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 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

The Model Player i has: private value v i ; private budget b i Seller has K “positions” ; worth of position j to player i is  j  v i –  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 u i (slot j, payment p) =  j  v i - pif p < b i negativeO/W

The Model Player i has: private value v i ; private budget b i Seller has K “positions” ; worth of position j to player i is  j  v i –  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 u i (slot j, payment p) =  j  v i - pif p < b i negativeO/W

Related Work 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). show envy-freeness add truthfulness on top

Related Work 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? show envy-freeness add truthfulness on top

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 ; v 1 = 20, v 2 = 10, v 3 = 7 p = 0 all players compete p = 7 player 3 drops

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 ; v 1 = 20, v 2 = 10, v 3 = 7 p = 0 all players compete p = 7 player 3 drops p = 8 p solves:  1  v 2 - p =  2  v 2 – 7  p = (  1 -  2 ) v player 2 drops

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 ; v 1 = 20, v 2 = 10, v 3 = 7 p = 0 all players compete p = 7 player 3 drops p = 8 player 2 drops Result: player 1 wins slot 1 and pays 8 player 2 wins slot 2 and pays 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 (with budget):  1 = 1.1,  2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 all players compete p = 7 player 3 drops ??

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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 all players compete p = 7.5 player 1 drops p = 7.6 player 2 drops Possible alternative: player 3 wins slot 1 and pays 7.6 player 2 wins slot 2 and pays 7.5

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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 all players compete p = 7 player 3 drops ??

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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 all players compete p = 7 player 3 drops ?? However if p. 3 does not drop she can also end up with negative utility. Conclusion: no ex-post equilibrium

Solution: The Generalized Position Auction Example (with budget):  1 = 1.1,  2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 all players compete p = 7 SLOT 2 SLOT 1

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

Solution: The Generalized Position Auction Example (with budget):  1 = 1.1,  2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 p = 7 SLOT 2 SLOT 1 p = 7 p = 7.5 p = 7.6 player 1 drops player 2 drops player 3 wins slot 1, pays 7.6

Solution: The Generalized Position Auction Example (with budget):  1 = 1.1,  2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 p = 7 SLOT 2 SLOT 1 p = 7 p = 7.5 p = 7.6 player 1 drops player 2 drops player 3 wins slot 1, pays 7.6 Auction for slot 2 resumes; players 1 & 2 participate

Solution: The Generalized Position Auction Example (with budget):  1 = 1.1,  2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7.5, b 2 = 7.6, b 3 = 9 p = 0 p = 7 SLOT 2 SLOT 1 p = 7 p = 7.5 p = 7.6 player 1 drops player 2 drops player 3 wins slot 1, pays 7.6 p = 7.5 player 1 drops player 2 wins slot 2, pays 7.5

The Generalized Position Auction SLOT ℓ (The direct version: players report types, and outcome is computed by the following algorithm) SLOT K..... SLOT (*) price ascent in auction ℓ stops when there are ℓ active players (*) player i remains in auction ℓ until price = min(b i, (  ℓ -  ℓ’ ) v i + p ℓ’ ) [ℓ’> ℓ : last slot in which player i was active when price stopped] pℓpℓ

The Generalized Position Auction SLOT ℓ (the direct version: players report types, and outcome is computed by the following algorithm) SLOT K..... SLOT (*) 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

Uniqueness Result turns out to be always identical to the extended DGS auction. (but different mechanism: ) –Different price path –Ours is slightly faster (nk 2 messages instead of nk 3 ) 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.

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  v i - P’ℓ =  ℓ  v i - Pℓ >  ℓ’  v i - Pℓ’ >  ℓ’  v i – P’ℓ’ contradicting envy-freeness of M’.

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.

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

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  v i - Pℓ >  ℓ’  v i - Pℓ’ >  ℓ’  v i – P’ℓ’

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  v i - Pℓ >  ℓ’  v i – P’ℓ’   ℓ  v i – (Pℓ +  ) >  ℓ’  v i – P’ℓ’

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  v i - Pℓ >  ℓ’  v i – P’ℓ’   ℓ  v i – (Pℓ +  ) >  ℓ’  v i – 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 + .

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  v i - Pℓ >  ℓ’  v i – P’ℓ’   ℓ  v i – (Pℓ +  ) >  ℓ’  v i – 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’’  v i – P’’ >  ℓ  v i – (Pℓ +  ) >  ℓ’  v i – P’ℓ’ which contradicts truthfulness of M’.

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