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I am a bachelor of education and recently, a game-theoretic solution concept piqued my research interest, at least for the purpose of research-writing as a personal hobby. In cooperative game theory, we have the so-called Shapley Value (and in the context of education, it is seen to be useful to "solve" the Freeloader Problem and the Fair Grade Allocation Problem when teachers assign marks or grades to students working together as a group).

Now, I am attempting to put forth a corollary of the Shapley axioms, and I am calling it the "Positive Correlation Axiom". This claim, I think, is intuitive and somehow "obvious". I am stuck on how I can establish such claim using a formal proof. So,

Suppose I am trying to show the existence of a positive correlation between $\textit{A}$ and $\textit{B}$, how can I prove such theorem?

Any help, be it a proof outline or a complete proof, is much appreciated.

R.L.D. Daguro
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  • I think "direct relationship" is informal and potentially ambiguous, because are affine relationships (e.g., $y=3x+7$) considered direct, or only direct proportionalities? Here's a recent tangentially-related reply (scroll to the last section) that I wrote. – ryang Dec 28 '21 at 07:51
  • @ryang Thank you for the comment. I think I should use the term "positive correlation" instead of "direct relationship" so as to avoid ambiguity. – R.L.D. Daguro Dec 28 '21 at 08:31
  • @R.L.D Daguro It is not clear at all to me what you are looking for. Can you provide a formal statement of what you mean by "positive correlation" between A and B. Are A and B random variables and you want to show correlation? Or is it that, "there exists a function $f$ such that $f(A)=B$? What is the domain of A and B? Can you give an example? – GameTheorist1982 Dec 28 '21 at 12:25
  • @GameTheorist1982 thank you for the feedback, what I mean by A is a player's marginal contribution and B is the same player's Shapley value. – R.L.D. Daguro Dec 29 '21 at 01:17

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