Goals• Become familiar with current research

topics• Practice finding research topics. Learn

what is publishable.• Gain the ability to analyze articles• Practice being able to respond to

questions in real time.• Improve writing skills• Learn to avoid being a lightning rod.

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Examples of research abilities• Pixar story – animators inspired each

other, there was a synergy in the group• Joe could listen to a presentation,

understand it, and find the key elements as to what needed to be done.

• Tom wanted to run the obvious experiments and get the obvious answers.

• Mike was king of “gold plating” – but couldn’t distinguish what to spend time on

2

You pick articles

• Class will consist of– Take turns being lead. Lead selects article of

the day.– Brief overview of article – by lead– Submitted article critique– Discussion of research points

3

Article Criteria• An article you have not studied extensively already.• Recent article (last five years) from a respected source.

Consult ACM and IEEE digital libraries (available for free on campus)

• Related to coalitions in multi agent systems• Articles that build a foundation for your project• One that is selected from reading several articles in your

area of interest.– One you understand. (You become the authority for the

discussion.)– One with merit: You need to be able to answer basic questions

such as (a) what are the major contributions? (b) what are interesting ways of extending the research?

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Cooperative/coalitional game theory

A composite of slides taken from Vincent Conitzer and Giovanni Neglia

(Modified by Vicki Allan)

5

coalitional game theory• There is a set of agents N• Each subset (or coalition) S of agents can work

together in various ways, leading to various utilities for the agents

• Cooperative/coalitional game theory studies which outcome will/should materialize

• Key criteria:– Stability: No coalition of agents should want to deviate from

the solution and go their own way– Fairness: Agents should be rewarded for what they

contribute to the group• (“Cooperative game theory” is the standard name (distinguishing it from

noncooperative game theory, which is what we have studied in two player games). However this is somewhat of a misnomer because agents still pursue their own interests. Hence some people prefer “coalitional game theory.”) 6

Transferable utility• Suppose that utility is transferable: If you get more intrinsic

benefit from a coalition, you can give some of your utility to another agent in your coalition (e.g. by making a side payment)

• Then, all that we need to specify is a value for each coalition, which is the maximum total utility for the coalition– Value function also known as characteristic function– Def. A game in characteristic function form is a set N of players together

with a function v() which for any subset S of N (a coalition) gives a number v(S) (the value of the coalition)

• The payout is a vector of utilities that sums to the value is possible

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Suppose we have the following characteristic function

Coalition Value of coalition

a 2

b 3

c 4

ab 6

ac 6

abc 12

8

How do we decidehow to pay each member for theircontribution?

Suppose we have the following characteristic function

Coalition Value of coalition

Possible payout

a 2 2

b 3 3

c 4 4

ab 6

ac 6

abc 12

9

Suppose we have the following characteristic function

Coalition

Value of coalition

Possible payoutBad if some of individual payouts doesn’t equal value of coalition

What about?

a 2 2 2

b 3 3 3

c 4 4 7

ab 6

ac 6

abc 1210

Let’s try a different exampleCoalition Value of

coalition

a 2

b 3

c 4

ab 10

ac 10

abc 14

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How do we decidehow to pay each member for theircontribution?

Suppose we have the following characteristic function

Coalition Value of coalition

Possible Payoffs

a 2 4

b 3 4

c 4 6

ab 10

ac 10

abc 14

12

Who would pull out?

Schedule for Article Review

9/6/2011 Antariksha9/8/2011 Pedro

9/13/2011 Kamilia

9/15/2011 David

9/20/2011 Stan

9/22/2011 Ahmadreza

9/27/2011 Chenguang9/29/2011 Steena

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Core• Outcome is in the core if and only if: every coalition receives a

total utility that is at least its original value and no smaller• A coalition is tempted to pull out if…

– For any coalition C, v(C) ≥Σi in Cu(i) • For example,

– v({1, 2, 3}) = 12, – v({1, 2}) = v({1, 3}) = v({2, 3}) = 8,– v({1}) = v({2}) = v({3}) = 0

• Now the outcome (4, 4, 4) is possible; it is also in the core (why?) and is the only outcome in the core.

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Emptiness & multiplicity• Example 2:

– v({1, 2, 3}) = 8, – v({1, 2}) = v({1, 3}) = v({2, 3}) = 6,– v({1}) = v({2}) = v({3}) = 0

• Now the core is empty! Notice, the core must involve the grand coalition (giving payoff for each).

• Conversely, suppose we have the following characteristic function– v({1, 2, 3}) = 18, – v({1, 2}) = v({1, 3}) = v({2, 3}) = 10,– v({1}) = v({2}) = v({3}) = 0

• Now lots of outcomes are in the core – (6, 6, 6), (5, 5, 8), …

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Issues with the core• When is the core guaranteed to be non-empty?• What about uniqueness? • What do we do if there are no solutions in the

core? What if there are many?

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Superadditivity• A characteristic function, v, is superadditive if for all

coalitions A, B with A∩B = Ø, v(AUB) ≥ v(A) + v(B)• Informally, the union of two coalitions can always act

as if they were separate, so should be able to get at least what they would get if they were separate

• Usually makes sense• Previous examples were all superadditive• Given this, it is always efficient for grand coalition to

form• Without superadditivity, finding a core is not possible.

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Convexity• v is convex if for all coalitions A, B, v(AUB)-v(B) ≥

v(A)-v(A∩B)• In other words, the amount A adds to B (in the union)

is at least as much it adds to the intersection.• Example. I have math skills. I join a group of people who are working on

projects. Each person in the group, who needs math expertise is benefitted by my joining. Thus, the larger the group I join, the better chance I will add value to the group.

• One interpretation: the marginal contribution of an agent is increasing in the size of the set that it is added to. The term marginal contribution means the additional contribution. Precisely, the marginal contribution of A to B is v(AUB)-v(B)

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Convexity• v is convex if for all coalitions A, B, v(AUB)-v(B)

≥ v(A)-v(A∩B)• Previous examples were not convex (why?) • v is convex if for all coalitions A, B, v(AUB)-v(B)

≥ v(A)-v(A∩B). • For example,

– v({1, 2, 3}) = 12, – v({1, 2}) = v({1, 3}) = v({2, 3}) = 8,– v({1}) = v({2}) = v({3}) = 0

– What makes it non-convex?19

• Let A = {1,2} and B={2,3}• v(AUB)-v(B) = = v{1,2,3} - v{2,3} = 12 – 8• v(A)-v(A∩B) = v{1,2} – v{2} = 8 - 0• Problem 4 < 8

Definition• v is convex if for all coalitions A, B, v(AUB)-v(B) ≥ v(A)-

v(A∩B)

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Convexity• In convex games, core is always nonempty. (Core

doesn’t require convexity, but convexity produces a core.)

• One easy-to-compute solution in the core: agent i gets u(i) = v({1, 2, …, i}) - v({1, 2, …, i-1})– Marginal contribution scheme- each agent is rewarded by what it ads to

the union.– Works for any ordering of the agents

Visit with your neighbor to explain why this method of computing a core solution always works.

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The Shapley value [Shapley 1953]• In dividing the profit, sometimes agent is given its marginal

contribution (how much better the group is by its addition)

• Example: A piano needs to be moved. The owner will pay $20 to have it moved. It takes two people to move it. Joe is trying to move it by himself. He can’t do it. You tell him, “I’ll help you move it, but I expect $18, as you couldn’t move it without me.”

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The Shapley value [Shapley 1953]• The simple marginal contribution scheme (of the previous

slide) is unfair because it depends on the ordering of the agents

• One way to make it fair: average over all possible orderings• Let MC(i, π) be the marginal contribution of i in ordering π• Then i’s Shapley value is ΣπMC(i, π)/(n!)• The Shapley value is always in the core for convex games• … but not in general, even when core is nonempty, e.g.

– v({1, 2, 3}) = v({1, 2}) = v({1, 3}) = 1,– v = 0 everywhere else

23

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Example: v({1, 2, 3}) = v({1, 2}) = v({1, 3}) = 1, (Note v({2,3})=0)

v = 0 everywhere else

1 2 3123

132

213

231

312

321avg

Compute the Shapley value for each.

Is the solution in the core?

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Example: v({1, 2, 3}) = v({1, 2}) = v({1, 3}) = 1, v = 0 everywhere else

1 2 3123 0 1 0

132 0 0 1

213 1 0 0

231 1 0 0

312 1 0 0

321 1 0 0

avg 2/3 1/6 1/6

Compute the Shapley value for each.

Is the solution in the core?

Briefly describe the characteristics of “core” and

“Shapley”

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Axiomatic characterization of the Shapley value

• The Shapley value is the unique solution concept that satisfies:– (Pareto) Efficiency: the total utility is the value of the grand

coalition, Σi in Nu(i) = v(N) [No one can be made better without someone being made worse.]

– Symmetry: two symmetric players (add the same amount to coalitions they join) must receive the same utility

– Dummy: if v(S {i}) = v(S) for all S, then i must get 0– Additivity: if we add two games defined by v and w by

letting (v+w)(S) = v(S) + w(S), then the utility for an agent in v+w should be the sum of her utilities in v and w• most controversial axiom as feel agent may deserve a discount (for

example, participant i’s cost-share of a runway and terminal is it’s cost-share of the runway plus his cost-share of the terminal)

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Computing a solution in the core• Can use linear programming (a formal mathematical

method):– Variables: u(i)– Distribution constraint: Σi in Nu(i) = v(N)– Non-blocking constraints: for every S, Σi in Su(i) ≥ v(S)

• Problem: number of constraints exponential in number of players (as you have values for all possible subsets)

• … but is this practical?

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Theory of cooperative games with sidepayments

• It starts with von Neumann and Morgenstern (1944)

• Two main (related) questions:– which coalitions should form?– how should a coalition which forms divide

its winnings among its members?• The specific strategy the coalition will

follow is not of particular concern...• Note: there are also cooperative games

without sidepayments29

The Shapley value: computation

• A faster way• The amount player i contributes to

coalition S, of size s, is v(S)-v(S-i)• This contribution occurs for those

orderings in which i is preceded by the s-1 other players in S, and followed by the n-s players not in S

• ki = 1/n! S:i in S (s-1)! (n-s)! (v(S)-v(S-i))

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The Shapley value has been used for cost sharing. Suppose threeplanes share a runway. The planes require 1, 2, and 3 KM to land.Thus, a runway of 3 must be build, but how much should each pay?

Instead of looking at utility given, look at how much increased cost was required.

1 2 3123132213231312321avg

The Shapley value has been used for cost sharing. Suppose threeplanes share a runway. The planesrequire 1, 2, and 3 KM to land.Thus, a runway of 3 must bebuild, but how much should each pay?

Instead of looking at utility given, look at how much increased cost was required.