Incentive Compatible Mechanisms for Supply Chain Formation

1. Incentive Compatible Mechanisms for Supply Chain Formation Y. Narahari hari@csa.iisc.ernet.inhttp://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 E-Commerce Lab, CSA, IISc

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

3. Talk Based on 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 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. Y. Narahari, N. Hemachandra, and Nikesh Srivastava. Incentive Compatible Mechanisms for Decentralized Supply Chain Formation. Proceedings of IEEE CEC 2007. E-Commerce Lab, CSA, IISc

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

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

6. Some Observations Players are rational and intelligent Conflict and cooperation are both relevant Some of the information is common knowledge 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 E-Commerce Lab, CSA, IISc

7. Simple Example: The Supply Chain Partner Selection Problem SCP EM1 EM2 B B A C A C Let us say it is required to select the same partner at the two stages E-Commerce Lab, CSA, IISc

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

9. Current Art • 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 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) E-Commerce Lab, CSA, IISc

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

12. Strategic form Games S1 U1 : S R Un : S R Sn N = {1,…,n} Players S1, … , Sn Strategy Sets S = S1 X … X Sn 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 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 S1 = S2 = {H,T} E-Commerce Lab, CSA, IISc

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

15. Example 3: Hawk - Dove Models the strategic conflict when two players are fighting over a company/territory/property, etc. E-Commerce Lab, CSA, IISc

16. Example 4: Indo-Pak Conflict Models the strategic conflict when two players have to choose their priorities 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 E-Commerce Lab, CSA, IISc

18. Nash Equilibrium • (s1*,s2*, … , sn*) is a Nash equilibrium if si* is a best response for player ‘i’ against the other players’ equilibrium strategies Prisoner’s Dilemma (C,C) is a Nash Equilibrium. In fact, it is a strongly dominant strategy equilibrium E-Commerce Lab, CSA, IISc

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

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

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

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

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

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

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

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

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

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

30. Underlying Bayesian Game Type sets Private Info: Costs S0,S1,…,Sn Strategy Sets Announcements N = {0,1,..,n} 0 : Planner 1,…,n: Partners N = {0,1,..,n} 0 : Planner 1,…,n: Partners Payoff functions A Natural Setting for Mechanism Design E-Commerce Lab, CSA, IISc

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

33. Social Choice Function and Mechanism S1 Sn θ1 θn Outcome Set Outcome Set g(s1(.), …,sn() X f(θ1, …,θn) X Є Є (S1, …, Sn, g(.)) x = (y1(θ), …, yn(θ), t1(θ), …, tn(θ)) A mechanism induces a Bayesian game and is designed to implement a social choice function in an equilibrium of the game. E-Commerce Lab, CSA, IISc

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

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

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

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

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

39. We say that mechanismimplements SCF in dominant strategy equilibrium if We say that mechanism implements SCF in Bayesian Nash equilibrium if Implementing an SCF • Dominant Strategy Implementation • Bayesian Nash Implementation • Observation Dominant Strategy-implementation Bayesian Nash- implementation Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic Theory”, Oxford University Press, New York, 1995. 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) If the direct revelation mechanism has a dominant strategy equilibrium in which • Bayesian Incentive Compatibility (BIC) If the direct revelation mechanism has a Bayesian Nash equilibrium in which E-Commerce Lab, CSA, IISc

41. Outcome Set Project Choice Allocation I0, I1,…, In : Monetary Transfers x = (k, I0, I1,…, In) K = Set of all k X = Set of all x E-Commerce Lab, CSA, IISc

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

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

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

45. SCF is AE if for each , satisfies An SCF is ex post efficient in quasi-linear environment iff it is AE + BB SCF is BB if for each , we have 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) • Lemma 1 E-Commerce Lab, CSA, IISc

46. A Dominant Strategy Incentive Compatible Mechanism • Letf(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient. • Let the payments be : Groves Mechanism E-Commerce Lab, CSA, IISc

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

48. A Bayesian Incentive Compatible Mechanism • Letf(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient. • Let types of the agents be statistically independent of one another dAGVA Mechanism E-Commerce Lab, CSA, IISc

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

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