# Sponsored Search Auctions#

Used to auction ad slots on websites. Model:

There are \(N\) slots with slot \(j \in N\) having associated click-through rate \(\alpha_j\) which is assumed to only depend on \(j\) (so for instance, quality of the ad itself does not matter);

There are \(M\) advertisers with advertiser \(i \in M\) having a value of \(v_i\) per click;

An allocation is a function \(A: N \to M\) that matches the \(i\)th slot to the \(A(i)\)th advertiser;

Define social welfare to be

\[W(A) = \sum_j v_{A(j)} \alpha_j.\]

What is the optimal allocation \(A\) to maximize \(W\)?

(General Second Price Auction)

General Second Price Auction

Ask each advertiser for a bid \(b_i\);

Assign highest bid to first slot, second highest bid to second slot, etc;

For each slot \(j\), advertiser \(A(j)\) pays \(b_{A(j+1)}\).

With the special case of one slot \(j=1\), GSP is equivalent to the second price auction.

The allocation rule in step two maximizes $\(\sum_j b_{A(j)}\alpha_j.\)$ However, GSP is not strategyproof: some advertisers may prefer to win a lower slot at an even lower price than a higher slot at a high price.

## Vickrey-Clarke-Groves Mechanism#

(Externality)

The externality of agent \(i\) is the difference in utility for all other agents when \(i\) is present versus when \(i\) is not.

Suppose \(v_1 > v_2 > v_3\) and there are two slots. In a welfare-optimal outcome, \(1\) gets \(v_1 \alpha_1\) for the top slot, \(2\) gets \(v_2 \alpha_2\) for the top slot, and \(3\) gets nothing. If \(1\) were not present, \(2\) gets \(v_2 \alpha_1\) and \(3\) gets \(v_3 \alpha_2\). Thus, \(1\)’s externality is

(Vickrey–Clarke–Groves Mechanism)

Ask each bidder for their valuation;

Find the welfare-maximizing allocation with respect to solicited bids in step \(1\);

Allocate slots via the welfare-maximizing allocation;

For each bidder \(i\):

Find the allocation that maximizes welfare for all agents other than \(i\);

Set \(i\)’s payment equal to the difference between how satisfied everyone else is when \(i\) is present versus when \(i\) is not.

The VCG auction is dominant strategy incentive compatible.

Proof. Bidder \(i\)’s payoff is

However, \(\sum_{k \neq i} b_k(X^{-i})\) does not depend on the bid \(i\) submits so \(i\)’s maximization problem is equivalent to maximizing \(v_i(X) + \sum_{k \neq i} b_k(X)\). Since the VCG mechanism already chooses the socially optimal outcome, it is a dominant strategy for all individuals to truthfully report.

In the context of sponsored search auctions, the assignment rule is still the same (highest bidder gets first slot, second highest bidder gets second slot, etc.) but payments are different. For slot \(\alpha_j\), advertiser \(A(j)\) pays

Another nice property of VCG auctions is that it is envy-free:

(Envy-Free)

An assignment \((A,p)\) consisting of an allocation rule and prices is envy-free if for all advertisers \(i,j\), advertiser \(i\) does not envy advertiser \(j\): \(i\)’s utility is (weakly) greater than \(i\)’s utility if they got \(j\)’s slot and paid \(j\)’s price.

In the context of sponsored search auctions, this means that

for all \(i,j \in M\).

## Unnatural Equilibria#

Suppose there is a single item and there are two bidders with \(v_A = 10, v_B = 9\). One equilibrium is \(b_A = 7, b_B = 100\). Then, \(B\) wins the auction and pays \(\$7\). This is an equilibrium (exercise for the reader to check this).

While no bidder has a profitable deviation, this outcome is not envy free: \(A\) would rather get \(B\)’s outcome than their current outcome. Adding the envy free requirement gets the following:

There is a correspondence between the following:

Envy-free equilibria of the GSP action;

Competitive market equilibria;

Stable matchings between buyers and (price,good) pairs.

As such, we can get two immediate corollaries:

We can efficiently find equilibria using deferred acceptance with prices.

The buyer-optimal (seller-worst) equilibrium corresponds to VCG payments.

## GSP vs VCG in Practice#

History:

In late 1990’s: Overture runs first price auctions;

Early 2000’s: Google, Yahoo, Bing start running GSP auctions;

Late 2000’s: Facebook runs auctions using VCG;

Late 2010’s: Google switches back to first price auctions.

Why the switch back to first price auctions? Many advertisers participate in different auctions using the same bids, and given fixed bids a first price auction makes more money than GSP, which makes more money than VCG.

Another trend: auctions are sensitive to bidder collusion.

2005: Bidding software must be authorized by search engine (easy to prevent collusion);

Early 2010’s: most ad bidding is through a small number of agencies (bad for competition and revenue);

Late 2010’s: ML based auto-bidders on Google, Bing, etc.

What about now? General move away from explicit auction rules:

Auction details are highly optimized and hard to understand (a lot of things are ML);

Big tech companies often know vales better than bidders (so they provide in-house bidders);

Advertisers exploit the fact that different companies have to compete (less incentive to explain rules).