Coase theorem analyzed by three-person games

Kazuhisa Nishi*

Abstract

The Coase theorem is well known in the field of law and economics. In this paper Coase theorem is analyzed in the framework of game theory. First, its theorem is considered by two sets of a two-person game which have the different rules, and it is found to be proved under the suitable conditions of pay-off functions. Second, these games are then transformed into one set of a three-person game. The purpose of this transformation is to reformulate two games as one game which has the variable rule determined by the actions of players. The peculiarity in this game is that there are two normal players and one player which is named here as virtual player. Thus, Coase theorem is considered as a starting point providing a prototype of the rule-variable game.

Keywords Coase theorem, Non-cooperative game, Three-person game

1. Introduction

Coase theorem is one of the famous theorems in the field of law and economics [1,2]. The outline of Coase theorem is as follows. In the absence of transaction cost,the allocation of resources is independent on the conditions of legal right and duty. On the other hand, in the presence of transaction cost, the existence of legal regulations makes an effect on the allocation of resources, and suppresses the coming of transaction cost. Forthe improvement of the social efficiency, it should be responsible to the party with the lower activity of producing. Further, if there is some tactics avoiding the damage, itshould be responsible to the party with the lower cost of the tactics. Then, this theorem is very interesting from the viewpoints of environmental economics or ecology. However, since this theorem is rather derived by Coase’intuitive analysis on the interplay between law and economy, its theoretical background seems to be not so evident [3-5]. The situation of theorem can be identified as the game theory in which there are negotiating players and tactics under theconditions of legalregulations. The purpose of this study is to analyze Coase theoremby two sets of a two-person game [6] and to transform it into one set of a three-person game which has the variable rules.

2. Two-person game

Let G =＜q11,q12…,q21, q22…, ＞be a two person game in normal form, where qii’is a finite set of strategies for player i and i(q1i’, q2i’) (i’ = 1, 2…) is a player i’s payoff function. Of these strategies, the specified variables which are player’s activity and providing the compensation are identified. The compensation is caused by the harm due to players’ actions. It is then assumed in this game that negotiation for the compensation is possible. Also, the so-called external effects such as legal regulations are regarded as imposing the restrictive rule to game theory.

Definition 2

(1)The strategy variables of player’s actions are m1(q11), m2(q21) and take the value of 1(0) in the presence (absence) of player’ action.

(2)The strategy variables corresponding to the compensation are p1(q12), p2(q22) and take the value of 0≦p≦1 determined by the bargaining.

(3)The harm of player 1 caused by player 2’s action is D and there exists no harm of player 2 from player 1.

(4)The payoffs of player 1, 2 are R1R2 in the absence of harm, respectively.

First let us consider the case without legal regulations. Expanding the payoff functions of player 1 and player 2 into the first order of m1m2, these are represented as

1(q1i, q2i) = m1F1(q1j, q2j) + m2F2(q1j, q2j) + m1m2F3 (q1j, q2j) + F4 (q1j, q2j), (j≧2) (1)

2 (q1i,q2i) =m1G1(q1j, q2j) + m2G2(q1j, q2j) + m1m2G3 (q1j, q2j) + G4 (q1j, q2j). (j≧2) (2)

In the case of D = 0, the pay-off functions are

1 = F1(q1j, q2j) = R1, 2 = G2 (q1j, q2j) =R2. (3)

In the case of D > 0 and m1= m2= 1,

F2 (q1j, q2j) + F3 (q1j, q2j) = －D, G1(q1j, q2j) + G3(q1j, q2j) =0. (4)

From these conditions, the pay-off functions are represented as

1(q1i, q2i) = m1(R1+f1(q1j, q2j) + m2F2(q1j, q2j) + m1m2(－D+ f3 (q1j, q2j)) + F4 (q1j, q2j), (5)

2 (q1i, q2i) = m1G1(q1j, q2j) + m2(R2 + g2(q1j, q2j)) + m1m2G3 (q1j, q2j) + G4 (q1j, q2j). (6)

Sincethe compensationCp is generated in the case of m1= 1, m2= 0 or m1= 0, m2= 1, the values of Cp are given by

Cp1 = －f1(q1j, q2j)－F4 (q1j, q2j) = G1(q1j, q2j) + G4 (q1j, q2j), (7)

Cp2 = －g2 (q1j, q2j)－G4 (q1j, q2j) = F2(q1j, q2j) + F4 (q1j, q2j). (8)

Here Cp1, Cp2are assumed to be linear functions of p1, p2which arestrategyvariables determined by the results of bargaining. Finally, the pay-off functions are

1(q1i, q2i) = m1R1－m1(p1 + －m1－m2)F4 (q1j, q2j) + m2(p2 + )

+ m1m2(－D+ f3 (q1j, q2j)), (9)

2 (q1i, q2i) = m2R2 + m1(p1 + －m2(p2 + ) + －m1－m2)G4 (q1j, q2j)

+ m1m2G3 (q1j, q2j). (10)

In the presence of legal regulations, the payoff functions change as follow as

1L(q1i, q2i) = m1R1－m1(p1 + －m1－m2)G4 (q1j, q2j) + m2(p2 + )

+ m1m2F3 (q1j, q2j)), (11)

2L(q1i, q2i) = m2R2 + m1(p1 + －m2(p2 + ) + －m1－m2)F4 (q1j, q2j)

+ m1m2(－D+ g3 (q1j, q2j)). (12)

Lemma 2

(1)There exist Nash equilibrium points in pay-off functions 1, 2, 1L, 2L.

(2)The sum of pay-off functions (1 + 2 or 1L + 2L) is the same value independent of the presence of legal regulations.

(3)Nash equilibrium points of these pay-off functions are Pareto optimum.

Proof 2

Sincef3 (q1j, q2j) + G3 (q1j, q2j) = F3 (q1j, q2j)+ g3 (q1j, q2j), the following relation is derived.

1(q1i, q2i)2 (q1i, q2i)1L(q1i, q2i)2L(q1i, q2i) =m1R1+m2R2－m1m2D (13)

It is easily seen from (13) that the sum of pay-off functions can be only determined by the variables m1, m2. This suggests that the sum is independent of the value of p1, p. However, in order to prove Coase theorem, Pareto optimum shouldbe indicated in Nash equilibrium points. The equilibrium points q11*, q21* depend on the relative values of R1, R2 and D. Here insteadof obtaining the equilibrium points directly, assuming the existence of the equilibrium points, in the sum of pay-off functions, Pareto optimum will be confirmed in some values of system parameters. This result is corresponding to Coase theorem I.

Next consider the effect of the transaction cost which was neglected in the above formulation.Inthe absence of legal regulations, the payoff functions are presented as

1(q1i, q2i) = m1R1－m1(p1 + －m1－m2)F4 (q1j, q2j) + m2(p2 + )

+ m1m2(－D+ f3 (q1j, q2j))－m1T, (14)

2 (q1i, q2i) = m2R2 + m1(p1 + －m2(p2 + ) + －m1－m2)G4 (q1j, q2j)

+ m1m2G3 (q1j, q2j)－m2T, (15)

where Tindicates the transaction cost.In the presence of legal regulations, the payoff functions are the same as to the case without transaction cost. In this case, pay-off fuctions indicate that the transaction cost in the presence of legal regulations disappears. In this case the sum of two players’ payoff is not necessary independent on thecondition of legal regulations because of the transaction costoccurring. The equilibrium points q1*, q2*depend on the relative values of R1, R2, D and T. Even if 1L(q1i, q2i) = max 1L(q1i, q2i) and 2L(q1i, q2i) = max 2L(q1i, q2i), the maximum sum of payoff will be generally dependent on the legal regulations. As a result, the sum of payoff as the social efficiency will determine the party (player) owing the legal duty. This case corresponds to Coase theorem II.

3. Three-person game

Let G =＜q11, q12…, q21, q22…,qv1, qv2…,v＞ be a three person game in normal form, where qii’is a finite set of strategies for player i and i (q1i’, q2i, qvi’’) (i’ = 1, 2…) is a player i’s payoff function. In this game, there are two normal players and one player which is named here as virtual player. Of these strategies, the specified variables which are player’s action and providing the compensation are identified. The compensation is caused by the harm due to players’ actions. It is then assumed in this game that negotiation for the compensation is possible.

Definition 3

(1)The strategy variables of player’s actions are m1(q11), m2(q21) and L(q31), and take the value of 1(0) in the presence (absence) of player’ action. The action of virtual player is to determine the presence (absence) of the regal regulations.

(2) The strategy variables corresponding to the compensation are p1(q12), p2(q22) and take the value of 0≦p≦1 determined by the bargaining.

(3) The harm of player 1 caused by player 2’s action is D and there exists no harm of player 2 from player 1.

(4) The payoffs of player 1, 2 are R1R2 in the absence of harm, respectively.

(5) The virtual player gains Rv1Rv2 from player 1,2.

For simplicity, let us consider the case without transaction cost. Expanding the payoff functions of players into the first order of m1m2, L, these are represented as

1= m1F1(q1j, q2j, qvj) + m2F2(q1j, q2jqvj) + LF3 (q1j, q2jqvj) + m1m2F4(q1j, q2jqvj)

+ m1LF5 (q1j, q2jqvj)+ m2LF6 (q1j, q2jqvj) + F7(q1j, q2jqvj), (j≧2) (16)

2 =m1G1(q1j, q2j, qvj) + m2G2(q1j, q2jqvj) + LG3 (q1j, q2jqvj) + m1m2G4 (q1j, q2jqvj)

+ m1LG5 (q1j, q2jqvj)+ m2LG6 (q1j, q2jqvj) + G7 (q1j, q2jqvj), (j≧2) (17)

v=m1H1(q1j, q2j, qvj) + m2H2(q1j, q2jqvj) + LH3 (q1j, q2jqvj) + m1m2H4 (q1j, q2jqvj)

+ m1LH5 (q1j, q2jqvj)+ m2LH6 (q1j, q2jqvj) + H7 (q1j, q2jqvj). (j≧2) (18)

Lemma 3

(1) There exist Nash equilibrium points in pay-off functions 1, 2, v.

(2) The sum of pay-off functions (1 + 2) is the same value independent of the value ofL.

(3) Selection of the legal regulations in a game can be determined by the actions of player 1 and 2.

Proof 3

Of these pay-off functions, 1 and2can be reducing to the expression of (9)～(12) in the cases of L = 1 or 0.Then, noting that R1→R1－Rv1R2→R2－Rv2 (16)～(17) are represented as

1 = m1(R1－Rv1)－m1(p1 + －m1－m2)((1－L)F4 (q1j, q2j) + LG4 (q1j, q2j))+ m2(p2 + )

+ m1m2((1－L) (－D+ f3 (q1j, q2j)) + LF3 (q1j, q2j))), (19)

2 = m2(R2－Rv2) +m1(p1 + －m2(p2 + ) + －m1－m2)((1－L)G4 (q1j, q2j)+LF4 (q1j, q2j))

+ m1m2(L(－D+ g3 (q1j, q2j)) + (1－L)G3 (q1j, q2j))). (20)

Following the case of proof 2, the similar relation is derived independent of the value of L

1(q1i, q2i)2 (q1i, q2i) = m1(R1－Rv1)+ m2(R2－Rv2)－m1m2D (21)

Also, vcan be defined to be 0 in the case of m1 = m2 = 0. Then, (18) is represented as

v=m1Rv1+m1K1(q1j, q2j, qvj) +m2Rv2+m2K2(q1j, q2j, qvj) + m1m2H4 (q1j, q2j, qvj)

+m1LH5(q1j,q2j,qvj)+m2LH6(q1j,q2j,qvj) (22)

Here let us assume the configuration of virtual player’ pay-off function. For simplicity, the other strategies such as q1j, q2j, qvj(j≧2) are neglected. The following assumptions are considered to reflect the fairness.

Assumptions

(1)The terms related to m1 except m1Rv1in pay-off function v is an increasing function of m1.

(2)The terms related to m2 except m2Rv2 in pay-off function v is a decreasing function of m2.

Thus, the pay-off function is given by

v=m1Rv1+m2Rv2+ m1(K1+LH5)+m2(K2+LH6) + m1m2H4, (23)

where the assumptions indicate K1+LH5+m2H4 > 0, K2+LH6+m1H4< 0.As seen easily, if m1H5+ m2H6> 0 under the assumptions, the equilibrium point of virtual player is L = 1. Similarly, if m1H5 + m2H6 0, the equilibrium point of virtual player is L = 0. This concludes that the selection of the legal regulations in this game can be determined by the actions of player 1 and 2.

4. Conclusion

Coase theorem is investigated by game theoretical approach. It is considered by two sets of a two-person game which have the different rules, and is found to be proved under the suitable conditions of pay-off functions. These two-person games are transformed into one set of a three-person game. The formula considered here suggests that there are possible two-person games which have the variable rules determined by the actions of two players.

References

[1] R. H. Coase, The problem of social cost, J. Law Econ. 3 (1960) 1-44.

[2] G. J. Stigler, The theory of price (3rd ed.), New York: Macmillan. (1966).

[3] J. Lee, H. Sabourian, Coase theorem, complexity and transaction costs, J. Econ. Theory 135 (2007) 214-235.

[4] E. Guzzini, A. Palestrini, Coase theorem and exchange rights in non-cooperative games, Eur. J. Law Econ.33 (2012) 83-100.

[5] F. E. Guerra-Pujol, Modeling the Coase theorem, Eur. J. Legal Stud. 5(2012).

[6] K. Nishi, A game theoretical analysis of Coase theorem, Game Theory Workshop in Tokyo. (2013)

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