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Computational Analysis of Position Auctions. Authors: David Robert Martin Thompson Kevin Leyton-Brown Presenters: Veselin Kulev John Lai. Motivation. Many different models of ad auctions Each model is partially understood Multiple equilibria e.g. Locally-envy free equilbria
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Computational Analysis of Position Auctions Authors: David Robert Martin Thompson Kevin Leyton-Brown Presenters: Veselin Kulev John Lai
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) • 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
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
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
AGG Example (cont.) • AGG Representation • 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
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
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?