across multiple bands, allowing bidders to assemble their desired spectrum portfolio without being exposed to aggregation risk, i.e. the risk of winning an unwanted subset of the target portfolio.1 In particular, the Combinatorial Clock Auction (CCA) format has now been widely used for spectrum auctions (see textbox further down for an overview of the CCA).

However, even though the CCA has a number of desirable properties, its sealed bid aspect is often met with objections from bidders.

The main objection to the CCA is that bidders may not ‘win’ the bids submitted in final clock round. These bids may be replaced in the winner determination by an alternative combination of bids, especially if in the final clock round there was excess supply of some of the lots. This means that, in some cases, bidders cannot be absolutely certain to win the package they bid for at the end of the open stage. A bidder can narrow the range of potential outcomes by submitting very few bids in the supplementary bids round (or in the extreme case, only maintaining or increasing its final clock bid), but by doing this the bidder may risk winning nothing. Alternatively, the bidder can reduce the risk of leaving the auction empty-handed, by making more bids for alternative packages that might fit better with the bids made by other bidders. However, in this case the bidder cedes control over the final outcome to the auction mechanism, which can select any of these bids as a winning bid. Many bidders have placed great emphasis on being able to work out how to ensure that they will win the lots for which they bid in the last clock round, but despite the constraints on supplementary bids that arise from the clock rounds, determining the bid amount that will guarantee this can be difficult.

A further reason for bidders objecting to the format is the pricing rule. Bidders cannot easily predict how much they will eventually pay for their winning bid. Even though the pricing rule guarantees that bidders will only pay the amounts they would have had to bid in order to win, so that bidding at valuations is a non-regret strategy, this is not always a feasible option. In particular, bidders facing budget constraints may simply be unable to bid their full valuation for alternative packages, and hence will need to select their targets carefully on the basis of their expectations of final prices. Even though prices are ultimately likely – but not certain – to be lower than bid amounts, this does not mean that bidders could simply bid above budget (leaving aside completely the governance issues that would be associated with doing so). This can be a problem, especially when looking at long-term spectrum licences for a sizeable amount of mobile spectrum, which can account for a substantial proportion of enterprise value.

Finally, being able to submit a rich set of alternative, mutually exclusive bids is very desirable in theory, as it guarantees to bidders that they will win a package that maximises their surplus (the difference between their bid and the price actually paid). In practice, some bidders appear to prefer alternative bidding strategies based on acquiring their preferred package within their budget (rather than ensuring that they win the portfolio that yields the greatest difference between valuation and price), effectively running down a preference list once the price for their preferred package exceeds the budget. Such bidders are only willing to switch to alternative packages when they cannot afford their preferred target, but such preference cannot be expressed when submitting alternative, mutually exclusive bids.

Generally, many bidders seem to prefer auction formats (such as the Simultaneous Multi-Round Ascending auction traditionally used for spectrum awards) that give them an opportunity to revise their bids in light of a provisional outcome, where they have a chance of bidding back until they are happy that they cannot do any better.

The auction design proposed below provides bidders with this opportunity, whilst incorporating desirable combinatorial features associated with the CCA.

Combining an iterative ‘pay-your-bid’ approach with a combinatorial bidding framework

In a simple ascending price, open-outcry auction, bidders can gradually increase their bid as the auction progresses, and will always have the opportunity to adjust their bids in light of price changes. This gives bidders full control over the combination of lots on which they end the auction, and the price that they pay. A key aspect of the mechanism is that bidders are able to increase their bid if they are outbid, so that they can “bid back” if they are not satisfied with the provisional outcome.

Although bidders pay the amount of their bid, the auction ends at the point at which the winners are bidding just enough to outbid the losers. In practical terms, this is aligned with the opportunity cost price rule used in the CCA. Therefore, provided that bidding is progressive, with bidders having to outbid competitors until the process comes to a close, a pay-your-bid rule will yield prices that are similar to those obtained using an opportunity cost price rule.

At the same time, an important aspect of the supplementary bids round in the CCA is that it allows bidders to express their demand for a greater range of packages than they can bid on during the clock rounds. This allows bidders to express their flexibility to switch across alternative packages depending on their relative prices, and gives the auctioneer the information needed to minimise the potential clash between bidders’ demands. It also allows bids for packages to reflect complementarities amongst lots, because package prices do not need to be simply a linear combination of individual lot prices. Therefore, it seems reasonable to allow bidders to submit alternative bids, but with the proviso of giving them tighter control over what set of bids they submit, and at what point in time.

Our proposed format achieves this by replacing the supplementary bids round with an opportunity for bidders to submit bids for alternative packages during the clock rounds, subject to constraints that link these alternative bids to the prevailing round prices. As in the CCA, it is reasonable to require that such bids must satisfy revealed preference constraints arising from the bids submitted by the bidder in earlier rounds.

Allowing bidders to submit additional bids during the clock rounds means that bidders have the option of bidding for alternative packages, but without the concern that if they do not make such bids they might end up winning nothing. This is because, while there is excess demand at round prices, bidders will have the opportunity to make these bids later in the auction (unless the auction closes before these bidders need to move to alternative packages).

Auction mechanics for the CMRA

The bidding process is similar to the clock phase of a CCA. However, unlike in the CCA, bidders would be allowed to submit multiple bids in any round, provided that these meet the constraints that would apply in the supplementary bids round of a CCA.

Specifically, in any round, bidders would be required to submit a headline bid, at round prices. The headline bid can be for zero lots (with a corresponding bid amount of zero) if the bidder does not wish to acquire any lots at round prices. If the bidder does not submit any bids, its headline bid will be for zero lots by default.

The bidder may also submit additional bids subject to the following restrictions:

• the bid amount for any additional bid cannot exceed the cost of the included lots at round prices, and must be at least the cost of the lots at reserve prices;
• if the bid is for zero lots, then the bid amount must be zero; and
• the set of bids submitted by the bidder (including any bids submitted in earlier rounds, the headline bid and any additional bids submitted in the round) must satisfy revealed preference constraints in order to ensure that bidders have good incentives to make their headline bids on the most preferred package at round prices, and not to withhold demand until later in the auction.2

By requiring bidders to submit a headline bid at round prices we make sure that the bidder must either be willing to accept round prices in one of its bids, or alternatively accept the possibility that it may not win any lots (as its bid for zero lots may then be accepted). At the same time, bidders cannot exceed the bid amounts determined by round prices, so bidding must be progressive and bidders are anchored to common prices.

A supplementary bids round would then not be required – instead the auction would end when accepting exactly one bid from each bidder (which might be a zero bid, provided that the bidder has submitted such a bid) is an optimal solution. To check this, we:

• calculate the highest value that could be obtained by accepting at most one bid from each bidder; and
• compare this with the highest value that can be obtained by accepting exactly one bid from each bidder (if such a feasible combination exists).

Note that in some cases it may not be possible to accept exactly one bid from each bidder given the lots available (i.e. for all possible sets of bids that includes one bid from each bidder there is excess demand). In this case the auction must continue.

As the auction progresses, some bidders may reduce demand in their headline bids, or offer flexibility to reduce demand at a discount through additional bids. It may then be possible to accept exactly one bid from each bidder. However, any potential allocation of lots that involves accepting bids that are below round prices needs to generate a value that is greater than the best offers made so far.

The proposed closing rule has two essential features:

• the auction will not end with a bidder not winning anything unless the bidder has made a bid for zero lots; and
• a bidder’s headline bid can only be rejected if the bidder has provided an alternative to this.

Therefore, bidders can anticipate and control the risk that they do not win anything at all, or that they win fewer lots than they express demand for at round prices.

Bidding in the CMRA

The proposed auction design allows bidders to follow alternative bid strategies that give them more control over their risks. For instance, a bidder may submit many bids at all times with a view to maximise surplus. Making a headline bid for the surplus-maximising package and then for all other packages with a bid amount equal to the value of the package minus the surplus on the headline bid would achieve this. Alternatively, a bidder can only submit headline bids and switch across packages when it runs out of budget. In this case the bidder is guaranteed to win with the headline bid or any of the earlier bids (which should be preferable given that these bids were made at lower round prices). Bidders can also start by only making headline bids, and then introduce flexibility through additional bids as the auction progresses.

Key features of the CMRA

The proposed format allows bidders progressively to increase the range of bids from which they would be happy for the auction mechanisms to select one as a winning bid. A bidder does not have to bid on a large range of packages in order to minimise the risk of winning nothing, as the auction will not end until one of the bids from each bidder can be accommodated, and therefore only a bidder who has submitted a zero bid faces the prospect of leaving the auction winning nothing. Successful bidders pay the price of their bid. At the same time, the proposed CMRA format:

• eliminates aggregation risks in the same way as the CCA by guaranteeing bidders that they will never win a subset of any of the packages on which they bid;
• suppresses substitution risks (provided that bidders submit bids for their alternative targets);
• does not incentivise bidders to reduce demand in their headline bid to prevent prices from increasing, as they always retain the ability to win a smaller package at a lower price through making additional bids (provided that one of these can be accommodated in a winning combination); and it
• helps budget-constrained bidders to manage their bids, allowing them to pursue their primary targets while they have budget for them and contract to smaller packages only when larger packages become too expensive.

The CMRA can also accommodate further restrictions on the winner determination outcome, such as spectrum floors or other requirements that permissible outcomes have to meet (though we note that, as in the CCA, introducing such outcome restrictions may require substantial changes to the activity rules to avoid that they are leveraged strategically).

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1. Combinatorial auctions allow bidders to make bids for combinations of lots (‘packages’) under the guarantee that they will only get all or none of the lots included in their bid. This is useful in auctions where bidders may wish to acquire several lots that are complementary. For example, where spectrum is being offered in small blocks, individual lots are often of little or no value on a standalone basis. Auction formats in which bidders may end up winning such individual lots are risky for bidders. For example, in the standard simultaneous multi-round ascending auction, a bidder may be the highest bidder on some lots at the point where it cannot afford to bid any longer for other lots that would complete the portfolio. [↩]
2. The activity rules would be the same as those used in the Irish Multiband Award (2012). The rules for the MBSA are available from ComReg’s website. [↩]