![Page 1: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/1.jpg)
Polyhedral Clinching Auctions and the AdWords Polytope
Renato Paes Leme
(Cornell University)
Joint work with Gagan Goel and
Vahab Mirrokni (Google NYC)
![Page 2: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/2.jpg)
Creating an Ads campaign …
![Page 3: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/3.jpg)
Creating an Ads campaign …
![Page 4: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/4.jpg)
Creating an Ads campaign …
![Page 5: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/5.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
![Page 6: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/6.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
nice and well studied auction with good game-theoretic properties but without budgets…
![Page 7: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/7.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
budget layer
![Page 8: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/8.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
engineering fix to adapt the original auction to the budgeted setting. Original game theoretic analysis is now lost.
budget layer
![Page 9: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/9.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
budget layer
![Page 10: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/10.jpg)
How to deal with budgets in practice ?
VCG, GSP, …
control loop Polyhedral Clinching Auction
Goal: Design an auction for AdWords that supports budgets natively, i.e., budgets are built in the game theoretic analysis
![Page 11: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/11.jpg)
What do we mean by budgets ?
![Page 12: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/12.jpg)
Classical quasi-linear utility function:
![Page 13: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/13.jpg)
Classical quasi-linear utility function:
Budget constrained utility function:
![Page 14: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/14.jpg)
Classical quasi-linear utility function:
Budget constrained utility function:
Very well understood: VCG, affine maximizers, …
![Page 15: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/15.jpg)
Classical quasi-linear utility function:
Budget constrained utility function:
Very well understood: VCG, affine maximizers, …
Surprisingly little is known.
![Page 16: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/16.jpg)
Goal: Design auctions for budget constrained agents
![Page 17: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/17.jpg)
Goal: Design auctions for budget constrained agents • Agents report values and budgets • Mechanism decides on allocation and payments for each player
![Page 18: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/18.jpg)
Goal: Design auctions for budget constrained agents • Agents report values and budgets • Mechanism decides on allocation and payments for each player • Requirements:
![Page 19: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/19.jpg)
Goal: Design auctions for budget constrained agents • Agents report values and budgets • Mechanism decides on allocation and payments for each player • Requirements:
(feasible set)
![Page 20: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/20.jpg)
Desirable properties • Incentive Compatibility:
assumption: budgets Bi are public
• Individual rationality:
• Pareto optimality:
An outcome (x,p) is Pareto-optimal if there is no (x’,p’) such that u’i ≥ ui, Σp’i ≥ Σpi and at least one of them is strict.
![Page 21: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/21.jpg)
Our main contribution Solve this problem for a large class of feasible sets P.
![Page 22: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/22.jpg)
Our main contribution Solve this problem for a large class of feasible sets P : (scaled) polymatroids.
Show this is impossible to be extended to general polytopes.
![Page 23: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/23.jpg)
Our main contribution Solve this problem for a large class of feasible sets P : (scaled) polymatroids.
Show this is impossible to be extended to general polytopes.
Conjecture: scaled polymatroids are the largest class for which this is possible. (we supply evidence for that)
![Page 24: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/24.jpg)
What do we know about budgets? [Dobzinski, Lavi, Nisan, FOCS’08] :: auction for one divisible good [Fiat, Leonardi, Saia, Sankowski, EC’11] :: auction for matching markets
![Page 25: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/25.jpg)
What do we know about budgets? [Dobzinski, Lavi, Nisan, FOCS’08] :: auction for one divisible good [Fiat, Leonardi, Saia, Sankowski, EC’11] :: auction for matching markets based on the clinching auctions framework [Ausubel, AER’97]
![Page 26: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/26.jpg)
How does it fit in our goal ? [Dobzinski, Lavi, Nisan, FOCS’08] P = Uniform Matroid [Fiat, Leonardi, Saia, Sankowski, EC’11] P = Transversal Matroid
![Page 27: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/27.jpg)
How does it fit in our goal ? [Dobzinski, Lavi, Nisan, FOCS’08] P = Uniform Matroid [Fiat, Leonardi, Saia, Sankowski, EC’11] P = Transversal Matroid
For AdWords and other more complicated markets, we need to solve it for more generic feasibility constraints P
![Page 28: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/28.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P.
![Page 29: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/29.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P. • Incentive compatibility • Individual Rationality • Budget Feasibility • Pareto Optimality
![Page 30: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/30.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P.
for a submodular function f.
![Page 31: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/31.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P. Our auction only needs oracle access to the submodular function f. Our auction has a natural geometric flavor.
![Page 32: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/32.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P. Many applications Auctions for network design, queuing systems, video on demand, matching markets, internet advertisement, …
![Page 33: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/33.jpg)
Our Results We provide an auction with all the desirable properties for any polymatroid P. Many applications Auctions for network design, queuing systems, video on demand, matching markets, internet advertisement, …
![Page 34: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/34.jpg)
Our results The set of that can be obtained this way form a polymatroid. We call it the AdWords Polytope. General model: • multiple slots • multiple keywords • easy to generalize
![Page 35: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/35.jpg)
Also on Sponsored Search with Budgets Independently, [Colini-Baldeschi, Henzinger, Leonardi, Starnberger, 2012] design an auction for sponsored search with one keyword, multiple slots and budgets.
![Page 36: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/36.jpg)
Our auction
polytope of feasible allocations price
clock
![Page 37: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/37.jpg)
Our auction
![Page 38: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/38.jpg)
Our auction
![Page 39: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/39.jpg)
Our auction
![Page 40: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/40.jpg)
Our auction
![Page 41: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/41.jpg)
Our auction In each step compute demands at price if ; and o.w.
![Page 42: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/42.jpg)
Our auction In each step compute demands at price Compute clinched amount
if ; and o.w.
![Page 43: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/43.jpg)
Computing clinched amounts What is the allocations that are still feasible at this point?
![Page 44: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/44.jpg)
Our auction: how to implement clinch ? How much can I allocate to 1 without harming player 2?
![Page 45: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/45.jpg)
Our auction: how to implement clinch ?
![Page 46: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/46.jpg)
Our auction: how to implement clinch ?
Clinching step
![Page 47: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/47.jpg)
Our auction: how to implement clinch ?
Clinching step
Theorem: Clinching as defined above results in a feasible allocation. If P is a polymatroid, δi can be computed efficiently using submodular minimization.
![Page 48: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/48.jpg)
Our auction: how to implement clinch ?
Clinching step
Theorem: Clinching as defined above results in a feasible allocation. If P is a polymatroid, δi can be computed efficiently using submodular minimization. [in practice there are more efficient algorithms for each case]
![Page 49: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/49.jpg)
Summary of the proof • Show clinching is well-defined and can
be computed efficiently
• Characterize Pareto-optimal outcomes for polymatroidal environments
• Show that the auction produces an outcome satisfying the characterization
![Page 50: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/50.jpg)
Extensions and Limits Going beyond polymatroids…
![Page 51: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/51.jpg)
General convex environment One budget-constrained player For a single budget constrained player (and many other unconstrained ones), it is possible do design an auction for any convex environment.
![Page 52: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/52.jpg)
What about 2 budget constrained players ? Weak impossibility: There is no auction following the clinching framework beyond (scaled) polymatroids.
![Page 53: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/53.jpg)
What about 2 budget constrained players ? Weak impossibility: There is no auction following the clinching framework beyond (scaled) polymatroids. Stronger impossibility: There exists a class of polytopes, for which no auction exists satisfying all the desirable properties.
![Page 54: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/54.jpg)
What about 2 budget constrained players ? Weak impossibility: There is no auction following the clinching framework beyond (scaled) polymatroids. Stronger impossibility: There exists a class of polytopes, for which no auction exists satisfying all the desirable properties.
No hope of an auction for a general polyhedral environment.
![Page 55: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/55.jpg)
Impossibility for decreasing marginals Single divisible good: Decreasing marginal valuations
![Page 56: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/56.jpg)
Impossibility for decreasing marginals Single divisible good: Thm: No auction with all the desirable properties for one divisible good with decreasing marginals.
Strengthens previous impossibility results of [Lavi, May’11] and [Fiat et al’11]
![Page 57: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/57.jpg)
Summary
Clinching auction for polymatroids
![Page 58: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/58.jpg)
Summary
Clinching auction for polymatroids
Characterization of Pareto Optimal Auctions
in general polyhedral environments
![Page 59: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/59.jpg)
Summary
Clinching auction for polymatroids
Impossibility for decreasing-marginals
and budgets
Characterization of Pareto Optimal Auctions
in general polyhedral environments
Impossibility for general polytopes
![Page 60: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/60.jpg)
Summary
Clinching auction for polymatroids
Impossibility for decreasing-marginals
and budgets
Characterization of Pareto Optimal Auctions
in general polyhedral environments
Impossibility for general polytopes
Positive results for one budget-constr
agent and general environments
![Page 61: Polyhedral Clinching Auctions and the AdWords Polytope](https://reader035.vdocuments.us/reader035/viewer/2022071602/613d5b34736caf36b75c5c14/html5/thumbnails/61.jpg)
Thanks !