Tuesday, January 17, 2012

The evolution of cooperation

“Under what conditions will cooperation emerge in a world of egoists without central authority?”

Robert Axelrod starts with this impressive sentence in his book “the evolution of cooperation”, which exerted significant influence on many fields: prevention of war, social evolution, cooperation among animals, human history, evolutionary game theory, networks of trust and reciprocity that build social capital, microeconomics, science fiction, etc.

Axelrod’s tit-for-tat in the iterated prisoner’s dilemma situation

By using iterated Prisoner’s Dilemma situation, Axelrod starts tackling the question. There are player A and B, and they can choose either defect (D) or cooperate(C). Under the circumstances, there are four outcomes (using the set up of his paper):

  • R if both cooperate
  • S if A cooperates and B defects
  • T if A defects and B cooperates
  • P if A defects and B defects

The author conducted his famous computer tournament, and found that in the iterated game, tit-for-tat is the best strategy. Tit-for-tat is the strategy to choose C in the first, and then mimic the strategy that other player chooses in the previous game. Then, if the players are all incentivized to maximize one’s own payoff, then they will adopt tit-for-tat, he claims. Thus, Axelrod’s analysis starts from the assumption that the player chooses tit-for-tat.

Let’s say player A and B adopt tit-for-tat strategy. Then, the pay off would be:

R + wR +w^2R + w^3R …

= R / (1- w)

where w = discount factor

Axelrod compared the payoff with other two strategies. One is the strategy to always choose D and the other is to alternate D and C (these two are the only potential invaders of tit-for-tat). Then, the payoff of the player will be the following:

  • (T + wP)/(1-w)
  • (T + wS)/(1-w^2) = wS + w^2T + w^3S…

Thus, if and only if the below conditions are satisfied, the tit-for-tat strategy is invaded by the other strategies:

  • w≧(T-R)/(T-P)
  • w≧(T-R)/(R-S)

If the probability of meeting certain types of players is statistically distributed, then we can provide the expected payoff of each strategy and come up with the conditions on which tit-for-tat thrives.

The result implies that tit-for-tat would be the best strategy in the following cases:

  • The future is important (i.e. discount factor is enough high)
  • The players interact frequently (i.e. the number of iteration is large)
  • The probability to meet tit-for-tat (or “nice”) player is high

In chapter 3, he provides several additional propositions on the strategy’s robustness and stability in the prisoner’s dilemma situation. Then, he concludes:

“Thus cooperation can emerge even in a world of unconditional defection. The development cannot take place if it is tried only by scattered individuals who have no chance to interact with each other. But cooperation can emerge from small clusters of discriminating individuals, as long as these individuals have even a small proportion of their interactions with each other. Moreover, if nice strategies (those which are never the first to defect) come to be adopted by virtually everyone, then those individuals can afford to be generous in dealing with any others. By doing so well with each other, a population of nice rules can protect themselves against clusters of individuals using any other strategy just as well as they can protect themselves against single individuals. “


The author applied his arguments in various fields. One was biology. In the paper published in “Science”, Robert Axelrod and William D. Hamilton (co-author) pointed out the certain cooperation shown in mutualistic symbioses: the fungus and alga that compose a lichen; the ants and ant-acacias, where the trees house and feed the ants which, in turn protect the tees; and the fig wasps and fig tree, where wasps, which are obligate parasites of fig flowers, serve as the tee’s sole means of pollination and seed set. They argue that symbioses mainly illustrate the other recent extension of evolutionary theory, the theory of reciprocation.

They said, their contribution in this area is as follows:

  • They provided the probabilistic game theory model in a biological context, formalizing the strategic possibilities inherent in the situations
  • Their analysis of the evolution of cooperation considers not just the final stability of a given strategy, but also the initial viability of a strategy in an environment dominated by non-cooperating individuals
  • Their applications include behavioral interaction at the microbial level

Criticism from economists

When I read this book, I thought that some conclusions are not quite new in economics, or maybe it even exaggerates results of the game. As imagined, there were many criticisms from economists, as Ken Binmore summarized in his article. He says:

“Other game theorists may protest at my recognizing someone who knew no game theory at the time he made his contribution and still resolutely ignores game-theoretic commentary on his work, but it is inescapable that the evolutionary ideas pioneered by Axelrod now provide the standard approach to the equilibrium selection problem in game theory. But it is necessary to insist that to recognise Axelrod as a pioneer in evolutionary equilibrium selection is to endorse neither his claims for the strategy TIT-FOR-TAT, nor his unwillingness to see what theory can do before resorting to complicated computer simulations.

In game theory, there already are fruitful theoretical results in iterated game, and Axelrod deliberately or indeliberately seems to ignore them. His usage of the famous “computer tournament” and of the examples seem to be arbitrary, as the arguments are lacking theoretical grounds. That said, the criticism does not destroy his entire contribution to the world, I believe. He provided the framework to think about the evolution from the perspective of game theory.


It is a book of optimism. We can imagine what will happen in the future of cooperation. As written in this book, the more probability of continuing the game, the more cooperations. As ICT development is making us closer and more related to each other, there will be more chances of cooperation in the future. Actually, that is what’s going on in some spheres such as collaborative consumption, open source programming, etc.

My another remark is about problem solving. Technological advancement provides problems to solve. In this book, thanks to the development of computational power, the author could assess the strength of tit-for-tat strategy. Whenever one seeks for great problems, it may wise to look at the set of available technologies.


  • “The evolution of cooperation” (Academic paper published in Science, Robert Axelrod and William D. Hamilton)
  • “The evolution of cooperation” (Book, Robert Axelrod)
  • Review by Ken Binmore, http://jasss.soc.surrey.ac.uk/1/1/review1.html

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