Using Dialogue Games to Form Coalitions with Self-Interested Agents

Luke RileyDepartment of Computer Science

University of LiverpoolL.J.Riley@Liverpool.ac.uk

Supervisors: Katie Atkinson & Terry Payne

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1. Coalition Formation in Cooperative Game Theory.

2. Coalition Formation in Argumentation.

3. The Issues and Problems Between these Two Approaches.

4. My Research.

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1. Coalition Formation in Cooperative Game Theory (CGT)

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Background

N-person cooperative games (coalition games) were proposed in 1944 by von Neumann & Morgenstern [1]:

Where...

[1] J. von Neumann and O. Morgenstern. The Theory of Games and Economic Behavior. Princeton University Press, 1944.

Characteristic Function:

Agent set:

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In its most traditional style the CGT outcome of a coalition game is:

Where...

CS = a set of coalitions (the coalition structure)

x = a vector of each individual agent's payoff in the game.

Solving a Coalition Game

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Finding a Stable Outcome – The Core

A Coalition Structure is core-stable if no subset of agents can benefit from defecting to another coalition.

The core [2] is the set where:

e.g. Example 1: Given a coalition game

where N = {1,2}, v({1}) = v({2}) = 5 and

v({1,2}) = 20 the proposed core outcome is

<{1,2}, x(10,10) >

e.g. Example 2: Given a coalition game

where N = {1,2}, v({1}) = v({2}) = 5 and

v({1,2}) = 20 the proposed core outcome is

<{1,2}, x(15,5) >

Yet core payoffs can sometimes be unfair

[2] D. Gillies. Some theorems on n-person games. PhD thesis, Princeton University, 1953.

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Epsilon-Core

Solution [3] →

The epsilon value can be seen as the cost of deviating.

Also the core can sometimes be empty

e.g. Example 3: Given a coalition game where N = {1,2,3}, forall subsets C if |C| = 2 then v(C) = 1 else v(C) = 0

e.g. Example 4: Given the coalition game of

example 3, the payoff vector x(1/3,1/3,1/3) is

1/3-core stable.

[3] Shapley, Lloyd S. and Shubik, M. Quasi-cores in a monetary economy with non-convex preferences , Econometrica (The Econometric Society) 34(4): 805–827, 1966.

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1. Coalition Formation in Argumentation

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Dung's Initial WorkDung showed that Argumentation

Frameworks were natural ways to represent n-person games, for example theorem 6 of [4]:

[4] P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995.

x(3,4,8)

x(3,3,5) x(3,3,3)

The AF represents 3 possible payoff vectors of the coalition game:

v({1}) = v({2}) = v({3}) = 3

v({1,3}) = 8

v({2,3}) = 12

or v(C) = 0

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Amgoud's Further ResearchAmgoud in [5] extended this research, where

she highlighted:

How to always find a solution to a coalition game

Outlines how agents can collaboratively build AFs for coalition games

How a dialogue game can be used to check if a certain coalition was in the best coalition structure

[5] L. Amgoud. An argumentation-based model for reasoning about coalition structures. In ArgMAS, pages 217-228, 2005.

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2. The Issues of Joining the Two Approaches

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Various IssuesCGT: Lacks flexible communication protocols

to form stable coalitions.

CGT: Generally does not take into account the computation and communication costs of finding stable coalition structures from a MAS perspective.

Arg: There is little research showing how payoff vectors are found and justified by MAS.

Arg: No research on how to stabilise coalitions games, using the epsilon-core

Arg: Only some limited direct mapping between the argumentation models and the CGT coalition game types (e.g. static, dynamic, skill games,...)

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My Current Research Question

How can self-interested agents make use of argumentation within

their communication to enable them to form a stable optimal

coalition structure with an approximately fair payoff

distribution?

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3. The Proposed Method

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Dialogue Games & Argumentation Schemes

Dialogue Games can be used to build argumentation frameworks in real time, where agents can assert and retract arguments.

Argumentation schemes are patterns of reasoning that when instantiated provide presumptive justification for the particular conclusion of the scheme

e.g:...

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Approximately fair payoffs

AFs can easily represent the core

...But the core can be unfair

Solution – restrict the payoffs allowed:

agents have to propose an equal split of v(C) or each agent should be given at least the same value it can get from a coalition of agents willing to defect

Agents can object to a proposed payoff by finding a better one for itself.

Once a core payoff is found, the dialogue stops

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Dialogue Games & Argumentation SchemesI have devised a dialogue game [6] to find an

optimal coalition structure with a restricted core payoff

Moves:

e.g:

[6] L. Riley, K. Atkinson, and T. Payne. Coalition structure generation for self interested agents in a dialogue game. Technical Report ULCS-12-004, University of Liverpool, 2012.

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Core example

Move Coalition {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

Coalition value

4 3 2 14 18 5 7

1 [3] [9/9]

2 [2] [10/4]

3 [3] [11/7]

FINISHCoalition Structure of move 3 is {{1,3}, {2}}, the payoff vector is x(11,3,7) and is core stable

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Epsilon-Core Example

Move e value Coalition {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

value 5 5 5 18 20 22 10

5 0 [5] [11/11]

1 value 6 6 6 17 19 21

6 1 [6] [7/12]

2 value 7 7 7 16 18 20

7 2 [7] [8/8]

3 value 8 8 8 15 17 19

8 3 [8] [9.5/9.5]

4 value 9 9 9 14 16 18

FINISH

Coalition Structure of move 8 is {{1},{2,3}}, the payoff vector is x(8,9.5,9.5) and is 3-core stable

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Potential Future

Modify argumentation scheme and attack relations so that other coalition games can be modeled (e.g stochastic, dynamic, skill games,...).

Optimise process: Combine mechanism design approach of [7] with efficient distribution methods of [8].

[7] Tuomas Sandholm, Kate Larson, Martin Andersson, Onn Shehory and Fernando Tohmé, Coalition structure generation with worst case guarantees, Artificial Intelligence, Volume 111, Issues 1–2, July 1999, Pages 209-238.

[8] T. Rahwan. Algorithms for Coalition Formation in Multi-Agent Systems. PhD thesis, University of Southampton, 2007.

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Thanks For Listening

Questions?