The cascade auction –
a mechanism for deterring collusion in auctions
Uriel Feige
Weizmann Institute
Work done at Microsoft Herzeliya
Joint work with Gil Kalai and Moshe Tennenholtz
Ad auctions – market for impressions
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Market for impressions
• End users (“impressions”) – items
• Advertisers / networks– bidders
(wish to target ads at the right end users)
• Publisher/Exchange – auctioneer
• Different advertisers have different valuations for different
impressions
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Motivating Example:
classical failure in ad exchanges
• In current ad exchanges each ad impression is sold using
second price auction.
• Ad networks serve as mediators on behalf of the bidders.
• Advertisers submit bids through their ad networks.
• Say network 𝐴 has two advertisers 𝐴1 and 𝐴2 with bids 10 and
8 resp., while network 𝐵 has a single advertiser 𝐵1 with bid 5.
• 𝐴 forwards 10 to the exchange, and 𝐵 forwards 5.
• Second price auction at the exchange will declare 𝐴 the
winner with price 5.
The publisher gets less than the second price.
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Our goal
Design an auction mechanism that strikes a
good balance between:
• Bidding truthfully is a dominant strategy for the
advertisers (as in second price auctions).
• The collusion implied by using mediators does
not lower the revenue of the seller significantly
below the “fair price”: that of second price
auction without collusion.
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Our Suggested Mechanism:
The Cascade Auction
• Parameter 𝑘 ≥ 1.
• Probabilities 𝑝1 , 𝑝2 ,…., 𝑝𝑘
𝑘
𝑝𝑗 ≥ 𝑝𝑗+1
𝑗=1 𝑝𝑗 = 1
• Floor price 𝑐0 (equals 0 in this talk).
• Let 𝑏1 , 𝑏2 ,……, 𝑏𝑘+1 be the 𝑘 + 1 highest bids
(above floor price) in decreasing order.
• For 1 ≤ 𝑗 ≤ 𝑘, the bidder of bid 𝑏𝑗 wins the
auction with probability 𝑝𝑗
(If 𝑏𝑗 corresponds to a floor price rather than an
actual bid, the item is not allocated.)
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The Cascade Auction (cont.)
• When the winner is the 𝑗-th highest bid the
payment made by the winner is the VickreyClarke-Groves (VCG) payment
𝑝𝑗−1 𝑘+1
𝑙>𝑗 𝑏𝑙 ( 𝑝𝑙−1 - 𝑝𝑙 )
• In other words, the expected payment
associated with winning by the 𝑗-th highest
bid is the expected loss it causes to the rest of
(lower) bids:
𝑘+1
𝑙>𝑗 𝑏𝑙 ( 𝑝𝑙−1
- 𝑝𝑙 )
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Common sense
• Bidders can bid truthfully, because of VCG
prices.
• Advertisers will expect ad networks to forward
their bids to the exchange network, even if the
ad network sees a higher bid.
• If 𝑝1 is sufficiently high, revenue will be similar
to that of second price auction without
collusion.
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Rest of the talk
• An exercise in mathematical modelling:
provide a formal model and formal theorems
in support of what our common sense already
tells us.
There are some subtleties that need to be
handled:
• A bidder may want to provide more than one
bid, and then there is no dominant strategy.
• What are the strategic goals of mediators?
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Multiple Bids
• A seller has one item for sale. There are
multiple buyers.
• The cascade auction mechanism allows a
single bidder to submit multiple bids.
• Bids 𝑖, 𝑏𝑖 specify which buyer made the bid
and a positive bid value.
• Tie-breaking among different buyers is
random.
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Collusion and agents
• A collusion is a set of buyers that coordinate their
bids.
• We model collusion by a notion of a mediator
(e.g. an ad network).
• An mediator is a bidding algorithm, and the
algorithm of the mediator is effectively a contract
that the mediator offers to buyers.
• The contract says that the buyers may provide
their inputs to the bidding algorithm, and the
mediator will bid on their behalf the output of
the algorithm.
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Collusion and agents (cont.)
• Each mediator is a function of its incoming bids to at
most 𝑘 bids on behalf of the buyers.
• Buyers may submit bids directly to the seller, and also
through one or more mediators. We refer to a buyer
submitting bids only directly to the seller as being
independent.
• If a mediator submits a bid 𝑖, 𝑏𝑖 and the bid wins,
the item goes to buyer 𝑖. The mediator is not allowed
to instead give the item to a sibling of 𝑖 who is using
its services.
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The cascade auction as a
multi-player game
• Given the set of mediators, we get a game in
strategic form, where the players are the
buyers 𝑁 = 1,2, … . , 𝑛
• An action of a player is a pair 𝐴𝑖 , 𝐴𝐶 , where
𝐴𝑖 is a set of bids it submits directly to the
seller, and 𝐴𝐶 =(𝐴𝐶1 ,…., 𝐴𝐶𝑚 ) is the 𝑚 sets of
bids it submits to the 𝑚 mediators.
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Are mediators also players?
Yes, but our analysis will circumvent the need to
model their strategic behavior (the type of
contracts that they offer).
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Naive buyers
A buyer is naive if he is independent and he
submits a single bid.
Proposition: There is a strictly dominant action
for a naïve buyers, and this action is to bid his
value.
Proof: follows from VCG prices.
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Non-Naïve Buyers
Proposition: In a cascade auction with 𝑘 = 2, if a
buyer has value 𝑣 and the two top bids by other
buyers are 𝑏1 and 𝑏2 , then submitting two
independent bids is preferable over one
independent bid if and only (𝑣 + 𝑏2 )/2 > 𝑏1 .
The two bids will then be 𝑣 and 𝑏1 + 𝜀.
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Naive mediator
• The naive mediator asks each buyer for his bids.
• The naive mediator sorts all bids that he receives in order of
decreasing value, determines the value of the 𝑘th highest bid, and
passes to the seller those bids having at least this value.
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Naive mediators
Proposition: If in the cascade auction all mediators are
naive, then for every buyer, in every undominated action,
1. The buyer does not submit an independent bid.
2. If the buyer submits multiple bids, all these bids are
submitted through the same naive mediator.
3. At least one bid that the buyer submits to the naïve
mediator is the true value for the buyer, and the other
bids (if any) are not higher.
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Notation for rest of talk
• Consider the realistic case 𝑘 = 2 and let 𝑝=𝑝1 .
• Denote the three highest values that buyers have by
𝑣1 > 𝑣2 > 𝑣3 (for simplicity we assume that there are no
ties). Without loss of generality, these values are held by
buyers 1, 2 and 3 respectively.
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Naïve mediators
Theorem: In the cascade auction with 𝑘 = 2, if all
mediators are naive and if every buyer uses an undominated
action, then the expected revenue of the seller is at least
(2𝑝 − 1) 𝑣2 .
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Arbitrary mediators
The need to model the strategic behavior of
mediators (the type of contracts that they offer)
is circumvented by considering pure Nash
equilibria, modeling the possibility of a player to
leave a mediator and become independent.
It can be shown that the cascade auction always
has pure Nash equilibria (regardless of the
mediators).
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Nash Profiles (best response)
Theorem: In every pure Nash profile,
if 𝑣2 < 𝑣1 /2 the expected revenue of the seller
is at least 𝑣2 , and if 𝑣2 ≥ 𝑣1 /2 the expected
revenue of the seller is at least
𝑣1
2𝑝 − 1 max( , 𝑣3 )
2
The expectation is over randomness of the
auction, not of the players!
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Semi best response
A weaker (and hence more general notion) than Nash
equilibrium. Applies even when players do not know the
input of other players to the mediators.
Given the set of bids received by the seller, 𝐴𝑖 , 𝐴𝐶 is a
semi-best response if no independent action 𝐴′𝑖 results in
an action 𝐴′𝑖 , 𝐴𝐶 that offers the buyer higher expected
payoff than 𝐴𝑖 , 𝐴𝐶 does.
A profile in which each buyer plays a semi-best response
to the others’ actions is a Semi-Nash profile.
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Arbitrary mediators
Theorem: In every semi-Nash profile the
expected revenue for the seller is at least
2𝑝 − 1 min[𝑣2 , max
𝑣1
, 𝑣3
2
]
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Conclusion
• Introduced a sealed bid auction of a single
item in which the winner is chosen at random
among the highest 𝑘 bids according to a fixed
probability distribution, and the price for the
chosen winning bid is the VCG price.
• Our analysis suggests that this type of auction
gives higher revenues compared to second
price auction in cases of collusion/mediation,
as common in ad exchanges.
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Principles of analysis
• Informative special cases: naïve buyer, naïve
mediator.
• Multiple bids by the same buyer.
• Independent bids allow reasoning about outcome
without specifying mediators.
• Undominated actions circumvent difficult (and
uninformative) equilibrium analysis.
• Nash and semi-Nash profiles as a way of
quantifying over all reasonable contracts offered
by mediators.
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