In this model, the criterion of a rational donor is maximization of the probability of simultaneous survival of the recipient and the donor, which is given by the product of their probabilities of survival.

Therefore, the donor here assesses the threat to himself as being equal to the threat to the recipient. We assume that the sole threat to both individuals is a low amount of funds.

In this case, the donor's criterion when deciding on the subsidy amount is the following function:

where the initial income d and the extinction zone boundaries b0 and b1 are parameters of the problem

Suppose that the donor has enough income to ensure the survival of both individuals:

We fix parameters b0 and b1 and introduce function a(d), which gives the optimal subsidy as a function of the initial income. The domain of this function is the interval (b0 + b1, +∞).

If the two agents have the same extinction zone boundary, i.e. if b0 = b1, the optimal strategy is obviously equal distribution of income between the two agents:^{[1]}

Figure 62: The subsidy amount versus the donor's initial income d0 in the absolute solidarity model where the two individuals are equally resilient (b0 = b1)

If b0 ≠ bv the solution to the problem is to prefer the more threatened individual. The following two figures show the subsidy amount as a function of the donor's initial income. In Figure 63 the recipient is more resilient, while in Figure 64 the donor is more resilient.

Figure 63: The subsidy amount versus the donor's initial income d0 in the absolute solidarity model where the recipient is more resilient (b0 < b1).

Figure 64: The subsidy amount versus the donor's initial income d0 in the absolute solidarity model where the donor is more resilient (b0 > b1).

Logically, therefore, the donor's behaviour is determined not only by the specific altruistic criterion, but also by the situation of each individual (in our case specifically by their relative resilience or threat of extinction, i.e. by the relationship between parameters b0 and b1). In some situations the donor will strongly prefer himself, but in other situations he is capable of increasing his own personal threat of extinction by providing a subsidy.

10.3.2 MODEL OF MINIMIZATION OF THE RISK OF SIMULTANEOUS EXTINCTION OF BOTH INDIVIDUALS: CRUEL ALTRUISM

In this model, the donor maximizes the probability of survival of at least one (any) member of the community. This probability can be expressed as follows:

[donor's survival probability] + [recipient's survival probability] - [probability of simultaneous survival of both agents]:^{[2]}

Suppose that the donor has enough income to ensure the survival of at least one of the two agents:

In this case, we again model the donor's decision by the optimization

where the domain of function a(d) (which gives the optimal subsidy as a function of initial income given the fixed extinction zone boundaries) is this time the interval (min (b0, b1, +∞).

The donor has two possible strategies and will choose the most advantageous with respect to the probability of survival of at least one agent:

Strategy I: ensure that the more resilient agent survives and the less resilient one does not (by not providing a subsidy),

Strategy II: ensure that both agents survive.

If the two agents have the same subsistence level, i.e, if b0 = b1 = b, the donor will have to choose which agent (out of the two equally resilient ones) will survive. Suppose that the recipient will be the survivor. For income d<2b the donor will leave his entire income to the recipient and will die himself, because such a low income is insufficient to ensure the survival (with a non-zero probability) of both agents. He will therefore, of course, choose Strategy I:

for

However, even if the donor's income d allows for the simultaneous survival of both members, i.e. if d > 2b, Strategy I may be more advantageous because it will provide a higher probability of survival of at least one agent. This will be the case for a lower-than-boundary income level, which in our case is four times the extinction zone boundary dh = 4b. At this boundary income level, the maximized probability is the same for both strategies (we denote the probability of survival of at least one agent for Strategy I and income d by pI(d) and the same for Strategy II by pII(d)):

The donor prefers Strategy I if income is below the boundary level (i.e. if d ≤ dh). If it is above this level (i.e. if d > dh) the donor's optimal strategy is fundamentally different. Instead of providing equal support to both, he will transfer all funds to a selected member. At the boundary point, the path of the optimum is discontinuous:

Figure 65 shows the subsidy amount as a function of the donor's initial income for the case where b0 = b1 = b:

Figure 65: The optimal subsidy a versus the donor's income d in the model of minimization of the risk of simultaneous extinction of both agents—the case of equally threatened agents; below the boundary income level dh the donor sacrifices himself

If the two individuals are equally resilient and, by contrast, the recipient is sacrificed, the optimal subsidy path a(d) changes only in that the subsidy is zero when income is below the boundary level:

Figure 66: Ditto with the recipient being sacrificed below the boundary income level

If the donor is the more resilient, i.e. if b0 < b1, the plot of the optimal subsidy a against income d is similar as in the previous figure, the only difference being that the boundary point moves to the right, i.e. the switch to Strategy II occurs at higher income d. We illustrate this in Figure 67 for the case where b0=b and b1= 2b.

Figure 67: Ditto for the case where the recipient is more threatened

When trying to minimize the risk of extinction of both agents, a rational donor will therefore behave discontinuously at the boundary^{[3]} point dh = 9b. If his income falls, say, from level d = 10b, there will be a sudden and drastic change in his behaviour at level d = 9b and his originally totally altruistic transfer of two-thirds of his income will change: the donor (according to the logic of the assumed criteria of maximization of the probability of survival of at least one agent) will completely stop providing the subsidy, thereby condemning the subsidized agent to death.

[1] At the optimum it must hold that the marginal transfer of funds from the first agent to the second will reduce the latter's probability of extinction to an equal extent as it increases the for mer's probability of extinction. In other words, the derivatives of the probability of extinction with respect to the amount of funds obtained must be equal. From this we can derive the optimal ratio in which the donor divides disposable income d.

[2] Subtraction of the probability of simultaneous survival of both is necessary because the first two addends both contain it and so otherwise it would be counted twice.

[3] As in the previous case, the probability of survival of at least one agent at the boundary point d = 9b is the same for Strategies I and II: pI(9b) = pII(9b) = 8/9. For d < 9b Strategy I is better, while for d > 9b Strategy II is better.

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