1 / 22

Welfare and Profit Maximization with Production Costs

Welfare and Profit Maximization with Production Costs. A. Blum, A. Gupta, Y. Mansour , A. Sharma. Model. Multiple buyers Arbitrary valuation Multiple products Mechanism: prices Online setting Goals: Maximize Welfare Maximize Profit. Main Focus: Production cost increases. MODEL.

birch
Download Presentation

Welfare and Profit Maximization with Production Costs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Welfare and Profit Maximization with Production Costs A. Blum, A. Gupta, Y. Mansour, A. Sharma

  2. Model • Multiple buyers • Arbitrary valuation • Multiple products • Mechanism: prices • Online setting • Goals: • Maximize Welfare • Maximize Profit Main Focus: • Production cost • increases

  3. MODEL 25₪ 5₪ 7 ₪ 9₪ 21₪ payment SW

  4. cost 25₪ 6₪ 9₪ 9₪ 33₪

  5. Production Cost • CS literature • Unlimited supply • Fixed production cost • Limited Supply • Phase transition • Two extreme alternatives

  6. Production cost: ECON 101 • Increasing marginal production cost S D

  7. This Work • non-decreasing production cost • Posted prices • Online setting $1.99

  8. Our Results: Welfare Maximization • SW = value - cost • Simple algorithm • Price the kthitem of product j by the cost of the (2k)thitem of product j • Constant competitive ratio in many cases • Linear • Polynomial • Logarithmic • Fails for limited supply

  9. Our Results: Welfare Maximization • SW = value - cost • Convex production cost: • Logarithmic competitive ratio • Handles limited supply • [BGN]

  10. Our results: Profit Maximization • Profit = revenue - cost • Logarithmic competitive ratio • Combining: • Social Welfare maximization, multiple buyers • Revenue Maximization, single buyer • Similar to [AAM]

  11. General Structural Theorem • Fix a pricing scheme π • Consider a product prices cost prices opt Items sold Alg profit

  12. General Structural Theorem • Fix a pricing scheme π • Sum the area across products • If ΣjBLUE< αΣjBROWN+ β • Then SW(alg) > (SW(opt)- β)/ α

  13. General Structural Theorem • PROOF • Buyer b buys Sb in π and Ob in opt. • Consider prices πbwhen buyer comes • At the prices of πb : vb (Sb) – πb (Sb) ≥ vb(Ob) – πb (Ob) • Summing over buyers Σbvb (Sb) – πb(Sb) ≥ Σbvb(Ob) – πb(Ob) SW(alg) + C(alg) - P(alg) ≥ SW(opt) + C(opt) - P(opt)

  14. General Structural Theorem • P(alg) - C(alg) = profit(alg) • Maximize the Regret term P(opt) - C(λ) • Fix prices πb to be final prices • Only increases P(opt) • NOW: the term P(opt) - C(λ) = ΣiΣj P(j) – Ci(j) is exactly the sum of the BLUEareas

  15. Twice the index algorithm • For each item j, • The price of k-th copy is cost(2k) • NOTE: increasing cost implies price ≥ cost • Technically Need to compare BLUE vsBROWN Linear Cost: prices cost

  16. Twice the index algorithm • For each item j, • The price of k-th copy is cost(2k) • NOTE: increasing cost implies price > cost • Technically Need to compare BLUE vsBROWN Performance: • Linear cost: c(x)=ax+b α = 1/6β = Σjcj(2)-cj(1) • Polynomial cost c(x)=axd α = 1/2d β = 2(d+2)d+1Σjcj(2) • Logarithmic cost c(x)=ln(x+1) α = ln(3/2)/2 β = 3|J|

  17. Twice the index algorithm • When does it fail?! • Limited supply: • k items with fixed cost • Pricing: • First k/2 at cost • Last k/2 infinite • Very poor SW • Good SW: [BGN] prices cost

  18. Convex Functions • Multiple discrete prices • Enough items for in each price level • Run limited supply per price level • E.g., [BGN] • Smooth shifts between the price levels. • Limit the jump • Assume a given upper bound on values • Umax in [Z/ε, Z]

  19. Convex Functions • Two types of items • Many copies sold • The last completely sold price interval gives the required performance • Few items sold • More problematic • Introduces additive loss • Uses the convexity THEOREM: B=O(log(mn)) C= cost of first B items SW(alg) < [SW(opt) – C]/B

  20. Profit Maximization • Given: • SW maximization algo. A • Approx. ratio α1 , β1 • Single buyer profit max algo. B • Approx ratio α2 , β2 • Output: Profit Max Algorithm • Approx ratio O(α1 α2) , O(β1/(α1 α2)+(mβ2)/α1) • Similar to [AAM]

  21. Profit Maximization • Algorithm • With prob ½ use the prices of A. • If A gets high revenue we are done • With prob. ½ use sum of prices of A and B • B is memoryless (works for a single buyer) • If A gets high revenue we are done • Otherwise: there is a significant welfare left • A maximizes the SW • So B can get a fraction of the remaining SW.

  22. Conclusion • Changing cost of production • Interpolates between the extreme • Well studied in Economics • Reasonable competitive ratio • Constant for many interesting cases • Simple pricing algorithms • Future work: • Offline, better ratios • Decreasing prices, initial results • Beyond convex cost

More Related