Last update: 06/7/14
My Research
I study Mathematical Biology. It is a branch of theoretical biology. I study biological phenomena using mathematics. In particular I'm interested in the following three topics:
  1. Evolutionary game theory
  2. Evolution of cooperation
  3. Theoretical models of animal behavior
1 Evolutionary game theory

In evolutionary game theory, it is often assumed that the focal population is infinite and is well-mixed. Recent pioneering studies have shown that we obtain unexpected results in the absence of the assumption of infiniteness. Suppose that the population size is given by N (finite) and consider a matrix game with two pure strategies, A and B, which payoff matrix is given by
.

Fixation probability of strategy A, denoted by rho_A, is defined as the probability that an A-mutant in the population of N-1 B strategiests eventually takes over the whole population.

If A and B were neutral, rho_A would be exactly equal to the reciprocal of the population size, 1/N. Therefore if rho_A is large than 1/N, we can say that strategy A is favored by natural selection. Nowak et al.(2004) found

is the condition for adaptive evolution of strategy A.

I investigated the consequences of removing the other assumption, well-mixed population. Generally, the structure of the popualtion is given by a (finite and connected) graph.

As a result, I find that strategy A is favored by natural selection on a graph of degree k (= a graph, each of which vertex has k edges) if

holds. (Ohtsuki et al. (2006) Nature 441, 502-505)BAs an example, let us study the following Prisoner's Dilemma game:
.

Applying the result above to this game, we find that

is the decisive condition for the evolution of cooperation on a graph of degree k. In other words, cooperation evolves if the benefit-cost ratio of cooperation exceeds the number of neighbors.
2 Evolution of cooperation

The evolution of cooperation is one of the biggest problems in evolutionary biology. The theory of kin selection by Hamilton (1964) succeeded in explaining altruistic behavior among relatives. Reciprocal altruism, first proposed by Trivers (1971), explains cooperation between non-relatives who interact repeatedly. This scenario has been theoretically confirmed by the triumph of the well-known Tit-For-Tat (TFT) strategy.

An interesting question if cooperation can evolve between non-relatives who interact only once. Internet auction is a good example of this. Participants are not related. Also it is rare that one meets the same person more than once.

Indirect reciprocity refers to the mechanism how cooperation is transferred thorugh person to person via reputation. One does cooperate with those who have a good reputation, but he refuses to cooperate those who have a bad reputation. Cheaters are selectively removed in this way.

I study the evolution of norms in indirect reciprocity. A norm is a rule of whom one regards as good and whom one regards as bad. The simplest norm is the one which regards those who cooperated as good and those who defected as bad. It is called scoring (Nowak&Sigmund 1998). Surprisingly, however, cooperation cannot evolve under this norm.

Suppose that player A is a bad person who does not cooperate at all. Taking this into account, player B refuses to help A. Under scoring, however, B is labelled as bad bacause B did not cooperate with A; B's intention does not matter. Here lies the problem in scoring.

I have theoretically shown by a formal ESS analysis that the concept of justified defection (e.g. a good person who refused to help a bad person remains good because he punished a bad one) is imperative in indirect reciprocity. I found eight norms, which is called "leading eight", that maintain the highest level of cooperation by indirect reciprocity (Ohtsuki&Iwasa (2004) Journal of Theoretical Biology 231, 107-120).
3 Theoretical models of animal behavior

Currently I'm working on the following topics.
back to top