# Cooperative Game and the Law

**DOI:**https://doi.org/10.1007/978-1-4614-7883-6_635-1

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## Keywords

Cooperative Game Theory Shapley Value Grand Coalition Noncooperative Game Approach Coase Theorem## Definition

While noncooperative game theory applied to the law is now a subfield of law and economics literature, cooperative game theory has a more strange history. Many of the founding fathers of the cooperative game theory (Shapley, Shubik, Owen, Aumann) were interested in legal examples to illustrate their games; however, law and economics literature has not systematically investigated the meaning of cooperative game theory for the law and is still mostly noncooperative oriented. The aim of the entry is to draw a general picture of what cooperative game theory may add to the law and economics literature. We focus on the positive and normative aspects of cooperative game theory, and we provide illustrative examples in different fields (private law, public law, regulation, theory of the law).

### Cooperative Game Theory and the Law: A Missed *Rendezvous*?

In their famous book *Game Theory and the Law* published in 1994, Baird, Gertner, and Picker said nothing about cooperative games and the law (Baird et al. 1994). The authors mainly focused on noncooperative games and gave to the Nash equilibrium the most preeminent role to understand strategies of legal players (plaintiffs, defendants, and judges). Most of the subfields of law and economics have been deeply changed by the use of game theory in place of the Chicago-style price theory. Compared with noncooperative game theory, cooperative game theory is still underestimated. From a historical perspective, here lies a paradox. While most of the classical games studied by cooperative branch of game theory have obvious consequences for the law (the ownership game by Shapley and Shubik 1967; the bankruptcy game by Aumann and Maschler 1985; the airport games by Littlechild and Thomson 1977), no structured paradigm on law/economics/cooperative games has emerged compared to the noncooperative game approach. The first reason lies in the fact that the legal examples modeled with cooperative game theory were not explicitly oriented in a law and economics perspective. The second reason is that, at the very beginning of the law and economics movement, leaders of the field, like Posner or Calabresi, were almost exclusively interested in the efficiency of the common law (maximization of wealth) which is not an issue addressed by cooperative game theorists.

Much has changed in recent times. A lot of new and original models focus on unprecedented applications that can be envisaged through cooperative game theory as applied to private law, public law, regulation, and legal theory. These legal-oriented models may overlap applications in public economics or industrial organization (imperfect competition, public goods, matching, and networks). We have chosen to focus on some of the most suggestive applications of cooperative game theory for the law.

As we would like to avoid non-useful technical complexities, we deal with the most basic and simple games to show how they renew our ideas on what economics may add to the law. More importantly, we insist on the twofold features of cooperative game theory which can be considered from a positive point of view (how people cooperate and share the surplus created by cooperation) and from a normative view (how a judge or an arbitrator should settle a case when several people are in conflict about how to share a joint surplus).

First, definition and notation are introduced in order to better understand what cooperative game theory is. Second, we deal with more contemporary works using cooperative game to highlight regulation and public law issues. Third, we show that cooperative game approach is useful to renew private law and mainly torts. Last, we give some intuitions at a more abstract level to see how the legal theory could be influenced by the cooperative game approach.

### Definition, Notation, and Meaning

A transferable utility game (TU game) is a couple (*N,v*) with *N* the set of players and *v* the characteristic function. The two basic blocks of cooperative game theory are the coalitions and the characteristic function. The Grand coalition {1, 2, …*n*} is the coalition of all the players. The singletons {i} are coalitions restricted to single players. There are also all the subsets of players between singletons and *N*. With *n* players, there are 2^{n} − 1 coalitions. The worth of the Grand coalition is *v(N)* and is equal to the worth to be shared among players. The worth *v(i)* is what player *i* could get if he decided to behave on his own. More generally, the worth *v(S)*, with *S* a coalition of players, is what the coalition *S* is able to get for its members. The characteristic function associates to each coalition its worth but says nothing about how this worth is then shared among the members of *S*. Mostly, a hypothesis of transferability holds. Transferability is a serious assumption and states that utility is measurable in terms of money: side payments are possible among the players.

Once coalitions and characteristic function are defined, then the most important challenge for cooperative game theory is to solve the game that is to say to find a vector of payment (x_{1}, x_{2},…x_{n}) for the players. At first sight, infinity of vectors could work. But, two rationality requirements will be added. First is collective rationality condition: the sum of the payments should be necessarily equal to the worth of the Grand coalition (no more, no less). The vector of payment should be feasible (it is impossible to give players more than what is created by the Grand coalition) and should ensure that resources are not wasted (it would be irrational to give them less than the surplus jointly created). Second is the individual rationality condition: the payment of a player *i* should be more important that its worth *v(i)*: without this condition, a single player would have no incentive to cooperate with others. The vectors which satisfy these two conditions are called imputations.

*C(N,v)*is defined as follows:

Behind the core is the idea of stability. In case of non-empty core, no individual nor any coalition has an interest to leave the Grand coalition for their own. On the contrary, an empty core means no guarantee on the stability of cooperation among players: some of them have an incentive to leave the Grand coalition and get more, out from the Grand coalition. Cooperative game theorists have then studied the conditions under which the core is non-empty (e.g., when games are convex – that is to say *v*(*S*) + *v*(*T*) ≤ *v*(*S*∪*T*) + *v*(*S*∩*T*) for all *S* and *T* – the core is non-empty). Still, the set of the core may be large enough.

It is possible to define other concept solutions with an axiomatic perspective. A trade-off arises between “not enough” and “too much” axioms. If few axioms are required, it will be easy to find allocations which solve the game, but the set of solutions is likely to be too large. If too much axioms are required, the set of solution is likely to be empty. Each axiom is debatable on rational and normative grounds.

A particularly interesting rule to solve a game is the Shapley value. The Shapley value may be explained in two alternative ways. First, the Shapley value depends on the marginal contribution of players to the coalitions. For a player *i*, its marginal contribution to a coalition *S* is the difference between the worth of the coalition *v(S)* and the worth of the coalition *v(S\i)*. The Shapley value allocates to player *i* his average marginal contribution. Second, the Shapley value – defined on any cooperative game – follows four axioms which characterize uniquely this rule. The first is efficiency (the worth of the Grand coalition should be shared); the second is the null player axiom (players who contribute zero to any coalition get nothing back); the third is the additivity axiom (for a game that is the sum of two games, the Shapley value for the former is the sum of the Shapley values for the latters); and the fourth is symmetry (two substitutable players should receive the same payoff).

Beyond the core and the Shapley value, other sharing rules are discussed in the literature (e.g., the nucleolus). More recently, new paths in cooperative game theory have been developed: games with a priori unions (Owen 1977) or graph-restricted games (Myerson 1977). In these games, a structure of cooperation is defined through a network of preexisting links, and the value is calculated on this structure.

### Cooperative Games, Antitrust, and Regulation: From Stability to Fairness

A contemporary perspective on cooperative game theory and the law would consider that public regulation is one of the main fields where cooperative game theory is useful. In part I, we have shown that cooperative game theory may be considered from a twofold perspective: first, it models “situations in which the players may conclude binding agreements that impose a particular action or a series of actions on each player” (Maschler et al. 2013); second, axiomatization of solution concepts indicates how a judge or an arbitrator should allocate the value among the players. The first perspective could be said positive and the second normative. As soon as public agencies aim at regulating the behaviors of individual or firms implied in a common activity, cooperative game theory has something to say about the following: (1) Is cooperation among players stable? (2) How will be (or should be) the surplus due to cooperation shared?

A first subfield is antitrust. Collusion and anticompetitive practices may be considered as cooperation among players: colluders coordinate their actions in order to get monopoly profits. Some of antitrust scholars assert that cooperative game theory is irrelevant insofar as collusion being illegal, there is no way to conclude binding agreements; on the contrary others consider that the non-emptiness of the core is useful to better understand the stability of cartels and the efficiency of anticompetitive practices (Telser 1985; criticized in Wiley 1987). Analyzing the famous *Addyston Pipe* case, one of the most famous cases in antitrust according to Judge Bork, Bittlingmayer and Telser argue that cartelization may be useful for the firms to share fixed costs (Bittlingmayer 1982). This cooperation avoids the inefficiencies in sectors where marginal cost is under the average cost. This idea is followed by contemporary literature on oligopoly games. Both Cournot and Bertrand oligopolies are concerned. The main issue addressed is the stability of collusion. Some of anticompetitive behaviors (as prohibitive restrictive agreements or concerted practices) imply that players coordinate their strategies to get their best outcome and may organize side payments. In some models, a hypothesis of sharing the best technology is done, but it is not a necessary condition (Lardon 2017). The stability of the agreement among members of the oligopoly is one of the key elements of this literature. The core is consequently the most studied solution: if the core is non-empty, stability of the cartel is expected; on the contrary, the emptiness of the core implies that the cartel is unstable insofar as some players or groups of players have an incentive to leave the Grand coalition. Zhao applies this reasoning to the sugar cartel in the USA at the end of the nineteenth century. For this author, the increasing of the cartel organizational costs due to the Sherman Act made the core empty and lead to instability (Zhao 2014). What is interesting in this literature of oligopoly games is to work out different scenario implying different blocking rules. Indeed, when a player or a group of players decide to leave the Grand coalition, the interaction between them and the remaining players becomes strategic, and several strategies are conceivable. For example, they may behave to maximize their own utility or to minimize the utility of the other players. The conclusions drawn are particularly interesting for antitrust authorities regarding the best way to enforce antitrust law and to enhance instability of collusion.

A second subfield is concerned with public regulation, public utilities, and facilities at large. In many contexts covered by public regulation, agents (firms, individuals, or groups like municipalities) cooperate and are looking for the best way to share their costs: fisheries conferences, commons, pools, water or telecom networks, facilities that will be jointly used, and water from a river at a national or international level are some of the numerous examples of such situations. Take the example of the airport fees analyzed by Littlechild and Thomson (1977) which deals with how to share the cost of a common facility (a runaway). Assume three aircraft companies which cooperate to build a new runaway. Due to the size of the planes they own, the first company needs a small airstrip, the second a medium one, and the third a large one. The cost of each airstrip is *c1 < c2 < c3*. If firms do not cooperate, they bear a total cost of *c1 + c2 + c3*, while cooperation leads to a total cost equals to *c3* (we suppose that there is no congestion and the largest airstrip is enough to land all the aircrafts whatever their size is). The core is non-empty but large enough and let unsolved the precise cost paid by each firms. A regulator could use more specific sharing rule. A natural solution would say that the total cost *c3* should be divided as follows: the smallest part of the airstrip is used by all the companies: *c1* should be equally divided among the three companies. The additional cost *(c2*−*c1*) should be paid only by the companies 2 and 3. The additional cost (*c3*−*c2*) should be paid by the third firm (the only firm which needs a large airstrip). This intuitive solution is the Shapley value of the airport game. As such, the Shapley value is a fair compromise that a regulator could implement to sharing the costs among firms.

Sometimes, legislators and regulators play a major role in cooperation. This is the case when regulated firms are required by the law to cooperate. The REACH legislation in the European Union (Registration, Evaluation, Authorization and Restriction of Chemicals) is very illustrative. The European Union has decided in 2006 to make a systematic evaluation of the toxicity of chemicals. The number of chemicals is so high that it is impossible for public authorities to evaluate by themselves their danger. At the same time, firms privately own information regarding the toxicity of chemicals (scientific reports, experiences, etc.). REACH legislation creates SIEF which are forum to organize the exchange of information among firms. The difficulty lies in the side payments: some firms may provide very valuable data, some add information already get by others, some have no information at all, etc. Based on the costs of replication of data, Dehez and Tellone (2013) provide a cooperative game model and advocate the Shapley value as a fair compromise to calculate payments/rewards of each participant to a SIEF. Beal and Deschamps follow this path and discuss different rules in order to compare their axiomatic properties (Beal and Deschamps 2016).

Last, and more recently, a third important subfield using cooperative game theory has grown: the implementation of public matching algorithms. Regulators may face complex issues of matching the two sides of the market. In a famous paper, Roth studied the National Resident Matching Program in the USA that aims at matching hospitals with resident doctors (Roth 1984). The NRMP is a centralized “clearinghouse” introduced in the 1950s to allocate resident doctors to hospitals. Roth discovered that the algorithm used in the NRPM is very close to a Gale-Shapley algorithm (Gale and Shapley 1962) that aims at finding stability allocation (stable matches means that no matched couple would like to break up and forms new matches to be better off). The key aspect from the regulator perspective is that there are several stable sets with different normative properties regarding the side of the market which is favored. Such public algorithms are now implemented by the law in many fields including health care, education (universities and applicants), labor markets, etc.

### Cooperative Game Theory and Private Law

Private law – property, torts, and contracts – is also a field studied by cooperative game theory. First, property rights are concerned. In a famous paper on land ownership, Shapley and Shubik (1967) show how different types of property (feudal world, private property, village commune, corporate and joint ownership, etc.) may be modeled in terms of cooperative games. Shapley and Shubik show how the characteristic function has to be changed to well describe different types of property rights.

Second, tort law has been studied through cooperative game theory. Here too, literature is divided in a positive branch and a normative one. From the positive point of view, cooperative game theory has added to a better understanding of the Coase theorem. The Coase theorem is well known in law and economics: in case of zero transaction costs, competition, and perfect delineation of property rights, agents may reach a mutually advantageous agreement. The level of externality is independent of the initial distribution of property rights and maximizes the value of the production (the result is invariant and efficient). Aivazian and Callen (1981) reconsidered the issue and show that the Coasean result is not robust (Coase 1981) when there are more than two players (e.g., several polluters and one victim): the emptiness of the core leads to the impossibility to reach a stable agreement (players have incentive to block any agreement with another coalition). More recently, Aivazian and Callen (2003); Gonzalez et al. (2016); Gonzalez and Marciano (2017), and others provided more complex examples with several victims. The results converge: the Coase theorem does not hold insofar as the delineation of property right influences the emptiness of the core. More importantly, positive transaction costs lead unexpected results: some authors consider that positive transaction costs make the stability of agreement more likely (precisely because renegotiation is costly), and others think not.

In a different perspective, Dehez and Ferey use cooperative game theory from a purely normative point of view as a description of legal adjudication (Ferey and Dehez 2016a, b). The cases studied are tort cases implying multiple tortfeasors who jointly cause a unique harm to a victim or a group of victim. They first consider the Shapley value to estimate the part of the damage to be paid by each tortfeasor, and then they show that the principles of the American Restatement are consistent with the Shapley value principles. Concept solutions are here considered from a normative perspective to better solve conflicts among defendants about their respective shares of responsibility and to describe the behaviors of judges.

In the same vein, a third interesting example regarding private law is about debt and insolvability. In a famous model, Aumann and Maschler (1985) address the issue of the burden of insolvability of a firm. In that case, suppose that players are creditors and get claims against a firm (the debtor) which is unable to pay all its debts back. Suppose *E* be the total amount of value available (the value of the assets left) and suppose *d1*, *d2*, and *d3* the claims of the debtors 1, 2, and 3 with *d1 + d2 + d3 > E*. In that case, the worth of each coalition *S* is defined as the maximum that *S* can get once all the others creditors (*N\S*) get their claims back, *d(N\S)*. Formally, *v(S) = Max (0, d(N\S))*. The authors compare different solution concepts and show that the rules advocated in the Talmud is close to the nucleolus. Their paper is not explicitly oriented in a law and economics perspective, but they suggest that cooperative game theory is very useful to better understand the deep reasons of legal and/or moral reasoning.

Many other examples from private law could be added: patents (several firms cooperate to get a patent), condo (several people have to share the costs of a condo, Crettez and Deloche 2014), or even the contracts used by bitcoin miners when they pool together to find the relevant hash and have to share the bitcoins get in common. These examples show how fruitful cooperative game theory is for the law.

## Conclusion: Cooperative Games and Legal Theory

Law seems to be a quite natural field of application of cooperative game theory. We have provided many examples of legal topics studied with cooperative game theory. To conclude, we would like to add some more speculative views. Is cooperative game theory descriptive, predictive, or normative (Aumann 1985)? Sure, in some cases, cooperative game theory is useful to predict how agents will behave; in other cases, the normative aspect of cooperative game theory is more interesting and provides some axiomatic solutions that help judges or arbitrators to settle conflicts. As Aumann states, “Normative aspects of game theory may be subclassified using various dimensions. One is whether we are advising a single player (or group of players) on how to act best in order to maximize payoff to himself, if necessary at the expense of the other players; and the other is advising society as a whole (or a group of players) of reasonable ways of dividing payoff among themselves. The axis I’m talking about has the strategist (or the lawyer) at one extreme, the arbitrator (or judge) at the other” (Aumann 1985).

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