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Dec 10, 2022 •299 likes •1.69k views

This Talk We give a simple allocation and pricing mechanism whose Nash equilibrium solves a very large optimization problem This Talk We give a simple allocation and pricing mechanism whose Nash equilibrium solves a very large optimization

  1. Directly Applying VCG Pros 1. Result applies in very general settings 2. Allocation of Adverts is provably optimal Cons 1. Advertisers submit their entire utility function 2. Utility isn't for a single adauction but for all adauctions 3. Platform needs to solve a massive optimization

  2. Directly Applying VCG Pros 1. Result applies in very general settings 2. Allocation of Adverts is provably optimal Cons 1. Advertisers submit their entire utility function 2. Utility isn't for a single adauction but for all adauctions 3. Platform needs to solve a massive optimization This talk: We deal with these issue by appropriately decomposing this massive optimization.

  3. A Massive Optimization

  4. A Massive Optimization Max Utility

  5. A Massive Optimization Max Utility Mean click-rate

  6. A Massive Optimization Max Utility Mean click-rate Per impression Assignment Constraints

  7. A Massive Optimization Max Utility Mean click-rate Per impression Assignment Constraints small

  8. A Massive Optimization Max Utility Mean click-rate Per impression Assignment Constraints small Large!

  9. A Massive Optimization Max Utility Mean click-rate Per impression Assignment Constraints small Large! Uncountably infjnite!!

  10. A Massive Optimization ● even if we knew all the parameters, it's impossible to solve this optimization ofg-line

  11. A Massive Optimization ● even if we knew all the parameters, it's impossible to solve this optimization ofg-line ● Still … maybe we can solve a lot of small optimizations...

  12. A Small Optimization When a search occurs, solve:

  13. A Small Optimization Assignment Problem When a search occurs, solve:

  14. A Small Optimization Assignment Problem When a search occurs, solve:

  15. A Small Optimization Assignment Problem When a search occurs, solve: Lots of polynomial time algorithms:

  16. A Small Optimization Assignment Problem When a search occurs, solve: Lots of polynomial time algorithms: Hungarian ; Hopcroft-Karp ; Bertsekas' Auction ...

  17. Optimization Decomposition Solve the big optimization with many little optimizations

  18. Optimization Decomposition Solve the big optimization with many little optimizations 1. Substitution:

  19. Optimization Decomposition Solve the big optimization with many little optimizations 1. Substitution: 2. Separability:

  20. Optimization Decomposition Solve the big optimization with many little optimizations 1. Substitution: 2. Separability: MAIN IDEA: Substitute utility for Separate out the resulting optimization

  21. Optimization Decomposition Solve the big optimization with many little optimizations 1. Substitution: 2. Separability: MAIN IDEA: Substitute utility for Separate out the resulting optimization THE RESULT: A massively distributed VCG Mechanism

  22. Our Results

  23. A Preliminary Calculation

  24. A Preliminary Calculation

  25. A Preliminary Calculation

  26. A Preliminary Calculation Assignment LF-Transform Problem

  27. A Preliminary Optimization Result

  28. A Preliminary Optimization Result To solve the Massive Optimization

  29. A Preliminary Optimization Result To solve the Massive Optimization Advertiser's must signal average prices

  30. A Preliminary Optimization Result To solve the Massive Optimization Advertiser's must signal average prices Platform solves Assignment when each search occurs

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