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Posts Tagged ‘Prisoner’s Dilemma’

One of the problems that Swarm Intelligence research faces is a precise definition[1]. Many words that are associated with SI are generally Emergence, Self-Organization, Collective Intelligence etc. There is no general mathematical definition to it yet. This lack of a credible and workable framework has made research in this field ad-hoc. In simple words we need a theory to swarming. Is there a theory to swarming behavior or swarm intelligence? We basically have analogies. There is no theory yet i guess.

There have been efforts to rectify this and i will cover this in a post sometime soon.

However we also definitely need a theory to explain altruism in social insects (In a swarm), in animals or in Humans. I basically got interested in Altruism due to my interest in Swarm Intelligence and social insects.

bees_500x432.jpg

Why does a honey bee do the ultimate sacrifice and lay down its life when it feels its kin is in danger? Why do walruses adopt orphans? How is it that dogs can adopt off-springs of cats, other dogs, and even tigers? There are scores of examples that show such altruistic behavior.There are variants obviously to the altruism that we are talking about.

Altruism as i have mentioned implicitly is a social behavior. A behavior is social if it has implications for both the actor and the recipient. Social behaviors can broadly be categorized depending on whether the actor or the recipients are benefited. Altruism is the category when the fitness level of the actor is reduced after a action and that of the recipient is increased. A selfish behavior contrasts the above exactly. Other two types are mutually beneficial and spiteful. Mutually beneficial is the type in which the fitness of both are increased and spiteful is the reverse of it.

In The God Delusion, Richard Dawkins summarizes some kinds of altruism and separates them and also gives some explanation.

1. The first kind he points out is altruism towards our kin, with individuals with whom we have common genes. The honey bee example in most probability fits into this “category”. Also like how we are evolved to be kind towards our kids. The genes that “code” for such behavior towards our kins people who share genes with us are more likely to survive. This is given by Hamilton’s Kin Selection Theory[2]. It states that altruism is favored when:
rb-c>0

Where “c” is the fitness cost to the altruist, “b” is the fitness benefit to the recipient and “r” is a measure of their genetic relatedness.

2. The second kind is reciprocal altruism. The example of a buffalo / crow pair fits in here. This form does not require any sharing of genes. Individuals from VERY different species can actually exist in a symbiotic relationship through this altruistic form. Each individual contributes something that the other individual can not obtain on its own. Within a species, like in us humans, this trait has become more specific and has evolved into the form that we tend to do good to people who do/ can do good things for us.

3. Dawkins then also talks about the importance of building a good reputation.

4. One of the most interesting reasons that he mentions and which is very true IMO is that excessive altruism may be a show of superiority in some manner. That is that it can be because the person can afford to be “altruistic” This can in some way be understood as a payoff in a game theory scenario.

One interesting approach to the questions above is discussed in a nice paper that i read over the past week.

In this paper the two researchers, in their model consider a large and panmictic (unstructured) population where individuals interact pairwise in successive rounds.

Assumptions:

  1. Number of rounds of interaction for an individual follows a geometric distribution with a parameter ω, which is a probability that an individual will interact with an individual after a round of interaction.
  2. The Focal Individual can interact with two classes of individuals, one closely related genetically to it and the other not so closely related.
  3. X is the probability of interacting non-randomly with an individual of the related class.
  4. 1-X is the complimentary probability. 1− X interactions occur randomly with any member of the population.

All repeated rounds of interactions take place with the same partner. During each round of interaction the FI invests I• into helping with I• varying between 0 and 1. This investment incurs a cost CI• to the FI and generates a benefit BI• . A fraction ζ of the benefit generated by helping directly return to the FI and the complementary fraction 1−ζ goes to the partner[3].

The fecundity of FI can either be positive or negative depending on the value of the term:

ζBI– CI

The fecundity of the partner will always be positive unless the FI gets all the benefit of the act (i.e. ζ=1 ) or if it does not invest in helping at all (I• = 0). We noted earlier that the FI is interacting with two classes of individuals. And therefore the fecundity with both these classes of individuals will be different.

To generalise the relative fecundity of the FI interacting with a j class individual will be given by:

form-1.jpg

We assume that the individual follows a “tit for tat” kind of a approach i.e. that the investment level into helping at a given round depends linearly on the partner’s investment at the previous round. Hence, the investment depends on three traits:

  1. the investment on the first round τ
  2. The response slope β on the partner’s investment for the preceding round.
  3. The memory m (varying between zero and one) of the partner’s investment at the preceding round.

“m” is the probability of not making an error in assignment by considering that a partner has not co-operated in the previous move when he infact has. τ and β can evolve. The paper gives a well put and terse presentation of the idea to generate a formula and then considers cases when co-operation and when altruism can evolve. I highly recommend this paper. It is a wonderful paper and a must read! I thoroughly enjoyed it. It can be obtained here.
The same problem has also been approached by Richard Dawkins comprehensively in the following video that was shown on the BBC. This video has been obtained courtesy of richarddawkins.net

In the video Dawkins starts off with how his seminal book “The Selfish Gene” was misunderstood and how it lead to the phrase “Nice Guys Finish Last”, this phrase was coined by Garrett Hardin to sum up the selfish gene idea. Then how do these “nice guys/altruistic agents” survive? Should not have natural selection wiped them off? Dawkins then argues that the idea of selfish genes can actually give rise to co-operative and altruistic behavior*. It leads to the development of a pre-wired programme or strategy for achieving some desired goal. This is analogous to human strategy in situations. Dawkins then moves to game theory and gives a wonderful explanation of the Prisoners Dilemma and concepts like the tragedy of the commons and tries to explain how altruistic behavior (coupled with a tit for tat kind learning behavior) can actually fetch best pay-offs. It is this “selfish” advantage that actually leads to altruistic behavior. And finally suggests that Hardin’s idea could be slightly modified after this analysis to “Nice Guys Finish First”, which also is the name of the programme.
To sum up, this wonderful video leads to the same conclusion in part as the paper described above does.

That is that the following conditions lead to the evolution of
altruism and cooperation
:

1. Direct benefits to the individual performing a cooperative act.

2. Direct or indirect information allowing a better than random guess about whether a given individual will behave cooperatively in repeated reciprocal interactions.

3. Altruism or cooperation can evolve if the cost-to-benefit ratio of altruistic and cooperative acts is greater than a threshold value. The cost-to-benefit ratio can be altered by coercion, punishment and policing which therefore act as mechanisms facilitating the evolution of altruism and cooperation. [3] Dawkins explained this idea by the PD example in the video.

However integrating the four ideas mentioned at the start of the post is a problem. Can it be possible to do so? This can have massive implications in understanding social insect behavior and swarm intelligence among many others.

* Co-operative behavior may not always be altruistic.

References:

[1] Foundations of Swarm Intelligence: From Principles to Practice, Mark Fleischer, Institute for Systems Research, University of Maryland College Park.

[2] The genetical evolution of social behaviour. Hamilton, W. D. 1964. I. Journal of Theoretical Biology 7:1-16.

[3] The evolution of cooperation and altruism. A general framework and a classification of models. Laurent Lehmann and Laurent Keller.

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