1. Online Learning for Online Pricing Problems Maria Florina Balcan

2. Algorithmic –Customers’ shopping lists / valuations known to the algorithm. (Seller knows market well) Incentive-compatible auction –Customers submit lists / valuations to mechanism, which decides who gets what for how much. Must be in customers’ interest to report truthfully. On-line pricing –Customers arrive one at a time, buy what they want at current prices. Seller modifies prices over time. Three versions (easiest to hardest)

3. Adaptive algorithms for pricing a single good. (Connections to experts and bandit problems) $500 a glass

4. Pricing a single good Say you are selling lemonade (or a cool new software tool, or tickets to the world’s fair). Protocol #1: for t=1,2,…T –Seller posts price p t –Buyer arrives with valuation v t –If v t ¸ p t, buyer purchases and pays p t, else doesn’t. –v t revealed to algorithm. Protocol #2: same as protocol #1 but without last step. Assume all valuations in [1,h] $500 a glass $1 $5.00 a glass Goal: do nearly as well as best fixed price in hindsight.

5. Pricing a single good Say you are selling lemonade (or a cool new software tool, or tickets to the world’s fair). Protocol #1: for t=1,2,…T –Seller posts price p t –Buyer arrives with valuation v t –If v t ¸ p t, buyer purchases and pays p t, else doesn’t. –v t revealed to algorithm. Bad algorithm: “best price in past” –What if sequence of buyers = 1, h, 1, …, 1, h, 1, …, 1, h, … –Alg makes T/h, OPT makes T. Ratio of h worse!

6. Pricing a single good Say you are selling lemonade (or a cool new software tool, or tickets to the world’s fair). Protocol #1: for t=1,2,…T –Seller posts price p t –Buyer arrives with valuation v t –If v t ¸ p t, buyer purchases and pays p t, else doesn’t. –v t revealed to algorithm. Good algorithm: “combining expert advice” –Define one expert for each price p = (1+ ² ) i 2 [1,h]. –Best price of this form gives profit ¸ OPT/(1+ ² ). –Run RWM algorithm. Get expected gain at least: OPT/(1+ ² ) 2 - O( ² -1 h log( ² -1 log h)) [extra factor of h coming from range of gains]

7. Pricing a single good Say you are selling lemonade (or a cool new software tool, or tickets to the world’s fair) Good algorithm: “combining expert advice” –Define one expert for each price p = (1+ ² ) i 2 [1,h]. –Best price of this form gives profit ¸ OPT/(1+ ² ). –Run RWM algorithm. Get expected gain at least: OPT/(1+ ² ) 2 - O( ² -1 h log( ² -1 log h)) [extra factor of h coming from range of gains] Only arbitrarily small constant factor worse, with O(h log log h) additive loss! Can’t hope to do much better: e.g., if only one high bidder dominates the rest.

8. Pricing a single good Say you are selling lemonade (or a cool new software tool, or tickets to the world’s fair). What about Protocol #2? [just see accept/reject decision] –Now we can’t run RWM directly since we don’t know how to penalize the experts! –Called the “adversarial bandit problem” –How can we solve that? $1 $5.00 a glass

9. Pricing a single good Exponential Weights for Exploration and Exploitation (exp 3 ) RWM Exp3 Distrib p t Expert i ~ q t $1.25 Gain g i t Gain vector ĝ t qtqt q t = (1- ° )p t + ° unif ĝ t = (0,…,0, g i t /q i t,0,…,0) OPT 1. RWM believes gain is: p t ¢ ĝ t = p i t (g i t /q i t ) ´ g t RWM 3. Actual gain at t is: g i t = g t RWM (q i t /p i t ) ¸ g t RWM (1- ° ) 2.  t g t RWM ¸ /(1+ ² ) - O( ² -1 nh/ ° log n) OPT · nh/ °

10. Pricing a single good Exponential Weights for Exploration and Exploitation (exp 3 ) RWM Exp3 Distrib p t Expert i ~ q t $1.25 Gain g i t Gain vector ĝ t qtqt q t = (1- ° )p t + ° unif ĝ t = (0,…,0, g i t /q i t,0,…,0) OPT 1. RWM believes gain is: p t ¢ ĝ t = p i t (g i t /q i t ) ´ g t RWM 3. Actual gain is at t: g i t = g t RWM (q i t /p i t ) ¸ g t RWM (1- ° ) 2.  t g t RWM ¸ /(1+ ² ) - O( ² -1 nh/ ° log n) OPT · nh/ ° 3.5. Actual overall gain >=  °  /(1+ ² ) - O( ² -1 nh/ ° log n) OPT

11. Pricing a single good Exponential Weights for Exploration and Exploitation (exp 3 ) RWM Exp3 Distrib p t Expert i ~ q t $1.25 Gain g i t Gain vector ĝ t qtqt q t = (1- ° )p t + ° unif ĝ t = (0,…,0, g i t /q i t,0,…,0) OPT 1. RWM believes gain is: p t ¢ ĝ t = p i t (g i t /q i t ) ´ g t RWM 3. Actual gain is at t is: g i t = g t RWM (q i t /p i t ) ¸ g t RWM (1- ° ) 2.  t g t RWM ¸ /(1+ ² ) - O( ² -1 nh/ ° log n) OPT 4. E[ ] ¸ OPT. OPT Because E[ĝ j t ] = (1- q j t )0 + q j t (g j t /q j t ) = g j t, so E[max j [  t ĝ j t ]] ¸ max j [ E[  t ĝ j t ] ] = OPT. · nh/ °

12. Pricing a single good Exponential Weights for Exploration and Exploitation (exp 3 ) RWM Exp3 Distrib p t Expert i ~ q t $1.25 Gain g i t Gain vector ĝ t qtqt q t = (1- ° )p t + ° unif ĝ t = (0,…,0, g i t /q i t,0,…,0) OPT Conclusion ( ° = ² ) : E[Exp3] ¸ OPT/(1+ ² ) 2 - O( ² -2 h log(h) loglog(h))

13. Algorithmic Problem, Single-minded Bidders n item types (coffee, cups, sugar, apples), with unlimited supply of each. m customers. Say all marginal costs to you are 0 [revisit this in a bit], and you know all the (L i, w i ) pairs. Each customer i has a shopping list L i and will only shop if the total cost of items in L i is at most some amount w i (otherwise he will go elsewhere). What prices on the items will make you the most money? Easy if all L i are of size 1. What happens if all L i are of size 2?

14. Algorithmic Pricing, Single-minded Bidders A multigraph G with values w e on edges e. Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: APX hard [GHKKKM’05]. 10 40 15 20 30 5 10 5

15. A Simple 2-Approx. in the Bipartite Case Goal: assign prices on vertices p v ¸ 0 as to maximize total profit, where: Set prices in R to 0 and separately fix prices for each node on L. Set prices in L to 0 and separately fix prices for each node on R Take the best of both options. Algorithm Given a multigraph G with values w e on edges e. Proof simple ! OPT=OPT L +OPT R 40 15 25 35 15 25 5 LR

16. A 4-Approx. for Graph Vertex Pricing Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: Randomly partition the vertices into two sets L and R. Ignore the edges whose endpoints are on the same side and run the alg. for the bipartite case. Algorithm Proof In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R. Given a multigraph G with values w e on edges e. simple ! 10 40 15 20 30 5 10 5

17. Algorithmic Pricing, Single-minded Bidders, k-hypergraph Problem What about lists of size · k? –Put each node in L with probability 1/k, in R with probability 1 – 1/k. –Let GOOD = set of edges with exactly one endpoint in L. Set prices in R to 0 and optimize L wrt GOOD. Let OPT j,e be revenue OPT makes selling item j to customer e. Let X j,e be indicator RV for j 2 L & e 2 GOOD. Our expected profit at least: Algorithm 10 15 20

18. On-line Pricing Customers arrive one at a time, buy or don’t buy at current prices. In (full information) auction model, we know valuation info for customers 1,…,i-1 when customer i arrives. In posted-price model, only know who bought what for how much. Goal - do well compared to best fixed set of item prices.

19. On-line Pricing Can run separate online auctions over items in L, customers in GOOD (customers who want exactly one item in L). Guarantee: perform comparably to best fixed set of item prices (for pts in L, people in GOOD). Our O(k)-approx. alg. can be naturally adapted to the online setting, by using results of [BH’05] and [BKRW’03] for the online digital good auction. Let OPT i be the best profit achievable (from item i) using a fixed price for item i from customers in GOOD whose bundle contain item i. Can use [BH’05] auction -- the expected profit of the online auction for item i is

20. On-line Pricing Can run separate online auctions over items in L, customers in GOOD (customers that who want exactly one item in L). Let OPT i be the best profit achievable (from item i) using a fixed price for item i from customers in GOOD whose bundle contain item i. Using the [BH’05] auction, the expected profit of the online auction for item i is: Overall, we achieve profit at least: Profit of the offline approx. alg.