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With nonnegative claims $c_i$ and an estate $0\le a\le {\sum }_i{c}_i$, a coalition's guaranteed worth is the estate left after all outside claims are met, or zero if those claims exhaust it. Writing the total deficit $d={\sum }_i{c}_i-a$ gives $v(S)=({\sum }_{i\in S}{c}_i-d{)}_{+}$. For disjoint coalitions, $(p+q-d{)}_{+}\ge (p-d{)}_{+}+(q-d{)}_{+}$, so this is a superadditive coalitional game. Its core of a cooperative game consists exactly of efficient allocations with $0\le x_i\le c_i$: the upper bounds are equivalent to the complementary-coalition constraints, and the lower bounds together with efficiency imply every other coalition constraint.

Players hold finite sets of resources, and their coalition value is the number of distinct resources they collectively cover. The value is generally subadditive rather than superadditive: overlapping resources are counted only once. It is a superadditive coalitional game exactly when the players' resource sets are pairwise disjoint.

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.

In a transferable utility game, the characteristic function of a coalitional game assigns each coalition $S$ the total payoff $v(S)$ that its members can attain or guarantee together, normalized by $v(\varnothing )=0$. If disjoint coalitions $S,T$ can combine their separate strategies without interference, their union can secure both payoffs, givingThis is a superadditive coalitional game. The pooling assumption explains the intended assertion, but superadditivity is not automatic for an abstract characteristic function: the two-player values $v(\{1\})=v(\{2\})=v(\{1,2\})=1$, with empty value zero, are a counterexample. Thus the word “always” requires either that pooling assumption or a definition restricting the class of games to superadditive ones.

For the given bankruptcy game, the total deficit is $19-15=4$, so the coalition value is $({\sum }_{i\in S}{c}_i-4{)}_{+}$. Its computed values areIt is indeed superadditive: for disjoint claim sums $p,q\ge 0$, $(p+q-4{)}_{+}\ge (p-4{)}_{+}+(q-4{)}_{+}$. If both right-hand terms are positive the difference is $4$; if only one is positive, adding the other claim sum cannot reduce the value; if neither is positive the right side is zero.