The title ‘cooperative games’ would be better termed games in coalitional form. The theory of games originally developed different conceptual forms, together with their associated solution concepts, namely, games in extensive form, in strategic form, and in coalitional form (von Neumann and Morgenstern 1944). The game in strategic form is sometimes referred to as the game in normal form, while that in coalitional form is also referred to as the game in characteristic form.

The game in extensive form provides a process account of the detail of individual moves and information structure; the tree structure often employed in its description enables the researcher to keep track of the full history of any play of the game. This is useful for the analysis of reasonably well-structured formal process models where the beginning, end and sequencing of moves is well-defined, but is generally not so useful to describe complex, loosely structured social interaction.

A simple example shows the connections among the three representations of a game.

Consider a game with two players where the rules prescribe that Player A moves first. He must decide between two moves. After he has selected a move, Player B is informed and in turn selects between two moves. After B has selected a move the game ends and depending upon the history of the game each player obtains a payoff. Figure 1a shows this game in extensive form. The vertex labelled 0 indicates the starting point of the game. It is also circled to indicate the information structure. Figure 2a shows a game whose only difference from the game in Fig. 1a is that in the latter Player B when called upon to select a move does not know to which of the choice points in his information set the game has progressed. In the game in Fig. 1a, when Player B makes his choice he knows precisely if Player A has selected move 1 or 2. Each vertex of the game is a choice point except the terminal vertices. Several vertices may be enclosed in the same information set. The player who ‘owns’ a particular information set is unable to distinguish among the choice points in a set. An arc (or branch of a tree) connecting a choice point with another choice point or a terminal point is a move. The moves emanating from any choice point are indexed so that they can be identified.

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Cooperative Games, Fig. 1

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Cooperative Games, Fig. 2

The final nodes at the bottom of the tree are not choice points but points of termination of the game and the numbers displayed indicate the value of the outcome to each player. The first number is the payoff to Player A and the second to Player B.

The extensive form may be reduced to the strategic form by means of strategies. A strategy is a plan covering all contingencies. Figure 1b shows that the moves and strategies for PA are the same, choose 1 or 2. But PB has four strategies as he can plan for the contingency that PB selects 1 or 2. A sample strategy 1, 1; 2, 1 may be read as: ‘If PA selects 1, select 1; if PA selects 2, select 1’.

The progression from extensive form to strategic form entails loss of fine structure. Details of information are no longer available. There are many extensive forms other than Fig. 1a which are consistent with Fig. 1b.

A further compression of the game representation beyond the strategic form may be called for. At the level of bargaining or diplomacy details of strategy may be of little importance. Instead emphasis is laid upon the value of cooperation. The cooperative or coalitional form represents the game in terms of the jointly optimal outcomes obtainable by every set of players. If payoffs are comparable and side-payments are possible the gain from cooperation can be represented by a single number. If not then the optimal outcomes attainable by a set S of players will be a Pareto optimal surface in s = |S| dimensions (where |S| is the number of elements in S).

A game in cooperative form with side-payments can be represented by a characteristic function which is a superadditive set function. We use the symbol Γ(N, υ) to stand for a game in coalitional form with a set N of players and a characteristic function v defined on all of the 2n subsets of N (where n = |N|). The condition of superadditivity is a reasonable economic assumption in a transactions cost-free world. \( \upsilon (S)+\upsilon (T)\le \upsilon \left(S\cup T\right) \) where S ∩ T = θ states that the amounts obtained by two independent coalitions S and T will be less than or at most equal to the amount that they could obtain by cooperating and acting together.

Returning to Figs. 1b and 2b we can reduce them to coalitional form by specifying how to calculate \( \upsilon \left(\theta \right),\upsilon \left(\overline{1}\right),\upsilon \left(\overline{2}\right)\mathrm{and}\;\upsilon \left(\overline{1,2}\right) \). The notation ‘\( \overline{1,2} \)’ reads as the set consisting of the players whose names are 1 and 2.

Let \( \overline{S}=N\hbox{--} S \) be the complement to S. The worst that could happen to S is that \( \overline{S} \) acts as a unit to minimize the joint payoff to S. Applying this highly pessimistic view to the games in Figs. 1b and 2b letting PA = 1 and PB = 2 we obtain the following:

$$ {\displaystyle \begin{array}{l}\upsilon \left(\theta \right)=0,\mathrm{the}\,\, \mathrm{coalition}\,\, \mathrm{of}\,\, \mathrm{no}\,\, \mathrm{one}\,\, \mathrm{obtains}\,\, \mathrm{nothing},\hfill \\ {}\mathrm{by}\ \mathrm{convention}.\hfill \\ {}\upsilon \left(\overline{1}\right)=0,\upsilon \left(\overline{2}\right)=0\hfill \\ {}\upsilon \left(\overline{1,2}\right)=12\hfill \end{array}} $$

Although the extensive and strategic forms of these games differ, they coincide in this coalitional form. More detail has been lost. The coalitional form is symmetric but the underlying games do not appear to be symmetric. The pessimistic way of calculating υ(S) may easily overlook the possibility that it is highly costly for \( \overline{S} \) to minimize the payoff to S. Thus it is possible that υ(S) does not reflect the threat structure in the underlying game. Prior to carrying out further game theoretic analysis on a game in characteristic function form the modeller must decide if the characteristic function is an adequate representation of the game. Harsanyi and Selten have suggested a way to evaluate threats (see Shubik 1982).

Applications

Depending upon the application, the extensive, strategic or coalitional forms may be the starting point for analysis. Thus in economic applications involving oligopoly theory one might go from economic data to the strategic form in order to study Cournot-type duopoly. Yet to study the relationship of the Edgeworth contract curve to the price system one can model the coalitional form directly from the economic data without being able even to describe an extensive or strategic form.

In any application, the description of the game in coalitional form is a major step in the specification of the problem. After the coalitional form has been specified a solution is applied to it. There are many solution concepts which have been suggested for games in coalitional form. Among the better known are the core, the value, the nucleolus, the kernel, the bargaining set and the stable set solutions. Only the core and value are noted here (for an exposition of the other see Shubik 1982).

The core of an n-person game in characteristic function form was originally investigated by Gillies and adopted by Shapley as a solution. The value was developed by Shapley (1951) and has been considered in several modifications to account for the presence or absence of threats and sidepayments.

We define \( \alpha =\left({\alpha}_1,{\alpha}_2,\dots, {\alpha}_n\ \right) \) where αi ≥0 for all i∈N and \( {\Sigma_i}_{\in N}{\alpha}_i=\upsilon (N) \) to be an imputation for the game Γ(N, υ) It is an individually rational division of the proceeds from total cooperation. The core is the set of imputations such that \( {\Sigma_i}_{\in S}{\alpha}_i\ge \upsilon (S) \) for all S ⊂ N. It is, in some sense, the set of imputations impervious to countervailing power. No subset of players can effectively claim that they could obtain more by acting by themselves. The core may be empty. An exchange economy with the usual Arrow–Debreu assumptions modelled as a game in coalitional form always has a core, and the imputation (or imputations) selected by the competitive equilibria of an exchange economy are always in the core of the associated market game.

The Shapley value is intuitively the average of all marginal contributions that an individual i can make to all coalitions. He developed the explicit formula to calculate the value imputation for any game in coalitional form with sidepayments. It is

$$ {\phi}_i=\sum_{i\in S}\sum_{S\subset N}\frac{\left(n-s\right)!\left(s-l\right)!}{n!}\left[\upsilon (S)-\upsilon \left(S/i\right)\right] $$

The term \( \upsilon (S)-\upsilon \left(S/i\right) \) measures the marginal contribution of i to the coalition S. The remaining terms provide the count of all of the ways the various coalitions involving i can be built up. For exchange economies with many traders a relationship between the competitive equilibria and the value can be established (for further discussion, see Shubik 1984).

Many situations involving voting can be modelled as a game in coalitional form where the characteristic function takes only two values, 0 and 1. Such games are called simple games (Shapley 1962). Shapley and Shubik (1954) suggested the use of the value to provide a power index for committee voting. The basic observation is that the power of a player increases in a nonlinear manner as the number of votes he controls increases. The value applied to a simple game provides an index of this power.

Cooperative games provide a way to carry out an analysis of many problems of interest to the social sciences without concern for the detail of the structure of process. Von Neumann and Morgenstern aptly noted that the difficulties to be encountered in the development of theories of dynamics in the social sciences were so large that the development of a primarily static theory of games in cooperative form was called for as a first step, bearing in mind that the eventual form of a theory of dynamics might have little resemblance to the statics. Some forty years after their seminal work much still remains to be done in the development of games in coalitional form.

See Also