Computational Analysis of Position Auctions
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Authors: David Robert Martin Thompson Kevin Leyton-Brown
Presenters: Veselin Kulev John Lai
Computational Analysis of Position Auctions
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Motivation
Many different models of ad auctions Each model is partially understood
Multiple equilibria e.g. Locally-envy free equilbria
Hard to theorize about full set of equilibria
Use computational techniques to fill in the gap
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Outline
Different auction types, preference types Action graph games Experimental setup Experimental results Discussion
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Auction Types
Generalized First Price (GFP) ith highest bid is allocated slot payment is exactly the submitted bid
Unweighted Generalized Second Price (uGSP) ith highest bid is allocated slot i payment is the (i+1)st highest bid
Weighted Generalized Second Price (wGSP) each bid bj is multiplied by a bidder-specific weight wj
order bids by bj * wj = effective bid for j ith highest effective bid is allocated slot i (call this agent k) payment is the (i + 1)st effective bid / wk
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Preference Types
Two Dimensions to Vary CTR: click through rate model Value: how much the user values a click
Edelman et. al. (EOS) CTR: decreasing in position, same across bidders Value: same value for all clicks, regardless of position
Varian (V) CTR: separable into position-specific and bidder-specific components; ctr(pos i, bidder j) = ctr(i) * score(j) Value: same as EOS (constant for all clicks)
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Preference Types (cont.)
Blumrosen et. al. (BHN) CTR: same as V (decreasing, bidder-specific but separable) Value: value per click increasing in rank; higher positions are valued more highly
Benisch et. al (BSS) CTR: same as EOS (decreasing, bidder-independent) Value: single peaked in position; strictly decreasing from peak
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Preference Types Summary
CTR Independent of Bidder
CTR is separable ( ctr(p, b) = ctr(p) * qual(b))
Value is Independent of Position
EOS V
Value Increases with Position
? BHN
Value is Single Peaked
BSS ?
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Formal Description
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Questions
EOS locally envy-free equilibria are efficient and VCG-revenue dominant how often does wGSP have efficient, VCG-revenue dominant? what happens in other equilibria?
V any symmetric equilibrium (globally envy free) is efficient and VCG-revenue dominant how often does wGSP have efficient, VCG-revenue dominating equilibria?
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Questions (cont.)
BHN there are preferences where wGSP has no efficient NE how often does wGSP have no efficient NE? How
much welfare is lost?
BSS wGSP can be arbitrarily inefficient how often does wGSP have no efficient NE? How
much social welfare is lost?
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AGG Example
Single Item First Price Auction Two bidders with values v1 = 4 and v2 = 6 Discretize and bounds bids
B2=1 B2=2 B2=3 B2=4 B2=5 B2=6
B1=1 ½(3) 0 0 0 0 0
B1=2 2 ½(2) 0 0 0 0
B1=3 1 1 ½(1) 0 0 0
B1=4 0 0 0 0 0 0
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AGG Example (cont.)
AGG Representation
b2 < 1 b2=1 b2 > 1
1 3 ½(3) 0
b2 < 2 b2=2 b2 > 2
2 2 ½(2) 0
b2 < 3 b2=3 b2 > 3
3 1 ½(1) 0
AGG size not dependent on number of possible v2 bids or discretization
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Action Graph Games
normal form representation can be very large strict independencies
Payoff for agent A is always independent of agents B’s action
context-specific independencies Payoff for agent A is independent of action of agent B for some subset of actions for A and B e.g. First Price Auction: Payoff for agent A is independent of agent B’s action if agent B bids less than agent A
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Why AGG?
compact size (exponentially smaller) does not increase with more agents AGG structure can be leveraged computationally polynomial time algorithm (in the compact size) for computing expected utility of a strategy
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Function Nodes
nodes that are not actions, but are computed based on actions can be useful to decrease the in-degree of action nodesif each player affects the function nodes independently, can still find expected utility in polytimeExample: GSP
payoff depends on the number of bids higher than you, but not the identity of those bids
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AGG Examples (cont.)
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Experimental Setup
Weakly dominated strategies removed Strategies where bidder bids higher than
value Strategies where agent has bids j > i,
where the allocation for the agent is the same for all bids of other agents
Happens when weights are very different Impact on locally envy-free? Uniform Sampling
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Experimental Results
EOS Approximately efficient Did not beat VCG revenue even in best
equilibria uGSP = wGSP more efficient than GFP Ambiguous revenue results (wGSP v. GFP)
V Approximately efficient Did beat VCG revenue Dominated GFP, uGSP in efficiency Revenue only better than GFP, uGSP in medium
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wGSP v. VCG Revenue
Edelman only examines locally envy-free equilibria (other equilibria might exist)
Bid interval may be empty Discretization Bids could be higher than bidder’s value
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wGSP v. VCG (EOS)
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Experimental Results
BHN wGSP had frequent, complete failures of
efficiency Discretized VCG also suffered from this wGSP had higher welfare than GFP, uGSP Ambiguous revenue results
BSS Similar to BHN
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Experimental Results Summary wGSP generally efficient Ambiguous revenue results (compared to
VCG); lower for EOS, higher for V, ambiguous for BHN, BSS
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Conclusion / Discussion
wGSP has comparable performance to VCG
Can leverage computation to help examine equilibria under different assumptions / mechanisms
What do the “other” equilibria look like? Which equilibria are selected in practice?
(hard to know)
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Conclusion / Discussion
How are weights computed? What happens if weights used by wGSP are not perfectly accurate?
Analysis is for single keyword auctions; do bidders actually optimize at this level?
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AGG Examples (cont.)