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