1. E-Commerce Lab, CSA, IISc 1 Incentive Compatible Mechanisms for Supply Chain Formation Y. Narahari hari@csa.iisc.ernet.inhari@csa.iisc.ernet.in http://lcm.csa.iisc.ernet.in/hari Co-Researchers: N. Hemachandra, Dinesh Garg, Nikesh Kumar September 2007 E-Commerce Lab Computer Science and Automation, Indian Institute of Science, Bangalore

2. E-Commerce Lab, CSA, IISc 2 OUTLINE 1.Supply Chain Formation Problem 2.Supply Chain Formation Game 3.Incentive Compatible Mechanisms for Network Formation  SCF-DSIC  SCF-BIC 4.Future Work

3. E-Commerce Lab, CSA, IISc 3 Talk Based on 1.Y. Narahari, Dinesh Garg, Rama Suri, and Hastagiri. Game Theoretic Problems in Network Economics and Mechanism Design Solutions. Research Monograph to be published by Springer, London, 2008 2.Dinesh Garg, Y. Narahari, Earnest Foster, Devadatta Kulkarni, and Jeffrey D. Tew. A Groves Mechanism Approach to Supply Chain Formation. Proceedings of IEEE CEC 2005. 3.Y. Narahari, N. Hemachandra, and Nikesh Srivastava. Incentive Compatible Mechanisms for Decentralized Supply Chain Formation. Proceedings of IEEE CEC 2007.

4. E-Commerce Lab, CSA, IISc 4 The Supply Chain Network Formation Problem Supply Chain Planner Echelon Manager

5. E-Commerce Lab, CSA, IISc 5 Cold Rolling PicklingSlittingStamping Master Coil 1 2 6 2 3 7 3 4 4 5 6 Suppliers 7 Forming a Supply Network for Automotive Stampings

6. E-Commerce Lab, CSA, IISc 6 Some Observations Players are rational and intelligent Some of the information is common knowledge Conflict and cooperation are both relevant Some information is is private and distributed (incomplete information) Our Objective: Design an “optimal” Network of supply chain partners, given that the players are rational, intelligent, and strategic

7. E-Commerce Lab, CSA, IISc 7 Simple Example: The Supply Chain Partner Selection Problem SCP EM2EM1 ACAC BB Let us say it is required to select the same partner at the two stages

8. E-Commerce Lab, CSA, IISc 8 1.Let us say SCP wants to implement the social choice function: f (x1, x2) = B; f (x1, y2) = A 2.If its type is x2, manager 2 is happy to reveal true type 3.If its type is y2, manager 2 would wish to lie 4.How do we make the managers report their true types? Preference Elicitation Problem Supply Chain Planner Echelon Manager 2 x1: A>B>C x2: C>B>A Echelon Manager 1 y2: B>C>A

9. E-Commerce Lab, CSA, IISc 9  W.E. Walsh and M.P. Wellman. Decentralized Supply Chain Formation: A Market Protocol and Competitive Equilibrium Analysis. Journal of Artificial Intelligence, 2003  M. Babaioff and N. Nisan. Concurrent Auctions Across the Supply Chain. Journal of Artificial Intelligence, 2004  Ming Fan, Jan Stallert, Andrew B Whinston. Decentralized Mechanism Design for Supply Chain Organizations using Auction Markets. Information Systems Research, 2003.  T. S. Chandrashekar and Y. Narahari. Procurement Network Formation: A Cooperative Game Approach. WINE 2005 Current Art

10. E-Commerce Lab, CSA, IISc 10 Complete Information Version Choose means and standard deviations of individual stages so as to : subject to A standard optimization problem (NLP)

11. E-Commerce Lab, CSA, IISc 11 1.How to transform individual preferences into social decision (SCF)? 2.How to elicit truthful individual preferences (Incentive Compatibility) ? 3.How to ensure the participation of an individual (Individual Rationality)? 4.Which social choice functions are realizable? Incomplete Information Version Supply Chain Planner Echelon Manager 2 Type Set 1 Type Set 2 Echelon Manager 1

12. E-Commerce Lab, CSA, IISc 12 Strategic form Games S1S1 SnSn U 1 : S R U n : S R N = {1,…,n} Players S 1, …, S n Strategy Sets S = S 1 X … X S n Payoff functions (Utility functions) Players are rational : they always strive to maximize their individual payoffs Players are intelligent : they can compute their best responsive strategies Common knowledge

13. E-Commerce Lab, CSA, IISc 13 Example 1: Matching Pennies Two players simultaneously put down a coin, heads up or tails up. Two-Player zero-sum game S 1 = S 2 = {H,T} (1,-1)(-1,1) (1,-1)

14. E-Commerce Lab, CSA, IISc 14 Example 2: Prisoners’ Dilemma

15. E-Commerce Lab, CSA, IISc 15 Example 3: Hawk - Dove 2 1 H Hawk D Dove H Hawk 0,020,5 D Dove 5,2010,10 Models the strategic conflict when two players are fighting over a company/territory/property, etc.

16. E-Commerce Lab, CSA, IISc 16 Example 4: Indo-Pak Conflict Pak India HealthcareDefence Healthcare 10,10-10, 20 Defence 20, -100,0 Models the strategic conflict when two players have to choose their priorities

17. E-Commerce Lab, CSA, IISc 17 Example 5: Coordination In the event of multiple equilibria, a certain equilibrium becomes a focal equilibrium based on certain environmental factors CollegeMG Road College 100,1000,0 MG Road 0,05,5

18. E-Commerce Lab, CSA, IISc 18 Nash Equilibrium (s 1 *,s 2 *, …, s n * ) is a Nash equilibrium if s i * is a best response for player ‘i’ against the other players’ equilibrium strategies (C,C) is a Nash Equilibrium. In fact, it is a strongly dominant strategy equilibrium Prisoner’s Dilemma

19. E-Commerce Lab, CSA, IISc 19 Mixed strategy of a player ‘i’ is a probability distribution on S i is a mixed strategy Nash equilibrium if is a best response against, Nash’s Theorem Every finite strategic form game has at least one mixed strategy Nash equilibrium

20. E-Commerce Lab, CSA, IISc 20 John von Neumann (1903-1957) Founder of Game theory with Oskar Morgenstern

21. E-Commerce Lab, CSA, IISc 21 Landmark contributions to Game theory: notions of Nash Equilibrium and Nash Bargaining Nobel Prize : 1994 John F Nash Jr. (1928 - )

22. E-Commerce Lab, CSA, IISc 22 Defined and formalized Bayesian Games Nobel Prize : 1994 John Harsanyi (1920 - 2000)

23. E-Commerce Lab, CSA, IISc 23 Reinhard Selten (1930 - ) Founding father of experimental economics and bounded rationality Nobel Prize : 1994

24. E-Commerce Lab, CSA, IISc 24 Pioneered the study of bargaining and strategic behavior Nobel Prize : 2005 Thomas Schelling (1921 - )

25. E-Commerce Lab, CSA, IISc 25 Robert J. Aumann (1930 - ) Pioneer of the notions of common knowledge, correlated equilibrium, and repeated games Nobel Prize : 2005

26. E-Commerce Lab, CSA, IISc 26 Lloyd S. Shapley (1923 - ) Originator of “Shapley Value” and Stochastic Games

27. E-Commerce Lab, CSA, IISc 27 Inventor of the celebrated Vickrey auction Nobel Prize : 1996 William Vickrey (1914 – 1996 )

28. E-Commerce Lab, CSA, IISc 28 Roger Myerson (1951 - ) Fundamental contributions to game theory, auctions, mechanism design

29. E-Commerce Lab, CSA, IISc 29 MECHANISM DESIGN

30. E-Commerce Lab, CSA, IISc 30 Underlying Bayesian Game N = {0,1,..,n} 0 : Planner 1,…,n: Partners Type sets Private Info: Costs S 0,S 1,…,S n Strategy Sets Announcements Payoff functions N = {0,1,..,n} 0 : Planner 1,…,n: Partners A Natural Setting for Mechanism Design

31. E-Commerce Lab, CSA, IISc 31 Mechanism Design Problem 1.How to transform individual preferences into social decision? 2. How to elicit truthful individual preferences ? O: Opener M: Middle-order L : Late-order Greg Yuvraj Dravid Laxman O<M<L L<O<MM<L<O

32. E-Commerce Lab, CSA, IISc 32 The Mechanism Design Problem  agents who need to make a collective choice from outcome set  Each agent privately observes a signal which determines preferences over the set  Signal is known as agent type.  The set of agent possible types is denoted by  The agents types, are drawn according to a probability distribution function  Each agent is rational, intelligent, and tries to maximize its utility function  are common knowledge among the agents

33. E-Commerce Lab, CSA, IISc 33 Social Choice Function and Mechanism f(θ 1, …,θ n ) θ1θ1θ1θ1 θnθnθnθn X Є S1S1S1S1 g(s 1 (.), …,s n () SnSnSnSn X Є x = (y 1 (θ), …, y n (θ), t 1 (θ), …, t n (θ)) (S 1, …, S n, g(.)) A mechanism induces a Bayesian game and is designed to implement a social choice function in an equilibrium of the game. Outcome Set

34. E-Commerce Lab, CSA, IISc 34 Two Fundamental Problems in Designing a Mechanism  Preference Aggregation Problem  Information Revelation (Elicitation) Problem For a given type profile of the agents, what outcome should be chosen ? How do we elicit the true type of each agent, which is his private information ?

35. E-Commerce Lab, CSA, IISc 35 Information Elicitation Problem

36. E-Commerce Lab, CSA, IISc 36 Preference Aggregation Problem (SCF)

37. E-Commerce Lab, CSA, IISc 37 Indirect Mechanism

38. E-Commerce Lab, CSA, IISc 38 Equilibrium of Induced Bayesian Game A pure strategy profile is said to be dominant strategy equilibrium if A pure strategy profile is said to be Bayesian Nash equilibrium Dominant Strategy-equilibrium Bayesian Nash- equilibrium  Dominant Strategy Equilibrium (DSE)  Bayesian Nash Equilibrium (BNE)  Observation

39. E-Commerce Lab, CSA, IISc 39 Implementing an SCF We say that mechanism implements SCF in dominant strategy equilibrium if We say that mechanism implements SCF in Bayesian Nash equilibrium if Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic Theory”, Oxford University Press, New York, 1995. Dominant Strategy-implementation Bayesian Nash- implementation  Observation  Bayesian Nash Implementation  Dominant Strategy Implementation

40. E-Commerce Lab, CSA, IISc 40 Properties of an SCF  Ex Post Efficiency For no profile of agents’ type does there exist an such that and for some  Dominant Strategy Incentive Compatibility (DSIC)  Bayesian Incentive Compatibility (BIC) If the direct revelation mechanism has a dominant strategy equilibrium in which If the direct revelation mechanism has a Bayesian Nash equilibrium in which

41. E-Commerce Lab, CSA, IISc 41 Outcome Set Project Choice Allocation I 0, I 1,…, I n : Monetary Transfers x = (k, I 0, I 1,…, I n ) K = Set of all k X = Set of all x

42. E-Commerce Lab, CSA, IISc 42 Social Choice Function where,

43. E-Commerce Lab, CSA, IISc 43 Values and Payoffs Quasi-linear Utilities

44. E-Commerce Lab, CSA, IISc 44 Policy Maker Quasi-Linear Environment project choiceMonetary transfer to agent 1 Valuation function of agent 1

45. E-Commerce Lab, CSA, IISc 45 Properties of an SCF in Quasi-Linear Environment  Ex Post Efficiency  Dominant Strategy Incentive Compatibility (DSIC)  Bayesian Incentive Compatibility (BIC)  Allocative Efficiency (AE)  Budget Balance (BB) SCF is AE if for each, satisfies SCF is BB if for each, we have  Lemma 1 An SCF is ex post efficient in quasi-linear environment iff it is AE + BB

46. E-Commerce Lab, CSA, IISc 46 A Dominant Strategy Incentive Compatible Mechanism 1.Let f(.) = ( k(.),I 0 (.), I 1 (.),…, I n (.) ) be allocatively efficient. 2.Let the payments be : Groves Mechanism

47. E-Commerce Lab, CSA, IISc 47 VCG Mechanisms (Vickrey-Clarke-Groves) Vickrey Auction Generalized Vickrey Auction Clarke Mechanisms Groves Mechanisms Allocatively efficient, individual rational, and dominant strategy incentive compatible with quasi-linear utilities. Allocatively efficient, individual rational, and dominant strategy incentive compatible with quasi-linear utilities.

48. E-Commerce Lab, CSA, IISc 48 A Bayesian Incentive Compatible Mechanism 1.L et f(.) = ( k(.),I 0 (.), I 1 (.),…, I n (.) ) be allocatively efficient. 2.Let types of the agents be statistically independent of one another 3. dAGVA Mechanism

49. E-Commerce Lab, CSA, IISc 49 BIC AE WBB IR SBB dAGVA DSIC EPE GROVES MOULIN

50. E-Commerce Lab, CSA, IISc 50 CASE STUDY

51. E-Commerce Lab, CSA, IISc 51 Casting stage Machining Stage Transportation stage C1 C4 M1 M5 T1 T6 ……… X1 X2 X3 Manager 1 Manager 2 Manager 3 SCP X=X1+X2+X3 Mechanism Design and Optimization

52. E-Commerce Lab, CSA, IISc 52 Information Provided by Service Providers Partner IdMeanStandard Deviation Cost P1131.0105 P1231.570 P1320.555 P1421.040 Partner Id MeanStandard Deviation Cost P2130.7535 P2221.0040 P2321.2535 P2420.7550 P2511.0070 Information for Manager 1 Information for Manager 2

53. E-Commerce Lab, CSA, IISc 53 Contd.. Partner Id MeanStandard Deviation Cost P3110.2520 P3210.515 P3320.7512 P3421.0010 P3521.259 P3621.508 Information for Manager 3

54. E-Commerce Lab, CSA, IISc 54 Centralized Framework EchelonPayment 120.388.2 210.2161101.00 310.148142.0 Solution of the Mean Variance Allocation Optimization problem in a centralized setting

55. E-Commerce Lab, CSA, IISc 55 Solutions in the Mechanism Design Setting EchelonPayment 1257.00 2269.80 3210.80 EchelonPayment 114.30 213.02 319.00 EchelonPayment 171.5 265.10 394.60 SCF-DSIC EchelonPayment 1128.70 2117.18 3170.28 EchelonPayment 1143.00 2130.20 3189.20 SCF- BIC with belief Probability 0.5 SCF- BIC with belief Probability 0.9 SCF- BIC with belief Probability 1.0

56. E-Commerce Lab, CSA, IISc 56 Future Work… Non-Linear Supply Chains Deeper Mechanism Design Solutions Cooperative Game Approach

57. E-Commerce Lab, CSA, IISc 57 To probe further… Y. Narahari, N. Hemachandra, Nikesh Srivastava. Incentive Compatible Mechanisms for Decentralized Supply Chain Formation. IEEE CEC 2007. Y. Narahari, Dinesh Garg, Rama Suri, and Hastagiri Prakash. Emerging Game Theoretic Problems in Network Economics: Mechanism Design Solutions, Springer, To appear: 2007 Andreu Mascolell, Michael Whinston, and Jerry Green. Microeconomic Theory. Oxford University Press, 1995 Roger B. Myerson. Game Theory: Analysis of Conflict. Harvard University Press, 1997.

58. E-Commerce Lab, CSA, IISc 58 Questions and Answers … Thank You …

59. E-Commerce Lab, CSA, IISc 59 Game Theory Mathematical framework for rigorous study of conflict and cooperation among rational, intelligent agents Market Buying Agents (rational and intelligent) Selling Agents (rational and intelligent)