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C-agents: a notion that allows extension of Multi-Agent Systems (MAS) to SoS

Eduardo Alonso

Systems and Control Research Centre

School of Mathematics, Computer Science and Engineering (MCSE)

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What is a System

• A collection of elements with a structure• That is preserved under transformations• No matter how we change it, some properties remain invariant• That is, they are conserved

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Example from classical mechanics

• System defined by Lagrangian• Motion minimizes “action”• Principle of Least Action, Fermat• Euler-Lagrange equation • Energy is conserved

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System, the maths

• Invariance is defined in terms of symmetries• Noether’s theorems, Pierre Curie• Sets are not enough• Group: a set with an operation

– Associativity– Identity– Inverse

• Rotations and reflections of equilateral triangle• Klein’s Erlangen program, Lorenz groups, SU(3) x SU(2) x U(1)

group

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Multi-systems: Types

• Collection, no structure • System, structure• Objects, no autonomy• Agents, autonomy

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• C&O nuts, valves, screws• S&O (CoS) engine • C&A (MAS) “Mad Max” drivers• S&A (SoS) traffic network

MAS: From agents to players

• An agent perceives the state of the world

• An agent executes actions• It receives a reward• A sequence of actions makes a

policy• Find optimal policy

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• Players execute moves• Each agent receives a payoff

or return • A set of choices or moves, one

per agent, is called a joint strategy

• Find stable joint strategy

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Prisoner 1

Prisoner 2

NC

NC 1y, 1y

C

C

free, 3y

3y, free

2y, 2y

Nash equilibrium

• No agent should have an incentive to individually deviate from agreed-upon strategies

• Once a strategy is adopted, under the assumption that one agent uses it, the other agent(s) cannot do better by using a different strategy

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Problems

• There are situations in which there is no Nash equilibrium (Matching Pennies), or where there are more than one Nash equilibrium (Battle of the Sexes)

• Sometimes, the Nash equilibrium is clearly irrational (Prisoners’ Dilemma)

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Prisoner 1

Prisoner 2

NC

NC 1y, 1y

C

C

free, 3y

3y, free

2y, 2y

C-Agents and SoS• States defined as joint strategies • Actions are normative actions –

create a deontic structure and enforce joint strategies such as {NC,NC}

• C- agent’s reward function, can take many forms and metrics, from the addition of individual utilities (for instance, all vehicles get to their destination quickly and safely) to the maximization of global utilities (Pareto optimality)(traffic volume, congestion, pollution, casualties)

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• An agent perceives the state of the world

• An agent executes actions• It receives a reward• A sequence of actions makes a

policy• Find optimal policy

Optimal control: Pontryagin’s Minimum Principle

Emergence: Order within Chaos

• The important thing is that the system’s properties emerge from the provision of new solutions, through a C-agent, to individuals’ problems, not all the way around (bottom-up)

• As with {NC,NC}, C-agent’s actions change a sub-optimal joint strategy into an optimal one

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Hierarchies

• Notice that there is no central controller setting the game, the players, their actions or their outcomes. And there is no global utility function as such that is being maximized

• The controller only intervenes to “solve problems” and to guarantee that the individuals maximize their utility functions

• Importantly, hierarchies are formed

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Symmetry breaking

• Also notice that in a “pure” MAS-group the elements are the joint strategies, and the operations each agent’s turn

• No matter how the operations are combined, the Nash equilibrium is the solution

• We need to break the symmetry!

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Back to systems• Collections of systems in thermodynamics and statistical physics• Quantum Field Theory, the theory of interaction

• Not just composition of systems• Emergence (up)• Symmetry breaking (down)• Formation and evolution of

complex hierarchies• Principle of Maximum Entropy production

dissipation of energy

strive for equilibrium/stability• Systems of Systems

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What’s the right formalization?

• Collection of groups?

• Groupoid– A group with many objects– The operation is only partial– Associative where defined– Inverse defined on paths– Up to isomorphism

• Variable symmetries• More symmetries• Hierarchies, n-categories

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On-going project:

How to formalize MAS-SoS (emergence, hierarchical structure, symmetry breaking) using abstract algebra

Thank you for you attention

Questions?

Eduardo Alonso

E.Alonso@city.ac.uk

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