Academic excellence for business and the professions 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) 1

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 2

Example from classical mechanics System defined by Lagrangian Motion minimizes “action” Principle of Least Action, Fermat Euler-Lagrange equation Energy is conserved 3

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 4

Multi-systems: Types Collection, no structure System, structure Objects, no autonomy Agents, autonomy 5 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 6 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

7 Prisoner 1 Prisoner 2 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 8

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) 9

10 Prisoner 1 Prisoner 2 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) 11 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 12

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 13

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! 14

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 15

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 16

On-going project: How to formalize MAS-SoS (emergence, hierarchical structure, symmetry breaking) using abstract algebra Thank you for you attention Questions? Eduardo Alonso 17