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Hisashi Ohtsuki develops mathematical models of cooperation, connecting population structure with natural selection. His research asks how costly help can persist when individuals that withhold help can still receive its benefits.^[ref]

In a 2006 study, Ohtsuki and colleagues represented individuals by graph vertices and their interactions by graph edges. They calculated when a single cooperator had a fixation probability above its neutral benchmark. The result depended on how reproduction filled vacant sites, as well as on the graph. This helped make network reciprocity a quantitative mechanism: local interaction patterns and the updating rule can alter whether helping is favoured. The analysis used weak selection and large-population approximations, so the resulting threshold must be read with those assumptions.^[ref]

Martin A. Nowak works on mathematical models of evolution, including the evolution of cooperation. His 2006 synthesis separated five mechanisms: kin selection, direct reciprocity, indirect reciprocity, network reciprocity and group selection. Each mechanism changes which interactions contribute to an individual's evolutionary success.^[ref]

A simple direct reciprocity calculation shows how repeated encounters can change incentives. Suppose cooperation costs $c$, benefits the partner by $b>c$, and an encounter continues with probability $w$. Two cooperating reciprocators obtain expected payoff $(b-c)/(1-w)$. An unconditional defector against a strategy that stops helping after a defection obtains $b$ initially and no later benefit. The first payoff exceeds the second precisely when $w>c/b$. With $b=3$, $c=1$ and $w=1/2$, these payoffs are four and three.

This calculation depends on the specified repeated-interaction model and retaliation rule. It identifies a mechanism by which cooperation can be favored, without asserting that a single inequality describes every biological system. His institutional research profile also lists mathematical work on infection and cancer.^[ref]

Portrait source: www.oeb.harvard.edu/people/martin-nowak .

Network reciprocity is the promotion of cooperation by the pattern of interactions in a biological population. When cooperators interact preferentially with other cooperators, their help can sustain local clusters. A graph makes the mechanism precise: individuals occupy graph vertices, and graph edges specify who interacts with whom. Recognition of a previous helper is not required.

In the donation game, let each cooperator pay a cost $c$ per neighbour and supply benefit $b$ to that neighbour. On a regular graph with four neighbours per individual, a cooperator surrounded by four cooperators receives payoff $4b-4c$; an isolated cooperator surrounded by nonhelpers receives $-4c$. Thus the same behaviour has different consequences in different local arrangements.

Hisashi Ohtsuki and colleagues derived the approximate condition $b/c>k$ for natural selection to favour cooperation on large regular graphs of degree of a vertex $k$, under weak selection. Their updating rule first removed a random individual, then let its neighbours compete for the vacancy according to biological fitness. For $k=4$, $b=5$ and $c=1$ satisfy the asymptotic condition. The rule concerns fixation probability under stated assumptions; changing the order of death and reproduction can change the result.^[ref]