On compensation schemes for data sharing within the European REACH legislation

Abstract

Article 30 of Regulation (EC) No 1907/2006 concerns the sharing of data between users of a chemical substance. We study this bargaining problem by means of a special class of games in coalitional form called data games (Dehez and Tellone in J Public Econ Theory 15:654–673, 2013). For such problems, compensation schemes specify how the data owners should be compensated by the agents in needs of data. On the class of data games, the Core, the Nucleolus and the Shapley value provide relevant compensation schemes. We provide four comparable axiomatic characterizations of the set of all (additive) compensation schemes belonging to the Core, of the Nucleolus, of the Shapley value and of the Full compensation mechanism, a compensation scheme exclusively designed for data sharing problems. The axioms reflects principles of various theories of justice.

Appendix

In order to demonstrate the logical independence of the axioms imposed in Propositions 2 to 7, we exhibit the following compensation schemes.

Proposition 2

• The Shapley value $\text {Sh}$ satisfies Pooling, Compensation, Reasonableness but not Exclusive data promotion.

• The compensation scheme f on $\cal{C}$ defined by $f=100\times \eta$ satisfies Pooling, Compensation and Exclusive data promotion but not Reasonableness.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\eta _i(N,C_{(D,O)})+\sum _{k\in D^E(O)}\frac{c_k}{n},\quad i\in N, \end{aligned}$$

  satisfies Pooling, Reasonableness and Exclusive data promotion but not Compensation.

Proposition 3 and 4

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})={\bf 0}_n$ satisfies Compensation, Pooling, Exclusive data promotion and Monotonicity with respect to the cost of exclusive data but not Equal concessions.

• The Shapley value $\text {Sh}$ satisfies Compensation, Pooling and Equal concessions but neither Exclusive data promotion nor Monotonicity with respect to the cost of exclusive data.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f(N,C_{(D,O)})=(|D\backslash D^E(O)|+1)\times \eta (N,C_{(D,O)}) \end{aligned}$$

  satisfies Compensation, Equal concessions, Exclusive data promotion and Monotonicity with respect to the cost of exclusive data but not Pooling.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\eta _i(N,C_{(D,O)})+|D^E(O)|,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Equal concessions, Exclusive data promotion and Monotonicity with respect to the cost of exclusive data but not Compensation.

Proposition 5

• The Nucleolus $\eta$ satisfies Compensation, Pooling, Equal treatment of equals and Equal concessions but not Invariance to enlarging the owner set.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})={\bf 0}_n$ satisfies Compensation, Pooling, Equal treatment of equals and Invariance to enlarging the owner set but not Equal concessions.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\text {Sh}_i(N,C_{(D,O)})+|D|,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Equal treatment of equals, Equal concessions, Invariance to enlarging the owner set but not Compensation.

• For any $N\in U,$ consider any compensation vector $a\in {\mathbb {R}}^n\backslash \{{\bf 0}_n\}$ such that $\sum _{i\in N}a_i=0.$ The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\sum _{k\in D:o_k<n}\frac{c_k}{n}-\sum _{k\in D_i(O):o_k<n}\frac{c_k}{o_k}+a_i\times |\{k\in D:o_k=n\}|, \quad i\in N, \end{aligned}$$

  satisfies Pooling, Compensation, Equal concessions and Invariance to enlarging the owner set but not Equal treatment of equals.

• Consider the data game $(N^{\prime},C_{(\{h,k\},O^{\prime})})\in \cal{C}$ such that $n^{\prime}\ge 3,$ $O^{\prime}(h)=\{1,3\}$ and $O^{\prime}(k)=\{2,3\}.$ The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})=\text {Sh}(N,C_{(D,O)})$ if $(N,C_{(D,O)})\ne (N^{\prime},C_{(\{h,k\},O^{\prime})})$ and such that $f_1(N^{\prime},C_{(\{h,k\},O^{\prime})})=\text {Sh}_1(N^{\prime},C_{(\{h,k\},O^{\prime})})+c_h/2,$ $f_2(N^{\prime},C_{(\{h,k\},O^{\prime})})=\text {Sh}_2(N^{\prime},C_{(\{h,k\},O^{\prime})})+c_k/2,$ $f_3(N^{\prime},C_{(\{h,k\},O^{\prime})})=\text {Sh}_3(N^{\prime},C_{(\{h,k\},O^{\prime})})-c_h/2-c_k/2$ and $f_i(N^{\prime},C_{(\{h,k\},O^{\prime})})=\text {Sh}_i(N^{\prime}C_{(\{h,k\},O^{\prime})})$ for each $i\in N^{\prime}\backslash \{1,2,3\}$ satisfies Compensation, Equal treatment of equals, Equal concessions and Invariance to enlarging the owner set but not Pooling.

Proposition 6

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\sum _{k\in D\backslash D_i(O)}\frac{(o_k)^2c_k}{n-o_k}-\sum _{k\in D_i(O):O(k)\ne N}o_kc_k,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Compensation, Equal treatment of equals, Strong bargaining position but not Invariance to entry of new owners.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})={\bf 0}_n$ satisfies Pooling, Compensation, Equal treatment of equals, Invariance to entry of new owners, but not Strong bargaining position.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\text {FCM}_i(N,C_{(D,O)})+|D\backslash D^E|,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Equal treatment of equals, Strong bargaining position, Invariance to entry of new owners but not Compensation.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})=\text {FCM}(N,C_{(D,O)})$ if $O(k)=O(h)$ for all $k,h\in D$ and $f(N,C_{(D,O)})=\text {Sh}(N,C_{(D,O)})$ otherwise satisfies Compensation, Equal treatment of equals, Strong bargaining position, Invariance to entry of new owners, but not Pooling.

• For all $(N,C_{(\cal{D},O)})$ and all $k\in \cal{D}$ with $O(k)\ne N,$ define numbers $a_i^k,$ $i\in N\backslash O(k),$ that are not all null and such that $\sum _{j\in N\backslash O(k)}a_j^k=0.$ Then, the compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\text {FCM}_i(N,C_{(D,O)})+\sum _{k\in D\backslash D_i(O)}a_i^k,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Compensation, Strong bargaining position, Invariance to entry of new owners but not Equal treatment of equals.

Proposition 7

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\sum _{k\in D\backslash D_i(O)}\frac{o_kc_k}{n-1}-\sum _{k\in D_i(O):O(k)\ne N}\frac{(n-o_k)c_k}{n-1},\quad i\in N, \end{aligned}$$

  satisfies Pooling, Compensation, Equal treatment of equals, Weak bargaining position but not Invariance to entry of new acquirers.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})={\bf 0}_n$ satisfies Pooling, Compensation, Equal treatment of equals, Invariance to entry of new acquirers, but not Weak bargaining position.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\text {FCM}_i(N,C_{(D,O)})+|D^E|,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Equal treatment of equals, Weak bargaining position, Invariance to entry of new acquirers but not Compensation.

• The compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by $f(N,C_{(D,O)})=\text {FCM}(N,C_{(D,O)})$ if $O(k)=O(h)$ for all $k,h\in D$ and $f(N,C_{(D,O)})=\text {Sh}(N,C_{(D,O)})$ otherwise satisfies Compensation, Equal treatment of equals, Weak bargaining position, Invariance to entry of new acquirers, but not Pooling.

• For all $(N,C_{(\cal{D},O)})$ and all $k\in \cal{D}$ with $O(k)\ne N,$ define numbers $a_i^k,$ $i\in O(k),$ that are not all null and such that $\sum _{j\in O(k)}a_j^k=0.$ Then, the compensation scheme f on $\cal{C}$ defined, for all $(N,C_{(D,O)})\in \cal{C},$ by

  $$\begin{aligned} f_i(N,C_{(D,O)})=\text {FCM}_i(N,C_{(D,O)})+\sum _{k\in D_i(O)}a_i^k,\quad i\in N, \end{aligned}$$

  satisfies Pooling, Compensation, Weak bargaining position, Invariance to entry of new acquirers but not Equal treatment of equals.