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Is anyone not  familiar with the children's game of Rock-Paper-Scissors? Unlikely. But did you know that this apparently simple game sports a dedicated World RPS Society with its own world championships? Maybe even more surprising might be that lizards and even bacteria are also addicted to this game.

Bacteria (Escheria coli): A growing colony of bacteria faces fierce competition for space and resources. To keep the lead, Escheria coli produces a toxin to efficiently reduce the number of competitors. Concurrently, the bacteria obviously have to produce an antidote that makes them immune themselves. After some time the vicious toxic strain may have eliminated all competitors and this renders the toxin production utterly useless. Consequentially, a mutant strain that only produces the antidote but not the toxin saves energy that can be dedicated to reproduction and hence it will eventually replace the toxic strain. But now that no toxins are around, the production of the antidote becomes equally useless and another mutant strain can take over that dedicates all ressources to reproduction. This leads to a population bacteria that is again vulnerable to toxins and the cycle may repeat itself endlessly: Rock beats Scissors beats Paper beats Rock... Actually this is not exactly what occurs in nature. For bacteria in solution, the amplitude of the frequency oscillations of the three types keeps increasing such that eventually one bacterial strain dies out and another follows shut. However, in spatially extended settings, this Rock-Scissors-Paper-type dominance can lead to traveling waves and was suggested to promote biodiversity (Kerr et al, Nature 2002).

Lizards (Uta stansburiana): Maybe an even more intriguing instance of Rock-Scissors-Paper-type cycles have been reported in the mating behavior of side-blotched lizards (Uta stansburiana). Males display three different kinds of mating strategies: monogamous males guard their female, polygamous males defend a territory and keep a female harem and opportunistic 'sneaker' males sire offspring in secretive copulations. A population dominated by monogamous males can be invaded by the polygamous type because they may produce more offspring but once the polygamous males prevail, they become vulnerable to the sneaking type because of the increasing efforts to guard their terrotory and harem. Finally, once sneaking males abound, the monogamous type is favoured again. Incidentially, and to make the life of field biologists a little easier, the three types of males can be easily distinguished by their throat colors: blue-throats are monogamous, orange-throats are dominant and maintain large territories and yellow-throats are the sneakers: Blue beats Yellow beats Orange beats Blue... (Zamudio & Sinervo, PNAS 2000).

Volunteering: As a last example, Rock-Paper-Scissors-type cyclic dominance of strategies can arise whenever the participation in Prisoner's Dilemma type interactions is voluntary. Such optional participation is implemented by considering a third strategic type, the loners. Loners are risk averse and do not engage in Prisoner's Dilemma interactions but rather rely on small but fixed earnings. Voluntary participation can foil exploiters and overcome the social dilemma. This simple but effective mechanism does not require sophisticated strategic behavior or cognitive abilities but operates efficiently under full anonymity. Thus, it provides an escape hatch out of some social traps. Populations where cooperators abound are prone to exploitation and invasion by defectors but once defectors increase, the loners' option becomes increasingly attractive and as soon as loners dominate cooperation can become favourable again: Cooperator beats Loner beats Defector beats Cooperator... (Hauert et al, Science 2002).

Volunteering in public goods games is discussed in a separate interactive tutorial. For an overview, the sections on Prisoner's Dilemma interactions and Public Goods games and on cooperation in structured populations are good starting points. The specifics of optional interactions are covered in a section on volunteering.

Example 5	The Rock-Scissors-Paper game in well-mixed populations. Color code: Paper Scissors Rock
	Rock-Scissors-Paper The replicator dynamics predicts endless cycles in the limit of infinite population sizes. However, the dynamics is structurally unstable, i.e. the interior fixed point on the simplex S3 with equal proportions of all three strategies is a neutrally stable center surrounded by closed orbits. Perturpations arising from the finiteness of more realistic populations are sufficient to destabilize the system resulting in trajectories that spiral outwards untill they eventually hit the boundary such that only a single strategy survives.
	Rock-Scissors-Paper The dynamics is quite different if individuals adopt always the strategy that would perform best under the current conditions, the so-called best-reply dynamics. The best-reply dynamics has the same fixed points as the replicator dynamics but now the interior fixed point is stable. Trajectories converge in piecewise linear streches to the fixed point.

Example 6	The Rock-Scissors-Paper game in lattice populations. Color code: Paper Scissors Rock   New paper New scissors New rock
	Travelling waves In spatially structured populations where each individual is confined to a lattice site and interacts only within its local neighborhood, the cyclic dominance leads to traveling waves across the lattice and fascinating spatio temporal patterns.
	Spirals Selecting deterministic updating (synchronous lattice updates where individuals adopt the best neighboring strategy) and a symmetrical initial configuration can result in even more hypnotizing patterns - use at your own risk.

Selected publications on recent research results:

• Kerr, B., Riley, M. A., Feldman, M. W. & Bohannan, B. J. M. (2002) Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature 418, 171-174.
• Zamudio, K. R. & Sinervo, B. (2000) Polygyny, mate-guarding, and posthumous fertilization as alternative male mating strategies, Proc. Natl. Acad. Sci. USA 97, 14427-14432.
• Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games, Science 296, 1129-1132.