I'm considering a cooperative game where players want to share resources in a fair manner.
You need to apply Nash solution (axiomatically founded), it is for cooperative games. Equilibrium is an indvidualistic concept applied in strategic games.
For a "fair" allocation, probably neither is applicable. Certainly Nash equilibrium, a noncooperative concept, is not. But for Nash bargaining theory, the outcome depends on threats or outside options in ways that would be difficult to characterize as fair. Herve Moulin's extensive work may be useful to you. He regards the Shapley Value as having fairness properties. I do not, and would suggest that you investigate cake-cutting or non-envious allocations, but Moulin's book may be helpful on those as well. See Fair Division and Collective Welfare by Hervé Moulin
OK, if you need find something appropriate for cooperative games then I would reccomend you see books by Moulin. There are several types of solutions, main: Core, Nuclears, Shaply's Value and depending on context of problem you are studying you have to chooze a suitable one. As soon as you want to find a "fair" allocation I can suppose Shaply's Value is the best or even Nash Bargaining Solution. NBS well correspond to the task of a pie division where coalitions profits are not revealed. Good luck.
Following up on Valeriy's comments -- His list of the "main" solutions is the same one I would give. I personally prefer the nucleolus as it can be used to determine a point within the core when the core has many values. But it is not clear that any provide a condition of "fairness," since all depend, in various ways, on outside options or strategic positioning (Shapley value). Instead of game theory, I suggest that that Maha Jebalia explore the mathematical-economic ideas of Duncan Foley:
Foley, Duncan K. (1967), “Resource Allocation in the Public Sector,” Yale Economic Essays v. 7, (Spring) pp. 73-76.
Pazner, E. A. and D. Schmeidler (1978), “Egalitarian Equivalent Allocations: A New Concept of Economic Equity,” Quarterly Journal of Economics v. 92, (Nov.) pp. 671-687.
Varian, H. R. (1974), “Equity, Envy and Efficiency,” Journal of Economic Theory v. 9, pp. 63-91.
Varian, H. R. (1975), “Distributive Justice, Welfare Economics, and the Theory of Fairness,” Philosophy and Public Affairs v. 4, pp. 223-247.
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