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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.