Sharing, gift-giving, and optimal resource use in hunter-gatherer society

Abstract

Hunter-gatherer societies are characterized by decentralized decision making and shared access to resources. Goods are distributed via reciprocal exchange, sharing, and gift-giving, in the end resulting in relatively equal distribution. Work effort, however, is not distributed equally; the best hunters exert a disproportionate share of productive effort. We argue that these features of the hunter-gatherer economy are interrelated, and are consistent with the view that customary gift-giving and sharing arrangements manage usage of open-access resources. In particular, sharing rules can induce optimal hunting effort, while gift-giving can serve to elicit information about hunter productivity.

Appendices

Appendix

Derivation of sharing rules

Recall the first-order condition associated with expected welfare maximization for optimal group efforts, Eq. (5):

$$\begin{aligned} A(X)+\sum x_j A'(X)-c'_j(x_j)=0, \quad i=1,2,3,\dots ,N \end{aligned}$$

(A.1)

Equation (10) describes optimal efforts under the sharing rules:

$$\begin{aligned} \begin{aligned} \left( A(X)+x_i A'(X)\right) \left( 1 - \sum _{k \ne i} t_{ik} \right) + \sum _{k \ne i} t_{ki} A'(X)x_k - c_i'(x_i) =0, \\ i=1,2,3,\dots ,N \end{aligned} \end{aligned}$$

(A.2)

The task is to choose sharing instruments $t_{ik}$ in (A.2) so that (A.1) is replicated. Algebraically, this requires that the sharing rules satisfy the N equations:

$$\begin{aligned} \begin{aligned} \left( A(X)+x_i A'(X)\right) \left( 1 - \sum _{k \ne i} t_{ik} \right) + \sum _{k \ne i} t_{ki} A'(X)x_k \\ = A(X)+\sum x_jA'(X),\quad i=1,2,3,\dots ,N \end{aligned} \end{aligned}$$

(A.3)

The system cannot be solved as-is for sharing instruments as there are $n(n+1)$ sharing instruments yet only n equations in (A.3). If, however, $n=2$, sharing instruments are identified, and satisfy the equation system:

$$\begin{aligned} \begin{aligned} \left( A(X)+x_1 A'(X)\right) \left( 1 - t_{12} \right) + t_{21} A'(X)x_2 = A(X)+\sum x_1A'(X), \\ \left( A(X)+x_2 A'(X)\right) \left( 1 - t_{21} \right) + t_{12} A'(X)x_1 = A(X)+\sum x_2A'(X) \end{aligned} \end{aligned}$$

(A.4)

Solving (A.4) for the sharing rules gives:

$$\begin{aligned} t_{12}=-\frac{x_2^*A'(X^*)}{A(X^*)},\quad t_{21}=-\frac{x_1^*A'(X^*)}{A(X^*)} \end{aligned}$$

(A.5)

The sharing rules in (A.5) suggest a generalization to the N-hunter case, as the amount shared with hunter 2, say, depends directly upon hunter 2’s optimal effort $x^*_2$, and only indirectly on the efforts of hunter 1 through the aggregate optimal effort level $X^*$. Generalization to the rules of the form where the amount shared with a hunter depends upon the hunter’s optimal effort:

$$\begin{aligned} t_{.k}=-\frac{x_k^*A'(X^*)}{A(X^*)} \end{aligned}$$

(A.6)

is immediate. As shown in the text following proposition 1, these shares sum to less than one.

Derivation of the type-revealing gift schedule

Here we fill in some of the details of the derivation of the gift-giving rule that induces hunters to correctly report type, which is analogous to a cost-of-effort parameter. In the event that a hunter reports type $\hat{\theta }_i$, and it is the case that this report is larger than that of any other hunter—$\hat{\theta }_i > \hat{\theta }_j,j\in N, j\ne i$—hunter i wins the right to be atop the sharing scheme, but must pay out a gift obligation $g(\hat{\theta }_i)$. If, however, another hunter reveals a type that $\hat{\theta }_j > \hat{\theta }_i$, hunter i does not produce and simply collects a share $\frac{1}{n-1}$ of the gift from the highest type.

Let $P_{i}$ denote that probability that hunter i submits the highest bid. That is:

$$\begin{aligned} P_{i}=\text {Prob}[\hat{\theta _i}>\hat{\theta }_1,\hat{\theta }_2,...,\hat{\theta }_{i-1},\hat{\theta }_{i+1},...,\hat{\theta }_n] \end{aligned}$$

(B.1)

In equilibrium, hunters truthfully report types, so $P_i$ in (B.1) is the same as the chances of $\hat{\theta }_i$ is greater than $n-1$ i.i.d, uniform random variables. Therefore:

$$\begin{aligned} P_{i}=F(\hat{\theta }_i)^{n-1}=\left( \frac{\hat{\theta }_i}{\overline{\theta }}\right) ^{n-1} \end{aligned}$$

(B.2)

In the event that hunter i does not submit the highest productivity bid, which occurs with probability $1-P_{i}$, hunter i receives a gift equal to the expected value of $g(\theta )/(n-1)$ conditional on $\theta _i<\text {max}[\theta _{-i}]$ and $\theta$ being the maximum of $n-1$ uniform random variables. Since $\theta$ follows the distribution of the maximum of $n-1$ uniform random variables, it has density $(n-1)f(\theta )F(\theta )^{n-2}$. The relevant conditional expectation is then:

$$\begin{aligned} E\left[ \frac{g(\theta )}{n-1}|\theta >\hat{\theta }_i\right] =\frac{1}{1-P_{i}}\int _{\hat{\theta }}^{\overline{\theta }}\frac{g(\theta )}{n-1}(n-1)f(\theta )F(\theta )^{n-2}d\theta \end{aligned}$$

(B.3)

Equation (B.3) simplifies to:

$$\begin{aligned} E\left[ \frac{g(\theta )}{n-1}|\theta >\hat{\theta }_i\right] = \frac{1}{1-P_{i}}\int _{\hat{\theta }_i}^{\overline{\theta }}g(\theta )\frac{1}{\overline{\theta }}\left( \frac{\theta }{\overline{\theta }}\right) ^{n-2}d\theta \end{aligned}$$

(B.4)

Expected returns to hunter i from submitting a bid of $\hat{\theta }_i$, assuming other hunters truthfully reveal their types are then:

$$\begin{aligned} ER_i(\hat{\theta _i},\theta _i)=\left( \frac{\hat{\theta }_i}{\overline{\theta }}\right) ^{n-1}(\theta _i-g(\hat{\theta }_i))+\int _{\hat{\theta }_i}^{\overline{\theta }}\frac{g(\theta )}{\overline{\theta }}\left( \frac{\theta }{\overline{\theta }}\right) ^{n-2}d\theta \end{aligned}$$

(B.5)

Differentiating this expression with respect to $\hat{\theta _i}$ gives a first-order condition describing the optimal bid for a hunter of type $\theta _i$, given all other hunters correctly bid $\hat{\theta }_j=\theta _j$:

$$\begin{aligned} \frac{\partial ER_i(\hat{\theta _i},\theta _i)}{\partial \hat{\theta }_i}= & {} \frac{(n-1)}{\theta }\left( \frac{\hat{\theta }_i}{\overline{\theta }}\right) ^{n-2}(\theta _i-g(\hat{\theta }_i))- \nonumber \\&\left( \frac{\hat{\theta }_i}{\overline{\theta }}\right) ^{n-1}g'(\hat{\theta }_i)-\frac{g(\hat{\theta }_i)}{\overline{\theta }}\left( \frac{\hat{\theta }_i}{\overline{\theta }}\right) ^{n-2} \end{aligned}$$

(B.6)

Truthful revelation for agent i requires that $\hat{\theta }_i=\theta _i$. Substituting this into (B.6), setting the result to zero, and simplifying gives the following differential equation describing the function $g(\theta )$:

$$\begin{aligned} g'(\theta )+\frac{g(\theta )n}{\theta }-(n-1)=0 \end{aligned}$$

(B.7)

The solution to (B.7) is:

$$\begin{aligned} g(\theta )=C\theta ^{-n}+\frac{n-1}{n+1}\theta \end{aligned}$$

(B.8)

Where C in (B.8) is a constant of integration. Since $g(\theta )$ in (B.8) would explode for small values of n if C in (B.8) were not zero, we arrive at the gift function:

$$\begin{aligned} g(\theta )=\frac{n-1}{n+1}\theta . \end{aligned}$$