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
7 ₪ 9₪ 21₪ 25₪ 5₪
SWpayment
MODEL
25₪ 9₪ 9₪ 33₪ 6₪
cost
Production Cost
• CS literature– Unlimited supply• Fixed production cost
– Limited Supply• Phase transition
– Two extreme alternatives
Production cost: ECON 101
• Increasing marginal production cost
SD
This Work
• non-decreasing production cost
• Posted prices
• Online setting
1.99$
Our Results: Welfare Maximization
• SW = value - cost• Simple algorithm– Price the kth 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
Our Results: Welfare Maximization
• SW = value - cost• Convex production cost:– Logarithmic competitive ratio– Handles limited supply • [BGN]
Our results: Profit Maximization
• Profit = revenue - cost• Logarithmic competitive ratio• Combining:– Social Welfare maximization, multiple buyers– Revenue Maximization, single buyer
• Similar to [AAM]
General Structural Theorem
• Fix a pricing scheme π• Consider a product
Items sold
prices
cost
prices
Alg profit
opt
General Structural Theorem
• Fix a pricing scheme π• Sum the area across products• If ΣjBLUE < α ΣjBROWN + β• Then SW(alg) > (SW(opt)- β)/ α
General Structural Theorem
• PROOF• Buyer b buys Sb in π and Ob in opt.
• Consider prices πb when buyer comes
• At the prices of πb :vb (Sb) – πb (Sb) ≥ vb(Ob) – πb (Ob) • Summing over buyersΣb vb (Sb) – πb(Sb) ≥ Σb vb(Ob) – πb(Ob) SW(alg) + C(alg) - P(alg) ≥ SW(opt) + C(opt) - P(opt)
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) = Σi Σj P(j) – Ci(j) is exactly the sum of the BLUE areas
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
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+bα = 1/6 β = Σj cj(2)-cj(1)• Polynomial cost c(x)=axd
α = 1/2d β = 2(d+2)d+1Σj cj(2)• Logarithmic cost
c(x)=ln(x+1)α = ln(3/2)/2 β = 3|J|
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]
costprices
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]
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 itemsSW(alg) < [SW(opt) – C]/B
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]
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.
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