Mechanism Design Traditional Algorithmic Setting Mechanism Design Setting

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 1 10 3 s 10 1 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 1 10 3 s 10 1 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? 1 10 8. 9 3 s 10 1 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 • • ti is true type of agent i (private value) ai 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 ai o(a) = outcome of mechanism vi(o, ti) is value of outcome o to agent i pi(a) = payment to agent i u(i) = pi(a) + vi(o, ti) is utility of total outcome to i

Mechanism Properties • Dominant strategy – If for each agent i, there exists a reported value ai 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 vi(o, ti) • Creates optimal output assuming truthful values • Payment calculation – pi(a) = Σj≠ivj(o, tj) + hi(a-i) where hi() is an arbitrary function of a-i • Key point: pi(a) is not dependent on ai

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