to the model created of value $v(N)$.

However, in a modified game, **Definition 2** places bounds on the payoff to individual players instead on the sum of payoffs. We can scale the vector $\phi(N)$ by a positive constant $\alpha$ such that the resulting payoff vector $<x_1,\dots,x_n> = \alpha\phi(N)$ preserves the ratio of contribution between each player. Here, we introduce the concept of proportionality to categorise any payoff vector satisfying the following property:

A payoff vector $<x_1,x_2,\dots,x_n>$ is **proportional** for a given contribution measure vector $\mathcal{C}\in\mathcal{R}^n$ if and only if there exists a positive constant $\alpha$ such that $<x_1,x_2,\dots,x_n> = \alpha  \mathcal{C}$.

In the rest of the paper, we will assume the *shapley value* as our contribution measure because a payoff vector proportional to the *shapley value* preserves its desirable properties covered in Section 3.1.3.

Desirable properties of outcomes in modified cooperative game
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Recall that an outcome of a cooperative game is defined by the resulting coalition structure formed and payoff vector defined over the coalitions in the coalition structure (Section 3.1.1). Some outcomes satisfy certain desirable properties in data sharing. Here, we summarise these properties.

Given a modified game with a group of $n$ players, a characteristic function $v(.)$ 