Game theory in jurisprudence

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by Wojciech Zaluski


I Introduction

Game theory (GT) is a mathematical tool for studying situations where the decision of one person depends on what that person expects the other persons to do (the strategic situations); GT offers a classification of these situations and specifies how rational agents should act in them. The role of GT in legal analysis is both descriptive and normative. In its descriptive role, GT helps, first, to clarify the nature of law by treating it as a “device” that emerged for resolving collective action problems, and, second, to analyse specific legal rules by making predictions as to how they are likely to influence the way people behave; in these contexts, GT is strictly connected with the economic analysis of law: it assumes its basic tenet saying that the law should contribute to the maximization of social wealth. In its normative role, GT itself generates normative criteria for assessing legal rules and decisions. GT also sheds light on the nature of rights.

 

II Philosophical assumptions and the main solution concept of GT

The main philosophical assumptions of GT are: 

1) Agents are utility-maximizers – they act as though they wanted to maximize the satisfaction of their preferences defined over the consequence of their actions. Agents’ preferences are described in accordance with the theory of revealed preference, i.e. “deduced” from their choice behaviour. It is usually assumed that these preferences are self-regarding. 

2) Agents are perfectly rational – they behave in accordance with GT’s solution concepts and have common knowledge of their rationality.

These assumptions can be criticized on the following grounds. 

Ad 1) As experimental research (e.g. of Kahnemann and Tversky) shows, agents’ behaviour systematically violates this assumption. Furthermore, this assumption discounts the fact that agents may derive utility, not only from the consequences of their actions, but also from actions themselves (symbolic utility).

Ad 2) GT puts too strong demands upon agents’ capacity to act rationally and to gain information and thereby is likely to provide inaccurate predictions regarding their behaviour. 

In response to these objections, a defender of GT may point out that modern GT is a very flexible tool, because it, e.g., succeeds in modelling agents’ bounded rationality, i.e. the goal-directed behaviour of agents who are limited in their capacity to act rationally; provides models of situations where an agent possesses incomplete information (i.e. ignores the structure of the game – the strategies available to the players or their payoffs); does not imply that the solution concepts of GT model mental processes whereby agents make their decisions, but only that these decisions are likely to be the same as those which they would make were they to consciously use these concepts. 

The basic solution concept of GT is the concept of a Nash equilibrium (NE). A NE – to be reached by rational players – is a steady state of a game, i.e. one from which no player has an incentive to deviate given the strategies chosen by the other players. 

There are, however, strategic situations which (1) possess a unique NE, which is yet not Pareto-optimal[1]; (2) possess multiple NE each of which agents are likely to adopt; (3) possess multiple NE some of which agents are unlikely to adopt as resting on incredible threats or beliefs. The situations (3) testify to the weakness of the concept of a NE; accordingly, they can be eliminated by supplementing it with some additional requirements of rationality.[2] The situations (1) and (2), though, cannot be eliminated in this way, since their existence is a reflection of real dilemmas faced by agents rather than a result of some weakness of the concept of a NE; these situations are described in III 1., as they are intimately connected with the problem of the emergence of legal norms. 


III Descriptive role of GT

1. GT as a tool for describing the emergence of legal norms

It is plausible to argue that legal norms emerged as a “device” for resolving collective actions problems (CAPs), i.e. problems arising when agents have incentives to take actions that are not in their joint interest. GT gives a precise description of these problems and explains in what way they can be resolved by means of legal norms. 

CAPs modelled by the Prisoner’s Dilemma game

The simplest model of CAPs is a two-person version of the Prisoner’s Dilemma. Two players have a choice over two strategies: C – cooperate, and D – defect. A game is the Prisoner’s Dilemma if each player has the following preference-ordering over the possible results of their choices (the ordering is presented from the viewpoint of Player 1): T > R > P > S, where T is Temptation (arising if Player 1 defects, and Player 2 cooperates), R is Reward (arsing if both players cooperate), P is Punishment (arising if both players defect), and S is the Sucker’s result (arising if Player 1 cooperates, and Player 2 defects). The matrix is therefore as follows[3]:

Player 1/Player 2 C D
C R, R S, T
D T, S P, P

Let us give an example of this game. Tom and John are hot-tempered men, detesting each other. A tradesman visits their small town and offers each of them a gun. Thus, they face the following choice: to buy (D) or not to buy (C) a gun. For each player the best situation is that he plays D, while his opponent plays C (the player buying a gun gains advantage over the other player); each prefers the situation “both play C” to the situation “both play D” (if both play D, there will arise the danger of their life-and-death gun fight); and each most fears the situation in which he plays C, while his opponents plays D. The game is therefore the Prisoner’s Dilemma. 

The problem is that the only steady state of the Prisoner’s Dilemma, i.e. its NE, is {D, D}: no player is tempted to switch to the strategy C, given the strategy D of the other player.[4] This result, however, fails to be Pareto-optimal, since each player’s payoff would be increased if each of them chose C. The players can be motivated to choose C by a legal norm either providing sanctions for and thus raising the costs of playing D or prohibiting selling guns and thus eliminating the whole situation. 

The Prisoner’s Dilemma serves also as a model of multiple other situations, especially those where there is some scarce good and agents have to restrain their consumption of it if they want to avoid the worst – {D, D} – result – namely, a fast exhaustion of the good. 

CAPs modelled by coordination games

There exists a type of CAPs – free from an inherent antagonism of the Prisoner’s Dilemma – that can be modelled by coordination games, i.e. games with at least two NE. An example may be useful. Two players live on the same bank of the river which tends to cause floods. Thus, it will be in the interest of each of them to decide to build a levee, yet only if the adjacent player decides to build a levee as well.[5] Should each of these players build or not build a levee? Note that this interaction has the following two NE: first, in which both players build a levee, and second, in which they do not build a levee (these strategy combinations are in a NE, since no player is tempted to switch to a different strategy given the other player’s strategy). The problem is that even though the first NE is better for each of them than the second, they may fail to reach it, since each of them – fearing that the other player may decide not to build a levee – is likely to decide not to build a levee as well. However, the prospect of successful coordination is relatively high here, since the first NE is distinctly better than the second for both players. The coordination problems are more acute in games with two NE among which players are indifferent or which are valued differently by each of them. The role of law in all the above situations is to coordinate players’ moves (e.g., by making one of them obligatory) so that they may avoid a coordination failure and thus achieve the jointly utility-maximizing NE. 

The above analysis of CAPs suggests the following explanation of the emergence of legal norms: in order to overcome CAPs – omnipresent in social life – and thereby to secure a better realization of their interests, i.e. to achieve socially desirable results, agents agreed to “create” legal norms and the state vested with the power to guarantee their observance. This explanation needs to be supplemented by an account of the role of conventions (noted already by Hume) in resolving the CAPs modelled by coordination games. This account may run along two different lines. (1) Prior to passing legal norms secured by the state’s force, agents developed conventions, i.e. generally accepted modes of behaviour that helped them to overcome coordination problems. These conventions could be better or worse, yet any convention was better than the lack of convention, which was equivalent to the lack of coordination among agents. Now legal norms may be treated as incorporating and introducing stability to these conventions – as their reifications (as Hume insisted, the rules, e.g., for the transference and inheritance of property, or for the succession of monarchy became conventions as a result of some historical accident and were then consolidated as rules of justice). (2). One may say that the main role of legal rules is not to consolidate conventions but, rather, to bring about changes of inefficient conventions that cannot be achieved by the sum of individual actions freely chosen.[6]


2. GT as a tool for analysing specific legal problems

GT provides, not only precious insights into the nature of law, but also enables an analysis of specific legal problems from virtually all the branches of law. The importance of GT lies in that it helps to predict how agents are likely to behave in response to various rules and thereby enables shaping these rules so that they induce agents to act in a way that maximizes social wealth. If one wants to construct a model of a legal problem, one must properly choose a game form (a matrix, a tree or the so-called characteristic-function) as well as assumptions concerning agents’ rationality and information available to them. The problem of information may be especially intricate: it is, e.g., difficult to shape laws that provide agents with incentives to act in a way which makes everyone better off when agents or a court lack all necessary information, or when agents possess information yet are not inclined to disclose it; for such cases the role of GT is particularly important, since it helps examine effects that legal rules governing the transmittal of information may have on the way people interact with each other. 

Let us look at a concrete legal problem – the compliance problem in contract law. The problem reduces to the question of how to induce parties to comply with a contract which could not be executed at the moment it was signed. Note that for this contract to be executed one of the parties to it will have to fulfil his part of the contract without knowing whether the other will fulfil his part. This situation can be modelled by the Prisoner’s Dilemma, since, assuming that the parties are rational agents with self-regarding preferences, their preference-ordering will: T: one party does not perform the contract (strategy D), while another party does (strategy C) > R: both parties perform the contract > P: no party performs the contract > S: one party performs the contract, while another party does not. The awareness that the result of a contract which cannot be immediately executed is likely to be the NE {D, D} may prevent agents from concluding it. The role of law is to provide parties with incentives to conclude such contracts, e.g., by imposing sanctions on those agents who fail to comply with a contract they signed. It should not be thought, however, that without legal intervention agents would never conclude such contracts. A contract of this kind would be concluded if all potential parties to it had other-regarding preferences or a sense of fairness that would prevent them from reneging on it and if all these parties knew about this fact; these two conditions, however, are not easy to satisfy. GT also predicts that even agents with self-regarding preferences are likely to enter into such a contract with each other if they know that their interaction will be repeated (they care for reputation and the continuance of mutually beneficial interactions); a model of such interactions is the iterated Prisoner’s Dilemma. It should be stressed, however, that game-theoretic tools applied in the analysis of specific legal problems are usually much more sophisticated than the Prisoner’s Dilemma.


IV Normative role of GT

Bargaining theory – a branch of GT – helps to solve the bargaining problem (the problem of distributing the surplus of goods between parties who contributed to bringing it about) by providing unique solutions to it. Presumably the most plausible (i.e. satisfying a set of very plausible axioms) bargaining solution is the Nash solution, which prescribes the outcome that maximizes the product of the bargainers’ increments of utility in relation to their initial bargaining position. Bargaining solutions, however, are criticized, e.g., for operating on too narrow an informational base (encompassing only players’ utility functions and their initial bargaining positions), and for not being rules of distributive justice but merely rules of rationality devoid of truly ethical character. 

GT may also fulfil a descriptive role in this context[7]: it sheds light on how legal rules – by creating an appropriate bargaining environment – may affect the behaviour of parties in a bargain (GT shows, e.g., that legal rules introducing exit options, giving parties the right to terminate the bargaining game and receive some alternative payoff[8], provide them with incentives to reach an outcome prescribed by bargaining solutions); it also enables modelling bargainers’ possessing private information. 


V GT and rights

It is possible to represent individual rights in game forms – as admissible combinations of agents’ strategies. This approach to rights (assumed by, e.g., R. Nozick) is called independent, as it specifies each person’s rights without making any reference to her preferences or to the consequences of her choices. It implies that the priority of rights is total, i.e. that they cannot be overridden by any consequential analysis. Thus understood rights limit the range of available social alternatives between which agents are allowed to choose using a given mechanism of collective decision-making. This approach is treated as a way of bypassing Sen’s paradox of the Paretian liberal (since this approach does not invoke individual preferences, it cannot generate the conflict with the preference-based principle of Pareto-optimality). According to the alternative – integrated (social choice theory) – approach (assumed, e.g., by A. Sen), an individual right is an entitlement to determine a social preference. This approach allows making the consequential analysis of rights, thus depriving them of their total priority: the violation of rights is introduced – as a sort of consequence – into the evaluation of the states of affairs, which implies that trade-offs between right and non-right considerations are possible. 


Bibliography

1. R. Axelrod, The Evolution of Cooperation, Basic Books, New York1984.

2. D G. Baird, R H Gertner, R C. Picker, Game Theory and the Law, Harvard University Press, Cambridge, Mass., London 2000. 

Fundamental book about the role of game theory in legal analysis. It contains a rich bibliography concerning the application of game theory in particular branches of law.

3. K. Binmore, Playing Fair. Game Theory and the Social Contract Vol. I, The MIT Press, Cambridge, Mass., London 1995.

4. K. Binmore, Just Playing. Game Theory and the Social Contract Vol. II, The MIT Press, Cambridge, Mass., London 1995.

5. D. Fudenberg, J. Tirole, Game Theory, The MIT Press, Cambridge, Mass., London, England 1993.

A textbook for advanced readers with good mathematical background.

6. D. Gauthier, Morals by Agreement, Clarendon Press, Oxford 1986. 

7. R. Hardin, Morality within the Limits of Reason, The University of Chicago Press, Chicago, London 1988.

8. D.M. Kreps, Game Theory and Economic Modelling, Clarendon Press, Oxford 1990.

A textbook for beginners.

9. D. K. Lewis, Convention, Harvard University Press, Cambridge, Mass., London1969. 

10. J. F. Nash, The Bargaining Problem, “Econometrica” 18, pp.155-162, 1950.

11. H. Peyton Young, Equity. In Theory and Practice, Princeton University Press, Princeton, New Jersey 1995.

12. E. Ullmann-Magalit, The Emergence of Norms, Oxford University Press, Oxford 1978. 

13. J. Waldron, Law and Disagreement, Clarendon Press, Oxford 1995. 

14. W. Zaluski, Justice as Mutual Advantage from the Perspective of Game Theory, Zakamycze, Krakow 2005.


Related entries

LawEconomicsConsequentialist argumentation and reasoningConvention and conventionalism in the lawEfficiency.


Footnotes

[1] A pair of payoffs is Pareto-optimal if it cannot be improved upon, i.e. if no player can be made better off without making the other one worse off.

[2] The result are the refinements of a NE (e.g. the subgame perfect NE or the perfect Bayesian NE).

[3] The matrix form can be applied to situations where players make simultaneous decisions, sequential decisions without knowing what the other players have done, as well as to situations where there are many players who make sequential decisions, but, given that negotiations between them are costly, the decision of each player is likely to have little effect on the decisions of the other. 

[4] What’s more, each player’s best response to whatever the other player decides to do is to play D: if, say, Player 2 buys a gun, then also Player 1 should do it, since otherwise Player 2 will gain advantage over him; if Player 2 does not buy a gun, then Player 1 should buy it, since thereby he will gain advantage over Player 2; in game-theoretic parlance, it is said that in the Prisoner’s Dilemma both players have a strictly dominant strategy D.

[5] See Baird, Gertner, Picker [2000], pp. 50-68

[6] See Hardin 1988, p. 51. In this context, Hardin gives an example of Sweden which by means of a legal regulation switched at 5 a.m. September 3 of 1967 its driving convention from driving on the left to driving on the right.

[7] A useful game here is the Rubinstein bargaining model of alternating offers and counter-offers, which belongs to non-cooperative bargaining theory (unlike the Nash solution, which belongs to cooperative bargaining theory). 

[8] Such rules are, for example, the rule providing the right to specific performance, the perfect tender rule in contract law, different bankruptcy rules.

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