Authors: David Robert Martin Thompson Kevin Leyton-Brown Presenters: Veselin Kulev John Lai Computational Analysis of Position Auctions

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

Outline  Different auction types, preference types  Action graph games  Experimental setup  Experimental results  Discussion

Auction Types  Generalized First Price (GFP)  i th highest bid is allocated slot  payment is exactly the submitted bid  Unweighted Generalized Second Price (uGSP)  i th highest bid is allocated slot i  payment is the (i+1) st highest bid  Weighted Generalized Second Price (wGSP)  each bid b j is multiplied by a bidder-specific weight w j  order bids by b j * w j = effective bid for j  i th highest effective bid is allocated slot i (call this agent k)  payment is the (i + 1) st effective bid / w k

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)

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

Preference Types Summary CTR Independent of Bidder CTR is separable ( ctr(p, b) = ctr(p) * qual(b)) Value is Independent of Position EOSV Value Increases with Position ?BHN Value is Single PeakedBSS?

Formal Description

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?

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?

AGG Example  Single Item First Price Auction  Two bidders with values v1 = 4 and v2 = 6  Discretize and bounds bids B2=1B2=2B2=3B2=4B2=5B2=6 B1=1½(3)00000 B1=22½(2)0000 B1=311½(1)000 B1=

AGG Example (cont.)  AGG Representation b2 < 1b2=1b2 > 1 13½(3)0 b2 < 2b2=2b2 > 2 22½(2)0 b2 < 3b2=3b2 > 3 31½(1)0  AGG size not dependent on number of possible v2 bids or discretization

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

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

Function Nodes  nodes that are not actions, but are computed based on actions  can be useful to decrease the in-degree of action nodes  if each player affects the function nodes independently, can still find expected utility in polytime  Example: GSP  payoff depends on the number of bids higher than you, but not the identity of those bids

AGG Examples (cont.)

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

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

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

wGSP v. VCG (EOS)

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

Experimental Results Summary  wGSP generally efficient  Ambiguous revenue results (compared to VCG); lower for EOS, higher for V, ambiguous for BHN, BSS

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)

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?

AGG Examples (cont.)