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Position Auctions with Budgets: Existence and Uniqueness

Position Auctions with Budgets: Existence and Uniqueness. Ron Lavi Industrial Engineering and Management Technion – Israel Institute of Technology. Joint work with Itai Ashlagi, Mark Braverman, Avinatan Hassidim, and Moshe Tennenholtz. Overview.

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Position Auctions with Budgets: Existence and Uniqueness

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  1. Position Auctions with Budgets: Existence and Uniqueness Ron Lavi Industrial Engineering and Management Technion – Israel Institute of Technology Joint work with Itai Ashlagi, Mark Braverman, Avinatan Hassidim, and Moshe Tennenholtz

  2. Overview • Starting point: The elegant “generalized English auction”, of Edelman, Ostrovsky, and Schwarz, for position auctions • Private values, incomplete information • Truthful, envy-free, Pareto-efficient • Drawback: Not suitable for players with budget constraints • Realistic assumption • Our work: • “Extend” the auction to support budgets • New format exhibits all above desired properties • Outcome is equivalent to another “extension”, of the DGS auction (by Aggarwal, Muthukrishnan, Pal and Pal) • Turns out: This is the unique possible outcome satisfying above properties

  3. The Model • Player i has: private value vi ; private budget bi • Seller has K “positions” ; worth of position j to player i is j  vi • 1> 2> …. > K • Same model of EOS (2007), Varian (2007) • A player has quasi-linear utility if pays less than budget cap; negative utility otherwise: • Goal: auction that satisfies • Ex-post equilibrium: regardless of values, if others follow strategy, so do player i (has “no-regret”) [call this “truthful”] • Pareto-efficiency: cannot weakly improve all utilities • Envy-free: players do not want to switch positions+payments j  vi - p if p < bi negative O/W ui(slot j, payment p) =

  4. The Model • Player i has: private value vi ; private budget bi • Seller has K “positions” ; worth of position j to player i is j  vi • 1> 2> …. > K • Same model of EOS (2007), Varian (2007) • A player has quasi-linear utility if pays less than budget cap; negative utility otherwise: • Goal: auction that satisfies • Ex-post equilibrium: regardless of values, if others follow strategy, so do player i (has “no-regret”) [call this “truthful”] Proposition: envy-free  Pareto-efficient j  vi - p if p < bi negative O/W ui(slot j, payment p) =

  5. Related Work show envy-freeness • Extensions of DGS: • Van der Laan and Yang (2008) • Kempe, Mu’alem and Salek (2009) • Aggarwal, Muthukrishnan, Pal, and Pal (2009) • Hatfield and Milgrom (2005) – a more general setting for non-quasi-linearity, seems to subsume the above. Also viewed as an extension of DGS (as the authors note). • add • truthfulness on top

  6. Related Work show envy-freeness • Extensions of DGS: • Van der Laan and Yang (2008) • Kempe, Mu’alem and Salek (2009) • Aggarwal, Muthukrishnan, Pal, and Pal (2009) • Hatfield and Milgrom (2005) – a more general setting for non-quasi-linearity, seems to subsume the above. Also viewed as an extension of DGS (as the authors note). • Q: what if we try to extend the generalized English auction? • add • truthfulness on top

  7. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (no budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 p = 7 player 3 drops all players compete p = 0

  8. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (no budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 p solves:1  v2 - p = 2  v2 – 7  p = (1 - 2) v2 + 7 p = 8 player 2 drops p = 7 player 3 drops all players compete p = 0

  9. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (no budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 Result:player 1 wins slot 1 and pays 8player 2 wins slot 2 and pays 7 p = 8 player 2 drops p = 7 player 3 drops all players compete p = 0

  10. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 p = 7 player 3 drops ?? all players compete p = 0

  11. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (withbudget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 Possible alternative:player 3 wins slot 1 and pays 7.6player 2 wins slot 2 and pays 7.5 p = 7.6 player 2 drops p = 7.5 player 1 drops all players compete p = 0

  12. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 p = 7 player 3 drops ?? all players compete p = 0

  13. Budgets and the Generalized English Auction • The generalized English auction: • Price ascends; players drop (rename players in reverse drop order) • The i’th dropper wins slot i, pays price point of i+1 drop • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 However if p. 3 does not drop she can also end up with negative utility. Conclusion: no ex-post equilibrium p = 7 player 3 drops ?? all players compete p = 0

  14. Solution: The Generalized Position Auction • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 p = 7 all players compete p = 0 SLOT 1 SLOT 2

  15. Solution: The Generalized Position Auction • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 Player 3 no longer wants slot 2 Number of players interested in slot 2 is equal to slot number p = 7 p = 7 all players compete p = 0 SLOT 1 SLOT 2

  16. Solution: The Generalized Position Auction • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 player 3 wins slot 1, pays 7.6 p = 7.6 player 2 drops p = 7.5 player 1 drops p = 7 p = 7 p = 0 SLOT 1 SLOT 2

  17. Solution: The Generalized Position Auction Auction for slot 2 resumes; players 1 & 2 participate • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 player 3 wins slot 1, pays 7.6 p = 7.6 player 2 drops p = 7.5 player 1 drops p = 7 p = 7 p = 0 SLOT 1 SLOT 2

  18. Solution: The Generalized Position Auction • Example (with budget): 1 = 1.1, 2 = 1 ; v1= 20, v2= 10, v3= 7 b1= 7.5, b2= 7.6, b3= 9 player 2 wins slot 2, pays 7.5 player 3 wins slot 1, pays 7.6 p = 7.6 player 2 drops p = 7.5 p = 7.5 player 1 drops player 1 drops p = 7 p = 7 p = 0 SLOT 1 SLOT 2

  19. The Generalized Position Auction • (The direct version: players report types, and outcome is computed by the following algorithm) (*) price ascent in auction ℓ stopswhen there are ℓ active players (*) player i remains in auction ℓ untilprice = min(bi, (ℓ - ℓ’) vi + pℓ’) [ℓ’> ℓ : last slot in which player i was active when price stopped] pℓ . . . . . . . . . . . . . . . . . . . . SLOTℓ SLOTK SLOT1

  20. The Generalized Position Auction • (the direct version: players report types, and outcome is computed by the following algorithm) (*) when slot 1 is sold, auction for slot K resumes, for K-1 slots, with one less player. THM: this is truthful and envy-free . . . . . . . . . . . . . . . . . . . . SLOTℓ SLOTK SLOT1

  21. Uniqueness • Result turns out to be always identical to the extended DGS auction. (but different mechanism: ) • Different price path • Ours is slightly faster (nk2 messages instead of nk3) THM: Any mechanism that is truthful, envy-free, individually rational, and has no positive transfers, must yield the same outcome. • Holds even if values are public and only budgets are private.

  22. Proof Sketch • Use two properties of the generalized position auction: • If player i wins slot ℓ and declares smaller budget still > Pℓ then she still wins slot ℓ. • Slot prices are minimal among all mechanisms. • Let M denote our auction, and fix another mechanism M’ that satisfies all properties. Fix arbitrary tuple of types. Lemma: Let B={ s | Ps = P’s }. Then w(B) = w’(B). Proof: By contradiction i such that: (1) i = w(ℓ) = w’(ℓ’) (2) Pℓ = P’ℓ (3) Pℓ’ < P’ℓ’ ℓvi - P’ℓ = ℓvi - Pℓ >ℓ’vi - Pℓ’ > ℓ’vi – P’ℓ’ contradicting envy-freeness of M’.

  23. Proof Sketch • Use two properties of the generalized position auction: • If player i wins slot ℓ and declares smaller budget still > Pℓ then she still wins slot ℓ. • Slot prices are minimal among all mechanisms. • Let M denote our auction, and fix another mechanism M’ that satisfies all properties. Fix arbitrary tuple of types. Inductive claim: for slot ℓ = K,…,1: • Set of winners of slots 1,.., ℓ is the same for M,M’ • For slot s > ℓ: (a) Ps = P’s (b) w(s) = w’(s) • We need only prove (a) + (b) for some slot ℓ given correctness of inductive claim for slot ℓ+1.

  24. Proof Sketch Proof for (a) Pℓ = P’ℓ Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption. Claim: Pℓ’ = P’ℓ’ Note: This implies (a) since i in w’(B) implies i in w(B) implies Pℓ = P’ℓ

  25. Proof Sketch Proof for (a) Pℓ = P’ℓ Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption. Claim: Pℓ’ = P’ℓ’ Proof: Otherwise Pℓ’ > P’ℓ’ ℓvi - Pℓ >ℓ’vi - Pℓ’ > ℓ’vi – P’ℓ’

  26. Proof Sketch Proof for (a) Pℓ = P’ℓ Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption. Claim: Pℓ’ = P’ℓ’ Proof: Otherwise Pℓ’ > P’ℓ’ ℓvi - Pℓ > ℓ’vi – P’ℓ’  ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’

  27. Proof Sketch Proof for (a) Pℓ = P’ℓ Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption. Claim: Pℓ’ = P’ℓ’ Proof: Otherwise Pℓ’ > P’ℓ’ ℓvi - Pℓ > ℓ’vi – P’ℓ’  ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’ When player i declares budget = Pℓ +  she still wins slot ℓ in M, and thus wins some slot ℓ’’ < ℓ in M’. She pays P’’ < Pℓ + .

  28. Proof Sketch Proof for (a) Pℓ = P’ℓ Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption. Claim: Pℓ’ = P’ℓ’ Proof: Otherwise Pℓ’ > P’ℓ’ ℓvi - Pℓ > ℓ’vi – P’ℓ’  ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’ When player i declares budget = Pℓ +  she still wins slot ℓ in M, and thus wins some slot ℓ’’ < ℓ in M’. She pays P’’ < Pℓ + .Her utility in this case increases: ℓ’’vi – P’’ > ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’ which contradicts truthfulness of M’.

  29. Summary • Study position auctions with private values and private budget constraints. • Extend the generalized English auction to handle budgets, maintaining all its desired properties. • Prove that the result is the unique possible truthful mechanism that satisfies: • Envy-freeness • Individual Rationality • No Positive Transfers

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