Report

Negotiation A Lesson in Multiagent System Based on Jose Vidal’s book Fundamentals of Multiagent Systems Henry Hexmoor SIUC Negotiation: The Bargaining Problem • Interaction in order to agree on a deal • Approach is to exchange messages among agents – Objective is to reach a deal, that: 1. maximizes utilities, 2. avoids expiration, 3. avoids risk of conflict, and 4. avoid failure on deal Automated Negotiation • Knowledge and Decision making is distributed to local sites • Utilities are optimized without: 1. central aggregation, or 2. central reasoning Examples: • Large organizations, governments, societies Bargaining problem(Nash 1950) ui : R Where represents set of deals R represents real number of states • : the no deal deal u ( ) i.e., agent prefers no deal to • negative utility Pareto Optimal • A deal is Pareto Optimal if there is no other deal such that no one prefers it over • For two agents i and j Pareto Frontier j Space of possible deals A deal i Independence of Utility units Property • A Negotiation is independent of utility units if when U chooses and when given U ' 1u1 , 2u2 ,..., I uI : u U chooses i ui ( ' ) i ui ( ) Where e.g., money in different countries ' Symmetry Property • A negotiation protocol is symmetric if the solution remains the same as long as the set of utility function U is the same, regardless of which agent has which the utility. Rationality • A deal is individually rational iff i ui ( ) ui ( ) Irrelevant Alternatives Property • A negotiation to protocol is independent of irrelevant alternatives if it is true that when given the set of possible deals it chooses and when ' where ' it again chooses , assuming U stays constant Egalitarian Solution • Gains are equally shared and arg max ui ( ' ) 'E i Where E represents set of deals which equal payoff E ui ( ) u j ( ) ij Egalitarian solution for two… Egalitarian deal may not be Pareto Optimal uj ui Egalitarian Social Welfare solution • A deal that maximizes the utility received by the agent with the smallest utility Example: Helping the poor! arg max min ui ( ) i Every problem is guaranteed to have an egalitarian social welfare solution Utilitarian solution • A deal that the deal that maximizes the sum of all utilities arg max ui ( ) i • The utilitarian deal is a Pareto optimal deal. • There might be more than one utilitarian deals in the case of a tie. • The utilitarian deal violates the independence of utility units assumption. Nash Bargaining solution • A deal that maximizes the product of the utilities : • The Nash solution is 1. 2. 3. Pareto efficient, independent of utility units, symmetric, and 4. independent of irrelevant alternatives arg max ui ( ' ) ' . Kalai-smorodinsky • A deal that distributes utilities in proportion to the maximum that the agent can get. Human preferences for deals is complex! The Rubinstein Bargaining Process • Agents act only at discrete time steps. • In each time step, one of the agents proposes a deal to the other who either accepts it or rejects it. • If the offer is rejected then we move to the next time step where the other agent gets to propose a deal. • Once a deal has been rejected it is considered void and cannot be accepted at a later time. • The alternating offers models does not have a dominant strategy. • We assume that time is valuable. The agents’ utility for all possible deals is reduced as time passes. E.g., haggling over how to split an ice cream sundae which is slowly melting. Time matters • Introducing a discount factor ti = i’s discount coefficient at time t = 0 do it now or lose = 1 do it whenever . . . • The agents’ utility for every possible deal decreases monotonically as a function of time with a discount factor. Theorem • The Rubinstein alternating offers game where the agents have complementary linear utilities has a unique subgame perfect equilibrium strategy where 1 j – Agent i proposes a deal 1 i j and accept the offer j from j only if 1 – Agent j proposes a deal 1 and accept the offer i from i only if * i * j ui ( j ) ui ( *j ), i i j u j ( i ) u j ( i* ) Corollaries 1 j (1 ) * i * j * j * i i Monotonic Concession Protocol 1. i arg max ui ( ) 2. propose i 3. receive j proposal 4. if ui ( j ) ui ( i ) 5. then accept j 6. else j 'i such that uj ( 'i ) u j ( i ) and ui ( 'i ) ui ( ) 7. goto 2 Zeuthern strategy • Willingness to risk deal break down, riski ui ( i ) ui ( j ) ui ( i ) • Agent calculates the risks for both agents. • The agent with the smallest risk should concede just enough to get the deal agreed in one step. • Zeuthern strategy converges to Nash solution. One step negotiation (Rosenschein and Zlotkin, 1994) • Each agent then has two proposals: the one it makes and the one it receives. • The agents must accept the proposal that maximizes the product of the agents’ utilities. • If there is a tie then they coordinate and choose one of them at random. 1. E { | ' ui ( )u j ( ) ui ( ' )u j ( ' )} 2. i arg max ui ( ) E 3. propose i 4. receive j 5. if 6. ui ( j )u j ( j ) ui ( i )u j ( i ) then report error , j is not 7. else, coordinate with following strategy j to choose randomly between i and j Distributed Search 0 Deals that dominate Search through dominant deals as in a hill climbing strategy problems my exist Ad-hoc Negotiation Strategies • Faratin(Jenning’s student) deployed Multiagent Negotiation systems, such as ADEPT Task allocation problem T A mapping • Ci : S R agent i incurs a cost for performing task S Example: Postman problem: Trading letters to lower their costs Problems with lies . . .