Does suffering dominate enjoyment in the animal kingdom? An update to welfare biology

Abstract

Ng (Biol Philos 10(3):255–285, 1995. https://doi.org/10.1007/bf00852469) models the evolutionary dynamics underlying the existence of suffering and enjoyment and concludes that there is likely to be more suffering than enjoyment in nature. In this paper, we find an error in Ng’s model that, when fixed, negates the original conclusion. Instead, the model offers only ambiguity as to whether suffering or enjoyment predominates in nature. We illustrate the dynamics around suffering and enjoyment with the most plausible parameters. In our illustration, we find surprising results: the rate of failure to reproduce can improve or worsen average welfare depending on other characteristics of a species. Our illustration suggests that for organisms with more intense conscious experiences, the balance of enjoyment and suffering may lean more toward suffering. We offer some suggestions for empirical study of wild animal welfare. We conclude by noting that recent writings on wild animal welfare should be revised based on this correction to have a somewhat less pessimistic view of nature.

Appendix: Derivation of Eq. (5)

We derive Eq. (5) starting with the following maximization problem:

$$\begin{aligned} & \max_{{\left\{ {C_{E} , C_{S} } \right\}}} \ln (\alpha C_{E} + 1) + \ln (\upalpha{\text{C}}_{\text{S}} + 1) \\ & \quad \quad s.t.\quad M = \frac{1}{n + 1}C_{E} + \frac{n}{n + 1} C_{S} \\ \end{aligned}$$

This yields the Lagrangian:

$$L = \ln (\alpha C_{E} + 1) + \ln (\upalpha{\text{C}}_{\text{S}} + 1) + \mu (M(n + 1) - C_{E} - nC_{S} )$$

Leading to first-order conditions:

$$0 = \frac{\partial L}{{\partial C_{E} }} = \frac{\alpha }{{\alpha C_{E} + 1}} - \mu$$

$$0 = \frac{\partial L}{{\partial C_{S} }} = \frac{\alpha }{{\alpha C_{S} + 1}} - n\mu$$

Combining and rearranging terms gives:

$$C_{E} = nC_{S} + \frac{n - 1}{\alpha }$$

Plugging this into the budget leads to the following:

$$M(n + 1) = n C_{S} + \frac{n - 1}{\alpha } + nC_{S} = 2nC_{S} + \frac{n - 1}{\alpha }$$

$$C_{S} = \frac{M}{2n}(n + 1) - \frac{n - 1}{2n\alpha }$$

Combining this with the equation for $C_{E}$ in terms of $C_{S}$, we get:

$$C_{E} = \frac{M}{2}(n + 1) + \frac{n - 1}{2\alpha }$$

Now we can calculate the balance of suffering and enjoyment by taking suffering minus enjoyment per individual:

$$\begin{aligned} \frac{1}{n + 1}(nS(C_{S} ) - E(C_{E} )) & \quad = \frac{1}{n + 1}\left( {n\ln \left[ {\frac{M\alpha }{2n}(n + 1) - \frac{n - 1}{2n} + 1} \right] - \ln \left[ {\frac{M\alpha }{2}(n + 1) + \frac{n - 1}{2} + 1} \right]} \right) \\ & \quad = \frac{1}{n + 1}\left( {n\ln \left[ {\frac{1}{n}\left( {\frac{M\alpha }{2}(n + 1) + \frac{n + 1}{2}} \right)} \right] - \ln \left[ {\frac{M\alpha }{2}(n + 1) + \frac{n + 1}{2}} \right]} \right) \\ & \quad= \frac{1}{n + 1}\left( {\ln \left[ {\frac{1}{{n^{n} }}\left( {\frac{(M\alpha + 1)(n + 1)}{2}} \right)^{n - 1} } \right]} \right) \\ &\quad = \left( {\frac{1}{n + 1}} \right)\ln \left[ {\frac{{(\upalpha{\text{M}} + 1)^{{{\text{n}} - 1}} (n + 1)^{n - 1} }}{{2^{n - 1} n^{n} }}} \right] \\ \end{aligned}$$