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. Online scenario Seller posts prices Buyer comes – Buy a bundle Outcome – Seller gets payment – Buyer gets SW Next time step – Seller need to gets more items Changing production cost – Adjusts prices

4. 7 ₪ 9₪ 21₪ 25₪5₪ SW payment

5. 25₪ 9₪ 9₪ 9₪33₪ 6₪6₪

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

7. Production cost: ECON 101 Increasing marginal production cost S D

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

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

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

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

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

13. General Structural Theorem Fix a pricing scheme π Sum the area across products If Σ j BLUE < α Σ j BROWN + β Then SW(alg) > (SW(opt)- β)/ α

14. General Structural Theorem PROOF Buyer b buys S b in π and O b in opt. Consider prices π b when buyer comes At the prices of π b : v b (S b ) – π b (S b ) ≥ v b (O b ) – π b (O b ) Summing over buyers Σ b v b (S b ) – π b (S b ) ≥ Σ b v b (O b ) – π b (O b ) SW(alg) + C(alg) - P(alg) ≥ SW(opt) + C(opt) - P(opt)

15. General Structural Theorem P(alg) - C(alg) = profit(alg) Maximize the Regret term P(opt) - C(λ) Fix prices π b to be final prices – Only increases P(opt) NOW: the term P(opt) - C(λ) = Σ i Σ j P(j) – C i (j) is exactly the sum of the BLUE areas

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 vs BROWN Linear Cost: prices cost

17. 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 vs BROWN Performance: Linear cost: c(x)=ax+b α = 1/6 β = Σ j c j (2)-c j (1) Polynomial cost c(x)=ax d α = 1/2d β = 2(d+2) d+1 Σ j c j (2) Logarithmic cost c(x)=ln(x+1) α = ln(3/2)/2 β = 3|J|

18. 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] cost prices

19. 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 – U max in [Z/ε, Z]

20. 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

21. Profit Maximization Given: – SW maximization algo. A Approx. ratio α 1, β 1 – Single buyer profit max algo. B Approx ratio α 2, β 2 Output: Profit Max Algorithm – Approx ratio O(α 1 α 2 ), O(β 1 / (α 1 α 2 )+(mβ 2 )/α 1 ) Similar to [AAM]

22. 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.

23. 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