Non-Cooperative Representations of Cooperative Games

Abstract

Non-cooperative games in normal form are specified by a player set, sets of player strategies, and payoff functions. Cooperative games, meanwhile, are specified by a player set and a worth function that maps coalitions of players to payoffs they can feasibly achieve. Although these games study distinct aspects of social behavior, this paper proposes a novel attempt at relating the two models. In particular, cooperative games may be represented by a non-cooperative game in which players can freely sign binding agreements to form coalitions. These coalitions inherit a joint strategy set and seek to maximize collective payoffs. When these coalitions play against one another, the equilibrium payoffs for each coalition coincide with what is predicted by the worth function. This paper proves sufficient conditions under which cooperative games can be represented by non-cooperative games. This paper finds that all strictly superadditive partition function form (PFF) games can be represented under Nash equilibrium (NE) and rationalizability; that all weakly superadditive characteristic function form (CFF) games can be represented under NE; and that all weakly superadditive PFF games can be represented under trembling hand perfect equilibrium (THPE).

1. Introduction

From the outset, non-cooperative and cooperative games appear to offer entirely separate frameworks for understanding collective behavior. Non-cooperative games, as captured most simply by a normal-form game, comprise of a set of players, each endowed with some set of strategies and a payoff function, where the payoff is determined by the strategies that each player chooses to play. Research in non-cooperative game theory often seeks to address what strategies are likely to be chosen. Solution concepts in this field therefore pertain to the notion of equilibrium. Meanwhile, cooperative games abstract away the idea of strategic play and focus solely on the players and the outcomes. Two popular models of cooperative games include characteristic function form (CFF) games and partition function form (PFF) games. Both are uniquely identified (up to a zero-normalization) by a set of players and a worth function. For CFF games, the worth function is a mapping from subsets of the player set to a payoff. These subsets of the player set are often referred to in the literature as coalitions. For PFF games, this mapping is instead from tuples constructed from a coalition and the partition of the player set that contains it (Thrall & Lucas, 1963). These tuples are often referred to as embedded coalitions (Kóczy, 2018). Under the assumption of transferable utility, these payoffs are aggregable and divisible amongst members of a coalition. Research in cooperative game theory often attempts to develop algorithms in which these payoffs can be divided across players in ways that are fair and/or conducive to maximal cooperation. Solution concepts in this field therefore emphasize the notion of welfare in comparison.

This paper follows recent attempts to explore the relationships between these two frameworks despite their differences in specification and research objectives (Maskin, 2016). It is worth noting at this point that the conception of cooperative game theory was indeed marked by non-cooperative foundations. Von Neumann and Morgenstern (1944) studied n-player zero-sum games and assumed that any coalition formed from the player set would give rise to its complement. The worth function of the associated cooperative game was defined by the maximin payoff that each coalition obtains. In this regard, we can think of the worth function as a summary statistic of an underlying non-cooperative game, capturing the payoff outcomes that coalitions can feasibly achieve without needing to specify the strategies needed to generate such outcomes. Naturally, we may then ask whether this is true of all worth functions; that is, how can we characterize the set of cooperative games that have a non-cooperative representation?

To answer this question, this paper proposes an extension of von Neumann and Morgenstern’s characterization of cooperative games to consider general-sum games, not just zero-sum games, and a variety of non-cooperative solution concepts, not just maximin outcomes. In particular, we will say that a cooperative game has a non-cooperative representation under a particular equilibrium solution concept if there exists a normal-form game for which the following is true. After partitioning the player set of the normal-form game into coalitions and adjusting the strategy sets and payoff function accordingly so as to reflect the fact that players within any given coalition now play strategies as part of some joint action to maximize the collective payoff, the payoffs to each coalition at equilibrium always coincide exactly with what each coalition is assigned by the worth function.

To motivate this set-up, we can imagine these coalitions as forming after players sign binding agreements without cost to play strategies that serve the collective interests of the members of the coalition. These agreements to cooperate are often understood to bind under external enforcement rather than self-enforcement, so issues of credibility are not a problem here (Ray, 2007). We introduce the term composite game to designate the games resulting from coalition formation; they are composite in the sense that we may decompose the coalitions, strategies, and payoffs prior to aggregation to restore the original normal-form game. Evidently, the number of composite games that can be constructed from a normal-form game equals the number of partitions of its player set. A composite game may have a unique solution, or it may have multiple solutions. But in any case, the payoffs to each coalition at any solution must be the same as that given by the worth function. Thus, in the case of multiple solutions, payoffs to each coalition are identical across solutions. This ensures that there is no ambiguity as to what coalitions can feasibly achieve and that each coalition’s payoff as according to the worth function is well-defined. For consistency, the same non-cooperative solution concept should apply across all composite games. When all this holds, we say that the resulting cooperative game is represented under the pre-specified non-cooperative solution concept. Or equivalently, the non-cooperative solution concept admits a representation for the cooperative game.

This paper provides novel theorems regarding the existence and construction of representations for particular classes of cooperative games. Strictly superadditive PFF games are represented under Nash equilibrium (NE), as are weakly superadditive CFF games. Moreover, strictly superadditive PFF games are represented under certain generalizations and refinements of NE, namely rationalizability and trembling hand perfect Nash equilibrium (THPE), respectively. This paper further proves that THPE admits representations for weakly superadditive PFF games, the most general class of cooperative games studied here. This paper finally ends with a discussion of the assumptions and limitations of the approach set out here. This includes assumptions regarding the process of coalition formation as well as the imposition of strategic dominance in the construction of the representations.

2. Preliminaries

2.1. Notation and Definitions for Non-Cooperative Games

Take a normal-form game $\Gamma =(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$, where $N=\{1,2,\dots ,n\}$ is the player set. Let $\Pi \left(N\right)$ denote the set of all partitions of N and $\left[N\right]=\left\{\right\{1\},\{2\},\dots ,\{n\left\}\right\}$ denote the finest partition of this player set. We refer to $\left\{N\right\}$ as the partition containing the grand coalition, i.e., the coalition of all players. For any partition $\pi \in \Pi \left(N\right)$, we denote the size or cardinality of the partition as $\left|\pi \right|$. For any two partitions, $\pi ,{\pi }^{\prime }\in \Pi \left(N\right)$, we say that ${\pi }^{\prime }$ is a refinement of $\pi$, if for all coalitions $S\in \pi$, there exists a subset of ${\pi }^{\prime }$ denoted $\mathcal{P}$ such that ${\bigcup }_{T\in \mathcal{P}}T=S$. We say that ${\pi }^{\prime }$ is a strict refinement of $\pi$ if ${\pi }^{\prime }$ is a refinement of $\pi$ and $\pi \ne {\pi }^{\prime }$.

Let $X_i$ denote the finite set of pure strategies of player i. For any coalition $S\in \pi$, we let $X_S={\times }_{i\in S}{X}_i$ denote their joint pure strategy set. For any given strategy $\sigma$, we use subscript notation to denote the player to whose strategy set it belongs (this may be a coalition) and superscript notation as an index within that strategy set. We identify any dominant strategies with the asterisk superscript.

We adopt the convention $({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})\in {\times }_{S\in \pi }{X}_S$ to denote the strategy profile consisting of the joint action played by coalition S in the partition $\pi$, holding fixed the strategies played by the other coalitions. We denote $u_S({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})={\sum }_{i\in S}{u}_i({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})$ to be the payoff to coalition S at this strategy profile.

For convenience, we also define the projection functions ${\mathrm{proj}}_i:{\times }_{i\in N}{X}_i\to X_i$ and ${\mathrm{proj}}_S:{\times }_{S\in \pi }{X}_S\to X_S$ as a means of extracting the actions played within a given strategy profile by player i and coalition S, respectively. Thus, ${\mathrm{proj}}_i({\sigma }_1,\dots ,{\sigma }_n)={\sigma }_i$ and ${\mathrm{proj}}_S({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})={\sigma }_S$.

Definition 1

(Composite Games). Given Γ and some partition of the player set $\pi \in \Pi \left(N\right)$, we denote the corresponding composite game as ${\Gamma }_{\pi }=(\pi ,{\left(X_S\right)}_{S\in \pi },{\left(u_S\right)}_{S\in \pi })$.

Remark 1.

Note that Γ is a degenerate composite game of itself insofar as $\Gamma ={\Gamma }_{\left[N\right]}$.

Definition 2

(Composite Equilibria (CE)). Under a pre-specified non-cooperative solution concept, we say that the strategy profile $x\in {\times }_{i\in N}{X}_i$ is a composite equilibrium (CE) of Γ if there exists a partition $\pi \in \Pi \left(N\right)$ such that x solves the composite game ${\Gamma }_{\pi }$.

That is, the strategies in a composite equilibrium (CE) of $\Gamma$ played jointly or otherwise form the solution to a composite game of $\Gamma$. In fact, the same strategy profile may solve multiple composite games of $\Gamma$.

The following definition provides insight into sufficient means of identifying CE when the solution concept is NE.

Definition 3

(Potentially Strictly/Weakly Dominant Strategies (PSD/PWD)). Given some strategy profile $x\in {\times }_{i\in N}{X}_i$, we say that player i’s strategy ${\sigma }_i={\mathit{proj}}_i\left(x\right)$ is potentially strictly/weakly dominant (PSD/PWD) in x if there exists a coalition S containing player i such that the joint strategy ${\times }_{j\in S}{\sigma }_j$, where ${\sigma }_j={\mathit{proj}}_j\left(x\right)$ for all $j\in S$, is the strictly/weakly dominant strategy of coalition S. That is, ${\times }_{j\in S}{\sigma }_j={\sigma }_S^{\ast }$.

As the following lemmata demonstrate, we may be able to construct payoff functions ${\left(u_i\right)}_{i\in N}$ and strategy sets ${\left(X_i\right)}_{i\in N}$ of a game $\Gamma$ such that the CE of $\Gamma$ are strategy profiles in which all strategies are PSD and vice versa. In doing so, we will first need to introduce two functions.

Let ${\kappa }^{strict}:{\times }_{i\in N}{X}_i\times 2^N\to \mathbb{N}$ be the function that maps a strategy profile and subset of the player set to the number of PSD strategies played by players within this subset in the strategy profile. And let ${\rho }_x^{strict}:N\to 2^N$ be the function defined by

$${\rho }_x^{strict}\left(i\right)=\left\{\begin{array}{cc}S\hfill & \mathrm{if}i\in S\mathrm{and}\exists {\sigma }_S^{\ast }\in X_S\mathrm{s}.\mathrm{t}.{\times }_{j\in S}{\mathrm{proj}}_j\left(x\right)={\sigma }_S^{\ast }\hfill \\ \varnothing ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(1)

For a fixed strategy profile x, we interpret ${\rho }_x^{strict}$ as mapping a player to its corresponding coalition if the player’s strategy in x is PSD and to the empty set otherwise. Note that it need not be the case that ${\rho }_x^{strict}$ is well-defined for every strategy profile. This occurs when there exist distinct $S,T\in 2^N$, some player $i\in S\cap T$, and a strategy profile x such that ${\times }_{j\in S}{\mathrm{proj}}_j\left(x\right)={\sigma }_S^{\ast }$ and ${\times }_{j\in T}{\mathrm{proj}}_j\left(x\right)={\sigma }_T^{\ast }$. But when ${\rho }_x^{strict}$ is well-defined, we can infer from (1) the following relationship between ${\kappa }^{strict}$ and ${\rho }_x^{strict}$:

$${\kappa }^{strict}(x,S)=\left|S\right|-\sum _{i\in S}\mathbf{1}({\rho }_x^{strict}\left(i\right)=\varnothing ).$$

(2)

From (2), we can easily see that for all partitions of the player set $\pi \in \Pi \left(N\right)$

$${\kappa }^{strict}(x,N)=\sum _{S\in \pi }{\kappa }^{strict}(x,S).$$

(3)

Lemma 1.

Suppose a game $\Gamma =(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ is such that for all coalitions $S\in 2^N$, there exists a strictly dominant strategy ${\sigma }_S^{\ast }$ in their strategy set $X_S$. Then, for any CE of Γ under NE denoted x, ${\kappa }^{strict}(x,N)=\left|N\right|$.

Proof. 

If all coalitions have a strictly dominant strategy, then all composite games are dominant-solvable and admit a unique NE in which these strategies are played. It follows that all CE consist only of potentially dominant strategies. □

Lemma 2.

Suppose a game $\Gamma =(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ is such that ${\rho }_x^{strict}$ is well-defined for all strategy profiles. Then, a strategy profile $x\in {\times }_{i\in N}{X}_i$ is a CE of Γ under NE if ${\kappa }^{strict}(x,N)=\left|N\right|$.

Proof. 

It suffices to show that if all the strategies in x are PSD, ${\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}$ is a partition of N.

First, observe that if ${\kappa }^{strict}(x,N)=\left|N\right|$, no player is mapped to the empty set. Since each player must therefore be mapped onto some coalition that contains them, we have that $N\subseteq {\bigcup }_{S\in {\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}}S$. Moreover, ${\bigcup }_{S\in {\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}}S\subseteq N$ must hold as the codomain of ${\rho }_x^{strict}$ is $2^N$. It follows that ${\bigcup }_{S\in {\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}}S=N$.

Now suppose that ${\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}$ fails to be a partition as there exist coalitions $S,T\in {\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}$ such that S and T are not disjoint. Then, there exists a player j such that $j\in S\cap T$. This implies that player j’s strategy in x is part of the dominant strategies of both coalitions S and T. This contradicts the requirement that ${\rho }_x^{strict}$ be well-defined for all strategy profiles. □

2.2. Notation and Definitions for Cooperative Games

With the same player set as defined in the non-cooperative game, we can consider a cooperative game $(N,v)$, where v is the worth function. We refer to the tuple $(S,\pi )\in 2^N\times \Pi \left(N\right)$ as an embedded coalition if $S\in \pi$. $v(S,\pi )$ is the worth of the coalition S given the partition $\pi$ and is defined only for embedded coalitions.

The worth function v is weakly superadditive if for all pairs of embedded coalitions of the form $(S,\pi )$ and $(T,\pi )$—that is, we require the coalitions S and T to be disjoint and belong to the same partition—the following holds: $v(S,\pi )+v(T,\pi )\le v(S\cup T,\{S\cup T\}\cup \pi \setminus \{S,T\left\}\right)$. v is said to be strictly superadditive if the above always holds with strict inequality instead.

Given our assumption that players can form coalitions without incurring external costs, weak superadditivity is a minimal property to expect from worth functions, particularly for cooperative games represented by a non-cooperative game. In the underlying non-cooperative game, members of any given coalition can always play the same strategies as those played in a composite game prior to coalition formation but doing so jointly. Holding fixed the coalitional structure of the other players, this guarantees a value equal to the sum of their prior payoffs. In this regard, players are always weakly better off from forming coalitions.

When the coalitional structure of the other players does change, we may observe differences in what a coalition can feasibly achieve. v is said to exhibit a positive externality if there exist disjoint embedded coalitions of the form $(R,\pi )$, $(S,\pi )$, and $(T,\pi )$ such that $v(R,\pi )<v(R,\{R\}\cup \{S\cup T\}\cup \pi \setminus \{R,S,T\left\}\right)$. v exhibits a negative externality if there exist disjoint embedded coalitions for which this strict inequality is reversed. Note that from this definition, a worth function may exhibit both positive and negative externalities.

In general, intuition for these properties is best developed in the case where the cooperative game is represented by a non-cooperative game. For example, an externality in the worth function reflects how coalition formation/dissolution results in an aggregation/disaggregation of payoffs that incentivize a change in strategy, ultimately shifting the equilibrium to one that rewards different payoffs for all the players. Given this, we present a formal definition below of what it means for a cooperative game to admit a normal-form game representation.

Definition 4

(Representation). Given a non-cooperative solution concept to be applied throughout, the cooperative game $(N,v)$ is represented under the solution concept if there exists a normal-form game $(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ such that the solution to each of its composite games yields payoffs to each embedded coalition equal to its worth as defined in $(N,v)$. We say that $(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ is a (normal-form) representation of $(N,v)$ under the non-cooperative solution concept.

Remark 2.

Suppose that $(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ is a representation of $(N,v)$ under NE such that all its CE consist only of strictly dominant strategies. Then, by definition, $v(S,\pi )=u_S({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}}^{\ast })$ must hold for all embedded coalitions $(S,\pi )$.

3. Example Representations Under NE

As we shall show, the task of constructing a representation of a cooperative game under NE simplifies when we only consider possible candidates for which Lemmata 1 and 2 hold. When we assume such a normal-form game $\Gamma$, we can count the number of PSD strategies in a strategy profile—that is, apply ${\kappa }^{strict}$—to determine whether or not it is a CE of $\Gamma$. We can then define each player’s payoff function accordingly such that payoffs to coalitions at composite equilibria coincide with their worth in the cooperative game. Evidently, we must also ensure that these payoff functions defined at strategy profiles that are not composite equilibria do not contradict the requirements for Lemmata 1 and 2 to be applied. We hence proceed as follows.

For all players $i\in N$, we define their strategy set to be $X_i=\left\{{\sigma }_i^S\right|S\in 2^N,i\in S\}$. Critically, we assume that each coalition has a strictly dominant strategy of the form ${\sigma }_S^{\ast }={\times }_{i\in S}{\sigma }_i^S$. By extension, for any individual player, their strictly dominant strategy is indexed by ${\sigma }_i^{\ast }={\sigma }_i^i$. When this assumption holds, ${\rho }_x^{strict}$ is well-defined for all strategy profiles. Thus, as long as ${\left(u_i\right)}_{i\in N}$ is defined such that ${\sigma }_S^{\ast }={\times }_{i\in S}{\sigma }_i^S$ holds for each coalition and that ${\sigma }_S^{\ast }$ strictly dominates, we can invoke Lemmata 1 and 2.

It turns out that we can construct a parametric family of ${\left(u_i\right)}_{i\in N}$ that satisfies this requirement. Moreover, given a strictly superadditive PFF game, $(N,v)$, each ${\left(u_i\right)}_{i\in N}$ in the family can be made to satisfy $v(S,\pi )=u_S({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}}^{\ast })$ for each embedded coalition.

For any given strategy profile x, let $u_i\left(x\right)={\theta f}_i\left(x\right)+g_i\left(x\right)$ be the payoff function for each player i, where $\theta \in {\mathbb{R}}^{+}$ is a parameterizing constant. We define $f_i\left(x\right)$ and $g_i\left(x\right)$ by

$$\begin{array}{cc}\hfill f_i\left(x\right)& =\left\{\begin{array}{cc}-\left|N\right|\hfill & \mathrm{if}{\rho }_x^{strict}\left(i\right)=\varnothing \hfill \\ \left|N\right|-{\kappa }^{strict}(x,N)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \\ \hfill g_i\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \frac{v({\rho }_x^{strict}\left(i\right),{\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N})}{|{\rho }_x^{strict}\left(i\right)|}}\hfill & \mathrm{if}{\kappa }^{strict}(x,N)=\left|N\right|\hfill \\ |{\rho }_x^{strict}\left(i\right)|\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

Moreover, we denote $f_S\left(x\right)={\sum }_{i\in S}{f}_i\left(x\right)$ and $g_S\left(x\right)={\sum }_{i\in S}{g}_i\left(x\right)$ for all coalitions $S\in 2^N$.

At first glance, the motivation behind such definitions may appear somewhat opaque. Intuitively, we require some control over the payoffs in the events where the player plays a PSD strategy and where all players play PSD strategies. This is achieved by $f_i\left(x\right)$ and $g_i\left(x\right)$, respectively, and their moderation by the parameter $\theta$ allows the possibility of payoffs ${\left(u_i\right)}_{i\in N}$ to support the condition that ${\sigma }_S^{\ast }={\times }_{i\in S}{\sigma }_i^S$ holds for each coalition.

To illustrate this, let $\Gamma (N,v,\theta )=(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$ be a normal-form game in which the strategy sets and payoff functions are as defined before for some choice of worth function v and parameter $\theta$. We demonstrate that there exist values of $\theta$ for which $\Gamma (N,v,\theta )$ represents $(N,v)$ under NE when v is a strictly superadditive partition function for the cases where $\left|N\right|=2$ and $\left|N\right|=3$, these being the smallest non-trivial cooperative game and the smallest non-trivial cooperative game that can exhibit externalities, respectively.

Example 1

(Strictly Superadditive PFF Games for $\left|N\right|=2$). Consider the following parameterization of the worth function for a two-player zero-normalized game:

$$v\left(\right\{1\},\{\left\{1\right\},\left\{2\right\}\left\}\right)=0,v\left(\right\{2\},\{\left\{1\right\},\left\{2\right\}\left\}\right)=0,v\left(\right\{1,2\},\{\{1,2\}\left\}\right)=a,a>0.$$

Any two-player strictly superadditive PFF game is uniquely identified by the parameter a up to a zero-normalization. Figure 1 provides the payoff matrix for $\Gamma (N,v,\theta )$ with $(N,v)$.

Observe that when $\theta >max\{{\displaystyle \frac{a}{2}}-1,1-a\}$, ${\sigma }_1^{\left\{1\right\}}$, ${\sigma }_2^{\left\{2\right\}}$, and ${\sigma }_1^{\{1,2\}}\times {\sigma }_2^{\{1,2\}}$ are indeed the strictly dominant strategies of player 1, player 2, and coalition $\{1,2\}$, respectively. Further inspection of the payoffs at CE shows that $\Gamma (N,v,\theta )$ represents $(N,v)$ under NE for all $\theta >max\{{\displaystyle \frac{a}{2}}-1,1-a\}$.

Example 2

(Strictly Superadditive PFF Games for $\left|N\right|=3$). Now, consider the following parameterization of the worth function for a three-player zero-normalized game:

$$\begin{array}{c}v\left(\right\{1\},\{\left\{1\right\},\left\{2\right\},\left\{3\right\}\left\}\right)=0,v\left(\right\{2\},\{\left\{\right\{1\},\{2\},\{3\left\}\right\})=0,v(\left\{3\right\},\left\{\right\{1\},\{2\},\{3\left\}\right\})=0,\\ v\left(\right\{1,2\},\{\{1,2\},\left\{3\right\}\left\}\right)=a,v\left(\right\{3\},\{\{1,2\},\left\{3\right\}\left\}\right)=b,a>0,\\ v\left(\right\{1,3\},\{\{1,3\},\left\{2\right\}\left\}\right)=c,v\left(\right\{2\},\{\{1,3\},\left\{2\right\}\left\}\right)=d,c>0,\\ v\left(\right\{\{2,3\},\left\{\right\{1\},\{2,3\left\}\right\})=e,v(\left\{1\right\},\left\{\right\{1\},\{2,3\left\}\right\})=f,f>0,\\ v\left(\right\{1,2,3\},\{\{1,2,3\}\left\}\right)=g,g>max\{0,a+b,c+d,e+f\}.\end{array}$$

Any three-player strictly superadditive PFF game is uniquely identified by the parameters a, b, c, d, e, f, and g up to a zero-normalization. The payoff matrix for $\Gamma (N,v,\theta )$ with $(N,v)$ as defined is given in Figure 2.

Observe that ${\sigma }_1^{\left\{1\right\}}$, ${\sigma }_2^{\left\{2\right\}}$, ${\sigma }_3^{\left\{3\right\}}$, ${\sigma }_1^{\{1,2\}}\times {\sigma }_2^{\{1,2\}}$, ${\sigma }_1^{\{1,3\}}\times {\sigma }_3^{\{1,3\}}$, ${\sigma }_2^{\{2,3\}}\times {\sigma }_3^{\{2,3\}}$, and ${\sigma }_1^{\{1,2,3\}}\times {\sigma }_2^{\{1,2,3\}}\times {\sigma }_3^{\{1,2,3\}}$ are the strictly dominant strategies of players 1, 2, and 3 and coalitions $\{1,2\}$, $\{1,3\}$, $\{2,3\}$, and $\{1,2,3\}$, respectively, when the following holds: $\theta >max\{1,{\displaystyle \frac{a}{2}}-1,{\displaystyle \frac{c}{2}}-1,{\displaystyle \frac{e}{2}}-1,{\displaystyle \frac{-b}{3}},{\displaystyle \frac{-d}{3}},{\displaystyle \frac{-f}{3}},{\displaystyle \frac{g-3}{6}},{\displaystyle \frac{1-a}{2}},{\displaystyle \frac{1-c}{2}},{\displaystyle \frac{1-e}{2}},{\displaystyle \frac{a+2b-8}{4}},{\displaystyle \frac{c+2d-8}{4}},{\displaystyle \frac{e+2f-8}{4}},4-g\}$.

Again, by inspecting the payoffs at CE, we see that $\Gamma (N,v,\theta )$ represents $(N,v)$ under NE for these values of θ.

4. Theorems on the Existence of Representations of Cooperative Games

Theorem 1.

Any strictly superadditive PFF game $(N,v)$ is represented under NE.

Proof. 

Recall that for any given strictly superadditive PFF game $(N,v)$, we define the normal-form game $\Gamma (N,v,\theta )=(N,{\left(X_i\right)}_{i\in N},{\left(u_i\right)}_{i\in N})$, whereby

$$\begin{array}{ccc}\hfill X_i& =& \left\{{\sigma }_i^S\right|S\in 2^N,i\in S\},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill u_i\left(x\right)& =& \theta f_i\left(x\right)+g_i\left(x\right),\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill f_i\left(x\right)& =& \left\{\begin{array}{cc}-\left|N\right|\hfill & \mathrm{if}{\rho }_x^{strict}\left(i\right)=\varnothing \hfill \\ \left|N\right|-{\kappa }^{strict}(x,N)\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill g_i\left(x\right)& =& \left\{\begin{array}{cc}{\displaystyle \frac{v({\rho }_x^{strict}\left(i\right),{\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N})}{|{\rho }_x^{strict}\left(i\right)|}}\hfill & \mathrm{if}{\kappa }^{strict}(x,N)=\left|N\right|\hfill \\ |{\rho }_x^{strict}\left(i\right)|\hfill & \mathrm{otherwise}.\hfill \end{array}\right..\hfill \end{array}$$

(7)

Let us conjecture that there exists some $\theta \in {\mathbb{R}}^{+}$ such that for every coalition $S\in 2^N$, there further exists a strictly dominant strategy of the form ${\sigma }_S^{\ast }={\times }_{i\in S}{\sigma }_i^S$. Under this conjecture, ${\rho }_x^{strict}$ is well-defined for all strategy profiles, and by extension, $\Gamma (N,v,\theta )$ is well-defined.

For any embedded coalition $(S,\pi )$, consider the strategy profiles $x^{\ast }=({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}})$ and $x^{\prime }=({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})$, where ${\sigma }_S\ne {\sigma }_S^{\ast }$ and ${\sigma }_{\pi \setminus \left\{S\right\}}$ is fixed across both strategy profiles. To verify our conjecture that ${\sigma }_S^{\ast }$ is the strictly dominant strategy of coalition S, we shall show that $u_S\left(x^{\ast }\right)>u_S\left(x^{\prime }\right)$. For brevity, let us denote $k={\kappa }^{strict}(x^{\ast },N)$, $l={\kappa }^{strict}(x^{\prime },N)$, and $m={\kappa }^{strict}(x^{\prime },S)$. That is, k is the number of PSD strategies in $x^{\ast }$ played by players in the player set N; l is the number of PSD strategies in $x^{\prime }$ played by players in N; and m is the number of PSD strategies in $x^{\prime }$ played by players in the coalition S.

Under our conjecture that ${\sigma }_S^{\ast }$ is the strictly dominant strategy of S, ${\kappa }^{strict}(x^{\ast },S)=\left|S\right|$, and so we find accordingly that ${\kappa }^{strict}\left(x^{\ast },{\bigcup }_{T\in \pi \setminus \left\{S\right\}}T\right)=k-\left|S\right|$; that is, the number of players outside of the coalition S playing PSD strategies in $x^{\ast }$ must be the difference between the number of players in the entire player set playing PSD strategies in $x^{\ast }$ and the size of the coalition S. Observe also that ${\kappa }^{strict}\left(x^{\ast },{\bigcup }_{T\in \pi \setminus \left\{S\right\}}T\right)$ forms a weak lower bound on ${\kappa }^{strict}(x^{\prime },{\bigcup }_{T\in \pi \setminus \left\{S\right\}}T)$ since ${\sigma }_S$ may consist of individual player strategies that when combined with the strategies of players outside the coalition S are strictly dominant for that new coalition. Finally, since ${\kappa }^{strict}(x^{\prime },{\bigcup }_{T\in \pi \setminus \left\{S\right\}}T)$ is at most the difference between ${\kappa }^{strict}(x^{\prime },N)$ and ${\kappa }^{strict}(x^{\prime },S)$, it follows that $k-\left|S\right|\le l-m$.

Then, from the definition in (6), we have that $f_S\left(x^{\ast }\right)=\left|S\right|\left(\right|N|-k)$ and $f_S\left(x^{\prime }\right)=m\left(\right|N|-l)-\left(\right|S|-m)\left|N\right|$, and so $f_S\left(x^{\ast }\right)-f_S\left(x^{\prime }\right)=2\left(\right|S\left|\right|N|-m\left|N\right|)-\left|S\right|k+ml$. Since we have shown that $k-\left|S\right|+m\le l$, it follows that $f_S\left(x^{\ast }\right)-f_S\left(x^{\prime }\right)\ge \left(\right|S|-m)\left(2\right|N|-k-m)$. As the number of PSD strategies played by players in a coalition must clearly be bounded from above by the size of that coalition, $\left|S\right|\ge m$, $\left|N\right|\ge k$, and $\left|N\right|>m$. Thus, it must be that $f_S\left(x^{\ast }\right)-f_S\left(x^{\prime }\right)\ge 0$.

Now, suppose $x^{\ast }$ and $x^{\prime }$ are such that $f_S\left(x^{\ast }\right)=f_S\left(x^{\prime }\right)$. From the inequality derived above, this holds if and only if $m=\left|S\right|$ and $k=l$. In turn, this implies that ${\pi }^{\prime }={\left\{{\rho }_{x^{\prime }}^{strict}\left(i\right)\right\}}_{i\in N}$ is a strict refinement of ${\pi }^{\ast }={\left\{{\rho }_{x^{\ast }}^{strict}\left(i\right)\right\}}_{i\in N}$. Let us denote $\mathcal{P}={\left\{{\rho }_{x^{\prime }}^{strict}\left(i\right)\right\}}_{i\in S}$ to be the subset of ${\pi }^{\prime }$ for which ${\bigcup }_{T\in \mathcal{P}}T=S$. We consider the cases in which $k=\left|N\right|$ and $k<\left|N\right|$ separately.

In the first case when $m=\left|S\right|$ and $k=l=\left|N\right|$, we obtain from (7) that $g_S\left(x^{\prime }\right)={\sum }_{T\in \mathcal{P}}v(T,{\pi }^{\prime })$ and $g_S\left(x^{\ast }\right)=v(S,{\pi }^{\ast })$. By strict superadditivity, we require that ${\sum }_{T\in \mathcal{P}}v(T,{\pi }^{\prime })<v(S;{\pi }^{\ast })$, and so it must be that $g_S\left(x^{\prime }\right)<g_S\left(x\right)$ and $u_S\left(x^{\prime }\right)<u_S\left(x\right)$.

Alternatively, when $m=\left|S\right|$ and $k=l<\left|N\right|$, we have that $g_S\left(x^{\ast }\right)={\left|S\right|}^2$ and $g_S\left(x\right)={\sum }_{T\in \mathcal{P}}{\left|T\right|}^2$. Since $0<\left|T\right|<\left|S\right|$ for all $T\in \mathcal{P}$, it follows that ${\sum }_{T\in \mathcal{P}}{\left|T\right|}^2<({\sum }_{T\in \mathcal{P}}{\left|T\right|)}^2={\left|S\right|}^2$. Thus, ${g_S\left(x\right)<g}_S\left(x^{\ast }\right)$ and once again $u_S\left(x\right)<u_S\left(x^{\ast }\right)$.

Now, let us define the threshold value $\overline{\theta }\in {\mathbb{R}}^{+}$ by

$$\begin{array}{c}\hfill \overline{\theta }=max\left\{{\displaystyle \frac{g_S({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})-g_S({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}})}{f_S({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}})-f_S({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})}}\right|S\in 2^N,{\sigma }_S\in X_S,\\ \hfill {\sigma }_S\ne {\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}}\in X_{\pi \setminus \left\{S\right\}},f_S({\sigma }_S^{\ast },{\sigma }_{\pi \setminus \left\{S\right\}})\ne f_S({\sigma }_S,{\sigma }_{\pi \setminus \left\{S\right\}})\}.\end{array}$$

(8)

Note that $\overline{\theta }$ exists as the maximum is taken over a finite set; the size of the set is bounded by the number of distinct pure strategy profiles in $\Gamma (N,v,\theta )$. So, when $x^{\ast }$ and $x^{\prime }$ are such that $f_S\left(x^{\ast }\right)>f_S\left(x^{\prime }\right)$, $u_S\left(x^{\ast }\right)>u_S\left(x^{\prime }\right)$ as long as $\theta >\overline{\theta }$. At these values of $\theta$, the conjecture for $\Gamma (N,v,\theta )$ that every coalition has a strictly dominant strategy holds.

Since $\pi \left(x\right)={\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}$ is a bijection between the sets $\left\{x\right|{\kappa }^{strict}(x,N)=\left|N\right|\}$ and $\Pi \left(N\right)$, it follows from Lemmata 1 and 2 that there exists a bijection between the set of partitions and the set of CE of $\Gamma (N,v,\theta )$. Thus, for any embedded coalition $(S,\pi )$, there exists a unique CE of $\Gamma (N,v,\theta )$, x, such that ${\left\{{\rho }_x^{strict}\left(i\right)\right\}}_{i\in N}=\pi$. For such x, ${\kappa }^{strict}(x,N)=\left|N\right|$, and we have that $u_S\left(x\right)={\sum }_{i\in S}{\displaystyle \frac{v({\rho }_x^{strict}\left(i\right),\pi )}{|{\rho }_x^{strict}\left(i\right)|}}=v(S,\pi )$. We conclude that $\Gamma (N,v,\theta )$ is a representation of $(N,v)$ under NE for $\theta >\overline{\theta }$. □

The construction of $\Gamma (N,v,\theta )$ by (4)–(7) given any cooperative game $(N,v)$ and $\theta >\overline{\theta }$, for $\overline{\theta }$ defined in (8), can in fact be used to prove the existence of representations for other classes of cooperative games under other non-cooperative solution concepts, as the following theorems demonstrate.

Theorem 2.

Any weakly superadditive CFF game $(N,v)$ is represented under NE.

Proof. 

Consider the construction of $\Gamma (N,v,\theta )$ by (4)–(7), where we replace ${\kappa }^{strict}$ and ${\rho }^{strict}$ with ${\kappa }^{weak}$ and ${\rho }^{weak}$, which counts the number of PWD strategies in a strategy profile and maps a player to its coalition corresponding to its PWD, respectively. Observe that for any coalition S in $\Gamma (N,v,\theta )$, ${\sigma }_S^{\ast }$ is no longer guaranteed to be a strictly dominant strategy since weak superadditivity of the characteristic function implies that there may exist distinct strategy profiles $x^{\ast }$ and $x^{\prime }$ such that $f_S\left(x^{\ast }\right)=f_S\left(x^{\prime }\right)$ and $g_S\left(x^{\prime }\right)={\sum }_{T\in \mathcal{P}}v(T,{\pi }^{\prime })=v(S,{\pi }^{\ast })=g_S\left(x^{\ast }\right)$. By construction, this is the only case in which $u_S\left(x^{\prime }\right)<u_S\left(x^{\ast }\right)$ may not hold.

Here, $x^{\ast }$ and $x^{\prime }$ are both NE in the same composite game ${\Gamma }_{{\pi }^{\ast }}(N,v,\theta )$. Moreover, it must be true that ${\kappa }^{weak}(x^{\ast },N)={\kappa }^{weak}(x^{\prime },N)=\left|N\right|$. It follows that for any coalition $U\in {\left\{{\rho }_{x^{\ast }}^{weak}\left(i\right)\right\}}_{i\in N\setminus S}={\left\{{\rho }_{x^{\prime }}^{weak}\left(i\right)\right\}}_{i\in N\setminus S}$, their payoffs satisfy $f_U\left(x^{\ast }\right)=f_U\left(x^{\prime }\right)$, $g_U\left(x^{\ast }\right)=v(U,{\pi }^{\ast })=v\left(U\right)$, and $g_U\left(x^{\prime }\right)=v(U,{\pi }^{\prime })=v\left(U\right)$, as characteristic functions do not exhibit externalities. Hence, payoffs to coalitions are identical across both NE and remain equal to the values as determined by the worth function. □

Theorem 3.

Any strictly superadditive PFF game $(N,v)$ is represented under rationalizability.

Proof. 

Observe that under the conditions noted in Lemmata 1 and 2, an immediate corollary is that a strategy profile $x\in {\times }_{i\in N}{X}_i$ is a CE of $\Gamma$ under rationalizability if and only if ${\kappa }^{strict}(x,N)=\left|N\right|$. So, Theorem 3 follows directly from the proof of Theorem 1. □

Theorem 4.

Any weakly superadditive PFF game $(N,v)$ is represented under trembling hand perfect equilibrium.

Proof. 

As detailed in the proof of Theorem 2, we are no longer guaranteed that each coalition has a strictly dominant strategy when the worth function is weakly superadditive. However, it remains the case that each coalition has a weakly dominant strategy, which generates a unique trembling hand perfect equilibrium in each composite game. By construction, it remains the case that payoffs at these CE coincide with the coalitions’ values as defined by v. □

5. Discussion on the Construction of Representations

As the theorems above demonstrate, it is possible to provide a variety of rigorous micro-foundations for a large range of cooperative games. In doing so, this paper endeavors to bridge two seemingly disconnected fields in game theory. As part of this attempt, the term representation has been coined to denote such a bridge between cooperative and non-cooperative games. A non-cooperative game represents a cooperative game if it details the strategy sets needed to generate the worth of each coalition in such a way that is consistent with equilibrium. Conversely, a cooperative game can be thought of as representing a non-cooperative game if it summarizes the payoff outcomes from coalition formation and non-cooperative play.

To the best of the author’s knowledge, this exact vocabulary or something similar has not been used before in the literature. Despite this, there remains a long-standing historical precedence in the exercise of deriving cooperative games from non-cooperative foundations. For example, in their pioneering work of cooperative game theory, Von Neumann and Morgenstern (1944) imply a representation of characteristic function form games by zero-sum games where the solution concept is maximin. As part of the representation, they required that the worth of the coalition be defined by the maximin payoff they would receive when playing against the complementary coalition. They similarly made attempts to consider representations by n-player general-sum games. For example, it was suggested that a fictitious player be introduced such that the resulting $(n+1)$-player game was zero-sum. This fictitious player could not play strategies that influenced the non-cooperative outcome but could compensate players for their cooperation and thus could affect the cooperative outcome of the game. By applying the same procedure as before and equating each coalition’s worth with their maximin payoffs against their complement, Von Neumann and Morgenstern (1944) derived what they called an extended CCF game. Removing all coalitions containing the fictional player from the domain of the worth function led to the creation of the restricted CCF game.

Since then, various restrictions on the representation of cooperative games have been relaxed. In an early subsequent work, Aumann and Peleg (1960) also considered general-sum games under maximin outcomes but relaxed the assumption that strategic interaction was limited to a coalition and its complement. A coalition S was considered $\alpha$-effective for some payoff $u_S$ if there existed a strategy for S that guaranteed a payoff of $u_S$, regardless of the strategies played by its complement. If all coalitions were $\alpha$-effective, the characteristic function would be well-defined and would set the worth of each coalition S to be the largest such $u_S$. Unsurprisingly, such cooperative games became known as $\alpha$-CFF games. Aumann and Peleg (1960) also laid the foundations for the $\beta$-CFF games, analogously defined using minimax outcomes. Since maximin and minimax outcomes may not coincide in a general-sum game, there is no guarantee that an $\alpha$-CFF game is mathematically identical to a $\beta$-CFF game despite sharing the same non-cooperative representation.

It was not until seminal papers such as Hart and Kurz (1983) and Ichiishi (1981) that the solution concept used for general-sum games changed from maximin to NE. This mathematical formalization was later picked up by papers such as Chander and Tulkens (1997), which assumed that the worth of a coalition S would be their Nash equilibrium payoff when playing against the remaining players acting as singleton coalitions. Cooperative games derived in this way were named $\gamma$-CFF games.

This paper builds on these predecessors by considering representations not only of CFF-games but also of PFF-games. This broader class of games relaxes the assumption of independence of the worth function and the player partition, resulting in the possible accommodation of externalities from coalition formation and dissolution. Although CFF-games are perhaps a more popular choice of model in cooperative game theory, PFF-games arguably offer greater realism in being able to model situations in which coalitional payoffs are additionally influenced by the extent of cooperation among players outside the coalition. These situations span a broad range of economic topics such as multilateral trade (Strantza et al., 2020; Yi, 1996), imperfect competition (Abe, 2021; Chander, 2020), and public goods provision (Abe & Funaki, 2020). In many of these cases, players free-ride off of a more profitable environment when coalitions are few in number. It is worth nothing that no PFF-game that exhibits externalities can be represented by a non-cooperative game when the solution concept is maximin. The worth prescribed to a coalition would be the best collective payoff it can achieve regardless of the behavior of the other coalitions. By definition, this excludes the representation of externalities.

While the approach of this paper has been to provide greater generality to the respresentation of cooperative games, the results of this paper do not exist without its assumptions. We address the realism of some of these assumptions here. Firstly, we assume that cost-free agreements to form coalitions can be made or equivalently that enforcement of these agreements is costless. This is often a common assumption in the literature used to motivate cooperative games (Ray & Vohra, 1999). Although the assumption may appear on the surface to be unrealistic, it is worth noting that any costs incurred in coalition formation can be subsumed by the worth of the coalition. Consequently, this assumption is no more than a renormalization of the worth function and has no effect on the existence of representations. Secondly and similarly, we also assume external enforcement of these agreements, meaning that these agreements do not have to bind through a mechanism of self-enforcement such as credible threats. Again, this is a common assumption when motivating the worth function. But it is important not to confuse this with the assumptions that motivate the solution to a cooperative game. To take an obvious example, many variants of the core that have been proposed as solutions to PFF games rely on expectations as to how the partition of the player set looks following a deviation from the grand coalition (Bloch & Van den Nouweland, 2014). Clearly, this suggests that from the perspective of the players, any agreement to form the grand coalition is not truly binding. However, this does not detract from the idea that the worth of the grand coalition can be understood to be what players can feasibly achieve if they have a binding agreement towards acting in the collective best interests.

Finally, it is also worth emphasizing what this paper does not attempt to perform. Although the proof of existence of a representation relies on a specific construction involving every coalition having dominant strategies, it does not claim that all representations must have this structure, which is admittedly a rare occurrence in many economic situations. All it demonstrates is the very general sufficient conditions under which the cooperative game, which can be thought as being a reduced-form model of economic behavior, has a structural interpretation. It makes no comment as to whether or not this structural interpretation is suitable for the problem at hand. For example, the existence of a non-cooperative representation under a variety of non-cooperative solution concepts is proven in this paper, but clearly the appropriate choice of the solution concept depends on the economic setting. Minimax as a solution concept is, for instance, more sensible in the domain of zero-sum games than general-sum games.

To illustrate this, consider the example of oligopolistic competition where firms operate with different marginal costs of production. In both Cournot and Bertrand competition, any cartel can threaten to reduce the profits of the other firms to zero either by flooding the market with goods to drive the price down or by undercutting all their competitors to gain complete market share. Such threats are hardly credible, and yet the cooperative game represented under maximin outcomes treats firms as taking these threats seriously. In particular, all cartels bar the grand cartel guarantee themselves zero; only the grand cartel is unexposed to threats and thus guarantees itself monopoly profits. As noted in Ray (2007), such a worth function delivers cooperative solutions that make little sense from an economic perspective. For example, by the symmetry of the worth function, the Shapley value of this game allocates each firm an equal share of the monopoly profit, despite there being heterogeneity in production costs across firms. Perhaps more striking is that the core admits any non-negative imputation, so even one that allocates the minimal surplus to the most efficient producer will supposedly not face rejection.

6. Conclusions

In the original conception of cooperative game theory, Von Neumann and Morgenstern (1944) demonstrate that some CFF games can be represented by zero-sum games where the players can form coalitions and play against their complement. They defined the worth of that coalition to be their maximin payoff. This paper generalizes this notion in three different ways: it asks what can be said of cooperative games with externalities, i.e., PFF games; it asks what happens if the non-cooperative representation was general-sum instead of zero-sum; and it asks what happens if the worth function was not defined by maximin payoffs but instead payoffs at NE and at generalizations and refinements thereof. In particular, this paper proves the existence of non-cooperative representations of cooperative games under a variety of solution concepts through the construction of a particular equivalence class of non-cooperative games. The construction is such that every coalition has a dominant strategy, which admittedly is a very particular requirement. Further research could seek to identify other equivalence classes with less restrictive requirements on the strategy sets and payoff functions that can similarly represent cooperative games. Moreover, this paper also limits its attention to non-cooperative representations that assume complete and perfect information. Potential extensions of this work including researching Bayesian game representations of cooperative games: can cooperative games be represented under incomplete information and belief-dependent strategies? Such generalizations can help further bridge the gap between cooperative and non-cooperative game theory.