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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]

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]

Weak selection is a regime of an evolutionary model in which the focal trait or interaction changes biological fitness only slightly relative to a baseline. It permits calculations close to neutral evolution. In one common evolutionary game theory convention, a payoff $P$ changes biological fitness according towhere $w$ controls the contribution of the game. Weak selection requires the payoff-dependent change to be small on the scale of baseline biological fitness.^[ref]

For example, take $w=0.01$ and two competitors with payoffs $P_1=2$ and $P_2=3$. Their fitnesses are $f_1=1.01$ and $f_2=1.02$. If their chances to fill a vacancy are proportional to biological fitness, the second competitor wins with probabilityonly slightly above the neutral value $1/2$. This invented calculation shows the meaning of a small selective bias. A small numerical $w$ would not ensure the same approximation if the payoffs were made enormously larger. Weak selection is also distinct from small biological population size: the strength of genetic drift and the cumulative effect of selection depend on additional features of the model.