Variables

	xk = current position of particle k
	vk = current velocity of particle k
	pk = best position found by particle k
	Q(x) = quality of position x
	g = index of best position found so far
	i.e., g = argmaxk Q(pk)
	f1, f2 = random variables uniformly distributed over [0, 2]
	w = inertia

Velocity & Position Updating

	vk¢ = w vk + f1 (pk Ð xk) + f2 (pg Ð xk)
	w vk maintains direction (inertial part)
	f1 (pk Ð xk) turns toward private best (cognition part)
	f2 (pg Ð xk) turns towards public best (social part)
	xk¢ = xk + vk
	Allowing f1, f2 > 1 permits overshooting and better exploration (important!)
	Good balance of exploration & exploitation
	Limiting vk < vmax controls resolution of search

Improvements

	Alternative velocity update equation: vk¢ = c [w vk + f1 (pk Ð xk) + f2 (pg Ð xk)]
		= constriction coefficient (controls magnitude of vk)
	Alternative neighbor relations:
		star: fully connected (each responds to best of all others; fast information flow)
		circle: connected to K immediate neighbors (slows information flow)
		wheel: connected to one axis particle (moderate information flow)

Spatial Extension

	Spatial extension avoids premature convergence
	Preserves diversity in population
	More like flocking/schooling models

Some Applications of PSO

	integer programming
	minimax problems
		in optimal control
		engineering design
		discrete optimization
		Chebyshev approximation
		game theory
	multiobjective optimization
	hydrologic problems
	musical improvisation!

MillonasÕ Five Basic Principles of Swarm Intelligence

	Proximity principle:
		pop. should perform simple space & time computations
	Quality principle:
		pop. should respond to quality factors in environment
	Principle of diverse response:
		pop. should not commit to overly narrow channels
	Principle of stability:
		pop. should not change behavior every time env. changes
	Principle of adaptability:
		pop. should change behavior when itÕs worth comp. price

Kennedy & Eberhart on PSO

	ÒThis algorithm belongs ideologically to that philosophical school
	that allows wisdom to emerge rather than trying to impose it,
	that emulates nature rather than trying to control it,
	and that seeks to make things simpler rather than more complex.
	Once again nature has provided us with a technique for processing information that is at once elegant and versatile.Ó

Additional Bibliography

	Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G.,& Bonabeau, E.  Self-Organization in Biological Systems.  Princeton, 2001, chs. 11, 13, 18, 19.
	Bonabeau, E., Dorigo, M., & Theraulaz, G.  Swarm Intelligence: From Natural to Artificial Systems.  Oxford, 1999, chs. 2, 6.
	SolŽ, R., & Goodwin, B.  Signs of Life: How Complexity Pervades Biology.  Basic Books, 2000, ch. 6.
	Resnick, M.  Turtles, Termites, and Traffic Jams: Explorations in Massively Parallel Microworlds.  MIT Press, 1994, pp. 59-68, 75-81.
	Kennedy, J., & Eberhart, R.  ÒParticle Swarm Optimization,Ó Proc. IEEE IntÕl. Conf. Neural Networks (Perth, Australia), 1995.  http://www.engr.iupui.edu/~shi/pso.html.

IV. Cooperation & Competition

	Game Theory and the Iterated PrisonerÕs Dilemma

The Rudiments of Game Theory

Leibniz on Game Theory

	ÒGames combining chance and skill give the best representation of human life, particularly of military affairs and of the practice of medicine which necessarily depend partly on skill and partly on chance.Ó Ñ Leibniz (1710)
	ÒÉ it would be desirable to have a complete study made of games, treated mathematically.Ó  Ñ Leibniz (1715)

Origins of Modern Theory

	1928: John von Neumann: optimal strategy for two-person zero-sum games
		von Neumann: mathematician & pioneer computer scientist (CAs, Òvon Neumann machineÓ)
	1944: von Neumann & Oskar Morgenstern:Theory of Games and Economic Behavior
		Morgenstern: famous mathematical economist
	1950: John Nash: Non-cooperative Games
		his PhD dissertation (27 pages)
		Ògenius,Ó Nobel laureate (1994), schizophrenic