1. SECOND PART: Algorithmic Mechanism Design

2. Suggested readings Algorithmic Game Theory, Edited by Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani, Cambridge University Press. Algorithmic Mechanism Design for Network Optimization Problems, Luciano Gualà, PhD Thesis, Università degli Studi dell’Aquila, 2007. Web pages by Éva Tardos, Christos Papadimitriou, Tim Roughgarden, and then follow the links therein

3. Implementation theory Imagine a “planner” who develops criteria for social welfare, but cannot enforce the desirable allocation directly, as he lacks information about several parameters of the situation. A mean then has to be found to “implement” such criteria.

4. The implementation problem Given: An economic system comprising of self- interested, rational agents, which hold some secret information A set of system-wide goals Question: Does there exist a mechanism that can implement the goals, by enforcing (through suitable economic incentives) the agents to cooperate with the system by revealing their private data?

5. Mechanism Design Informally, designing a mechanism means to define a game in which a desired outcome must be reached However, games induced by mechanisms are different from games in standard form: Players hold independent private values The payoff matrix is a function of these types  each player doesn’t really know about the other players’ payoffs, but only about its one!  Games with incomplete information  Dominant Strategy Equilibrium is used

6. Mechanism Design Problem: ingredients N agents; each agent has some private information t i  T i (actually, the only private info) called type A set of feasible outcomes F For each vector of types t=(t 1, t 2, …, t N ), a social-choice function f(t)  F specifies an output that should be implemented (the problem is that types are unknown…) Each agent has a strategy space S i and performs a strategic action; we restrict ourself to direct revelation mechanisms, in which the action is reporting a value r i from the type space (with possibly r i  t i ), i.e., S i = T i

7. Example: the Vickrey Auction Assume that the system-wide goal is to allocate a job by a sealed-bid auction. The set of feasible outcomes is given by all the bidders. The social-choice function is to allocate to the bidder with lowest true cost: f(t)=arg min i (t 1, t 2, …, t N ) Each agent knows its cost for doing the job (type), but not the others’ one: T i = [0, +  ]: The agent’s cost may be any positive amount of money t i = 80: Minimum amount of money the agent i is willing to be paid r i = 85: Exact amount of money the agent i bids to the system for doing the job (not known to other agents)

8. Mechanism Design Problem: ingredients (2) For each feasible outcome x  F, each agent makes a valuation v i (t i,x) (in terms of some common currency), expressing its preference about that output Vickrey Auction: If agent i wins the auction then its valuation is equal to its actual cost=t i for doing the job, otherwise it is 0 For each feasible outcome x  F, each agent receives a payment p i (x) in terms of the common currency; payments are used by the system to incentive agents to be collaborative. Then, the utility of outcome x will be: u i (t i,x) = p i (x) - v i (t i,x) Vickrey Auction: If agent’s cost for the job is 80, and it gets the contract for 100 (i.e., it is paid 100), then its utility is 20

9. Mechanism Design Problem: the goal Implement (according to a given equilibrium concept) the social-choice function, i.e., provide a mechanism M=, where: g(r) is an algorithm which computes an outcome x=x(r) as a function of the reported types r p(x) is a payment scheme specifying a payment w.r.t. an output x such that x=f(t) is provided in equilibrium w.r.t. to the utilities of the agents.

10. Mechanism Design: a picture Agent 1 Agent N Mechanism p1p1 pNpN tNtN t 1t 1 r 1r 1 r Nr N Private “types” Reported types Payments Output which should implement the social choice function Each agent reports strategically to maximize its well-being… …in response to a payment which is a function of the output!

11. Game induced by a MD problem This is a game in which: The N agents are the players The payoff matrix is given (in implicit form) by the utility functions

12. Implementation with dominant strategies Def.: A mechanism is an implementation with dominant strategies if there exists a reported type vector r*=(r 1 *, r 2 *, …, r N * ) such that f(t)=x(r * ) in dominant strategy equilibrium, i.e., for each agent i and for each reported type vector r =(r 1, r 2, …, r N ), it holds: u i (t i,x(r -i,r i * )) ≥ u i (t i,x(r)) where x(r -i,r i * )=x(r 1, …, r i-1, r i *, r i+1,…, r N ).

13. Mechanism Design: Economics Issues QUESTION: How to design a mechanism? Or, in other words: 1. How to design g(r), and 2. How to define the payment functions in such a way that the underlying social- choice function is implemented? Under which conditions can this be done?

14. Strategy-Proof Mechanisms If truth telling is the dominant strategy in a mechanism then it is called Strategy-Proof Agents report their true types instead of strategically manipulating it Utilitarian Problems: A problem is utilitarian if its objective function is such that f(t) =  i v i (t i,x) The Vickrey Auction is utilitarian

15. Vickrey-Clarke-Groves (VCG) Mechanisms A VCG-mechanism is (the only) strategy-proof mechanism for utilitarian problems: Algorithm g(r) computes: x = arg min x  F  i v i (r i,x) Payment function: p i (x)= h i (r -i ) -  j≠i v j (r j,x) where h i (r -i ) is an arbitrary function of the types of other players What about non-utilitarian problems? We will see…

16. VCG-Mechanisms are Strategy-Proof Proof (Intuitive sketch): Payment given to agent i p i (x)= h i (r - i )-  j≠i v j (r j,x) and both the terms above are independent of the type, strategy and valuation of agent i So it is best for agent i to report its true value. Strategic behavior does not lead to a beneficial outcome. ■

17. Clarke Mechanisms This is a special VCG-mechanism (known as Clarke mechanism) in which h i (r - i )=  j≠i v j (r j,x(r - i ))  p i =  j≠i v j (r j,x(r - i )) -  j≠i v j (r j,x) In Clarke mechanisms, agents’ utility are always non-negative

18. Clarke mechanism for the Vickrey auction The VCG-mechanism is: x=arg min x  F  i v i (r i,x)  allocate to the bidder with lowest reported cost p i =  j≠i v j (r j,x(r - i )) -  j≠i v j (r j,x)  p i =  j≠i v j (r j,x(r - i ))  pay the winner the second lowest offer, and pay 0 the losers Let us convince ourself it is strategy-proof by case analysis. For a player i, let T=min j≠i {r j }: t i t i, he may still win (again being paid T), but he may also lose (if r i >T), by getting a null utility; t i >T: then, if r i >t i, he keeps on not to win, while if r i <t i, he may win, but he will be paid T<t i, by getting a negative utility. Remark: the difference between the second lowest offer and the lowest offer is unbounded (frugality issue)

19. VCG-Mechanisms: Advantages For System Designer: The goal, i.e., the optimization of the social-choice function, is achieved with certainty. For Agents: Agents have truth telling as the dominant strategy, so they need not require any computational systems to deliberate about other agents strategies

20. VCG-Mechanisms: Disadvantages For System Designer: The payments may be sub-optimal System has to calculate N+1 functions Once with all agents (for g(r)) and once for every agent (for the associated payment) 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.

21. Mechanism Design: Algorithmic Issues QUESTION: What is the time complexity of the mechanism? Or, in other words: What is the time complexity of g(r)? What is the time complexity to calculate the N payment functions? What does it happen if it is NP-hard to compute the underlying social-choice function?

22. Algorithmic mechanism design for graph problems Following the Internet model, we assume that each agent owns a single edge of a graph G=(V,E), and establishes the cost for using it  The agent’s type is the true weight of the edge Classic optimization problems on G become mechanism design optimization problems! Many basic network design problems have been faced: shortest path (SP), single-source shortest paths tree (SPT), minimum spanning tree (MST), minimum Steiner tree, and many others

23. Summary of forthcoming results Centralized algorithm Selfish-edge mechanism SPO(m+n log n) SPTO(m+n log n) MSTO(m  (m,n))  For all these problems, the time complexity of the mechanism equals that of the canonical centralized algorithm!