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A coverage coalitional game has a nonempty core of a cooperative game exactly when all resource sets are pairwise disjoint. Singleton coalition constraints require $x_i\ge |B_i|$, while efficiency requires ${\sum }_i{x}_i=|{\bigcup }_i{B}_i|$. Any overlap makes these inequalities inconsistent. When there is no overlap, the allocation $x_i=|B_i|$ is the unique core allocation and gives every coalition its value exactly.

Let $v:2^N\to \mathbb{R}$, $v(\varnothing )=0$, be a coalitional game with $n=|N|$. The Shapley value isEquivalently, choose a uniformly random player ordering and average each player's marginal contribution when that player joins its predecessors. The coefficient counts precisely the orderings with predecessor set $S$.

The standard axioms are efficiency, ${\sum }_i{\varphi }_i=v(N)$; symmetry, giving equal values to players with identical marginal contributions; the dummy-player property, giving ${\varphi }_i=v(\{i\})$ if every marginal contribution of $i$ equals $v(\{i\})$; and additivity, ${\varphi }_i(v+w)={\varphi }_i(v)+{\varphi }_i(w)$. The null-player form of the dummy axiom gives zero to a player contributing nothing. These properties characterize the Shapley value. They also follow directly from its formula: efficiency telescopes along each ordering, symmetry relabels orderings, constant dummy contributions average to themselves, and the sum is linear in $v$.

For balanced contributions of Shapley values, remove player $j$ from a uniformly random ordering. The resulting ordering of $N\setminus \{j\}$ is uniform, so it computes $i$'s Shapley value in the restricted game. If $j$ comes after $i$, removing it does not change $i$'s marginal contribution. If $j$ precedes $i$, and $S$ consists of $i$'s other predecessors, the difference isThere are $(|S|+1)!(n-|S|-2)!$ full orderings with exactly $S\cup \{j\}$ before $i$. ConsequentlyBoth the summand and coefficient are unchanged when $i,j$ are interchanged. ThusThe restriction keeps all coalition values for the remaining players, exactly matching the payoff comparison after a departure. The identity does not require the differences to be positive.

Now consider the coverage coalitional game formed by the finite book sets. For disjoint coalitions $S,T$, put $B(S)={\bigcup }_{i\in S}{B}_i$. ThenThus coverage is generally subadditive. It is a superadditive coalitional game precisely when the book sets are pairwise disjoint: disjointness makes the displayed formula additive, while a common book of players $i,j$ gives $v(\{i,j\})<v(\{i\})+v(\{j\})$. For example, $B_1=B_2=\{b\}$ gives values $1,1,1$ and disproves unconditional superadditivity. The printed clause must therefore have its disjointness condition apply to superadditivity as well as to core nonemptiness.

For the core of a coverage game, an allocation must satisfy efficiency ${\sum }_i{x}_i=v(N)$ and ${\sum }_{i\in S}{x}_i\ge v(S)$ for every coalition. In particular $x_i\ge |B_i|$. Summing givesThe reverse inequality always holds, with equality exactly when no book appears in two sets. Therefore overlap makes the core empty. If the sets are disjoint, $x_i=|B_i|$ gives every coalition exactly its value and is a core allocation; the singleton lower bounds and efficiency make it the only one. Hence superadditivity and a nonempty core each occur exactly in the pairwise-disjoint case.

Finally write the game as a sum over individual books. For a book $b$, let $K_b=\{k:b\in B_k\}$. Its contribution to coalition value is one if the coalition meets $K_b$, and zero otherwise. In a random ordering, player $i$ contributes this book precisely when $i\in K_b$ and is earliest among its $|K_b|$ knowers. Each knower is equally likely to be earliest. Additivity of the Shapley value therefore gives the Shapley value of a coverage gameThis formula remains valid when books overlap and the core is empty.

Decompose a coverage coalitional game into one-resource games and use additivity of the Shapley value. A player contributes a resource precisely when it is the earliest player in a random ordering among those holding that resource. Each holder has probability the reciprocal of the number of holders of being earliest. Summing these contributions proves the formula, regardless of overlaps or emptiness of the core.