ics 806-Week 4 & 5

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School of Computing and

Informatics, University of Nairobi

MSC COMPUTER SCIENCE

Distributed rational decision-making for multi-agent systems

Session Topics

1. Definitions

2. How humans make decisions

3. SMART technique

4. Decision making in group setting5. Game Theoretic Perspective

6. Voting

7. Auctions8. Bargaining

9. Negotiation

10.Coalition Formation

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Distributed rational decision-making for multi-

agent systemsSources: Sandholm(1999). Distributed rational decision making; Paul Goodwin, George

Wright (2005). Decision Analysis for Management Judgment. John Wiley & Sons Ltd,

Third Edition; Michael Wooldridge (2002). Introduction to multi-agent systems.

Decision- is a choice from various

alternatives.

Rationality optimization from point of view

of goalsDistribution- involvement of several

participants with no central of control

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ICS 806 - MULTI-AGENT SYSTEMS

Distributed rational decision-making for multi-agentsystems

How Human beings make decision

1. They approximate or apply heuristics-rules of

thumb usually when they lack tools for decision

making

2. They do so because they have limited in abilities to compute.

This is referred to as bounded rationality (Paul Goodwin,

George Wright (2005)).

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Types of Heuristics used by Human beings in decision making

Minimalist strategy: recognize all the options and simply select one atrandom.

The last strategy: use the choice that worked the last time a decision hadto be made in a similar situation.

Lexicographic strategy: decide on the best attribute and use it to make a

selection such as the price property in buying a car.

Semi lexicographic decision strategy: consider several attributes and

apply some rules to choose. For example one may consider cost,

engine capacity and some rules for buying a car.

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Types of Heuristics used by Human beingsin decision making (cont)

Elimination by aspect strategy: select an attributeand apply a cut off point. People also make

decisions using satisficing strategy in which they

wait for options that appear sequentially until theymeet their expectations.

Reason based choice strategy: options aredescribed using various attributes and the optionwith the preferred attributes is selected. The

attributes are used as a ground for selection.

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Factors that affect how people make choices

timeavailable to make the decision

effortthat a given strategy will involve

knowledgeof the decision maker about the environment,

accurate decisionconstraints

Need for justificationof choice to others

desire to minimize conflict(for example, the conflict between thepros and cons of moving to another job).

- decision makers may choose their strategies so as to

balance the effort involved in making the decision

against the accuracy that they wish to achieve.

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Some terminologyObjective- an indication of the preferred direction of movement.

Example of objectives are statements such as minimize marketlosses or maximize market share.

Attribute -a feature that is used to measure performance in relation

to an objective. For example, if the objective is :-maximize the

exposure of a television advertisement then the attribute may benumber of people surveyed who recall seeing the advertisement in

order to measure the degree to which the objective was achieved.

Proxy attribute- an attribute which is not directly related to the

objective. For example, a company may use the proxy attribute staff

turnover to measure how well they are achieving their objective of

maximizing job satisfaction for their staff.

Value-a numerical score to measure attractiveness an action to adecision maker where there is no element of risk nor uncertainty.

Utility- a numerical score to measure attractiveness of an action to adecision maker where there is element of risk or uncertainty.

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SMART decision making technique- multiple objectivesSimple Multi-attribute Rating Technique (SMART):

Stage 1: Identify the decision maker (or decision makers).Stage 2: Identify the alternative courses of action. These can be,represented by the different offices the owner can choose.

Stage 3: Identify the attributes which are relevant to the decision problem.The attributes which distinguish the different offices can be factors such

as rent, size and quality of working conditions.

Stage 4: For each attribute, assign values to measure the performance ofthe alternatives on that attribute. For example, how well do theConstructing a value tree offices compare when considering the quality

of the working conditions they offer?Stage 5: Determine a weight for each attribute. This may reflect how

important the attribute is to the decision maker

Stage 6: For each alternative, take a weighted average of the values

assigned to that alternative. This will give us a measure of how well an anitem such as officeperforms over all the attributes.

Stage 7: Make a provisional decision.Stage 8: Perform sensitivity analysisto see how robust the decision is tochanges in the figures supplied by the decision maker.

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SMART example ( where should an office be located?)

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Smart examples: Values

Aggregate benefits = [(weighti * valuei)] / [(weighti)]

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Basic axioms related to SMART(1) Decidability: The decision maker should be able to decide which of the

options s/he prefers.

(2) Transitivity: If the decision maker prefers A to B; and s/he also prefers Bto C; then s/he must therefore also prefer A to C.

(3) Summation: This implies that if the decision maker prefers A to B and Bto C, then the strength of preference of A over C must be greater than the

strength of preference of A over B (or B over C).

(4) Solvability: This is necessary for the method of obtaining a valuefunction. For example it can be the distance from the center of town

which had a value halfway between the worst and best distances.

(5) Finite upper and lower bounds for value: The values that turn out to be

the best option are not necessarily so wonderful and the worst option is

not so awful.

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Decision making in group settingGroups- managers or stakeholders

There are essentially two approaches to the problem:

1. mathematical and2. behavioral aggregation

(although the approaches can be combined).

Mathematical aggregation rely on techniques such as thecalculation of a simple average of the judgments of the

individual group members.

Behavioral aggregation relies on reaching a decision by membersof the group communicating with each other either in open

discussion or via a more structured communication process.

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Examples: mathematical aggregation

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Examples: Aggregating probabilities (p.330)

Note potential problems with using probabilities

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Aggregating preferences

Member Preferences

Jana A > B > C

Jing B > C > AJumbo C> A > B

Taking votes for A>B: A gets two and B gets 1.

Comparing B and C (B > C): B gets 2 and C gets 1.

Comparing C and A (C>A): C gets 2 and A gets 1. So thepreference is not transitive. This is known as

Condorcets paradox.

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Arrowshowed in his Impossibility TheoremThere is no aggregation procedure that can produce

preferences with the following properties:1. A transitive group preference order for the options

under consideration;

2. If every member of the group prefers one option toanother then so must the group.

3. The group choice between two options, A and B,

depends only upon the preferences of members

between these options and not on preferences for

any other option. This discourages lying.

4. There is no dictator. No individual is able to impose

his or her preferences on the group.

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Values and utilities may be aggregated based on

simple average.

As given in the example that follows:

Destination Member 1 Member 2 AverageNanyuki 150 100 125

Usenge 80 90 85

Witu 100 86 93

Decision: members choose to go to Nanyuki

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Structured group decision making process- minimize negative group effects such as domineering by single

individuals;- realize this by controlling inter-personal interaction and information

flow such as using the Delphi method;

Delphi method:

(1) Panelists provide opinions about the likelihood of futureevents, or when those events will occur, or what the impact of

such event(s) will be. These opinions are often given as

responses to questionnaires which are completed individually by

members of the panel.

(2) The results of this polling of panelists are then tallied and statisticalfeedback of the whole panels opinions (e.g. range or medians) are

provided to individual panelists before a re-polling takes place. At this

stage, anonymous discussion (often in written form) may occur so thatdissenting opinion is aired.

(3) The output of the Delphi technique is a quantified group consensus,

which is usually expressed as the median response of the group of

panelists. (Paul Goodwin, George Wright (2005) p.337)

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The game theoretic perspective (Wooldridge)

Utilities and Preferences

Consider only two agents: Ag = {i, j} that are self-interested

Let = {1, 2, .. n} be a set of the preferred outcomes by agentsWe capture preferences by utility functions:

ui : Ru

j: R

Utility functions lead to preference orderingsover outcomes:

i means ui() ui()j means ui() > uj()Environment behaviour given by state transformer function:

: Ac x Ac agent is action (C or D) agentjs action (C or D)

Here is a state transformer function:

(D,D) =1, (D,C) =2, (C,D) =3, (C,C) =4(This environment is sensitive to actions of both agents.)Here is another:

(D,D) =1, (D,C) =1, (C,D) =1, (C,C) =1(Neither agent has any influence in this environment.)

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Rational ActionSuppose we have the case where bothagents can

influence the outcome, and they have utility functions asfollows:

ui(1)=1, ui(2)=2, ui(3)=3, ui(4)=4uj(

1)=1, uj(

2)=2, uj(

3)=3, uj(

4)=4

With a bit of mixing of notation:

ui

(D,D)=1, ui

(D,C)=2, ui

(C,D)=3, ui

(C,C)=4uj(D,D)=1, uj(D,C)=2, uj(C,D)=3, uj((C,C)=4

Then agent is preferences are:

C, CiC, DiD,CiD,DC is the rational choicefori.(Because iprefers all outcomes that arise through Cover

all outcomes that arise through D.)

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Payoff MatricesWe can characterize the previous scenario in a payoff matrix whereAgent iis

the column playerand Agentjis the row player.

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Dominant StrategiesGiven any particular strategy s(eitherCorD)agent i, there will be a number of possible

outcomes.

We say s1 dominates s2 if every outcome

possible by iplaying s1, is preferred overevery outcome possible by iplaying s2.A rational agent will never play a dominated

strategy. So in deciding what to do, we candelete dominated strategies. Unfortunately,there isnt always a unique undominated

strategy.

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Nash Equilibrium

In general, we will say that two strategies s1and s2are in Nash equilibrium if:

Under the assumption that agent iplays s1,

agentjcan do no better than play s2; andUnder the assumption that agentjplays s2,

agent ican do no better than play s1.

Neither agent has any incentive to deviatefrom a Nash equilibrium.

Unfortunately: Not every interaction scenario

has a Nash equilibrium.

Some interaction scenarios have more than

one Nash equilibrium.

C titi d Z S I t ti

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Competitive and Zero-Sum InteractionsWhere preferences of agents are diametrically

opposed we have strictly competitivescenarios.

Zero-sum encounters are those where utilities

sum to zero:

ui( ) + uj() = 0for all , Zero sum impliesstrictly competitive.

Zero sum encounters in real life are very rare . .. but people tend to act in many scenarios as

if they were zero sum.

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The Prisoners Dilemma

Two men are collectively charged with a crimeand held in separate cells, with no way of

meeting or communicating. They are told

that:1. if one confesses and the other does not, the

confessor will be freed, and the other will be

jailed for three years;

2. if both confess, then each will be jailed for

two years.Both prisoners know that if neither confesses,

then they will each be jailed for one year.

Payoff matrix for prisoners dilemma:

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Payoff matrix for prisoners dilemma:

Top left: If both defect, then both get punishment for mutual defection.

Top right: Ificooperates andjdefects, igets suckers payoff of 1, whilejgets 4.Bottom left: Ifjcooperates and idefects,jgets suckers payoff of 1, while igets 4.

Bottom right: Reward for mutual cooperation.

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Prisoners Dilemma cont.The individual rationalaction is defect. This guarantees a payoff of no worse than

2, whereas cooperating guarantees a payoff of at most 1.

So defection is the best response to all possible strategies: both agents defect,

and get payoff = 2.

But intuitionsays this is notthe best outcome: Surely they should both cooperateand each get payoff of 3!

This apparent paradox is the fundamental problem of multi-agent interactions. It

appears to imply that cooperation will not occur in societies of self-interestedagents.

Real world examples

nuclear arms reduction (why dont I keep mine. . . )free rider systems public transport;

in Kenya- television licenses.

The prisoners dilemma is ubiquitous. Can we recover cooperation?

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Distributed rational decision making continues(Tuomas W. Sandholm(1999). Distributed rational decision making.)

Automated negotiation with self interested agents is

important because:

Technology push that has led to standardized

communication infrastructure such as : Intenet, NII,EDI, FIPA, Concordia, Voyager, Odyssey, Telescript,

Java. Agents can easily communicate even if they are

heterogeneous.There is a move to computerize operational decisions

such as Internet transactions for purchase of goods,

information or bandwidth.There is a trend towards virtual organizations in

which dynamic alliances of small enterprises occur and

take advantage of economics of scale.

M lti A t t i t i t ti

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Multi-Agents systems can assist in automating

negotiation at operative decision level.

The agents are self interested and may pursue different goals.

Usually protocols and a strategy (way to use the protocol) are imposed

by the designer. Agents have interaction protocol but each agent

chooses their own strategy.

For self interested agents strategies are chosen so that individual goals

are best met.

The protocols should therefore be designed based on non-cooperative

perspective.

The Protocols considered are: voting, auctions, contracting, bargaining,

and coalition formation. The protocols however affect social outcomes.Note that self interested agents are concerned with their own local goals

and not the global(social) good. This is why protocols are important for

them.

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Criteria for evaluating protocolsSocial welfare sum of all agents utilities; can be used to compare protocols;

Pareto efficiency a solution X is Pareto efficient if there is no other solution Xin which an agent is better off than in X; and there is no agent worse off in X

than in X.

Individual rationality- occurs when negotiated payoff is better than the payoff

without negotiation. The mechanism should be individual rational for everyagent.

Stability the mechanism or protocol should not be manipulated. Agents shouldbe motivated to behave in some desired ways. Some mechanisms may have

dominant strategies. Sometimes Nash equilibrium can be used where

appropriate.

Computational efficiency agents use mechanisms and computationaloverheads should be as minimal as possible; protocols with minimum

computational overheads are preferred.

Distribution and communication efficiency- distributed protocols are preferred tocentralized ones to avoid single point of failures; however, communication

overheads should be minimized.

V ti

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Voting

Voting occurs on some social choice such asa swimming pool, public health centre, etc.

All agents give inputs to a mechanism.

The mechanism then makes a choice that willapply to all agents. Voters can be truthful or

non truthful (insincere).

Truthful voters f f f

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Truthful voters- will rank feasible social outcomes faithfully.Suppose A ={agents or voters} and O ={feasible outcomes}

Let agent i A have a strict transitive preference relation ion social choices in O. Suppose > is a social choice rule

that takes as inputs agent preferences and produces as

outputs ordered social choice preferences. Then thefollowing features are desirable of the mechanism:

1. should exist for all agents;

2. should be defined for every pair o, w O;

3. should be asymmetric and transitive over O;

4. Outcome should be Pareto efficient i.e. o io, o, o O,then o > o

5. Irrelevant alternatives should have no effect; if all for agents ranks o io iff o i o then > should also rank similarly;

6. No agent should be a dictator in that o i o then o > o for all other

agents;

Arrows impossibility theorem

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Arrows impossibility theorem

There is no social choice rule that satisfies all the six

conditions above.

Note that all agents preferences are known.

Strategic (insincere) voters

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Strategic (insincere) voters-occur in situations where agents reveal their preferences.-assumption: preferences known are those that are truthfully revealed.

- However, the agents that can benefit from lying about their preferences will

do so.

So design protocols, with stability items included such as dominant strategy

or Nash equilibrium.

This will make agents behave in some ways that leads to some social good.

The overall outcome should be the same as if the agents revealed theirpreferences or types truthfully.

Theorem: Revelation Principle:suppose some protocol implements socialchoice function f(.) in Nash (or dominant strategy) equilibrium where agents

are not necessarily truthful. Then the social choice function f(.) is

implementable in Nash (or dominant strategy) via a single step where agents

reveal their types truthfully.

Theorem: Gibbard-Sattethwait

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Theorem: Gibbard Sattethwaitimpossibility theoremlet agent types i consist of preferences orderi on O. Let there beno restrictions on i so that agents rank their preferences in anyorder.

Let |O| >= 3. If social choice function f(.) is truthfully implementable

in a dominant strategy equilibrium then f(.) is dictatorial in thatsome agent gets most preferred outcome irrespective of the types

others reveal.

Restricting preference and using Groves-Clarke Tax mechanism

enables constructing protocols that cannot be manipulated. The

voters are taxed according to how their votes may affect the

outcome. Gibbard-Sattethwait impossibility theorem is thereforecircumvented. There are other ways of circumventing the theorem.

Auctions

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AuctionsAn auction takes place between an agent known as the auctioneerand a

collection of agents known as the bidders.

The goal of the auction is for the auctioneer to allocate the goodto one of thebidders. In most settings the auctioneer desires to maximize the price; bidders

desire to minimize price.

Auction settings

Private value auctionsThe value of goods depends only on the agents. Only the winning agents utilize

the goods and do not re-sell them. Agents are assumed to know the valuesexactly.

Common value auctionsThe value of the goods depends entirely on others value of the good. For

example, where the goods are re-sold such as in treasury bills etc, tea or coffee

auctions.

Correlated value auctionsValue of goods depend partly on agents own preferences and partly on value of

others such as in negotiation within a contract.

Auction parameters

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Auction parameters

Valueof goods

Bid types(English, sealed first price, Dutch, Vickrey),and bidding process.

Goods value type:private value public/common value;correlated value.

Winner determinationmay be first price; second price.

Bidding process:open cry; sealed bid.

Bidding type: one shot; ascending; descending.

Auction protocols

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Auction protocols

English AuctionsAre: first-price, open cry, ascending.

The dominant strategy in an English Auction is

for agent to successively bid a small amount

more than the current highest bid until it reachestheir valuation, then withdraw.

The English Auction is susceptible to: winnerscurse where the good worn may of lower value;and shills in which time is wasted in making

decisions.

Dutch Auctions

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Dutch Auctions

Are: open-cry; descending

The auctioneer starts by good at artificially highvalue;

auctioneer lowers the offer price until some agentmakes a bid equal to the current offer price;

the good is then allocated to the agent that made

the offer.

First-Price Sealed-Bid Auctions

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First-Price Sealed-Bid Auctions

Are: one-shot

There is a single round;

bidders submit a sealed bid for the good;

the good is allocated to the agent that madehighest bid;

and the winner pays price of highest bid.

Best strategy is to bid less than true valuation.

Vickrey Auctions

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Vickrey Auctions

Are: second-price; sealed-bid auctions.

Good is awarded to the agent that made the

highest bid; at the price of the second highestbid.

Bidding to your true valuation is dominantstrategy in Vickrey auctions.

Vickrey auctions susceptible to antisocialbehavior.

Some issues in Auctions

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Some issues in Auctions

All auction mechanisms are not collusion proof

Bidders can collude and lower final awards

Most vulnerable: English and Vickrey auctionsThe sealed first price, Dutch and Vickrey

auctions collusions are only possible if the

bidders know each other ahead of bidding.

Vickrey Auction: auctioneer can lie by

overstating the value of the second highest bid.

B i i

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Bargaining

Agent agree on outcomes that are mutuallybeneficial.

Monopoly: one agent gets all the benefits of theinteraction.

Perfect competition: no agent gets the benefits

of the interaction.

Types: axiomatic; strategic.

Axiomatic bargaining theory

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g g yConsists of postulates and solutions that satisfy the

axioms are sought.An example is the Nash bargaining axioms (for 2 agents)

that follow:

Let agent i {1, 2}, ui be the utility function for eachagent, ui: O R, and the set of feasible utility vectors beconvex ({(u1(o), u2(o))| o O)}). Let u* = (u1(o*), u2(o*)).Then the axioms are:

Invariance- numeric utility functions only representordinal preferences among outcomes.Anonymity agents are anonymous in that switching labels of

players does not affect the outcome.Independence of irrelevant alternatives- removing o and not o* does

not affect the solution o*.

Pareto efficiency- both agents cannot get utility than under u* =

(u1(o*), u2(o*)).

Strategic bargaining theory

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g g g yAgents make offers to each other in pre-specified order.

The bargaining process may take a finite number ofsteps.

One approach: use discounts on every step where one

agent concedes some value.

Another approach: agents incur bargaining costs every

step of the process and the one whose costs are higher

concedes.

Computations:

intra-agent deliberativesearch: agents locally generate

alternatives, evaluates them, counter-speculate and lookahead in negotiation process

Inter-agent committal search:agents make binding

agreements with each other concerning the outcome.

Negotiation (Wooldridge)

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- process of reaching agreements on matters of

common interestComponents of a negotiation setting:

1.A negotiation set: possible proposals that

agents can make.

2.A protocol. Strategies, one for each agent,

which are private.

3.A rule that determines when a deal has been

struck and what the agreement deal is.

4.Process: Negotiation usually proceeds in aseries of rounds, with every agent making a

proposal at every round.

Negotiation in Task-Oriented Domains

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Negotiation in Task Oriented Domains

(TOD)A TOD is a triple where:

Tis the (finite) set of all possible tasks;

Ag ={ 1, .., n} is set of participant agents;

c=(T) R+ defines costof executing each subset oftasks;

An encounteris a collection of tasks where

Ti Tfor each i Ag.

Deals in TODs

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Deals in TOD s

Given encounter a dealwill be an allocation ofthe tasks T1 U T2to the agents 1 and 2.

The costto iof deal is c(Di

), and will bedenoted costi().

The utilityof deal to agent iis:utilityi() = c(Ti) costi()

The conflict deal, , is the deal consisting ofthe tasks originally allocated.

Deal is individual rationalif it weakly dominates theconflict deal.

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The Monotonic Concession Protocol

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Rules of this protocol are as follows. . .

Negotiation proceeds in rounds.1. On round 1, agents simultaneously propose a deal from

the negotiation set. Agreement is reached if one agent

finds that the deal proposed by the other is at least asgood or better than its proposal.

2. If no agreement is reached, then negotiation proceeds to

another round of simultaneous proposals.

3. In round u+ 1, no agent is allowed to make a proposalthat is less preferred by the other agent than the deal it

proposed at time u.

4. If neither agent makes a concession in some round u >0, then negotiation terminates, with the conflict deal.

The Zeuthen Strategy

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The Zeuthen Strategy

Three problems:

What should an agents first proposal be? Its mostpreferred deal

On any given round, who should concede? The agentleast willing to risk conflict.

If an agent concedes, then how muchshould itconcede? Just enough to change the balance of risk.

Willingness to Risk Conflict

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Suppose you have conceded a lot.

Then:1. Your proposal is now near to conflict deal.

2. In case conflict occurs, you are not much worse off.

3. You are more willingto risk conflict.

An agent will be more willingto risk conflict if thedifference in utility between its current proposal and

the conflict deal is low.

The Contract NetIs a well known task sharing protocol for task allocation:

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Is a well known task-sharing protocol fortask allocation:1. Recognition; 2. Announcement; 3. Bidding; 4. Awarding; 5. Expediting

Coalition formation (Sandholm, Brad Spangler(2003). Coalition Building.
http://www.beyondintractability.org/action/author.jsp?id=24548http://www.beyondintractability.org/action/author.jsp?id=24548

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(http://www.beyondintractability.org/essay/coalition_building/) June 2003. )

A coalition is a temporary alliance or partnering

of agents in order to achieve a common purpose

or to engage in joint activity.

Coalition building is the process by which groups

(agents, individuals, organizations, or nations)

come together to form a coalition.

Building a successful coalition

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g

involves a series of steps.

Potential coalition members must be persuadedthat forming a coalition would be to their

benefit. This is done by demonstrating:

1. that the goals are similar and compatible,2. that working together will enhance agents'

abilities to reach their goals,

3. that the benefits of coalescing are greater thanthe costs.

Benefits of a coalition

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1.A coalition can bring more expertise and

resources to bear on complex issues, where the

technical or other resources of any one agent orother agent groups would not be sufficient.

2.A coalition will increase the impact of each agent's

effort.

3.A coalition will increase available resources. Eachagent will gain access to the contacts,

connections, and relationships established by

other agents.

Problems with working in a Coalition

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1. Member agents can get distracted.

2. A coalition may only be as strong as its weakest link. Eachmember agent will have different levels of resources

different internal limitations.

3. To keep a coalition together, it is often necessary to caterto one side more than another, especially when

negotiating tactics.

4. The democratic principle of one agent-one vote may notalways be acceptable to agents with a lot of resources.

The coalition must carefully define the relationships

between powerful and less-powerful agents.

5. Individual organizations may not get credit for their

contributions to a coalition. Members that contribute a lot

may think they did not receive enough credit.

Implementing multi-agent coalitions

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Implemented by: looking for some equilibrium

solution; or using characteristic function game.

Equilibrium solution: formulating the coalitionformation protocols and examining the way the

coalition evolves.

The characteristic function game (CFG): coalition

S has characteristic function sv .

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Example of coalition structure for 4 agents (Sandholm, p.44)

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End of Session (Week 5) Exercises

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1. Describe how human beings make decisions.

2. What is the SMART decision making technique ?

3. Construct an agent model for decision making in a group.

4. What is game theory ?

5. Construct an agent model for implementing voting.

6. Construct an agent model that implements auctions.

7. Construct an agent model that implements bargaining.

8. Construct an agent model that implements negotiation.

9. Construct an agent model that implements coalitions.

ics 806-Week 4 & 5

Documents

Transcript of ics 806-Week 4 & 5