# On Best response Bidding in GSP Auc5ons. By M. Cary et al., HBS working paper , CS 700 Commentary by Thomas Léauté and Luděk Cigler

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1 On Best response Bidding in GSP Auc5ons. By M. Cary et al., HBS working paper , 2008 CS 700 Commentary by Thomas Léauté and Luděk Cigler Outline 1. Context 2. The Balanced Bidding (BB) strategy 3. Convergence results 4. Conclusion and discussion 1

2 Outline 1. Context 2. The Balanced Bidding (BB) strategy 3. Convergence results 4. Conclusion and discussion Problem to Solve Online search engines keyword aucqons Generalized Second Price aucqons Allocate slots in decreasing order of bids Google, Yahoo!, MSN Repeated game with incomplete informaqon 2

3 Model and AssumpQons Only one keyword, mulqple slots No budget limit The model defines asynchronous rounds: Choose one bidder uniformly at random Only this bidder updates his bid Alls bids are public NotaQons k slots with click through rates θ 1 > > θ k n players with private valuaqons per click v 1 > > v n Given b i, p s (i) = price to pay to get slot s At this price, player i s expected uqlity is θ s v i p s i ( ( )) 10/19/08 γ i = θ i θ i 1 γ * = max i>1 θ i θ i 1 3

4 Outline Context The Balanced Bidding (BB) strategy Convergence results Conclusion and discussion Best Response Bidding Strategies Target slot that maximizes uqlity Bid within corresponding price range s * j = argmax{ θ s ( v j p s ( j) )} p s * s ( j) < b' j < p j s * 1 p 0 ( j) = + ( ) 10/19/08 4

5 The Balanced Bidding (BB) Strategy Target slot that maximizes uqlity Bid the desired price for next best slot that would make it just as profitable When targeqng best slot, bid: 10/19/08 s * j = argmax{ θ s ( v j p s ( j) )} θ s * v j j p s * j j ( ) s ( ) = θ s j * 1 v j b' j ( ) b' j = 1 2 v j + p 1 ( j) ( ) The Balanced Bidding (BB) Strategy Target slot that maximizes uqlity s * j = argmax{ θ s ( v j p s ( j) )} Bid the desired price for next best slot that would make it just as profitable with the convenqon: θ 0 = 2θ 1 s ( ) = θ s j * 1 v j b' j θ s * v j j p s * j j ( ) ( ) 10/19/08 5

6 The Balanced Bidding (BB) Strategy Balances 2 risks: retaliaqon from bidder above by underbidding aaack from bidder below by overbidding The Unique Fixed Point of BB Price of slot i = (i+1)st highest bid 2b 1 for j =1 b * * j = γ j b j +1 + ( 1 γ j )v j for 2 j k v j for k < j n Envy free equilibrium of the staqc game, with VCG payments Does BB converge to this fixed point? 6

7 Outline Context The Balanced Bidding (BB) strategy Convergence results Conclusion and discussion Outline of the Convergence Proof Goal: prove that BB converges (to its only fixed point) in the asynchronous case (BB async) Restricted Balanced Bidding strategy (RBB) do not target a beaer slot than the current one (does not mean you will not get a beaer slot) Prove that RBB converges in the synchronous case (RBB sync) to the same fixed point Derive proof of BB async 10/19/08 7

8 Summary of Convergence Results sync async RBB BB Summary of Convergence Results sync async RBB BB 8

9 Non convergence of BB sync Counter example: θ 1 =1 > θ 2 = 2 /3 > θ 3 =1/3 All (but D) target slot 3, then slot 1, then 3 again Importance of the highest bid convenqon Summary of Convergence Results sync async RBB BB 9

10 Convergence of RBB sync Convergence provably happens in 2 stages: Stage 1: convergence of the price of the worst slot the set of losers (and their bids) Stage 2: convergence of the order of the winners all winning bids and the prices of all slots RBB sync: Stage 1 (Lemma 2) The lowest slot price and the set of losers converge ajer a number of rounds: 1 γ * t 1 = 2 + log γ * ( ) v k v k +1 ( ) v k +1 10

11 Summary of Convergence Results sync async RBB BB From RBB sync to BB async Compute the expected number of asynchronous rounds needed to walk through the synchronous rounds in the RBB sync proof Stage 1: Each of the t 1 synchronous rounds requires all top (k+1) bidders to update their bids, which requires on average O(n log (k)) asynchronous steps O(n log (n k)) addiqonal asynchronous steps for the losers to bid their true valuaqons 11

12 Outline 1. Context 2. The Balanced Bidding (BB) strategy 3. Convergence results 4. Conclusion and discussion Take away Message In a repeated GSP aucqon, if all players asynchronously play the Balanced Bidding strategy, the outcome and payments almost surely converge to a desirable point. This desirable point corresponds to a Nash equilibrium of the staqc GSP aucqon, in which payments correspond to VCG. 12

13 Food for Thought Is BB an equilibrium? (tricky ques5on) QuesQon the model and the assumpqons: Bidder chosen uniformly at random All bids are made public Speed of convergence? ImplicaQons in Economics/Real Life? Is BB an Equilibrium? If all bidders play BB, their strategies converge to an equilibrium of the sta5c game. Is the BB strategy some form of equilibrium of the repeated game? If all other bidders play BB, am I beaer off playing BB too? 13

14 QuesQoning the Model The bidder who updates his bid is chosen uniformly at random All bidders implicitly bid (roughly) at the same speed What if there is a slow bidder? ImplicaQons on the convergence speed? QuesQoning the AssumpQons All bids are made public Is this realisqc? Only prices are made public; is it enough? Is this done in real Qme? Bidders reason on obsolete or averaged data Does it sqll converge? If so, influence on the speed of convergence? Find a strategy using only local informaqon 14

15 Speed of Convergence Dependence of the bound on k and n? Can equilibrium ever be reached in real life? 15

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