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Dynamic Position Auctions with Consumer Search Scott Duke Kominers Harvard University Algorithmic Aspects in Information and Management June 16, 2009 Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 1 / 17 Background

  1. Our Model Framework & Conventions Athey and Ellison (2008) Model N advertisers quality: q π ( π = 1 , 2 , . . . , N ) Interpretation: “probability of meeting a consumer’s need” Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  2. Our Model Framework & Conventions Athey and Ellison (2008) Model N advertisers quality: q π ( π = 1 , 2 , . . . , N ) Interpretation: “probability of meeting a consumer’s need” Distribution: F ( · ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  3. Our Model Framework & Conventions Athey and Ellison (2008) Model N advertisers quality: q π ( π = 1 , 2 , . . . , N ) Interpretation: “probability of meeting a consumer’s need” Distribution: F ( · ) (public) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  4. Our Model Framework & Conventions Athey and Ellison (2008) Model N advertisers quality: q π ( π = 1 , 2 , . . . , N ) Interpretation: “probability of meeting a consumer’s need” Distribution: F ( · ) (public) Sorted: q 1 ≥ q 2 ≥ · · · ≥ q N . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  5. Our Model Framework & Conventions Athey and Ellison (2008) Model N advertisers quality: q π ( π = 1 , 2 , . . . , N ) Interpretation: “probability of meeting a consumer’s need” Distribution: F ( · ) (public) Sorted: q 1 ≥ q 2 ≥ · · · ≥ q N . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  6. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  7. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  8. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  9. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  10. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers search cost s i per-click Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  11. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers search cost s i per-click search until need is met Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  12. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers search cost s i per-click search until need is met or until expected benefit < s i Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  13. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers search cost s i per-click search until need is met or until expected benefit < s i Distribution: G ( · ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  14. Our Model Framework & Conventions Athey and Ellison (2008) Model M < N positions Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously continuum of consumers search cost s i per-click search until need is met or until expected benefit < s i Distribution: G ( · ) (public) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

  15. Our Model Framework & Conventions Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  16. Our Model Framework & Conventions Our Dynamic Model Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  17. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  18. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  19. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting Sequential rounds Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  20. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting Sequential rounds Synchronous updating Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  21. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting Sequential rounds Synchronous updating Advertisers play a “best-response” strategy Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  22. Our Model Framework & Conventions Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting Sequential rounds Synchronous updating Advertisers play a “best-response” strategy Consumers ignorant of dynamics Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

  23. Our Model Framework & Conventions Balanced Bidding Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  24. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  25. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  26. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  27. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  28. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  29. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Athey and Ellison (2008) Envy-Free Equilibrium Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  30. Our Model Framework & Conventions Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  31. Our Model Framework & Conventions Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  32. Our Model Framework & Conventions Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  33. Our Model Framework & Conventions Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j among the positions below the current position which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  34. Our Model Framework & Conventions Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j among the positions below the current position which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  35. Our Model Framework & Conventions Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π targets the position j among the positions below the current position which maximizes utility chooses a bid b π to satisfy the envy-free condition: G (¯ q j ) · (1 − q π ) · ( q π − b π j +1 ) = G (¯ q j − 1 ) · ( q π − b π ) Unique fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

  36. Our Model Results Main Result Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  37. Our Model Results Main Result Theorem (Convergence Theorem) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  38. Our Model Results Main Result Theorem (Convergence Theorem) If all advertisers play the Restricted Balanced Bidding strategy Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  39. Our Model Results Main Result Theorem (Convergence Theorem) If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  40. Our Model Results Main Result Theorem (Convergence Theorem) If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  41. Our Model Results Main Result Theorem (Convergence Theorem) If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient. The dynamic model is “well-approximated” by the static model. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

  42. Our Model Results Parameters Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  43. Our Model Results Parameters γ j ( q ) = (1 − q ) G (¯ q j ) G (¯ q j − 1 ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  44. Our Model Results Parameters γ j ( q ) = (1 − q ) G (¯ q j ) G (¯ q j − 1 ) � � G (¯ q j ) γ ∗ ( q ) = (1 − q ) max j > 0 G (¯ q j − 1 ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  45. Our Model Results Parameters γ j ( q ) = (1 − q ) G (¯ q j ) G (¯ q j − 1 ) � � G (¯ q j ) γ ∗ ( q ) = (1 − q ) max j > 0 G (¯ q j − 1 ) γ ∗∗ = max 1 ≤ π ≤ N γ ∗ ( q π ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  46. Our Model Results Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  47. Our Model Results Lemma Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  48. Our Model Results Lemma At every round t > t 1 = 2 + log γ ∗∗ ((1 − γ ∗∗ )( q M − q M +1 ) / q M +1 ) : Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  49. Our Model Results Lemma At every round t > t 1 = 2 + log γ ∗∗ ((1 − γ ∗∗ )( q M − q M +1 ) / q M +1 ) : � b π > q M +1 π < M + 1 , b π = q π π ≥ M + 1 . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  50. Our Model Results Lemma At every round t > t 1 = 2 + log γ ∗∗ ((1 − γ ∗∗ )( q M − q M +1 ) / q M +1 ) : � b π > q M +1 π < M + 1 , b π = q π π ≥ M + 1 . Within t 1 rounds, the N − M lowest-quality advertisers “drop out” of contention. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

  51. Our Model Results Convergence of the M Positions Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  52. Our Model Results Convergence of the M Positions By the Lemma Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  53. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions converge to the fixed point after round t 1 . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  54. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions converge to the fixed point after round t 1 . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  55. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  56. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  57. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  58. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  59. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Next round, all advertisers in π ( P ) repeat their bids. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  60. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Next round, all advertisers in π ( P ) repeat their bids. If π ( P ) = { 1 , . . . , M } , then we are done. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  61. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Next round, all advertisers in π ( P ) repeat their bids. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  62. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Next round, all advertisers in π ( P ) repeat their bids. Look at the advertiser π �∈ π ( P ) with the lowest bid. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

  63. Our Model Results Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t 1 . Set of stable positions: P = { p + 1 , . . . , M } Set of advertisers in positions of P : π ( P ) Next round, all advertisers in π ( P ) repeat their bids. Look at the advertiser π �∈ π ( P ) with the lowest bid. Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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