cooperative game theory

A cooperative game is given by specifying a value for every coalition. Formally, the coalitional game consists of a finite set of players $N$, called the grand coalition, and a characteristic function $v\; :\; 2^N\; \backslash to\; \backslash mathbb\{R\}$ $2^N$ denotes the power set of $N$. from the set of all possible coalitions of players to a set of payments that satisfies $v(\; \backslash emptyset\; )\; =\; 0$. The function describes how much collective payoff a set of players can gain by forming a coalition.

Cooperative game theory is a branch of game theory that deals with the study of games where players can form coalitions, cooperate with one another, and make binding agreements. The theory offers mathematical methods for analysing scenarios in which two or more players are required to make choices that will affect other players wellbeing.{{Cite book |last=Javier Muros |first=Francisco |title=Cooperative Game Theory Tools in Coalitional Control Networks |publisher=Springer Cham |year=2019 |isbn=978-3-030-10488-7 |edition=1 |pages=9–11 |language=English}}

Common interests: In cooperative games, players share a common interest in achieving a specific goal or outcome. The players must identify and agree on a common interest to establish the foundation and reasoning for cooperation. Once the players have a clear understanding of their shared interest, they can work together to achieve it.{{cn|date=April 2024}}

Necessary information exchange: Cooperation requires communication and information exchange among the players. Players must share information about their preferences, resources, and constraints to identify opportunities for mutual gain. By sharing information, players can better understand each other's goals and work towards achieving them together.{{cn|date=April 2024}}

Voluntariness, equality, and mutual benefit: In cooperative games, players voluntarily come together to form coalitions and make agreements. The players must be equal partners in the coalition, and any agreements must be mutually beneficial. Cooperation is only sustainable if all parties feel they are receiving a fair share of the benefits.{{cn|date=April 2024}}

Compulsory contract: In cooperative games, agreements between players are binding and mandatory. Once the players have agreed to a particular course of action, they have an obligation to follow through. The players must trust each other to keep their commitments, and there must be mechanisms in place to enforce the agreements. By making agreements binding and mandatory, players can ensure that they will achieve their shared goal.{{cn|date=April 2024}}

Let $S\; \backslash subsetneq\; N$ be a non-empty coalition of players. The subgame $v\_S\; :\; 2^S\; \backslash to\; \backslash mathbb\{R\}$ on $S$ is naturally defined as

: $v\_S(T)\; =\; v(T),\; \backslash forall~\; T\; \backslash subseteq\; S.$

In other words, we simply restrict our attention to coalitions contained in $S$. Subgames are useful because they allow us to apply solution concepts defined for the grand coalition on smaller coalitions.

Characteristic functions are often assumed to be superadditive {{harv|Owen|1995|p=213}}. This means that the value of a union of disjoint coalitions is no less than the sum of the coalitions' separate values:

$v\; (\; S\; \backslash cup\; T\; )\; \backslash geq\; v\; (S)\; +\; v\; (T)$ whenever $S,\; T\; \backslash subseteq\; N$ satisfy $S\; \backslash cap\; T\; =\; \backslash emptyset$.

Larger coalitions gain more:

$S\; \backslash subseteq\; T\; \backslash Rightarrow\; v\; (S)\; \backslash le\; v\; (T)$.

This follows from superadditivity. i.e. if payoffs are normalized so singleton coalitions have zero value.

A coalitional game {{mvar|v}} is considered simple if payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".{{cite book|author1=Georgios Chalkiadakis|author2=Edith Elkind|author3=Michael J. Wooldridge|title=Computational Aspects of Cooperative Game Theory|url=https://books.google.com/books?id=bN9aC0uabBAC|date=25 October 2011|publisher=Morgan & Claypool Publishers|isbn=978-1-60845-652-9}}

Equivalently, a simple game can be defined as a collection {{mvar|W}} of coalitions, where the members of {{mvar|W}} are called winning coalitions, and the others losing coalitions.

It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set. However, in other areas of mathematics, simple games are also called hypergraphs or Boolean functions (logic functions).

• A simple game {{mvar|W}} is monotonic if any coalition containing a winning coalition is also winning, that is, if $S\; \backslash in\; W$ and $S\backslash subseteq\; T$ imply $T\; \backslash in\; W$.
• A simple game {{mvar|W}} is proper if the complement (opposition) of any winning coalition is losing, that is, if $S\; \backslash in\; W$ implies $N\backslash setminus\; S\; \backslash notin\; W$.
• A simple game {{mvar|W}} is strong if the complement of any losing coalition is winning, that is, if $S\; \backslash notin\; W$ implies $N\backslash setminus\; S\; \backslash in\; W$.
• If a simple game {{mvar|W}} is proper and strong, then a coalition is winning if and only if its complement is losing, that is, $S\; \backslash in\; W$ iff $N\backslash setminus\; S\; \backslash notin\; W$. (If {{mvar|v}} is a coalitional simple game that is proper and strong, $v(S)\; =\; 1\; -\; v(N\; \backslash setminus\; S)$ for any {{mvar|S}}.)
• A veto player (vetoer) in a simple game is a player that belongs to all winning coalitions. Supposing there is a veto player, any coalition not containing a veto player is losing. A simple game {{mvar|W}} is weak (collegial) if it has a veto player, that is, if the intersection $\backslash bigcap\; W\; :=\; \backslash bigcap\_\{S\backslash in\; W\}\; S$ of all winning coalitions is nonempty.
• A dictator in a simple game is a veto player such that any coalition containing this player is winning. The dictator does not belong to any losing coalition. (Dictator games in experimental economics are unrelated to this.)
• A carrier of a simple game {{mvar|W}} is a set $T\; \backslash subseteq\; N$ such that for any coalition {{mvar|S}}, we have $S\; \backslash in\; W$ iff $S\backslash cap\; T\; \backslash in\; W$. When a simple game has a carrier, any player not belonging to it is ignored. A simple game is sometimes called finite if it has a finite carrier (even if {{mvar|N}} is infinite).
• The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection. According to Nakamura's theorem, the number measures the degree of rationality; it is an indicator of the extent to which an aggregation rule can yield well-defined choices.

A few relations among the above axioms have widely been recognized, such as the following

(e.g., Peleg, 2002, Section 2.1{{Cite book | last1 = Peleg | first1 = B. | chapter = Chapter 8 Game-theoretic analysis of voting in committees | doi = 10.1016/S1574-0110(02)80012-1 | title = Handbook of Social Choice and Welfare Volume 1 | volume = 1 | pages = 395–423 | year = 2002 | isbn = 9780444829146 }}):

• If a simple game is weak, it is proper.
• A simple game is dictatorial if and only if it is strong and weak.

More generally, a complete investigation of the relation among the four conventional axioms

(monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic computabilitySee

a section for Rice's theorem

for the definition of a computable simple game. In particular, all finite games are computable.

has been made (Kumabe and Mihara, 2011{{Cite journal | last1 = Kumabe | first1 = M. | last2 = Mihara | first2 = H. R. | doi = 10.1016/j.jmateco.2010.12.003 | title = Computability of simple games: A complete investigation of the sixty-four possibilities | journal = Journal of Mathematical Economics | volume = 47 | issue = 2 | pages = 150–158 | year = 2011 | url = http://mpra.ub.uni-muenchen.de/29000/1/MPRA_paper_29000.pdf| arxiv = 1102.4037 | bibcode = 2011arXiv1102.4037K | s2cid = 775278 }}),

whose results are summarized in the Table "Existence of Simple Games" below.

The restrictions that various axioms for simple games impose on their Nakamura number were also studied extensively.{{Cite journal | last1 = Kumabe | first1 = M. | last2 = Mihara | first2 = H. R. | title = The Nakamura numbers for computable simple games| journal = Social Choice and Welfare | volume = 31 | issue = 4 | page = 621 | year = 2008 | doi = 10.1007/s00355-008-0300-5 | url=http://econpapers.repec.org/paper/pramprapa/3684.htm| arxiv = 1107.0439 | s2cid = 8106333 }}

In particular, a computable simple game without a veto player has a Nakamura number greater than 3 only if it is a proper and non-strong game.

Let G be a strategic (non-cooperative) game. Then, assuming that coalitions have the ability to enforce coordinated behaviour, there are several cooperative games associated with G. These games are often referred to as representations of G. The two standard representations are:Aumann, Robert J. "[http://www.ma.huji.ac.il/~raumann/pdf/The%20Core%20of%20a%20Cooperative.pdf The core of a cooperative game without side payments]." Transactions of the American Mathematical Society (1961): 539-552.

• The α-effective game associates with each coalition the sum of gains its members can 'guarantee' by joining forces. By 'guaranteeing', it is meant that the value is the max-min, e.g. the maximal value of the minimum taken over the opposition's strategies.
• The β-effective game associates with each coalition the sum of gains its members can 'strategically guarantee' by joining forces. By 'strategically guaranteeing', it is meant that the value is the min-max, e.g. the minimal value of the maximum taken over the opposition's strategies.

The main assumption in cooperative game theory is that the grand coalition $N$ will form.{{cite book |last1=Peters |first1=Hans |title=Game theory: a multi-leveled approach |url=https://archive.org/details/gametheorymultil00pete |url-access=limited |date=2008 |publisher=Springer |isbn=978-3-540-69290-4 |pages=[https://archive.org/details/gametheorymultil00pete/page/n127 123] |doi=10.1007/978-3-540-69291-1_17 }} The challenge is then to allocate the payoff $v(N)$ among the players in some way. (This assumption is not restrictive, because even if players split off and form smaller coalitions, we can apply solution concepts to the subgames defined by whatever coalitions actually form.) A solution concept is a vector $x\; \backslash in\; \backslash mathbb\{R\}^N$ (or a set of vectors) that represents the allocation to each player. Researchers have proposed different solution concepts based on different notions of fairness. Some properties to look for in a solution concept include:

• Efficiency: The payoff vector exactly splits the total value: $\backslash sum\_\{\; i\; \backslash in\; N\; \}\; x\_i\; =\; v(N)$.
• Individual rationality: No player receives less than what he could get on his own: $x\_i\; \backslash geq\; v(\backslash \{i\backslash \}),\; \backslash forall~\; i\; \backslash in\; N$.
• Existence: The solution concept exists for any game $v$.
• Uniqueness: The solution concept is unique for any game $v$.
• Marginality: The payoff of a player depends only on the marginal contribution of this player, i.e., if these marginal contributions are the same in two different games, then the payoff is the same: $v(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; =\; w(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; ),\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; i\; \backslash \}$ implies that $x\_i$ is the same in $v$ and in $w$.
• Monotonicity: The payoff of a player increases if the marginal contribution of this player increase: $v(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; \backslash leq\; w(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; ),\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; i\; \backslash \}$ implies that $x\_i$ is weakly greater in $w$ than in $v$.
• Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players $|N|$.)
• Symmetry: The solution concept $x$ allocates equal payments $x\_i\; =\; x\_j$ to symmetric players $i$, $j$. Two players $i$, $j$ are symmetric if $v(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; =\; v(\; S\; \backslash cup\; \backslash \{\; j\; \backslash \}\; ),\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; i,\; j\; \backslash \}$; that is, we can exchange one player for the other in any coalition that contains only one of the players and not change the payoff.
• Additivity: The allocation to a player in a sum of two games is the sum of the allocations to the player in each individual game. Mathematically, if $v$ and $\backslash omega$ are games, the game $(\; v\; +\; \backslash omega\; )$ simply assigns to any coalition the sum of the payoffs the coalition would get in the two individual games. An additive solution concept assigns to every player in $(\; v\; +\; \backslash omega\; )$ the sum of what he would receive in $v$ and $\backslash omega$.
• Zero Allocation to Null Players: The allocation to a null player is zero. A null player $i$ satisfies $v(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; =\; v(\; S\; ),\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; i\; \backslash \}$. In economic terms, a null player's marginal value to any coalition that does not contain him is zero.

An efficient payoff vector is called a pre-imputation, and an individually rational pre-imputation is called an imputation. Most solution concepts are imputations.

The stable set of a game (also known as the von Neumann-Morgenstern solution {{harv|von Neumann|Morgenstern|1944}}) was the first solution proposed for games with more than 2 players. Let $v$ be a game and let $x$, $y$ be two imputations of $v$. Then $x$ dominates $y$ if some coalition $S\; \backslash neq\; \backslash emptyset$ satisfies $x\_i\; >\; y\; \_i,\; \backslash forall~\; i\; \backslash in\; S$ and $\backslash sum\_\{\; i\; \backslash in\; S\; \}\; x\_i\; \backslash leq\; v(S)$. In other words, players in $S$ prefer the payoffs from $x$ to those from $y$, and they can threaten to leave the grand coalition if $y$ is used because the payoff they obtain on their own is at least as large as the allocation they receive under $x$.

A stable set is a set of imputations that satisfies two properties:

• Internal stability: No payoff vector in the stable set is dominated by another vector in the set.
• External stability: All payoff vectors outside the set are dominated by at least one vector in the set.

Von Neumann and Morgenstern saw the stable set as the collection of acceptable behaviours in a society: None is clearly preferred to any other, but for each unacceptable behaviour there is a preferred alternative. The definition is very general allowing the concept to be used in a wide variety of game formats.{{cn|date=April 2024}}

• A stable set may or may not exist {{harv|Lucas|1969}}, and if it exists it is typically not unique {{harv|Lucas|1992}}. Stable sets are usually difficult to find. This and other difficulties have led to the development of many other solution concepts.
• A positive fraction of cooperative games have unique stable sets consisting of the core {{harv|Owen|1995|p=240}}.
• A positive fraction of cooperative games have stable sets which discriminate $n-2$ players. In such sets at least $n-3$ of the discriminated players are excluded {{harv|Owen|1995|p=240}}.

{{main|Core (game theory)}}

Let $v$ be a game. The core of $v$ is the set of payoff vectors

: $C(\; v\; )\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbb\{R\}^N:\; \backslash sum\_\{\; i\; \backslash in\; N\; \}\; x\_i\; =\; v(N);\; \backslash quad\; \backslash sum\_\{\; i\; \backslash in\; S\; \}\; x\_i\; \backslash geq\; v(S),\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash right\backslash \}.$

In words, the core is the set of imputations under which no coalition has a value greater than the sum of its members' payoffs. Therefore, no coalition has incentive to leave the grand coalition and receive a larger payoff.

• The core of a game may be empty (see the Bondareva–Shapley theorem). Games with non-empty cores are called balanced.
• If it is non-empty, the core does not necessarily contain a unique vector.
• The core is contained in any stable set, and if the core is stable it is the unique stable set; see {{harv|Driessen|1988}} for a proof.

For simple games, there is another notion of the core, when each player is assumed to have preferences on a set $X$ of alternatives.

A profile is a list $p=(\backslash succ\_i^p)\_\{i\; \backslash in\; N\}$ of individual preferences $\backslash succ\_i^p$ on $X$.

Here $x\; \backslash succ\_i^p\; y$ means that individual $i$ prefers alternative $x$

to $y$ at profile $p$.

Given a simple game $v$ and a profile $p$, a dominance relation $\backslash succ^p\_v$ is defined

on $X$ by $x\; \backslash succ^p\_v\; y$ if and only if there is a winning coalition $S$

(i.e., $v(S)=1$) satisfying $x\; \backslash succ\_i^p\; y$ for all $i\; \backslash in\; S$.

The core $C(v,p)$ of the simple game $v$ with respect to the profile $p$ of preferences

is the set of alternatives undominated by $\backslash succ^p\_v$

(the set of maximal elements of $X$ with respect to $\backslash succ^p\_v$):

: $x\; \backslash in\; C(v,p)$ if and only if there is no $y\backslash in\; X$ such that $y\; \backslash succ^p\_v\; x$.

The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection.

Nakamura's theorem states that the core $C(v,p)$ is nonempty for all profiles $p$ of acyclic (alternatively, transitive) preferences

if and only if $X$ is finite and the cardinal number (the number of elements) of $X$ is less than the Nakamura number of $v$.

A variant by Kumabe and Mihara states that the core $C(v,p)$ is nonempty for all profiles $p$ of preferences that have a maximal element

if and only if the cardinal number of $X$ is less than the Nakamura number of $v$. (See Nakamura number for details.)

Because the core may be empty, a generalization was introduced in {{harv|Shapley|Shubik|1966}}. The strong $\backslash varepsilon$-core for some number $\backslash varepsilon\; \backslash in\; \backslash mathbb\{R\}$ is the set of payoff vectors

: $C\_\backslash varepsilon(\; v\; )\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbb\{R\}^N:\; \backslash sum\_\{\; i\; \backslash in\; N\; \}\; x\_i\; =\; v(N);\; \backslash quad\; \backslash sum\_\{\; i\; \backslash in\; S\; \}\; x\_i\; \backslash geq\; v(S)\; -\; \backslash varepsilon,\; \backslash forall~\; S\; \backslash subseteq\; N\; \backslash right\backslash \}.$

In economic terms, the strong $\backslash varepsilon$-core is the set of pre-imputations where no coalition can improve its payoff by leaving the grand coalition, if it must pay a penalty of $\backslash varepsilon$ for leaving. $\backslash varepsilon$ may be negative, in which case it represents a bonus for leaving the grand coalition. Clearly, regardless of whether the core is empty, the strong $\backslash varepsilon$-core will be non-empty for a large enough value of $\backslash varepsilon$ and empty for a small enough (possibly negative) value of $\backslash varepsilon$. Following this line of reasoning, the least-core, introduced in {{harv|Maschler|Peleg|Shapley|1979}}, is the intersection of all non-empty strong $\backslash varepsilon$-cores. It can also be viewed as the strong $\backslash varepsilon$-core for the smallest value of $\backslash varepsilon$ that makes the set non-empty {{harv|Bilbao|2000}}.

{{main|Shapley value}}

The Shapley value is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity.{{Cite journal|last=Young|first=H. P.|date=1985-06-01|title=Monotonic solutions of cooperative games|journal=International Journal of Game Theory|language=en|volume=14|issue=2|pages=65–72|doi=10.1007/BF01769885|s2cid=122758426|issn=0020-7276}} It was introduced by Lloyd Shapley {{harv|Shapley|1953}} who showed that it is the unique payoff vector that is efficient, symmetric, additive, and assigns zero payoffs to dummy players. The Shapley value of a superadditive game is individually rational, but this is not true in general. {{harv|Driessen|1988}}

Let $v\; :\; 2^N\; \backslash to\; \backslash mathbb\{R\}$ be a game, and let $x\; \backslash in\; \backslash mathbb\{R\}^N$ be an efficient payoff vector. The maximum surplus of player i over player j with respect to x is

: $s\_\{ij\}^v(x)\; =\; \backslash max\; \backslash left\backslash \{\; v(S)\; -\; \backslash sum\_\{\; k\; \backslash in\; S\; \}\; x\_k\; :\; S\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; j\; \backslash \},\; i\; \backslash in\; S\; \backslash right\backslash \},$

the maximal amount player i can gain without the cooperation of player j by withdrawing from the grand coalition N under payoff vector x, assuming that the other players in i's withdrawing coalition are satisfied with their payoffs under x. The maximum surplus is a way to measure one player's bargaining power over another. The kernel of $v$ is the set of imputations x that satisfy

• $(\; s\_\{ij\}^v(x)\; -\; s\_\{ji\}^v(x)\; )\; \backslash times\; (\; x\_j\; -\; v(j)\; )\; \backslash leq\; 0$, and
• $(\; s\_\{ji\}^v(x)\; -\; s\_\{ij\}^v(x)\; )\; \backslash times\; (\; x\_i\; -\; v(i)\; )\; \backslash leq\; 0$

for every pair of players i and j. Intuitively, player i has more bargaining power than player j with respect to imputation x if $s\_\{ij\}^v(x)\; >\; s\_\{ji\}^v(x)$, but player j is immune to player i's threats if $x\_j\; =\; v(j)$, because he can obtain this payoff on his own. The kernel contains all imputations where no player has this bargaining power over another. This solution concept was first introduced in {{harv|Davis|Maschler|1965}}.

The Harsanyi dividend (named after John Harsanyi, who used it to generalize the Shapley value in 1963{{Cite book|title=Papers in Game Theory|last=Harsanyi|first=John C.|date=1982|publisher=Springer, Dordrecht|isbn=9789048183692|series=Theory and Decision Library|pages=44–70|language=en|doi=10.1007/978-94-017-2527-9_3|chapter = A Simplified Bargaining Model for the n-Person Cooperative Game}}) identifies the surplus that is created by a coalition of players in a cooperative game. To specify this surplus, the worth of this coalition is corrected by subtracting the surplus that was already created by subcoalitions. To this end, the dividend $d\_v(S)$ of coalition $S$ in game $v$ is recursively determined by

$\backslash begin\{align\}$

d_v(\{i\})&= v(\{i\}) \\

d_v(\{i,j\})&= v(\{i,j\})-d_v(\{i\})-d_v(\{j\}) \\

d_v(\{i,j,k\})&= v(\{i,j,k\})-d_v(\{i,j\})-d_v(\{i,k\})-d_v(\{j,k\})-d_v(\{i\})-d_v(\{j\})-d_v(\{k\})\\

&\vdots \\

d_v(S) &= v(S) - \sum_{T\subsetneq S }d_v(T)

\end{align}

An explicit formula for the dividend is given by $d\_v(S)=\backslash sum\_\{T\backslash subseteq\; S\; \}(-1)^$

Harsanyi dividends are useful for analyzing both games and solution concepts, e.g. the Shapley value is obtained by distributing the dividend of each coalition among its members, i.e., the Shapley value $\backslash phi\_i(v)$ of player $i$ in game $v$ is given by summing up a player's share of the dividends of all coalitions that she belongs to, $\backslash phi\_i(v)=\backslash sum\_\{S\backslash subset\; N:\; i\; \backslash in\; S\; \}\{d\_v(S)\}/$

{{Main|Nucleolus (game theory)}}

Let $v\; :\; 2^N\; \backslash to\; \backslash mathbb\{R\}$ be a game, and let $x\; \backslash in\; \backslash mathbb\{R\}^N$ be a payoff vector. The excess of $x$ for a coalition $S\; \backslash subseteq\; N$ is the quantity $v(S)\; -\; \backslash sum\_\{\; i\; \backslash in\; S\; \}\; x\_i$; that is, the gain that players in coalition $S$ can obtain if they withdraw from the grand coalition $N$ under payoff $x$ and instead take the payoff $v(S)$. The nucleolus of $v$ is the imputation for which the vector of excesses of all coalitions (a vector in $\backslash mathbb\{R\}^\{2^N\}$) is smallest in the leximin order. The nucleolus was introduced in {{harv|Schmeidler|1969}}.

{{harv|Maschler|Peleg|Shapley|1979}} gave a more intuitive description: Starting with the least-core, record the coalitions for which the right-hand side of the inequality in the definition of $C\_\backslash varepsilon(\; v\; )$ cannot be further reduced without making the set empty. Continue decreasing the right-hand side for the remaining coalitions, until it cannot be reduced without making the set empty. Record the new set of coalitions for which the inequalities hold at equality; continue decreasing the right-hand side of remaining coalitions and repeat this process as many times as necessary until all coalitions have been recorded. The resulting payoff vector is the nucleolus.

• Although the definition does not explicitly state it, the nucleolus is always unique. (See Section II.7 of {{harv|Driessen|1988}} for a proof.)
• If the core is non-empty, the nucleolus is in the core.
• The nucleolus is always in the kernel, and since the kernel is contained in the bargaining set, it is always in the bargaining set (see {{harv|Driessen|1988}} for details.)

Introduced by Shapley in {{harv|Shapley|1971}}, convex cooperative games capture the intuitive property some games have of "snowballing". Specifically, a game is convex if its characteristic function $v$ is supermodular:

: $v(\; S\; \backslash cup\; T\; )\; +\; v(\; S\; \backslash cap\; T\; )\; \backslash geq\; v(S)\; +\; v(T),\; \backslash forall~\; S,\; T\; \backslash subseteq\; N.$

It can be shown (see, e.g., Section V.1 of {{harv|Driessen|1988}}) that the supermodularity of $v$ is equivalent to

: $v(\; S\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; -\; v(S)\; \backslash leq\; v(\; T\; \backslash cup\; \backslash \{\; i\; \backslash \}\; )\; -\; v(T),\; \backslash forall~\; S\; \backslash subseteq\; T\; \backslash subseteq\; N\; \backslash setminus\; \backslash \{\; i\; \backslash \},\; \backslash forall~\; i\; \backslash in\; N;$

that is, "the incentives for joining a coalition increase as the coalition grows" {{harv|Shapley|1971}}, leading to the aforementioned snowball effect. For cost games, the inequalities are reversed, so that we say the cost game is convex if the characteristic function is submodular.

Convex cooperative games have many nice properties:

• Supermodularity trivially implies superadditivity.
• Convex games are totally balanced: The core of a convex game is non-empty, and since any subgame of a convex game is convex, the core of any subgame is also non-empty.
• A convex game has a unique stable set that coincides with its core.
• The Shapley value of a convex game is the center of gravity of its core.
• An extreme point (vertex) of the core can be found in polynomial time using the greedy algorithm: Let $\backslash pi:\; N\; \backslash to\; N$ be a permutation of the players, and let $S\_i\; =\; \backslash \{\; j\; \backslash in\; N:\; \backslash pi(j)\; \backslash leq\; i\; \backslash \}$ be the set of players ordered $1$ through $i$ in $\backslash pi$, for any $i\; =\; 0,\; \backslash ldots,\; n$, with $S\_0\; =\; \backslash emptyset$. Then the payoff $x\; \backslash in\; \backslash mathbb\{R\}^N$ defined by $x\_i\; =\; v(\; S\_\{\backslash pi(i)\}\; )\; -\; v(\; S\_\{\backslash pi(i)\; -\; 1\}\; ),\; \backslash forall~\; i\; \backslash in\; N$ is a vertex of the core of $v$. Any vertex of the core can be constructed in this way by choosing an appropriate permutation $\backslash pi$.

Submodular and supermodular set functions are also studied in combinatorial optimization. Many of the results in {{harv|Shapley|1971}} have analogues in {{harv|Edmonds|1970}}, where submodular functions were first presented as generalizations of matroids. In this context, the core of a convex cost game is called the base polyhedron, because its elements generalize base properties of matroids.

However, the optimization community generally considers submodular functions to be the discrete analogues of convex functions {{harv|Lovász|1983}}, because the minimization of both types of functions is computationally tractable. Unfortunately, this conflicts directly with Shapley's original definition of supermodular functions as "convex".

Corporate strategic decisions can develop and create value through cooperative game theory.{{Cite journal |last=Ross |first=David Gaddis |date=2018-08-01 |title=Using cooperative game theory to contribute to strategy research |journal=Strategic Management Journal |volume=39 |issue=11 |pages=2859–2876 |doi=10.1002/smj.2936|s2cid=169982369 }} This means that cooperative game theory can become the strategic theory of the firm, and different CGT solutions can simulate different institutions.

• Consensus decision-making
• Coordination game
• Intra-household bargaining
• Hedonic game
• Linear production game
• Minimum-cost spanning tree game - a class of cooperative games.

• {{Citation

| first = Jesús Mario

| last = Bilbao

| title = Cooperative Games on Combinatorial Structures

| publisher = Kluwer Academic Publishers

| year = 2000

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{{Game theory}}

Category:Game theory

Category:Mathematical and quantitative methods (economics)