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Algorithmic Mechanism Design

Algorithmic Mechanism Design. S Kameshwaran Oct 16, 2002. Till now. Centralized  Decentralized Shortest Path Problem  Routing Problem Marriage Problem  Trading Problem Algorithms  Mechanisms Mechanism Design Game Theory. Till now.. Mechanism Design. Given:

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Algorithmic Mechanism Design

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  1. Algorithmic Mechanism Design S Kameshwaran Oct 16, 2002

  2. Till now.. • Centralized  Decentralized • Shortest Path Problem  Routing Problem • Marriage Problem  Trading Problem • Algorithms  Mechanisms • Mechanism Design • Game Theory

  3. Till now.. Mechanism Design • Given: • System comprising of self-interested, rational agents • Set of system wide goals • Mechanism Design • Does there exist a mechanism that can implement the goals? • Implementation of the goals depends on the individual behavior of the agents

  4. Till now.. Game Theory • Given a game (mechanism), predicts the outcome by analyzing the individual behavior of the players (agents) • Interactive Decision Theory • Game: • N players • Rules of encounter: Who should act when and what are the possible actions • Every possible outcome of the game

  5. Game Theory • Normal Form Games • N players • Si=Strategy set of player i • Single simultaneous move: each player i chooses a strategy siSi • Nobody observes others’ move • The strategy combination (s1, s2, …, sN) gives payoff (p1, p2, …, pN) to the N players • All the above information is known to all the players and it is common knowledge

  6. Equilibrium • An equilibriums*= (s1*, s2*, …, sN*) is a strategy combination consisting of a best strategy for each of the N players in the game • Equilibrium strategies are the strategies players pick in trying to maximize their individual payoffs, knowing that other players are also doing the same • What is the best strategy? depends on the game

  7. Dominant Strategy Equilibrium: Prisoner’s Dilemma

  8. Dominant Strategy Equilibrium: Prisoner’s Dilemma Strategy Set

  9. Dominant Strategy Equilibrium: Prisoner’s Dilemma Strategy Set Strategy Set

  10. Dominant Strategy Equilibrium: Prisoner’s Dilemma Strategy Set Payoffs Strategy Set

  11. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner I’s Decision:

  12. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner I’s Decision: • If II Don’t Confess then it is best to Confess

  13. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner I’s Decision: • If II Don’t Confess then it is best to Confess • If II Confess then it is best to Confess

  14. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner I’s Decision: • If II Don’t Confess then it is best to Confess • If II Confess then it is best to Confess • It is best to Confess for I, regardless of what II does: Dominant Strategy

  15. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner II’s Decision:

  16. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner II’s Decision: • If I Don’t Confess, then it is best to Confess

  17. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner II’s Decision: • If I Don’t Confess, then it is best to Confess • If I Confess, then it is best to Confess

  18. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Prisoner II’s Decision: • If I Don’t Confess, then it is best to Confess • If I Confess, then it is best to Confess • It is best to Confess for II, regardless of what I does: Dominant Strategy

  19. Dominant Strategy Equilibrium: Prisoner’s Dilemma • It is best for both I and II to Confess regardless of what other one does • Confess is a Dominant Strategy for both • (Confess, Confess) becomes the Dominant Strategy Equilibrium • Note: Its beneficial for both to Don’t Confess, but it is not an equilibrium as both have incentive to deviate

  20. Dominant Strategy Equilibrium: Prisoner’s Dilemma • Dominant Strategy Equilibrium is a strategy combination s*= (s1*, s2*, …, sN*), such that si* is a Dominant Strategy for each i • Dominant Strategy is the best response to any strategy of other players • It is good for agent as it need not deliberate about other agents’ strategies • All games need not have dominant strategy equilibrium

  21. A Beautiful Mind: Nash Equilibrium • Nash Equilibrium is a strategy combination s*= (s1*, s2*, …, sN*), such that si* is a best response to (s1*, …,si-1*,si+1*,…, sN*), for each i • (s1*, s2*, s3*) is a Nash Equilibrium (3 player game) iff • s1* is the best response of 1, if 2 chooses s2* and 3 chooses s3* • s2* is the best response of 2, if 1 chooses s1* and 3 chooses s3* • s3* is the best response of 3, if 1 chooses s1* and 2 chooses s2* • Note: It is a simultaneous game and nobody knows what exactly the choice of other agents • Nash Equilibrium assumes correct and consistent beliefs

  22. Nash Equilibrium: Battle of the Sexes

  23. Nash Equilibrium: Battle of the Sexes • (Prize Fight, Prize Fight) is a NE: Best responses to each other

  24. Nash Equilibrium: Battle of the Sexes • (Prize Fight, Prize Fight) is a NE: Best responses to each other • (Ballet, Ballet) is a NE: Best responses to each other

  25. Nash Equilibrium • In a NE no agent can unilaterally deviate from its strategies given others’ strategies as fixed • Topologically it’s a fixed point • There may be no, one or many NE • Agent has to deliberate about the strategies of the other agents • If the game is played repeatedly and players converge to a solution then it has to be NE • Dominant Strategy Equilibrium  Nash Equilibrium (but the converse is not always true)

  26. Mechanism Design • Games induced by mechanisms are different from the previous games: • The payoff/output matrix is not known to the players (i.e) the players don’t know about the other players’ utilities • Game of Incomplete information • NE  Bayesian NE • Dominant Strategy Equilibrium is used in Strategy-Proof Mechanism • BNE is used in Bayesian Nash Mechanisms

  27. Mechanism Design Problem: Type of an Agent • N agents, and each agent has some private information called its type, tiTi (set of all possible types) • Agent i knows only its type but not the others’ types. • Other agents know agent i’s set of possible types is Ti • Auction Game: • Each agent knows its value for the good but not others’ value. The type of the agent is its value. • Ti=[75, 100]: The agents may value the good anywhere between 75 and 100 (known to all agents) • ti=80: Exact value of the good to the agent i (not known to other agents)

  28. Mechanism Design Problem: Output Specification • O is the set of outcomes • Output Specification g: For a given set of type configuration (t1, t2, …, tN) , it specifies a valid outcome o • Auction Game: • O: Different winners of the object • g: arg maxi (t1, t2, …, tN) (allocate to the bidder with highest value)

  29. Mechanism Design Problem: Valuation and Utility • If o is the outcome, ti is the type, then i’s valuation is given by a real valued function: vi(o,ti) • Auction Game: If agent i wins the good then its valuation is equal to its value for the good, otw it is 0 • If pi is the payment made by the agent, then utility of the outcome o, with type tiis ui= vi(o,ti)+pi • Auction Game: If agent i’s value of the object is 100, and if it pays 90, then the utility is 100-90=10 • Agent’s motive: Maximize (expected) Utility

  30. Mechanism Design Mechanism Design Problem • Ti: Set of possible types of agent i, T = iTi • Output Specification g:TO • Valuation: vi(o,ti), Quasi-linear Utility: vi(o,ti)+pi Mechanism • M=<S, O, P> • S= iSi, where Si is the strategy of agent i (Strategy is a function of the type information) • Mechanism specifies: • Outcome o as a function of the strategy combination • Payment p as a function of the strategy combination

  31. Strategy-Proof Mechanism Design • If truth telling is the dominant strategy in a mechanism then it is called as Strategy-Proof Mechanism • Agents report their true valuation function instead of strategically manipulating it • Utilitarian Mechanisms: • A mechanism is utilitarian if its objective is to maximize the overall value of the system: maxo i vi(o,ti) • The auction game and the marriage problem are utilitarian

  32. Strategy-Proof Mechanism Design • Strategy-proof mechanism: • Mechanism: Utilitarian maxo i vi(o,ti) • Utility: Quasi-Linear (vi(o,ti)+pi) Then the following payment function ensures that the mechanism is Strategy-proof (truth telling is dominant strategy) pi = j<>i vi(o,ti)+hi(t-i) (VCG Mechanisms) • hiis an arbitrary function of types of other players

  33. Strategy-Proof Mechanism Design • Proof (Intuitive sketch): • Payment made by agent i pi = j<>i vi(o,ti)+hi(t-i) • Both the terms above are independent of the type, strategy and valuation of i • So it is best for i to report its true value. Strategic behavior may not lead to a beneficial outcome (similar to Vickrey Auctions)

  34. Strategy-Proof Mechanism Design: Advantages • For System Designer: • The motive (maximizing the sum of the valuation of all the agents) is achieved with certainty. • The outcome is pareto-efficient (in fact ex-post efficient) • For Agents: • Agents have truth telling as the dominant strategy, so they need not require any computational systems to deliberate about other agents strategies

  35. Strategy-Proof Mechanism Design: Disadvantages • For System Designer: • The payments may not be budget-balanced • Budget balance ipi=0 (There is no external source of money. It only gets exchanged among the agents) • In a market with several buyers and sellers, the total money collected from the buyers is given to the sellers. If the total money collected is less than that to be given, then market is at a lost • VCG is the only strategy-proof pricing scheme for utilitarian functions

  36. Strategy-Proof Mechanism Design: Disadvantages • For System Designer (contd..): • System has to calculate the utilitarian function N+1 times • Once with all agents and the once for each agent removed from the system • If the problem is hard to solve then the computational cost may be very heavy • For Agents: • Agents may not like to tell the truth to the system designer as it can be used in other ways.

  37. Algorithmic Mechanism Design • Algorithmic issues in finding the solution to the utilitarian function and the payments • Research Problem: What is the complexity of determining the payments to the agents? • For shortest path problem, calculating a payment to the agent requires to determine the shortest path without the edge belonging to that agent • Calculate the shortest path n+1 times where n is the number of edges in the optimal solution • Main result: The complexity of finding the payments to all agents is same as the complexity of solving one instance of the shortest path problem

  38. Algorithmic Mechanism Design • What if finding the optimum solution to the utilitarian mechanism is hard? • Approximation schemes • If the problem can only be solved approximately, then strategy-proofness breaks • Research Problems: • What is the best approximation ratio for a given mechanism? • What are the payment functions for the approximate mechanism so that it becomes strategy-proof?

  39. Impossibility Result No mechanism can be simultaneously efficient, strategy-proof and budget-balanced • One has to compromise on any of the above • Budget-balance is mandatory for any real world system (no system would like to run in loss) • Efficiency is also mandatory as agents may not prefer inefficient outcomes  Bayesian Nash Mechanisms

  40. Bayesian Nash Mechanisms • Advantages: • Lot of options in pricing: Any kind of pricing is fine as long as it is budget-balanced • Disadvantages: • No dominant strategy: Computationally taxing for agents • Proof of existence of BNE for a mechanism is difficult (proof heavily relies on fixed point theorems which require lot of nice behavior from the system parameters) • Maybe more than one BNE: Choosing the best may not be possible • Determining the BNE: No standard procedure

  41. Algorithmic Aspects • Open Problem: • What are the algorithmic aspects for the Bayesian Mechanisms? • One has to consider the computational capabilities of individual agents • Devising a method for proving the existence of BNE without the use of fixed point theory • General method for determining at least one/all of the BNE points

  42. References • Equilibrium Concepts: • Games and Information: An Introduction to Game Theory, Eric Rasmusen, Basil Blackwell Publishers, 1989 • Any Game Theory Book • Algorithmic Mechanism Design • Algorithmic Mechanism Design, Noam Nisan and Amir Ronen, 2001 • Algorithms for Selfish Agents, Noam Nisan, 2001

  43. Coming Up.. • 18/10/02: Algorithms, Games and Internet • Motivating the use of GT and Mechanism Theory for modeling the Internet mathematically by using several Internet applications like Multi-Cast Routing, Peer-Peer file sharing, etc • Several open problems in this area

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