Mechanism Design

Overview Incentives in teams (T. Groves (1973)) Algorithmic mechanism design (Nisan and Ronen (2000)) - Shortest Path - Task Scheduling

Framework Something needs to be done with the help of n agents Is there a way of inducing them to do it (might lack knowledge or control) The way if it exists is called a “mechanism” Assumption1 : The agents are rational Assumption2 : The agents are independent (no communication) The mechanism is said to be truthful if there is no incentive for an agent to lie A lie is defined as something the agent could do so that the goal is not achieved.

An Organization CEO Sub-unit 1Sub-unit 2Sub-unit n ………

Pay-Fire Incentive -Optimally performing employees are rewarded -Pay is independent of how other employees perform -Assumes that the CEO has complete information

An Organization CEO Sub-unit 1Sub-unit 2Sub-unit n ………

Own Profit Incentive -- Payment to player i is independent of the decisions of the others --But it is dependent on the messages --Why is there no advantage in lying ?

Profit Sharing --It is hard to remove message dependence without losing truthfulness --Truthful mechanism – Nobody has incentive to lie --Strongly truthful mechanism – Truth telling is the only dominant strategy --Dominant strategy – No unilateral incentive to deviate

Direct Revelation Mechanisms The message strategy space and state space (t) are the same m(x(t),p(t)) x(t) is a set of feasible outputs given t p(t) is a vector of payments to the agents g(t,x(t)) is the function to optimize m’(x’(t),p’(t)) is a c-approximation for m(x(t),p(t)) if g(t,x’(t))<= c. g(t,x(t))

VGC mechanisms VGC (Vickrey-Groves-Clarke) VGC mechanisms are truthful x(t) is feasible iff it maximizes g (so that we concern ourselves with providing the correct incentive structure.)

Shortest Path Each edge is an agent People want to send messages to other people People are at vertices Goal is to minimize cost Each edge has a cost = Payment to each edge = Complexity is O(m *n * log(m))

Task Scheduling k tasks n processors State of agent i = Goal is to minimize the completion time of the set of tasks (make-span) A task need not go to the agent that does it the fastest.

Min-Work Mechanism

Min-Work (contd.) Min-Work is truthful Nisan and Ronen show it is strongly truthful Min-Work is an n-approximation for make- span

Bounds on approximations

Proof Sketch T1T T1T2 e1 e1 1+e1 1

Randomized Mechanisms A probability distribution over a family of mechanisms that share the same set of strategies and outputs Optimize the G=E(g) Payments etc. are defined as expectations over payments

Randomly-Biased Min-Work

We will first show that the mechanism is truthful.

Weighted VGC Mechanisms The mechanism is truthful.

Proof Sketch T1T2OptRbmw 1(b+e) b1rnd b12 g(t,opt(t))=1 + b + e = 1 + 4/3 = 7/3, 7/4 * g(t,opt(t))=49/12 g(t,rbmw(t))=1/4(( b) + ( b) + ( ) + (b+1)) =1/4(9+3b) =13/4 = 3.25 <= 49/12

Mechanisms with Verification Assumption: Agents actions can be verified Routing, Task scheduling etc. Check the effect of such a simplifying assumption both on mechanism design and computation

Make-span with Verification

Generalized Compensation and Bonus Mechanisms ---Participation and Bonus Constraints

Computational Problems Exponential-time allocation algorithm Approximations tend to violate truthfulness (will discuss a theorem from Nisan and Ronen) If the no. of agents are fixed, and declarations are bounded a truthful polynomial time approximation mechanism exists. (Computing the exact solution is NP-hard)

Bounded Scheduling Problems

Rounding Mechanism Compensation using actual times Bonus using rounded times. All revelations that are rounded up to the same value as the true revelations are dominant strategies.

Extensions Repeated games e-dominant strategies Partial verification