# Data from Canadian 700MHz and 2500MHz spectrum auctions

The 700MHz (2014) and 2500MHz (2015) spectrum auctions generated revenues of 5,270,636,002 CAD from 302 licenses and 755,371,001 CAD from 97 licenses. Both auctions used a combinatorial clock auction (CCA) format involving an ascending clock phase followed by a sealed-bid supplementary stage where bids could be made on packages of products. Final prices were determined using Vickrey pricing with a core-adjustment. An activity rule was used which required bidders to make bids or lose eligibility to bid in later clock rounds, along with a revealed preference rule which allows the eligibility limit to be exceeded as long as consistency checks are satisfied. For full details on the auction formats see the official documentation (700MHz rules, 700MHz additional details, 2500MHz rules); and the record of bids placed is here for 700MHz and here for 2500MHz.

Bid consistency

The revealed preference rule prevents some inconsistent behavior but not all. By “truthful”, we mean bids that are true indications of subjective value, and by “consistent” we mean bids that are indications of some fixed set of valuations, possibly not the bidder’s actual valuations.

The following table gives the values of Afriat’s critical cost efficiency index (CCEI) for the 700MHz auction. Recall that for a CCEI value $$x$$, if $$x < 1$$ there is at least some intransitivity in preferences (i.e. inconsistent bidding) and $$1-x$$ can be interpreted as the fraction of expenditure wasted making inefficient choices (see this by S. Kariv for more).

Bidder CCEI (clock rounds) CCEI (clock and supp. rounds)
Bell 0.930 0.417
Bragg 0.880 0.420
Feenix 1 1
MTS 0.996 0.627
Novus 1 1
Rogers 0.998 0.742
TBayTel 1 1
Telus 0.970 0.488
Videotron 0.879 0.560

Kroemer et al. conclude for the 700MHz auction, “the numbers suggest that bidders deviated substantially from straightforward bidding” in the clock rounds. But “it is not unreasonable to believe that bidders tried to bid up to their true valuation in the supplementary stage” because of higher bid amounts compared to the clock rounds.

The next table gives CCEI values for the 2500MHz auction. We extend the definition of CCEI to apply to supplementary bids as in Kroemer’s paper.

Bidder CCEI (clock rounds) CCEI (clock and supp. rounds)
Bell 0.913 0.712
Bragg 0.920 0.530
Corridor 1 1
MTS 1 1
Rogers 1 1
SSi Micro 1 1
TBayTel 1 1
Telus 0.997 0.996
Videotron 1 1
WIND 1 1
Xplornet 1 0.578

Kroemer et al. (Sec. 5.2) also point out that the total number of bids submitted in the 700MHz auction was much smaller than the number of possible bids, which probably indicates untruthful bidding since an omitted package must have valuation less than or equal to its (low) opening price. The same observation holds for the 2500MHz auction. More exactly, the auction formats enforced a limit on the number of packages bidders were allowed to submit, which was in the hundreds, and bidders generally did not reach the limit.

Ideally, we would determine whether the bids made are consistent with a non-truthful strategy incorporating gaming and/or coordination. The papers Janssen and Karamychev - “Raising Rivals’ Cost in Multi-unit Auctions” and Janssen and Kasberger - “On the Clock of the Combinatorial Auction” derive Bayesian Nash equilibria under gaming preferences and conclude that GARP is not violated in equilibrium gaming strategies. We note that the assumptions in these game models do not include all features of the CCAs under consideration e.g. discrete products, many bidders, public aggregate excess demand, revealed preference rule, initial EP limit, supplementary package limit, 50% mid-auction deposits.

Bids, budgets, and final prices

Bidders may have a notion of a budget – the maximum they are willing to spend. But how should this correspond to the maximum they should bid? In general, bidders may end up paying the exact amount of their highest bid, but looking at the data we see bid prices and final prices can be very different in practice. The following tables show figures from both auctions that illustrate this difference. All prices are given in CAD.

700MHz auction: highest bid placed. Average ratio: 0.192. Max ratio: 0.766.

Bidder Max bid ($$M$$) Allocation stage final price ($$p$$) Ratio ($$p/M$$) Final clock bid
Bell 3,999,999,000 565,705,517 0.141 1,366,867,000
Bragg 141,894,000 20,298,000 0.143 38,814,000
Feenix 60,650,000 284,000 0.005 346,000
MTS 73,067,000 8,772,072 0.120 10,853,000
Novus 112,359,000 0 0 0
Rogers 4,299,949,000 3,291,738,000 0.766 3,931,268,000
TbayTel 7,683,000 0 0 0
Telus 3,750,000,000 1,142,953,484 0.305 1,313,035,000
Videotron 677,524,000 233,328,000 0.344 468,530,000

700MHz auction: (highest) bid placed on package eventually won. Average ratio: 0.447. Max ratio: 0.766.

Bidder Max bid on won package ($$W$$) Allocation stage final price ($$p$$) Ratio ($$p/W$$) Allocation stage Vickrey price
Bell 2,583,868,000 565,705,517 0.219 565,705,000
Bragg 51,000,000 20,298,000 0.398 20,298,000
Feenix 425,000 284,000 0.668 284,000
MTS 40,000,000 8,772,072 0.219 3,198,000
Novus N/A 0 N/A 0
Rogers 4,299,949,000 3,291,738,000 0.766 3,291,738,000
TbayTel N/A 0 N/A 0
Telus 1,607,300,000 1,142,953,484 0.711 1,142,953,000
Videotron 490,000,000 233,328,000 0.476 233,328,000

2500MHz auction: highest bid placed. Average ratio: 0.135. Max ratio: 0.277.

Bidder Max bid ($$M$$) Allocation stage final price ($$p$$) Ratio ($$p/M$$) Final clock bid
Bell 542,746,000 28,730,000 0.053 76,214,000
Bragg 35,935,000 4,821,021 0.134 12,091,000
Corridor 9,300,000 2,299,000 0.247 N/A
MTS 13,609,000 2,242,000 0.165 2,609,000
Rogers 304,109,000 24,049,546 0.079 52,343,000
SSi Micro 851,000 0 0 0
TBayTel 12,001,000 1,731,000 0.144 1,731,000
Telus 1,771,723,000 478,819,000 0.270 1,038,472,000
Videotron 749,128,000 66,552,980 0.089 231,851,000
WIND 22,609,000 0 0 0
Xplornet 91,974,000 25,472,454 0.277 57,839,000

2500MHz auction: (highest) bid placed on package eventually won. Average ratio: 0.235. Max ratio: 0.410.

Bidder Max bid on won package ($$W$$) Allocation stage final price ($$p$$) Ratio ($$p/W$$) Allocation stage Vickrey price
Bell 536,563,000 28,730,000 0.054 28,730,000
Bragg 19,000,000 4,821,021 0.254 3,536,000
Corridor 6,440,000 2,299,000 0.357 2,299,000
MTS 11,000,000 2,242,000 0.204 2,242,000
Rogers OR 24,049,546 N/A 21,252,000
SSi Micro N/A 0 N/A 0
TBayTel 12,001,000 1,731,000 0.144 1,731,000
Telus 1,771,723,000 478,819,000 0.270 478,819,000
Videotron 358,477,000 66,552,980 0.186 61,092,000
WIND N/A 0 N/A 0
Xplornet 62,200,000 25,472,454 0.410 22,917,000

Across both auctions, we see that bidders paid an average of 16% of their maximum bid placed, where each bidder is equally weighted.

Misc. notes

Researchers design approximation algorithms for winner and price determination in auctions (in e.g. this paper) because exact optimization can be intractable as the number of bids grows, at least in the worst case. However, in these recent auction instances, theoretical intractability did not present a problem because the solution was computable in a small amount of time. The 2500MHz allocation stage involved 2,239 bids and a GLPK-powered solver finds the winners and final prices in a couple minutes on a standard computer. Simulations involving well over 30,000 random bids still take a feasible amount of time.

In the 2500MHz auction, there were 4 pairs of package submissions where the lower-priced package had strictly higher quantities of products. In this case the lower-priced package is superfluous. This table shows the packages, submitted by Bell.

Price of larger package (CAD) Price of smaller package (CAD) Number of products difference
1 535,917,000 536,214,000 3
2 536,628,000 536,645,000 3
3 536,401,000 536,545,000 3
4 536,434,000 536,563,000 3

Acknowledgement: Thanks to Z. Gao for pointers.

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