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A value has cooperative-game additivity when adding two characteristic functions on the same player set adds their allocation vectors. Here $(v+w)(S)=v(S)+w(S)$ for every coalition. The sum of two simple cooperative games need not be simple; the axiom is naturally stated on transferable utility games.

Every nontrivial simple cooperative game that is a convex cooperative game is a unanimity game. If two coalitions win, the supermodular inequality forces their intersection to win as well. Intersecting the finite nonempty family of winning coalitions gives a winning set $K\ne \varnothing$. Every winning coalition contains $K$; conversely monotonicity makes every superset of $K$ win. Thus the game is $u_K$. Conversely, every unanimity game is a supermodular set function, as a direct check of the winning and losing cases shows.

For a monotone simple cooperative game with empty value $0$ and grand-coalition value $1$, the core of a cooperative game consists exactly of the nonnegative unit-total payoff vectors supported on veto players.

For the proof, singleton core conditions give $x_i\ge 0$. Every winning coalition must receive at least $1$, the full available total, so all players outside it receive zero. A positively paid player therefore belongs to every winning coalition and is a veto player. Conversely, a unit-total nonnegative allocation supported on veto players gives every winning coalition exactly $1$ and every losing coalition at least $0$. These are precisely the core inequalities. The core is thus nonempty if and only if a veto player exists.

An induced subgraph game extended by one self-loop weight ${\ell }_i$ per player. Each loop is counted once, so $v(\{i\})={\ell }_i$ andSingleton and pair coalition values determine all weights; larger coalition values must then agree with their sums. Signed weights are allowed. For example, two loops of weight $1$ joined by an edge of weight $-1$ represent the simple cooperative game in which any coalition meeting those two players wins.

For three distinct players $i,j,k$ in a normalized simple cooperative game, enumerate the predecessor coalitions in the Shapley value formula:Subtracting the formula with $i$ and $j$ exchanged proves the displayed identity. Every coalition value is zero or one, so $2({\varphi }_i-{\varphi }_j)$ is an integer. This rules out, for example, coordinates differing by $1/6$, although each may separately satisfy the Shapley value denominator in a simple game. The difference identity itself holds for arbitrary three-player transferable utility games; the integer restriction uses the zero-one values.

A simple cooperative game is a transferable utility game $(N,v)$ withA coalition of value $1$ is a winning coalition; a value-$0$ coalition loses. The displayed definition uses the usual monotone, nontrivial convention: the empty coalition loses and the grand coalition wins.

Yes; without self-loops, such a nontrivial simple cooperative game has exactly one positive edge, of weight $1$. For each pair,because singleton and loop contributions are zero. Summing all edge weights gives $v(N)=1$, so precisely one unordered pair $\{a,b\}$ has weight $1$, and all other pairs have weight zero.

Thus the winning coalitions are exactly those containing both $a$ and $b$. For any $0\le \lambda \le 1$, setThis payoff vector is efficient, since ${\sum }_i{x}_i=v(N)=1$. Every winning coalition receives $1$ and every losing coalition receives a nonnegative total, so all inequalities $x(S)\ge v(S)$ hold. Hence it belongs to the core of a cooperative game.

In fact these are all core allocations. Singleton conditions require $x_i\ge 0$, and the winning pair requires $x_a+x_b\ge 1$; efficiency then forces equality and zero payoffs outside the pair. The endpoints are both veto players, consistent with the general characterization of the core of a simple cooperative game.

No. Two positive self-loops and one negative edge give a simple cooperative game with an empty core of a cooperative game. Give players $1,2$ loop weights ${\ell }_1={\ell }_2=1$, put $w_{12}=-1$, and give every other loop and edge weight zero. For a coalition $S$,Thus a coalition wins exactly when it contains player $1$ or player $2$. This is monotone, has values in $\{0,1\}$, and satisfies $v(\varnothing )=0,v(N)=1$. Extra players for any $n\ge 3$ do not alter any coalition value.

The core singleton inequalities would require $x_1\ge 1$, $x_2\ge 1$ and $x_i\ge 0$ for every other player, while efficiency requires ${\sum }_i{x}_i=1$. These requirements are inconsistent. Equivalently, the two disjoint winning singleton coalitions leave no veto player. Signed weights are essential to this particular representation and are permitted in the induced subgraph game model.

There is exactly one such nontrivial game for every $n\ge 1$: the unanimity game requiring every player. PutIt is a simple cooperative game, and permuting players preserves every coalition value. Thus it has the required symmetry of players in a cooperative game. It is also a convex cooperative game: if both $S,T$ are winning, all four terms in the supermodular inequality have value $1$; if exactly one is winning, their union is $N$ and their intersection loses, giving equality $1=1$; if neither wins, the right-hand side is zero and the inequality follows from nonnegativity. For $n=1$ the same definition gives the sole player value $1$.

To prove uniqueness, first establish that convex simple games are unanimity games. If $S,T$ are winning coalitions, the supermodular inequality givesso their intersection is also winning. The family of winning coalitions is finite and nonempty. Its intersection $K$ is therefore winning and nonempty, since the empty coalition loses. Every winning coalition contains $K$, and monotonicity makes every superset of $K$ winning. Hence $v=u_K$.

If $K$ were a proper nonempty subset of $N$, take $i\in K$ and $j\notin K$, and let $T=N\setminus \{i,j\}$. Then $T\cup \{i\}=N\setminus \{j\}$ contains $K$ and wins, whereas $T\cup \{j\}=N\setminus \{i\}$ loses. This violates symmetry of players in a cooperative game, which requires those two coalition values to agree. Thus $K=N$, proving the claimed count. The “nontrivial” convention excludes the identically zero game; if that degenerate game were also called simple, it would supply one additional symmetric convex example.

For a simple cooperative game, the Shapley value is the probability that a player is pivotal in a uniformly random ordering. Symmetry gives one value for permanent members and another for nonpermanent members.

A particular nonpermanent member is pivotal exactly when all five permanent members and exactly three of the other nine nonpermanent members precede them. The predecessor set then has size eight. There are $\left(\genfrac{}{}{0ex}{}{9}{3}\right)$ such sets, each giving $8!6!$ orderings. Their Shapley value is thereforeEvery ordering has exactly one pivotal member, since the empty coalition loses and the full coalition wins. This proves efficiency directly: $5\alpha +10\beta =1$, where $\alpha$ is the value of each permanent member. HenceThe vector has five entries $421/2145$ and ten entries $4/2145$. As a check, a permanent member is pivotal when they are last among the permanent members and occupy a position from nine to fifteen; counting those orderings gives the same $\alpha$.

Yes. Give players $1,2,3$ weights $1,1,2$, respectively, and let the quota be three. This weighted voting game is a simple cooperative game whose minimal winning coalitions are $\{1,3\}$ and $\{2,3\}$. In orders $123,132,213,231$, player 3 supplies the winning marginal contribution. In order $312$ player 1 does, and in order $321$ player 2 does. Averaging the six marginal contribution vectors gives exactly $(1/6,1/6,4/6)$ as the vector of Shapley values.

No. In a simple cooperative game, a player's marginal contribution along any permutation is zero or one. With three players there are six equally likely permutations, so every Shapley value is an integer multiple of $1/6$, by the Shapley value denominator in a simple game. But $6(5/12)=5/2$ and $6(3/12)=3/2$ are not integers. The vector sums to one, but fails this necessary denominator condition.

Yes. Use the simple cooperative game in which a coalition wins exactly when it contains at least two players. In every permutation, precisely the second arrival changes a losing coalition into a winning one. Each player is second in two of the six permutations, so each Shapley value is $2/6=1/3$. This constructs the required vector while verifying that singleton coalitions lose and the grand coalition wins.

No. The denominator test from part ii is satisfied here, so a stronger obstruction is needed. Put $a_i=v(\{i\})$ and $b_{ij}=v(\{i,j\})$. In a three-player simple cooperative game, the Shapley value formula gives, for distinct $i,j,k$,Subtract the corresponding expression for ${\varphi }_j$:Each coalition value is zero or one, so this difference must be an integer multiple of $1/2$. This is the obstruction from pairwise Shapley differences in a three-player simple game. The proposed first two coordinates differ by $1/6$, which is not an integer multiple of $1/2$. Therefore the vector cannot occur, even though each coordinate separately passes the $1/6$ denominator test.

Let $W(C)={\sum }_{i\in C}{w}_i$. If the two disjoint winning coalitions exchange players, the aggregate weight is conserved:If both resulting coalitions lost, each weight would be strictly below $q$, making their sum strictly below $2q$. This is impossible. Thus every weighted voting game has swap robustness for disjoint winning coalitions, even when the weights or quota are negative.

The converse is false. The four-player prime-cardinality game from part (a) has swap robustness for disjoint winning coalitions: swapping one player each way preserves each coalition's cardinality, so in fact both winning coalitions remain winning. Part (a) proves it is not a weighted voting game.

Even imposing monotonicity would not repair the converse under the printed disjointness restriction. As a stronger counterexample, use five players and declare a coalition winning exactly when it contains player $1$ and contains either both $2,3$ or both $4,5$. This is a simple cooperative game, and all its winning coalitions contain player $1$, so no two are disjoint; the stated swap condition holds vacuously.

Nevertheless $\{1,2,3\}$ and $\{1,4,5\}$ win, while $\{1,2,4\}$ and $\{1,3,5\}$ lose. Both pairs contain every player the same total number of times. A weighted representation would therefore givea contradiction. This is a trade obstruction to weighted voting and pinpoints how weak the disjoint-only swap requirement is.

The probabilistic Banzhaf index averages a player's marginal contribution over all coalitions not containing that player, each given equal weight. In a simple cooperative game, it is the probability of changing a losing coalition into a winning one when every other player joins independently with probability $1/2$. Its sum over players need not equal the grand-coalition value: in a three-player majority game each player has index $1/2$, giving total $3/2$ despite $v(N)=1$. This convention differs from both the raw count of pivotal coalitions and the normalized Banzhaf index.

The name refers to John Banzhaf, whose 1965 paper addresses the mathematical analysis of weighted voting.^[ref]

The Shapley value averages each player's marginal contribution over uniformly random player orderings. Exactly $|S|!(n-|S|-1)!$ orderings have $S$ immediately before player $i$, giving the formula. The values sum to $v(N)$ by telescoping each ordering. In a simple cooperative game this is the probability of being pivotal. For a convex cooperative game, Shapley value belongs to the core of a convex game guarantees a stable allocation as well.

In an $n$-player simple cooperative game, every marginal contribution along a player permutation is zero or one. The Shapley value is its average over the $n!$ equally likely permutations. Thus $n!{\varphi }_i(v)$ counts the orders in which player $i$ changes a losing coalition into a winning one and is an integer. In lowest terms, the coordinate's denominator divides $n!$. This is a necessary constraint on a proposed Shapley value vector, not a sufficient one.

A finite transferable utility game gives each coalition $S\subseteq N$ a real value $v(S)$ that its members can distribute among themselves. The usual normalization is $v(\varnothing )=0$. An efficient payoff vector $x$ satisfies ${\sum }_{i\in N}{x}_i=v(N)$. Simple cooperative games model winning coalitions with values zero and one; convex cooperative games model increasing marginal contributions.

For a fixed nonempty coalition $K$, the simple cooperative game in which a coalition wins exactly when it contains $K$. Members of $K$ are exactly the veto players. It is a convex cooperative game: intersections of winning coalitions still contain $K$, and checking the remaining zero-one cases establishes the supermodular inequality. Full symmetry of players in a cooperative game forces $K=N$.