Computational Analysis of Position Auctions

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

2. 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

3. Outline • Different auction types, preference types • Action graph games • Experimental setup • Experimental results • Discussion

4. 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

5. 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)

6. 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

7. Preference Types Summary

8. Formal Description

9. 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?

10. 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?

11. AGG Example • Single Item First Price Auction • Two bidders with values v1 = 4 and v2 = 6 • Discretize and bounds bids

12. AGG Example (cont.) • AGG Representation • AGG size not dependent on number of possible v2 bids or discretization

13. 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

14. 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

15. 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

16. AGG Examples (cont.)

17. 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

18. 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

19. 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

20. wGSP v. VCG (EOS)

21. 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

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

23. 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)

24. 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?

25. AGG Examples (cont.)