Algorithms and Economics of the Internet (CSCI-GA.3033-003) Vahab Mirrokni, Google Research, New York Richard Cole, Courant Institute, NYU

AdWords Ad Auction •  2002: Google introduced AdWords Select and later AdSense o  Second price auction o  Advertiser in position s paid the bid of advertiser in position s+1. o  Less susceptible to gaming. o  Yahoo! switched to this mechanism later.

Model: Position Auction  •  Generalized Second Price (GSP) Position Auction •  Assigning agents a = 1, ...., A to slots s = 1, ..., S •  Valuation of agent a for slot s: uas = va.xs,  •  va is the expected profit per click •  Number the slots x1 > x2 > ... > xS •  xs can be interpreted as click-through rate for slot s. •  Set xs = 0 for all s>S and assume A>S. •  uas is the expected profit to advertiser a from appearing in position s.

Model (cont d) •  slots sold via auction: agent bids ba •  slot with the best click-through rate assigned to the agent with the highest bid, and so on. •  let vs be the value/click of agent assigned to s, the price will be the bid of the agent below him: ps = bs+1 •  the net profit of agent a for slot s: (va - ps)xs = (va- bs+1)xs. •  Note: We know that xs>xs+1 and bs>bs+1

Example Two slots and three bidders. CTR for slot 1 = 200, CTR for slot 2 = 100. Bidders 1, 2, and 3 have values per click of $10, $4, and $2, respectively. First-Price Auction: Bidder 2 bids $2.01, to guarantee that he gets a slot. Then bidder 1 will not want to bid more than $2.02. But then bidder 2 will want to revise his bid to $2.03 to get the top spot, bidder 1 will in turn raise his bid to $2.04, and so on.

Example under the GSP position auctions: Now, if all advertisers bid truthfully, the bids are $10, $4, $2. Payments will be $4 and $2. Truth-telling is an equilibrium. However, if CTR for slot 2 =199. Payoff when truthful for bidder 1: (10-4)*200 = $1200 But, if bidder 1 changes bid to $3: payoff higher: (10-2)*199 = $1592 Not truthful in general!

Vickrey–Clarke–Groves vs. Position Auction •  VCG: charges advertiser in position s the externality that he imposes on others by taking one of the slots away from them. •  payment of the advertiser in position s = the aggregate value of clicks that all other advertisers would have received if s were not present the aggregate value of clicks others receive when s is present.  •  If only one slot available: PA = VCG = second price auction. •  Both set each agent's payments based on allocation of bids of other players, not the agent's own bid. •  Unlike PA, Truth-telling is a dominant strategy under VCG. •   Bing, Yahoo and Google don't use VCG!

Same example under VCG Bidder 2 payment is $200, as before. Bidder 1 payment is $600: (was $800 in PA) $200 for the externality that he imposes on bidder 3 (by forcing him out of position 2) $400 for the externality that he imposes on bidder 2 (by moving him from position 1 to position 2 and thus causing him to lose (200 − 100) = 100 clicks per hour). Revenue is lower under VCG.

Nash Equilibrium In equilibrium, each agent should prefer her current slot to any other slot: Definition 1: A Nash equilibrium set of prices (NE) satisfies

If agent changes the bid slightly it normally won't affect her position or payment à a range of bids and prices that satisfy these inequalities. The analysis of the position auction is much simplified by examining a particular subset of Nash equilibria.

Example

Symmetric Nash Equilibrium Definition 2: A symmetric Nash equilibrium set of prices (SNE) satisfies: 

Note: Note that the inequalities characterizing an SNE are the same as the inequalities characterizing an NE for t>s. Why SNE?

Why SNE? (cont'd) Fact 1: If a set of prices is an SNE it is an NE.

Proof: Since pt-1 >= pt

For all s and t.

Why SNE? One step solution: it is only necessary to verify the inequalities for one step up or down in order to verify that the set of inequalities is satisfied.  Fact 2: If a set of bids satisfies the symmetric Nash equilibria inequalities for s+1 and s−1, then it satisfies these inequalities for all s. Proof: by example: Let s assume that SNE holds for slots 1,2 and 2,3; now we have to show it holds for slots 1 and 3:

Adding up both sides:

Why SNE? (cont'd) Fact 3: Non-negative surplus In an SNE, vs >= ps. Proof: Using inequalities defining an SNE, 

Since xs+1 = 0.

Why SNE? (cont'd) Fact 4: Monotone values: In an SNE, vs-1 >= vs for all s. Proof: by definition of SNE:

Adding the two inequalities: Note: Since agents with higher values are assigned to better slots, an SNE is an efficient allocation.

Why SNE? (cont'd) Fact 5: Monotone prices

Proof:  By SNE definition: Rearrange: Second part:

Bounds SNE à explicit characterization of equilibrium prices and bids: In equilibrium, each agent's bid is bounded above and below by a convex combination of the bid of the agent below him and a value- either his own value or the value of the agent immediately above him.  Nash equilibria  can be found by recursively choosing a set of bids that satisfy these inequalities. The solutions are:

NE and SNE revenue Summing equations on the bounds of the bids, gives us upper and lower bounds on total revenue in an SNE.  How do these bounds relate to the bounds for the NE? maximum revenue NE = value of upper recursion of SNE ≥ value of lower recursion of SNE ≥ min revenue NE  Fact 6: The maximum revenue NE yields the same revenue as the upper recursive solution to the SNE.

Summary: SNE Properties •  Stability: SNE is in NE •  Local Stability: Local Stability implies Global Stability. •  Efficiency: It results in the most efficient solution. •  Revenue: It gives maximum revenue among all NE. •  Revenue: SNE brings more revenue than VCG. •  Practice: It has been observed in practice as the output of ad auctions.

Applications to Ad Auctions •  Google ranks the ads by the product of a measurement of ad quality and advertiser bid. •  Click-through rate (zs) for advertiser a in position s is the product of this quality effect es and position effect, xs.  •  zs = es.xs •  Advertisers are ordered by es.bs and each advertiser pays the minimum amount necessary to retain his position.  •  Let qst be the amount that advertiser s would need to pay to be in position t.  •  ==> qst = bt+1 (et+1/es)

Bounds on Values Let ps = bs+1 .

Divide by xs – xt  

By fact 2: max and min are the neighboring positions -->

Bounds on Values (cont'd) We can recursively apply these inequalities

This shows that the incremental costs must decrease as we move to lower positions.

Bounds on Values (cont'd) Use procedure to estimate bounds on v we can empirically determine the relationship between the bids and the values in real cases.