1. SECOND PART: Algorithmic Mechanism Design

2. 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. Subfield of economic theory with an engineering perspective: Designs economic mechanisms like computer scientists design algorithms, protocols, systems, …

3. The implementation problem Given: An economic system comprising of self- interested, rational agents, which hold some private preferences A system-wide goal (social-choice function) Question: Does there exist a mechanism that can enforce (through suitable economic incentives) the selfish agents to behave in such a way that the desired goal is implemented?

4. An example: auctions t 1 =10 t 2 =12 t 3 =7 r 1 =11 r 2 =10 Social-choice function: the winner should be the guy having in mind the highest value for the painting The mechanism tells to players: (1)How the item will be allocated (i.e., who will be the winner) (2) The payment the winner has to return, as a function of the received bids t i : is the maximum amount of money player i is willing to pay for the painting If player i wins and has to pay p his utility is u i =t i -p r i : is the amount of money player i bids (in a sealed envelope) for the painting r 3 =7

5. A simple mechanism: no payment t 1 =10 t 2 =12 t 3 =7 r 1 =+  r2=+r2=+ r3=+r3=+ …it doesn’t work… ?!? Mechanism: The highest bid wins and the price of the item is 0

6. Another simple mechanism: pay your bid t 1 =10 t 2 =12 t 3 =7 r 1 =9 r 2 =8 r 3 =6 Is it the right choice? Mechanism: The highest bid wins and the winner will pay his bid The winner is player 1 and he’ll pay 9 Player i may bid r i < t i (in this way he is guaranteed not to incur a negative utility) …and so the winner could be the wrong one… …it doesn’t work…

7. An elegant solution: Vickrey’s second price auction t 1 =10 t 2 =12 t 3 =7 r 1 =10 r 2 =12 r 3 =7 every player has convenience to declare the truth! (we prove it in the next slide) I know they are not lying Mechanism: The highest bid wins and the winner will pay the second highest bid The winner is player 2 and he’ll pay 10

8. Theorem In the Vickrey auction, for every player i, r i =t i is a dominant strategy proof Fix i, t i, r -i, and look at strategies for player i. Let R= max j  i {r j } Case: t i ≥ R Declaring r i =t i gives utility u i = t i -R ≥ 0 (player wins) declaring r i > t i ≥ R gives the same utility (player wins) declaring R < r i < t i gives the same utility (player wins) Case: t i < R Declaring r i =t i gives utility u i = 0 (player loses) declaring r i < t i < R gives the same utility (player loses) declaring r i > R > t i yields u i = t i -R < 0 (player wins) declaring r i < R ≤ t i yields u i =0 (player loses) declaring t i < r i < R gives the same utility (player loses)

9. Mechanism Design Problem: ingredients N agents; each agent i, i=1,..,N, has some private information t i  T i (actually, the only private info) called type Vickrey’s auction: the type is the painting value the agents have in mind, and so T i is the set of positive reals A set of feasible outcomes X Vickrey’s auction: X is the set of agents (bidders)

10. Mechanism Design Problem: ingredients (2) For each vector of types t=(t 1, t 2, …, t N ), and for each feasible outcome x  X, a social-choice function f(t,x) measures the quality of x as a function of t. This is the function that the mechanism aims to implement (i.e., it aims to select an outcome x* that minimizes/maximizes it), but the problem is that types are unknown! Vickrey’s auction: f(t,x) is the type associated with x (namely, a bidder x), and the objective is to maximize f, i.e., allocate to the bidder with highest type 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 Vickrey’s auction: the action is to bid a value r i

11. Mechanism Design Problem: ingredients (3) For each feasible outcome x  X, each agent makes a valuation v i (t i,x) (in terms of some common currency), expressing its preference about that output x Vickrey’s auction: if agent i wins the auction then its valuation is equal to its type t i, otherwise it is 0 For each feasible outcome x  X, each agent receives/gives (this depends on the problem) a payment p i (x) in terms of the common currency; payments are used by the system to incentive agents to be collaborative. Vickrey’s auction: if agent i wins the auction then he returns a payment equal to -r j, where r j is the second highest bid, otherwise it is 0 Then, for each feasible outcome x  X, the utility of agent i coming from outcome x will be: u i (t i,x) = p i (x) + v i (t i,x) Vickrey’s auction: if agent i wins the auction then its utility is equal to u i = -r j +t i ≥ 0, where r j is the second highest bid, otherwise it is u i = 0+0=0

12. Mechanism Design Problem: the goal Given all the above ingredients, design a mechanism M=, where: g(r) is an algorithm which computes an outcome x=x(r)  X as a function of the reported types r p(x) =(p 1,…,p N ) is a payment scheme w.r.t. outcome x specifying a payment value for each player which implements the social-choice function f in equilibrium (according to a given solution concept, e.g., dominant strategy equilibrium, Nash equilibrium, etc.), w.r.t. players’ utilities. (In other words, there exists a reported type vector r * for which the mechanism provides a solution x(r * ) and a payment scheme p(x(r * )) such that players’ utilities u i (t i, x(r * )) = p i (x(r * )) + v i (t i, x(r * )) are in equilibrium, and f(t, x(r * )) is optimal (either minimum or maximum))

13. Mechanism Design: a picture System Agent 1 Agent N “I propose to you the following mechanism M= ” 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 in equilibrium w.r.t. agents’ utilities Each agent reports strategically to maximize its well-being… …in response to a payment which is a function of the output!

14. Implementation with dominant strategies Def.: A mechanism is an implementation with dominant strategies if the s.c.f. f is implemented in dominant strategy equilibrium, i.e., there exists a reported type vector r * =(r 1 *, r 2 *, …, r N * ) such that 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 ), and f(t,x(r*)) is optimized.

15. Strategy-Proof Mechanisms If truth telling is the dominant strategy in a mechanism then it is called Strategy-Proof or truthful  r * =t.  Agents report their true types instead of strategically manipulating it  The algorithm of the mechanism runs on the true input

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

17. Utilitarian problems: A problem is utilitarian if its social-choice function is such that f(t,x) =  i v i (t i,x) notice: the auction problem is utilitarian, and so they are all problems where the s.c.f. is separately-additive w.r.t. agents’ valuations, as in many network optimization problems… Good news: for utilitarian problems there is a class of truthful mechanisms A prominent class of problems

18. Vickrey-Clarke-Groves (VCG) Mechanisms A VCG-mechanism is (the only) strategy-proof mechanism for utilitarian problems: Algorithm g(r) computes: x = arg max y  X  i v i (r i,y) Payment function for player i: 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 reported types of players other than player i. What about non-utilitarian problems? Strategy- proof mechanisms are known only when the type is a single parameter.

19. Theorem VCG-mechanisms are truthful for utilitarian problems proof Fix i, r -i, t i. Let ř=(r -i,t i ) and consider a strategy r i  t i x=g(r -i,t i ) =g(ř) x’=g(r -i,r i ) u i (t i,x) = u i (t i,x’) = [h i (r -i )+  j  i v j (r j,x)] + v i (t i,x) [h i (r -i )+  j  i v j (r j,x’)] + v i (t i,x’) = h i (r -i ) +  j v j (ř j,x) +  j v j (ř j,x’) but x is an optimal solution w.r.t. ř =(r -i,t i ), i.e., x = arg max y  X  i v i (ř,y)  j v j (ř j,x) ≥  j v j (ř j,x’) u i (t i,x)  u i (t i,x’).

20. How to define h i (r -i )? Remark: not all functions make sense. For instance, what happens in our example of the Vickrey auction if we set for every agent h i (r -i )=-1000 (notice this is independent of reported value r i of agent i, and so it obeys the definition)? Answer: It happens that players’ utility become negative; more precisely, the winner’s utility is u i (t i,x) = p i (x) + v i (t i,x) = h i (r -i ) +  j≠i v j (r j,x) + v i (t i,x) = -1000+0+12 = -988 while utility of losers is u i (t i,x) = p i (x) + v i (t i,x) = h i (r -i ) +  j≠i v j (r j,x) + v i (t i,x) = -1000+12+0 = -988  This is undesirable in reality, since with such perspective agents would not partecipate to the auction!

21. The Clarke payments This is a special VCG-mechanism in which h i (r - i )=-  j≠i v j (r j,x(r - i ))  p i (x) = -  j≠i v j (r j,x(r - i )) +  j≠i v j (r j,x) With Clarke payments, one can prove that agents’ utility are always non-negative  agents are interested in playing the game solution maximizing the sum of valuations when i doesn’t play

22. The Vickrey’s auction is a VCG mechanism with Clarke payments Observe that auctions are utilitarian problem. Then, the VCG-mechanism associated with the Vickrey’s auction is: x=arg max y  X  i v i (r i,y) …this is equivalent to allocate to the bidder with highest reported cost (in the end, the highest type, since it is strategy-proof) p i = -  j≠i v j (r j,x(r - i )) +  j≠i v j (r j,x) …this is equivalent to say that the winner pays the second highest offer, and the losers pay 0, respectively Remark: the difference between the second highest offer and the highest offer is unbounded (frugality issue)

23. 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

24. VCG-Mechanisms: Disadvantages For System Designer: The payments may be sub-optimal (frugality) Apparently, the system may need to run the mechanism’s algorithm N+1 times: once with all agents (for computing the outcome x), 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.

25. 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 implement the underlying social-choice function? Question: What is the time complexity of the Vickrey auction? Answer: Θ(N), where N is the number of players. Indeed, it suffices to check all the offers, by keeping track of the lowest and second lowest one.

26. Algorithmic mechanism design and network protocols Large networks (e.g., Internet) are built and controlled by diverse and competitive entities: Entities own different components of the network and hold private information Entities are selfish and have different preferences  MD is a useful tool to design protocols working in such an environment, but time complexity is an important issue due to the massive network size

27. Algorithmic mechanism design for network optimization problems Simplifying 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  Classic optimization problems on G become mechanism design optimization problems in which the agent’s type is the weight of the edge! Many basic network design problems have been faced: shortest path (SP), single-source shortest paths tree (SPT), minimum spanning tree (MST), and many others

28. Some remarks In general, network optimization problems are minimization problems (the Vickrey’s auction was instead a maximization problem) However, we have: for each x  X, the valuation function v i (t i,x) represents a cost incurred by player i in the solution x (and so it is a negative function of its types) the social-choice function f(t,x) is negative (since it is the sum of negative functions), and so its maximization (as computed by the algorithm of the VCG-mechanism) corresponds to a minimization of the sum of absolute values of the valuation functions payments are from the mechanism to agents

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

30. Exercise: redefine the Vickrey auction in the minimization version t 1 =10 t 2 =12 t 3 =7 r 1 =10 r 2 =12 r 3 =7 I want to allocate the job to the true cheapest machine Once again, the second price auction works: the cheapest bid wins and the winner will get the second cheapest bid The winner is machine 3 and it will receive 10 job to be allocated to machines t i : cost incurred by i if he does the job if machine i is selected and receives a payment of p its utility is p-t i