Arpita Biswas. Speaker. PhD Student (Google Fellow) Game Theory Lab, Dept. of CSA, Indian Institute of Science, Bangalore

Transcription

1 Speaker Arpita Biswas PhD Student (Google Fellow) Game Theory Lab, Dept. of CSA, Indian Institute of Science, Bangalore address:

2 OUTLINE Game Theory Basic Concepts and Results Mechanism Design Cooperative Game Theory Real-World Applications

3 GAME THEORY Mathematical framework for rigorous study of conflict and cooperation among rational and intelligent agents

4 GAME THEORY Mathematical framework for rigorous study of conflict and cooperation among rational and intelligent agents the agent would always choose an action that maximizes her/his (expected) utility. competent enough to make any inferences about the game that a game theorist can make. can carry out the required computations involved in determining a best response strategy

5 GAME THEORY Mathematical framework for rigorous study of conflict and cooperation among rational and intelligent agents the agent would always choose an action that maximizes her/his (expected) utility. preferences of the players expressed in terms of real numbers competent enough to make any inferences about the game that a game theorist can make. can carry out the required computations involved in determining a best response strategy

6 PRISONER S DILEMMA The problem is as follows: Two individuals arrested for a robbery (witnessed by several people). The police suspects that they were guilty of a similar crime earlier, but were never caught. The prisoners are lodged in separate prisons and interrogation happens separately The police tells each prisoner that: a. If you are the only one to confess, you ll get a light sentence of 1 year while the other would be sentenced to 10 years in jail. b. If both of you confess, both of you would be sentenced for 5 years. c. If neither of you confess, then each of you would get 3 years in jail. The police also informs each prisoner that the same has been told to the other prisoner.

7 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess The utility matrix models the strategic conflict when two players have to choose their priorities

8 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess < N, (A i ) i N, (U i ) i N > N set of players A i set of actions for player i U i A 1 A N R The utility matrix models the strategic conflict when two players have to choose their priorities Action profile or Strategy profile

9 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess

10 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess

11 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess

12 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess

13 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess Nash Equilibrium

14 PRISONER S DILEMMA Bunty Bubly Confess Not Confess Two Players: Bunty Bubly Confess Two Actions: Confess Not Confess Not Confess Nash Equilibrium A strategy profile in which no player gains by changing only his/her own strategy (assuming no one else changes their strategy)

15 PROJECT COORDINATION GAME Alice Bob Deep Learning Website Designing Two Players: Alice Bob Deep Learning Two Actions: Deep Learning Project Website Designing Project Website Designing

16 PROJECT COORDINATION GAME Alice Bob Deep Learning Website Designing Two Players: Alice Bob Deep Learning Two Actions: Deep Learning Project Website Designing Project Website Designing

17 PROJECT COORDINATION GAME Alice Bob Deep Learning Website Designing Two Players: Alice Bob Deep Learning Two Actions: Deep Learning Project Website Designing Project Website Designing Nash Equilibria

18 PROJECT COORDINATION GAME Alice Bob Deep Learning Website Designing Two Players: Alice Bob Deep Learning Two Actions: Deep Learning Project Website Designing Project Website Designing Nash Equilibria Does there exist there any other Nash Equilibrium in this game?

19 PROJECT COORDINATION GAME Alice Bob Deep Learning Website Designing Two Players: Alice Bob Deep Learning Two Actions: Deep Learning Project (DL) Website Designing Project (WD) Website Designing Nash Equilibria Does there exist there any other Nash Equilibrium in this game? Alice: With probability 2/3 choose DL and with probability 1/3 choose WD Bob: With probability 1/3 choose DL and with probability 2/3 choose WD

20 MIXED STRATEGY NASH EQUILIBRIUM

21 EXISTENCE OF NASH EQUILIBRIA IN GAMES Does Nash Equilibrium always exist?

22 EXISTENCE OF NASH EQUILIBRIA IN GAMES Does Nash Equilibrium always exist?,nash Theorem, Every finite strategic form game has at least one mixed strategy Nash Equilibrium.

23 EXISTENCE OF NASH EQUILIBRIA IN GAMES Does Nash Equilibrium always exist?,nash Theorem, Every finite strategic form game has at least one mixed strategy Nash Equilibrium. Is there an efficient algorithm for computing a mixed Nash equilibrium?

24 EXISTENCE OF NASH EQUILIBRIA IN GAMES Does Nash Equilibrium always exist?,nash Theorem, Every finite strategic form game has at least one mixed strategy Nash Equilibrium. Is there an efficient algorithm for computing a mixed Nash equilibrium?,daskalakis et al., Finding mixed strategy Nash Equilibrium is PPAD complete

25 OTHER TYPES OF EQUILIBRIA Strongly Dominant Strategy Equilibrium (SDSE): An action profile a 1,, a n is called strongly dominant strategy equilibrium for a game < N, A i, U i >, if i N and b i A i *a i +, U i a i, b i > U i b i, b i b i A i. Bunty Bubly Confess Not Confess Confess Not Confess

26 OTHER TYPES OF EQUILIBRIA Weakly Dominant Strategy Equilibrium (WDSE): An action profile a 1,, a n is called weakly dominant strategy equilibrium for a game < N, A i, U i >, if i N and b i A i, U i a i, b i U i b i, b i b i A i and U i a i, b i > U i b i, b i for some b i A i. Bunty Bubly Confess Not Confess Confess Not Confess

27 OTHER TYPES OF EQUILIBRIA Very Weakly Dominant Strategy Equilibrium (VWDSE): An action profile a 1,, a n is called very weakly dominant strategy equilibrium for a game < N, A i, U i >, if i N and b i A i, U i a i, b i U i b i, b i b i A i. Bunty Bubly Confess Not Confess Confess Not Confess

28 NO DOMINANT STRATEGY EQUILIBRIA (PROJECT COORDINATION GAME) Alice Bob Deep Learning Website Designing Deep Learning Website Designing

29 OTHER CATEGORIES OF GAMES Repeated games Dynamic games Stochastic games Network games Multi-level games (Stackelberg games) Differential games... Analyzing these games show how agents can rationally form beliefs over what other agents will do, and (hence) how agents should act Useful for taking a profitable action as well as predicting behavior of others.

30 MECHANISM DESIGN How would you create the rules of a game to achieve a desired objective? Ans: Mechanism Design

31 MECHANISM DESIGN How would you create the rules of a game to achieve a desired objective? Ans: Mechanism Design Reverse Engineering of Games Art of designing the rules of a game to achieve a specific desired outcome Game Theory, along with Mechanism Design have emerged as an important tool to model, analyze, and solve decentralized design problems in engineering involving multiple autonomous agents that interact strategically in a rational and intelligent way.

32 CAKE CUTTING Courtesy: Google images

33 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

34 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

35 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

36 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

37 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

38 CAKE CUTTING I want no less than half the cake I want no less than half the cake Courtesy: Google images

39 CAKE CUTTING: MECHANISM DESIGN Solution: Cut and Choose Mother makes one of the kids cutter and the other chooser Cutter : Cuts the cake into two halves Chooser: Gets to select one of the haves The cutter can cut the cake to two pieces that she considers equal. Then, regardless of what the chooser does, she is left with a piece that is as valuable as the other piece. The chooser can select the piece which he considers more valuable. Then, even if the cutter divided the cake to pieces that are very unequal (in the chooser's eyes), the chooser still has no reason to complain because he chose the piece that is more valuable in his own eyes.

40 CAKE CUTTING: MORE THAN TWO KIDS I want the cake to be split into exactly 4 equal parts I want one a piece with at least one fourth of all the fruits I want at least oneeighth of strawberry cream and at least oneeighth of kiwi cream I want a piece with at least onefourth of all the kiwi pieces. Courtesy: Google images

41 FAIR DIVISION Courtesy: Google images

42 FAIR DIVISION Courtesy: Google images

43 DESIRABLE PROPERTIES OF A MECHANISM Allocative Efficiency: Allocation should maximize the sum of value obtained by all the players. Individual Rationality: Players do not loose anything by participating in the game or Voluntary Participation Dominant Strategy Incentive Compatibility: Strategy-proofness Non-Dictatorship: There is no agent for whom all outcomes turn out to be favored outcomes.

44 COOPERATIVE GAME THEORY There is an incentive to cooperate collusion, binding contract, side payment Players can form a group and cheat the system to get a better pay-off Questions of Interest What are the conditions for forming stable coalitions? When will a single coalition (grand coalition) be formed? What is a fair distribution of payoffs among players

45 SHAPLEY VALUE

46 EXAMPLE: PROJECT CONTRACT A single project worth Rs One contractor (C) Two laborers (A and B). The contractor alone cannot finish the project without the laborers. The laborers cannot get the project contract without the contractor. If the contractor gets the project, it can be completed with the help of at least one laborer.

47 EXAMPLE: PROJECT CONTRACT A single project worth Rs One contractor (C) Two laborers (A and B). The contractor alone cannot finish the project without the laborers. The laborers cannot get the project contract without the contractor. If the contractor gets the project, it can be completed with the help of at least one laborer. How to split the cost among the contractor and the two laborers? <100,100,100>?

48 EXAMPLE: PROJECT CONTRACT Recall:

49 EXAMPLE: PROJECT CONTRACT Recall:

50 EXAMPLE: PROJECT CONTRACT Recall:

51 EXAMPLE: PROJECT CONTRACT Recall: Shapley value split: <50, 50, 200>

52 OTHER SOLUTION CONCEPTS IN COOPERATIVE GAME THEORY Stable Sets Core Kernel Nucleolus The Gately Point...

53 OTHER SOLUTION CONCEPTS IN COOPERATIVE GAME THEORY Stable Sets Core Kernel Nucleolus The Gately Point... Non Cooperative Game Theory Mechanism Design Cooperative Game Theory

54 REAL-WORLD APPLICATIONS Resource Allocation: Find a fair split of resources among agents Procurement Auction: Design an auction that maximizes social utilities Crowdsourcing: Design a mechanism to complete as many task as possible with maximum quality. Online Education Platforms (MOOCs): Designing incentives to improve participation level of students and instructors. Social Network Analysis: Discovering influential nodes, providing incentives to ensure maximum spread of information over a network....

55 REAL-WORLD APPLICATIONS

56 REAL-WORLD APPLICATIONS

57 REAL-WORLD APPLICATIONS

58 REAL-WORLD APPLICATIONS

59 REAL-WORLD APPLICATIONS

60 SPONSORED (KEYWORD) SEARCH AUCTION Separate auction for every query: Positions awarded in order of bid (more on this later). Advertisers pay bid of the advertiser in the position below.

61 SPONSORED (KEYWORD) SEARCH AUCTION Separate auction for every query: Positions are assigned to ads in the order of their bids. The payment is typically a function of the bids of the advertisers.

62 SPONSORED (KEYWORD) SEARCH AUCTION Separate auction for every query: Positions are assigned to ads in the order of their bids. The payment is typically a function of the bids of the advertisers. Simple setting One ad slot and N competing advertisers Which ad to show and what should the advertiser pay? Solving this requires a mechanism comprising an allocation rule and a payment rule.

63 SPONSORED (KEYWORD) SEARCH AUCTION Separate auction for every query: Positions awarded in order of bid (more on this later). The payment is typically a function of the bids of the advertisers. Simple setting One ad slot and N competing advertisers Which ad to show and what should the advertiser pay? Solving this requires a mechanism comprising an allocation rule and a payment rule. VCG (Vickrey-Clarke-Groves) mechanism : * Allocation rule: Give the ad-slot to the advertiser with maximum valuation/bid * Payment rule: Take the second-highest bid value from the selected advertiser

64 PAY PER CLICK AUCTION Each advertiser pays a value only when an user clicks the ad Each ad is an arm and click probabilities are the stochastic rewards (parameters to be learnt) Additionally, each advertiser bids a value that s/he is willing to pay when an user click the ad (strategic parameter) Payment received is a function of both: 1. Click probabilities of the ads 2. Declared bids of the advertisers GOAL: Design a mechanism (allocation rule and payment rule) that ensures truthful elicitation of bids ( strategy proof-ness ) as well as maximizes the total payment received from the advertisers within a limited number of trials.

65 USEFUL LINKS The book followed for preparing this lecture: Game Theory and Mechanism Design by Prof. Y. Narahari.

66