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In allocation problems, a given set of goods are assigned to agents in such away that the social welfare is maximised, that is, the largest possible globalworth is achieved. When goods are indivisible, it is possible to use moneycompensation to perform a fair allocation taking into account the actualcontribution of all agents to the social welfare. Coalitional games provide aformal mathematical framework to model such problems, in particular the Shapleyvalue is a solution concept widely used for assigning worths to agents in afair way. Unfortunately, computing this value is a $\#{\rm P}$-hard problem, sothat applying this good theoretical notion is often quite difficult inreal-world problems.  We describe useful properties that allow us to greatly simplify the instancesof allocation problems, without affecting the Shapley value of any player.Moreover, we propose algorithms for computing lower bounds and upper bounds ofthe Shapley value, which in some cases provide the exact result and that can becombined with approximation algorithms.  The proposed techniques have been implemented and tested on a real-worldapplication of allocation problems, namely, the Italian research assessmentprogram, known as VQR. For the large university considered in the experiments,the problem involves thousands of agents and goods (here, researchers and theirresearch products). The algorithms described in the paper are able to computethe Shapley value for most of those agents, and to get a good approximation ofthe Shapley value for all of them.

Computing the Shapley Value in Allocation Problems: Approximations and Bounds, with an Application to the Italian VQR Research Assessment Program