Biology student Derrick Grunwald told me about the following tale of the game theory of bacteria. It turns out that certain bacteria come in two varieties, altruistic and selfish. The classification relates to their reaction to emitting a certain molecule. The selfish types emit the molecule and then immediately claim it for themselves using a receptor in another part of the creature. The altruists emit the molecule to the colony and receive the average emissions available, sort of like a public good. Apparently, this process of emitting and receiving creates fitness for the bacteria. Derrick tells me that there is an ideal amount of the molecule to receive, m*. A bacteria exposed to too much is unhealthy as is one exposed to too little.
The puzzle to biologists is how their can be altruistic bacteria. While other-regarding or even eugenic preferences are possible in higher primates, it seems a stretch to consider such motives in bacteria.
So how can we resolve this puzzle using game theory and what does this tell us about the nature of these bacterial colonies? First off, why are there volunteers in the first place? What possible benefit is there from volunteering? When a bacteria emits the molecule, it doesn't get the amount exactly right. Sometimes it emits too much, sometimes too little. On average, it's the correct amount, but individually it is not. Thus, the volunteer bacteria are engaging in a bit of risk pooling. By emitting in general, all of these errors average out in the colony and each individual absorbs just the right amount of the molecule thanks to the magic of the law of large numbers. A colony of selfish bacteria are choosing not to insure. This is obviously less fit than insuring, but does have some advantages of reliability.
Let us study "equilibria" of this game. Suppose that a colony consists entirely of volunteers and it is invaded by a small number of selfish types. The volunteers will still absorb approximately m* of the molecule while the selfish will absorb 2m*--way too much of the molecule. Thus, the selfish are "killed with kindness" by the volunteers. Hence, all volunteers comprises an equilibrium.
What about a colony of all selfish types? While less fit than the volunteers, this colony is also immune to invasion. If a small number of volunteers show up, they hardly affect the absorption of the selfish, which is still approximately m* on average, but the volunteers themselves get almost none of the molecule. Thus, they too die out. So all selfish is an equilibrium despite its inferiority to the insurance scheme worked out by the volunteers.
What about mixed colonies? This is possible, but unstable. If the colony consists of a fraction f of volunteer types, a co-existing equilibrium can arise. In this situation, selfish types are systematically exposed to too much of the molecule since they absorb some of the production of the volunteers. Volunteers systematically have too little of the molecule since the selfish types are "stealing" some. And if the fitness of the two types is approximately equal, they can coexist.
The instability arises from the following problem. If the co-existing colony is invaded by selfish types, this improves the fitness of the selfish, since the emissions of the volunteers are now more diluted by the additional selfish types, but reduces the fitness of volunteers since there are now more selfish types "stealing" the molecule. Hence, such a perturbation would cause the colony to eventually drift to an all-selfish society. By contrast, if the colony is invaded by volunteers, just the opposite occurs--volunteers become more fit while selfish become less fit. Only if invasions of various types are occurring often enough to bring the population back to the equilibrium fraction f, will the colony continue to coexist. This is unlikely if invasions occur randomly, even if both types of invasions are equally likely.
Thus, using game theory, the puzzle of altruistic bacteria may be understood as purely selfish behavior.