Mechanism Design Traditional Algorithmic Setting Mechanism Design Setting

Shortest Path Problem Traditional Formulation Input –Directed graph G = (V,A) –Two special nodes s,t –Weight on each arc w(u,v) Output –Shortest path p from s to t Solvable in polynomial time s t

Selfish Agent View Suppose each edge is a link in the internet and is controlled by separate entities Further suppose the weights are the costs for the edge to transmit a message from s to t What incentive does an agent have to transmit the message? We’ll pay them to send a message s t

Payments How much should we pay an edge that is not used? How much should we pay an edge that is used? What if the edge lies about its cost to transmit the message across its link? s t

Mechanism Design Formulation Traditional Formulation Input –Directed graph G = (V,A) –Two special nodes s,t –Weight on each arc w(e) Output –Shortest path p from s to t Solvable in polynomial time New Formulation Input –Graph G = (V,A) Assumption: Biconnected –Special nodes s,t –Arc agents with true costs w(e) Gather phase –Agents report costs w’(e) Mechanism –Chooses s-t path –Computes payments to each arc agent –Algorithms for path computation and payments are known to agents

Mechanisms New Formulation Input –Graph G = (V,A) Assumption: Biconnected –Special nodes s,t –Arc agents with true costs w(e) Gather phase –Agents report costs w’(e) Mechanism –Chooses s-t path –Computes payments to each arc agent –Algorithms for path computation and payments are known to agents Path choice mechanism –Find shortest path given reported costs using traditional s-t shortest path algorithm Payment mechanism –For each edge e, compute the following –0 if arc is not used –d G|w(e) =   : cost of shortest path without using arc e –d G|w(e) =0 : cost of shortest path with arc e assuming arc e costs 0 –p(e) = d G|w(e) =   d G|w(e) =0 Any incentive for e to lie?

Mechanism Design Input –Agents with selfish interests and private values –Other characteristics which are known Gather phase –Agents report type to mechanism Output phase –Provide solution to problem –Provide payments to (or collect payments from) agents

Notation t i is true type of agent i (private value) a i is type reported by agent i to mechanism a is the vector of all agent types reported a -i is the vector of types reported minus a i o(a) = outcome of mechanism v i (o,t i ) is value of outcome o to agent i p i (a) = payment to agent i u(i) = p i (a) + v i (o,t i ) is utility of total outcome to i

Mechanism Properties Dominant strategy –If for each agent i, there exists a reported value a i such that for all possible reported values of other agents a -i, u(i) is maximized –We assume that agents are rational and thus employ dominant strategies Truthful (strategyproof) mechanism –For each agent, reporting true private value is a dominant strategy Strongly truthful mechanism –For each agent, reporting true private value is the only dominant strategy

Mechanism Goals Rationality assumption –We assume that agents are rational and thus employ dominant strategies Truthful mechanism –We can assume that the agents reveal true values Output optimization –Create a solution, assuming truthful values, where value is optimal or approximately optimal Price optimization –Minimize amount paid to agents or maximize amount collected from agents Output and price computations –Should be done in polynomial time

Vickrey-Clarke-Groves (VCG) Mechanisms Objective function –Summation of all agents’ valuation functions Σ v i (o,t i ) Creates optimal output assuming truthful values Payment calculation –p i (a) = Σ j≠i v j (o,t j ) + h i (a -i ) where h i () is an arbitrary function of a -i Key point: p i (a) is not dependent on a i

Notes to add Monotone fcts Price of Anarchy nash equilibrium Price of Stability Drawbacks of VCG –The VCG framework is a general method for creating truthful mechanisms. We address the following four drawbacks: (1) VCG selects the outcome that maximizes the total social welfare, whereas often the decision-maker wants to maximize some other function. (2) In the case where the decision-maker is purchasing something, VCG must sometimes pay an unacceptably high premium to induce truthtelling. (3) Sometimes the decision-maker would like to use the VCG mechanism, but cannot because computing it is NP-hard. (4) VCG resists manipulation by single agents, but, in general, multiple agents could collude to cheat the mechanism. Quote from Aaron Archer