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) 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

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 ?

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=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

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

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 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

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.)