On the efficiency of the common law: an application to the recovery of rewards

Abstract

Richard Posner’s influence on the field of law and economics cannot be overstated. Among his many contributions, Posner offered an early conjecture that remains fascinating and controversial to this day: the idea that common law rules are more likely than legislative codes to be concerned with efficiency. In this paper, I compare the efficiency of a common law rule of contracting to the efficiency of a civil law rule. In common law jurisdictions, claimants must have knowledge of a reward in order to recover. In civil law jurisdictions, however, no such knowledge is required. I analyze the efficiency of each rule by examining the incentives created by each rule. In a finding that agrees with Posner’s hypothesis, I argue that the common law rule is more efficient. The model has a number of applications beyond contract default laws. I use the model to discuss three legal questions previously analyzed by Richard Posner: (1) incentivizing innovation; (2) the finders-keepers rule in property law; and (3) salvage rights in maritime law.

Appendix: Model with multiple searchers

In this Appendix, I illustrate that the simple result in the single searcher model carries over to situations with more than one searcher. There are socially valuable investments that are not made because the rewarder’s incentives to set a reward inducing investment do not align with the socially desirable conditions of investment. The possibility of independent discovery creates a wedge between the private and social incentives to invest in discovering property or socially valuable information, resulting in under-investment.

Consider a situation where a rewarder posts his reward and two independent and identical searchers simultaneously make a decision whether or not to invest. If neither searcher invests, the likelihood of the property or information being discovered, serendipitously, is $\rho _{0}$. If one searcher invests, but the other does not, the likelihood of the property or information being found increases by $\Delta \rho _{1}$. The searcher who makes the investment increases her probability of finding the property or information by $\Delta \rho _{1}$. If both searchers invest, then the probability of finding the property or information is greater than if just one searcher invests. The marginal increase in likelihood of discovery is $\Delta \rho _{2}$. I assume that the returns to investment are decreasing: $\Delta \rho _{1} > \Delta \rho _{2}$.

• Let $P_{1}$ be the probability of the property or information being found if one searcher is searching. That is, $P_{1} = \rho _{0} + \Delta \rho _{1}$.

• Let $P_{2}$ be the probability of the property or information being found if two searchers are searching. That is, $P_{2} = \rho _{0} + \Delta \rho _{1} + \Delta \rho _{2}$.

1.1 Socially optimal investment

The optimal number of searchers who invest in discovering the property or information will depend on the value of the property or information: If it is very valuable relative to the cost of investment, it is optimal to have both searchers searching; but, if the property or information is of low value and investment cost is high, it is optimal to induce neither searcher.

The expected benefit of neither searcher investing is $\rho _{0}V$. The expected benefit of having one searcher search is $\left( \rho _{0} + \Delta \rho _{1}\right) V - i$. The expected benefit of having both searchers search is $\left( \rho _{0} + \Delta \rho _{1} + \Delta \rho _{2} \right) V - 2i$. It follows that the additional searchers should only be added if the marginal social benefit of investment by the nth searcher, $\Delta \rho _{n}$, exceeds the cost of investment, i. The optimal number of searchers investing (“searchers”) is shown in the table below.

Optimal # of searchers
0 searchers	if $V < \frac{i}{\Delta \rho _{1}}$
1 searcher	if $V \in \left[ \frac{i}{\Delta \rho _{1}}, \frac{i}{\Delta \rho _{2}} \right)$
2 searchers	if $V \ge \frac{i}{\Delta \rho _{2}}$

Throughout this subsection, let $V^{*}_{1} = \frac{i}{\Delta \rho _{1}}$ and $V^{*}_{2} = \frac{i}{\Delta \rho _{2}}$. If $V < V^{*}_{1}$, no searchers should invest; if $V > V^{*}_{2}$, both searchers should invest. If $V \in \left[ V^{*}_{1}, V^{*}_{2} \right)$, it is socially optimal for only one searcher to invest.

1.2 Searchers’ decision to invest

With more than one searcher, a searcher’s decision to invest is slightly more complicated than in the single searcher case. Because I assume the searchers are identical and independent, if neither of them invest, then each of the searchers has a 50 % chance of finding the property or information first. Neither searcher is “luckier” than the other. That is, the probability that a given searcher will find the property or information serendipitously is $\frac{1}{2}\rho _{0}$. Also, if both searchers invest in finding the property or information, then they have a 50 % chance of finding the property or information before the other. The probability that a given searcher will find the property or information if both searchers invest is $\frac{1}{2} P_{2}$.

Once the rewarder posts the reward R, the two searchers play the non-cooperative simultaneous game shown below.

	Investment	No investment
Investment	$\biggl ( \frac{1}{2} P_{2} R - i \biggr )$, $\biggl (\frac{1}{2} P_{2}R - i \biggr )$	$\biggl ( (P_{1} - \frac{1}{2} \rho _{0})R - i \biggr )$, $\biggl ( \frac{1}{2} \rho _{0} R \biggr )$
No investment	$\biggl ( \frac{1}{2} \rho _{0} R \biggr )$, $\biggl ( (P_{1} - \frac{1}{2} \rho _{0} ) R - i \biggr )$	$\biggl ( \frac{1}{2} \rho _{0} R \biggr )$, $\biggl ( \frac{1}{2} \rho _{0} R \biggr )$

The equilibrium of this game turns on the value of R:

• If $R < \frac{i}{\Delta \rho _{1}}$, then neither searcher will invest. That is, if R is sufficiently low, there will be only one Nash equilibrium, $\{$ no investment, no investment $\}$, the bottom right corner of the matrix. Call $\underline{R} = \frac{i}{\Delta \rho _{1}}$. Note that $\underline{R} = V^{*}_{1}$.

• If $R \ge \frac{2i}{\Delta \rho _{1} + \Delta \rho _{2}}$, then both searchers will invest. That is, if R is sufficiently high, then there will be only one Nash equilibrium, $\{$ investment, investment $\}$, the top left corner of the matrix. Call $\overline{R} = \frac{2i}{\Delta \rho _{1} + \Delta \rho _{2}}$. It can be shown that $\overline{R} > \underline{R}$. Note that $\overline{R} < V^{*}_{2}$.

• If $R \in \left[ \frac{i}{\Delta \rho _{1}}, \frac{2i}{\Delta \rho _{1} + \Delta \rho _{2}} \right)$, there are three equilibria. There are two pure Nash equilibria, $\{$ investment, no investment $\}$, $\{$ no investment, investment $\}$, as well as a mixed strategy equilibrium. When the rewarder sets a reward between $\underline{R}$ and $\overline{R}$, the searchers play mixed strategies. In equilibrium, each searcher makes the investment with probability $\alpha \in [0,1)$, where $\frac{d \alpha }{d R} > 0$.⁴⁷

1.3 Rewarder’s offer of reward

As in the single searcher case, the rewarder’s incentives here are not aligned with the social optimum. In order to induce any investment, the reward needs to be set at or above $\underline{R}$. The rewarder will do this only if the value of the property or information is above $\underline{V}$. It follows that $\underline{V} > V^{*}_{1}$ if there is a chance of independent discovery.

The rewarder prefers not to induce investment—and pay out no reward—if the potential for serendipitous discovery is greater than the expected return from setting a reward:

$$\begin{aligned} \rho _{0}V > [V - R] \cdot \phi \end{aligned}$$

where $\phi$ is the probability that the property or information will be found if a reward, R, is set.⁴⁸ Therefore, the rewarder will only set a reward $R > \underline{R}$, inducing positive probability of investment by the searchers, if $V > \underline{V}$:

$$\begin{aligned} \underline{V} = \left( \frac{\phi }{\phi - \rho _{0}} \right) \underline{R} \end{aligned}$$

Recall that $\underline{R} = V^{*}_{1}$. Since $\frac{\phi }{\phi - \rho _{0}} > 1$ when $\rho _{0} > 0$, it is the case that $V > V^{*}_{1}$:

$$\begin{aligned} \underline{V} = \left( \frac{\phi }{\phi - \rho _{0}} \right) V^{*}_{1} \end{aligned}$$

As with the single searcher case, there are situations where it would be socially desirable to invest in finding the property or information, but the rewarder has no incentive to set a reward. This leads to under-investment in searching for property or information. Note, once again, there is no problem of under-investment if there is no independent or serendipitous discovery. If $\rho _{0} = 0$, then $\underline{V} = {V}^{*}_{1}$.

If the likelihood of independent discovery, $\rho _{0}$, is very high, then the rewarder will set no reward even if it is socially optimal to have two searchers invest. Recall that when $V = V^{*}_{2}$ it is socially optimal to have two searchers investing in finding the property or information. The rewarder, however, will prefer to set no reward if:

$$\begin{aligned} \rho _{0} V^{*}_{2}> & {} P_{2} \cdot \left[ V^{*}_{2} - \overline{R} \right] \\ \rho _{0} \left( \frac{i}{\Delta \rho _{2}} \right)> & {} \left( \rho _{0} + \Delta \rho _{1} + \Delta \rho _{2} \right) \cdot \left[ V^{*}_{2} - \overline{R} \right] \\ \rho _{0} \left( \frac{i}{\Delta \rho _{2}} \right)> & {} \left( \rho _{0} + \Delta \rho _{1} + \Delta \rho _{2} \right) \cdot \left[ \frac{i}{\Delta \rho _{2}} - \frac{2i}{\Delta \rho _{1} + \Delta \rho _{2}} \right] \\ \Longrightarrow \rho _{0}> & {} \frac{\Delta \rho _{1}^{2} - \Delta \rho _{2}^{2}}{2 \Delta \rho _{2}}\\ \end{aligned}$$

Even though it is socially optimal to have two searchers investing, if the probability of independent discovery is sufficiently great, it is privately optimal for the rewarder to set no reward. There is, again, a wedge driven between the private and social incentives.⁴⁹

1.4 Example

Take the example from Sect. 3.1. Now suppose there are two searchers, Barbara and Billie. The investment costs are $200 each. If neither makes the investment, the probability of Charlie being found is 50 %. If one searcher makes an investment, by taking the day off work, the probability of finding Charlie increases by 25–75 %. If both searchers invest, taking the day off work, the probability of finding Charlie increases by a further 10–85 %. Here, where Adam values Charlie at $1000, it is socially optimal to have one searcher search; it is socially wasteful to have both searchers searching. Adam, however, will not post a reward for the same reasons noted above: He prefers to see if the searchers will serendipitously find Charlie.