Information Sharing and Cooperative Search in Fisheries

Abstract

We present a dynamic game of search and learning about the productivity of competing fishing locations. Perfect Bayesian Nash equilibrium search patterns for non-cooperating fishermen and members of an information sharing cooperative are compared with first-best outcomes. Independent fishermen do not internalize the full value of information, and do not replicate first-best search. A fishing cooperative faces a free-riding problem, as each coop member prefers that other members undertake costly search for information. Pooling contracts among coop members may mitigate, but are not likely to eliminate free-riding. Our results explain the paucity of information sharing in fisheries and suggest regulators use caution in advocating cooperatives as a solution to common pool inefficiencies in fisheries.

Appendices

Appendix 1: Belief Updating

Three belief updating scenarios are possible in this model: no signal, one signal, or two signals about the profitability of a fishing site. Under the independence assumption—payoffs across sites are independent—a signal from site $k$ provides no new information about site $j$. As such, updated beliefs at site $j$ equal the prior belief, regardless of the information observed in the signal $s_{i,k}$.

Suppose, instead, that a single signal from site $j$ is observed. For prior beliefs $\mu _{i,j},\ \sigma ^2_{i,j}$, and single payoff signal $s_{i,j}$, the updated beliefs of fisherman $i$ at site $j$ are given as,

$$\begin{aligned} \mu '_{i,j}\left| s_{i,j}\right.&= \theta _{i,j}\mu _{i,j} + (1-\theta _{i,j})s_{i,j} \\ \sigma ^{2'}_{i,j}\left| s_{i,j}\right.&= \theta _{i,j} \sigma ^2_{i,j}, \end{aligned}$$

where $\theta _{i,j}=\sigma ^2/(\sigma ^2+\sigma ^2_{i,j})\in [0,1]$ weights the new information contained in the signal against prior beliefs. As uncertainty is resolved, more weight is placed on the prior.

Suppose now that two signals are available to fisherman $i$ from site $j$; one from fisherman $i$ and the other from $-i$. Then updated beliefs at $j$ are given as,

$$\begin{aligned} \mu '_{i,j}\left| s_{i,j},s_{-i,j}\right.&= \theta _{i,j}~\mu _{i,j} + (1-\theta _{i,j})~\bar{s}_j \\ \sigma ^{2'}_{i,j}\left| s_{i,j},s_{-i,j}\right.&= \theta _{i,j}\sigma ^2_{i,j}, \end{aligned}$$

where $\bar{s}_j$ denotes the average payoff signal observed at site $j$ and $\theta _{i,j}=\sigma ^2/(\sigma ^2+2\sigma ^2_{i,j})\in [0,1]$ again weights the new information (from two signals) against prior beliefs.

Appendix 2: Bayesian Nash Equilibrium

This section solves the date $t_2$ Bayesian game for independent risk neutral fishermen. We solve for the thresholds $s_i^U, s_i^L, s_{-i}^U$, and $s_{-i}^L$ from (4), which separate feasible signals into regions. There are four possible scenarios to consider depending on the period 1 site choices. We present the calculations for the case where both fishermen fish site 1 in the first fishing period. The analysis of the remaining cases follows analogously and is not repeated.

We begin by specifying how fishermen form beliefs about beliefs. Since both fishermen fished site 1 in the first period, $i$’s belief about $-i$’s private information should be a function of $s_{i,1}$, i.e., $i$’s private information obtained from fishing site 1. The best information available about $-i$’s beliefs comes from $i$’s updated beliefs. As such, each fisherman believes they share the same updated beliefs about sites 1 and 2.

Suppose $s_{-i}^U$ and $s_{-i}^L$ exist such that the representative fisherman’s counterpart plays the strategy outlined in Eq. (4). For the representative fishermen with $s_{i,1} \in [s_i^L,\ s_i^U]$ to play a mixed strategy at $t_2, a'_i \in (0,1)$, it must be the case that, based on his conjecture about his counterpart’s strategy, he is indifferent between fishing site 1 and site 2. Setting the expected payoff of fishing these sites equal and solving for $a'_{-i}$ finds,

$$\begin{aligned} a'_{-i} = \frac{\mu '_{i,1} - \mu '_{i,2} - \kappa + 2\kappa \Pr \left( s_{-i,1} < s_{-i}^U|s_{i,1}\right) }{2\kappa \Pr \left( s_{-i}^L \le s_{-i,1} \le s_{-i}^U | s_{i,1}\right) } \quad \forall \ s_{i,1} \in \left[ s^L_i,\ s^U_i\right] . \end{aligned}$$

(11)

Using the same method, we can calculate the mixed fishing strategy for the representative fisherman, $a'_i$.

In a BNE, each player’s action must be optimal subject to their Bayesian updated belief about the strategy of their rival. Using $a'_i$ and $a'_{-i}$, we derive four equations that define the parameters $s_i^U, s_i^L, s_{-i}^U$, and $s_{-i}^L$ that will satisfy a BNE.

The representative fisherman believes that his counterpart is indifferent between mixing and fishing site 1 when $a'_{-i}$ exactly equals one (his counterpart knows that this is the representative fisherman’s belief). For site 1 to be strictly dominant, it must be that site 1 is still preferred even under the worst case (when both fish the site). Using this fact, the updating rules in Appendix 1, and the strategy profile in Eq. (4), the representative fisherman’s upper signal threshold $s_i^U$ must satisfy

$$\begin{aligned} s_i^U = \frac{1}{1-\theta _{i,1}}\left\{ \mu _{i,2} - \theta _{i,1}\mu _{i,1} +\kappa \right\} \end{aligned}$$

(12)

where $\theta _{i,1}=\sigma ^2/(\sigma ^2+\sigma ^2_{i,1})$.

Similarly, fisherman $i$ believes his counterpart is indifferent between mixing and fishing site 2 when $a_{-i}$ exactly equals zero (his counterpart knows that this is the representative fisherman’s belief). For site 2 to be strictly dominant, it must be that site 2 is still preferred even under the worst case (when both fish the site). Using this fact and given the strategy profile, the representative fisherman’s lower signal threshold $s_i^L$ must satisfy

$$\begin{aligned} s_i^L = \frac{1}{1-\theta _{i,1}}\left\{ \mu _{i,2}-\theta _{i,1}\mu _{i,1} -\kappa \right\} . \end{aligned}$$

(13)

Using the same argument, we construct the following for fisherman $-i$:

$$\begin{aligned} s_{-i}^U = \frac{1}{1-\theta _{-i,1}}\biggl \{\mu _{-i,2} - \theta _{-i,1}\mu _{-i,1} +\kappa \biggr \} \end{aligned}$$

(14)

$$\begin{aligned} s_{-i}^L = \frac{1}{1-\theta _{-i,1}}\biggl \{\mu _{-i,2} - \theta _{-i,1}\mu _{-i,1} -\kappa \biggr \} . \end{aligned}$$

(15)

The thresholds that satisfy the BNE , $s_i^U, s_i^L, s_{-i}^U$, and $s_{-i}^L$, defined by Eqs. (12)–(15), must simultaneously hold.