We discussed in class that with n agents, there are n! permutations of agents which resultsin n! different deterministic serial dictatorship mechanisms. Every serial dictatorshipis strategyproof and results in a Pareto optimal outcome (ex post).Consider m >n items to be assigned to n agents. Propose a mechanism that:1. receives as input a full ranked list (strict ordering; no ties) of items from each agent,and based on this input computes an outcome,2. is Pareto optimal (aka Pareto efficient ex post),3. is strategyproof (no agent can benefit from misreporting),4. guarantees the outcome to be envy-free up to one item (EF1).For your answer, 1) justify that your proposed mechanism satisfies each of the itemsabove and 2) given an example that shows the steps of your proposed mechanism.

Pareto ideal A Pareto diagram is a reference chart. The lengths of the bars address recurrence or…

le discussed in class that with n agents, there are n! permutations of agents which n! different deterministic serial dictatorship mechanisms. Every serial dict. strategyproof and results in a Pareto optimal outcome (ex post). Consider m > n items to be assigned to n agents. Propose a mechanism that: 1. receives as input a full ranked list (strict ordering; no ties) of items from eac and based on this input computes an outcome, 2. is Pareto optimal (aka Pareto efficient ex post), 3. is strategyproof (no agent can benefit from misreporting),

le discussed in class that with n agents, there are n! permutations of agents which n! different deterministic serial dictatorship mechanisms. Every serial dict. strategyproof and results in a Pareto optimal outcome (ex post). Consider m > n items to be assigned to n agents. Propose a mechanism that: 1. receives as input a full ranked list (strict ordering; no ties) of items from eac and based on this input computes an outcome, 2. is Pareto optimal (aka Pareto efficient ex post), 3. is strategyproof (no agent can benefit from misreporting),

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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no handwritten Alice and Bob are playing a match to see who is the first to win n games, for some fixed n > 0. Suppose Alice and Bob are equally competent, that is, each of them wins a game with probability 1/2. Further, suppose that they have already played i + j games, of which Alice won i and Bob won j. Give an efficient algorithm to compute the probability that Alice will go on to win the match. For example, if i = n − 1 and j = n − 3, then the probability that Alice will win the match is 7/8, since she must win any of the next three games.
Exercise 3 Consider calculation of PageRank for a directed graph. Consider two jumping probabilities a and B such that a > B. For every node u, let Pa(u) be the PageRank of u calculated using a and let Pg (u) be the PageRank of u calculated using B. For each of the following statements, indicate whether the statement is true or false and provide a brief explanation for your answer. 1. The inequality Pa (u) > Pg (u) must hold for all nodes u. 2. The inequality Pa(u) > Pg (u) must hold for at least one node u.
Give an instance of STABLEMATCHING where the Propose-Dispose algorithm yields a matching where strictly more women get their most preferred man than men get their most preferred woman.
5.1. Trace the operation of Greedy Best-First Search, applied to the problem of getting to the goal state G from state S given below, for the following heuristic function: h,(N) = sum of the (Manhattan) distances of every tile to its goal position 341 25 876 State S 432 16 578 State G
3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A², then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A², then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k≥ 1. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of…