Evolutionary Analysis of the Assignment of Property Rights

Atsushi Tsuneki

doi:10.1515/ajle-2024-0125

Article

Evolutionary Analysis of the Assignment of Property Rights

Atsushi Tsuneki

Published/Copyright: March 21, 2025

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From the journal Asian Journal of Law and Economics

Abstract

This study aims to determine a theoretical basis for the assignment of property rights by applying an evolutionary game model in which two types of workers apply for their effort for the increase of the property value. By extending the evolutionary game approach to determine the theoretical basis supporting that the property rights institution is preceded by the evolutionarily developed human psyche, due originally to Eswaran and Neary, the resulting evolutionary stable set of preferences differs from them since the relative endowment effect hardwired depends on the productivity difference between the two parties.

Keywords: evolutionary game model; property right; endowment effect

JEL Classification: C7; K3

Corresponding author: Atsushi Tsuneki, College of Economics, Nihon University, 3-2 Kanda-Misakicho 1-chome, Chiyoda-ku, Tokyo 101-8360, Japan, E-mail: tsuneki.atsushi@gmail.com

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: Grant-in-Aid for Scientific Research for Challengi

Award Identifier / Grant number: Grant-in-Aid for Scientific Research(B) 18H00867

Acknowledgments

The author thanks Mukesh Eswaran, Tomotaka Fujita and Hugh Neary for their valuable suggestions regarding the concept of this study. I would like to sincerely thank the important comments from the reviewer and the generous support from the editor, Professor Jeong-Yoo Kim. Any remaining errors are the responsibility of the author.

Funding: This research is supported by JSPS Grant-in-Aid for Scientific Research for Challenging Exploratory Research (No. 26590046) and JSPS Grant-in-Aid for Scientific Research(B) (18H00867).
Conflicts of interest: The author declares no conflict of interest.
Availability of data and material: Not applicable.
Code availability: Not applicable.

Appendix

First, the preliminary results of the functions g (v,w) = ln (s^*(v,w))–e^*(v,w) and h (v,w) = ln (t^*(v,w))–f^*(v,w) are summarized, as defined in Equation (13). From Equations (5) and (6), they are

(A1) ∂ g ∂ v = w 1 − v − e * 2 v v + w , ∂ h ∂ w = v 1 − w − f * 2 w v + w ,

(A2) ∂ 2 g ∂ v ∂ w = 1 − v − 2 e * 4 v w v + w 2 , ∂ 2 h ∂ v ∂ w = 1 − w − 2 f * 4 v w v + w 2 .

As pointed out by Eswaran and Neary (2014:222),

(A3) ∂ 2 g ∂ v 2 < 0 , i f v ≤ 5 3 , ∂ 2 h ∂ w 2 < 0 , i f w ≤ 5 3 ,

(A4) Δ = ∂ 2 g ∂ v 2 ∂ 2 h ∂ w 2 − ∂ 2 g ∂ v ∂ w ∂ 2 h ∂ v ∂ w > 0 , if v < 1 and w < 1 .

Proof of Proposition 1:

Substituting (A1) into the first-order conditions in Equations (15) and (16) results in

(A5) w 2 a v + w = − 1 − v 1 − v − e * > 0 , v 2 b v + w = − 1 − w 1 − w − f * > 0 ,

which, in turn, implies the inequality conditions

(A6) 1 − v > 0 , 1 − v − e * < 0 , 1 − w > 0 , 1 − w − f * < 0 .

Therefore, the values v ˜ , w ˜ that satisfy Equations (15) and (16) are both less than unity.

From the inequalities (A6) and (A2), ∂ 2 g ∂ v ∂ w < 0 and ∂ 2 h ∂ v ∂ w < 0 at v , w = v ˜ , w ˜ . From this result and (A3), the slopes of the best-response functions are d v b r w d w = − ∂ 2 σ ∂ v ∂ w ∂ 2 σ ∂ v 2 < 0 and d w b r v d v = − ∂ 2 τ ∂ v ∂ w ∂ 2 τ ∂ w 2 < 0 at v , w = v ˜ , w ˜ . Therefore, v and w are strategic substitutes at v , w = v ˜ , w ˜ .
It is clear from Equations (15) and (16), and (A3) that the second-order conditions for the maximization problem (14) are satisfied for v ˜ , w ˜ . Therefore, it is a locally unique equilibrium. This means v ˜ , w ˜ is locally uninvadable because this concept requires only that v ˜ , w ˜ is a neighborhood-strict Nash equilibrium, as proven by Cressman (2009).
Using Equations (15) and (16) and substituting in (A1) renders

(A7) a − b = 1 2 v + w v 1 − f * 1 − w − w 1 − e * 1 − v .

Substituting (5) into (A7) shows that

(A8) a − b = 1 2 v + w v − w + v w v − w 1 − v 1 − w v + w

must be satisfied at v , w = v ˜ , w ˜ .

If a >b, suppose that v ˜ ≤ w ˜ . Then, the right-hand side of (A8) becomes negative, and the contradiction follows. Therefore, v ˜ > w ˜ must be met.

Similarly, a <b→ v ˜ < w ˜ . When a = b, suppose v ˜ ≠ w ˜ . Then, the left-hand side of (A8) is zero, and the right-hand side is non-zero, which means that a contradiction follows. Therefore, v ˜ = w ˜ is implied. In summary, we have a > b → v ˜ > w ˜ , a = b → v ˜ = w ˜ , and a < b → v ˜ < w ˜ . In proving the opposite implications, we first check directly from (A8) that v ˜ = w ˜ → a = b . By taking the contrapositives of a > b → v ˜ > w ˜ and a < b → v ˜ < w ˜ , and using v ˜ = w ˜ → a = b , we have v ˜ < w ˜ → a < b and v ˜ > w ˜ → a > b . Then, Equation (17) is established by summarizing all the results.

When using the definition of s in Equations (6) and (17), Equation (18) follows.

Proof of Proposition 2:

Denoting the second-order derivatives of σ and τ around v ˜ , w ˜ as σ_ij and τ_ij, respectively, where i,j = v,w,a,b,μ, the standard comparative static matrix using Equations (15) and (16) is σ v v σ v w τ w v τ w w d v d w = − σ v a d a 0 0 − τ w b d b . Inverting the left-hand side matrix, we have d v d w = 1 Δ τ w w − σ v w − τ w v σ v v − σ v a d a 0 0 − τ w b d b . From Equations (15) and (16), we have σ_vv<0 and τ_ww<0, using (A3). Similarly, from (A2) and (A6), σ_vw<0 and τ_wv<0. Additionally, from Equations (15) and (16), σ v a = 1 v − 1 > 0 , τ w b = 1 w − 1 > 0 .

Using the inverted matrix and the above inequalities, we now have

(A9) ∂ v ∂ a = − 1 Δ τ w w σ v a > 0 , ∂ w ∂ a = 1 Δ τ w v σ v a < 0 ,

(A10) ∂ v ∂ b = 1 Δ σ v w τ w b < 0 , ∂ w ∂ b = − 1 Δ σ v v τ w b > 0 , and

From (A9) and (A10), results 2.1 and 2.2 follow.

References

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Received: 2024-12-17

Accepted: 2025-03-10

Published Online: 2025-03-21

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https://doi.org/10.1515/ajle-2024-0125

Keywords for this article

evolutionary game model; property right; endowment effect