From Small to Large Numbers

Before proceeding from the two-person to the n-person model, it is perhaps useful to summarize the conceptual schemata that has been developed. From a

1. natural distribution, a

2. constitutional contract is negotiated, from which, in turn,

3. postconstitutional contract becomes possible, through a. private-goods trade (goods rival in consumption) and/or b. public-goods trade (goods nonrival in consumption).

When we introduce a large number of participants, negotiations may take place among subgroups or coalitions contained within the larger and more inclusive community membership. A ''natural distribution'' may, therefore, be conceived at any one of several levels of aggregation. At one extreme, which we might call the pure natural distribution, no coalitions exist, and

16. In the elementary discussion here, I am assuming that B is able to exclude A from the use of Z without excessive cost. If nonexclusion is inherent in Z, then no property right will be initially assigned in this good in the no-production setting. If Z must be produced, however, the discussion becomes applicable, even if, once produced, Z is nonexcludable. Exclusion will, in this case, take the form of not producing.

each person acts strictly on his own in the genuine Hobbesian ''warre'' of each against all. From this base, however, ''constitutional contracts'' maybe made among members of groups of any two or more persons, with internal assignments of rights, while as among the separate groups conflict continues. At such levels, the separate subgroups or coalitions (some of which may include only one member) are in a natural distribution fully analogous to that described for the two-person model. The process of contractual internaliza-tion may proceed as the subgroups become larger until some final negotiating process which incorporates all persons in the community within a single constitutional structure.

The final or ultimate constitutional contract will define the rights assigned to each person in the inclusive community. And each person will find his own position improved over that which he might have enjoyed in any one of the natural distributions noted above, because he will not have to exert or contribute effort to defense and predation, either as an individual on his own account or as a contributing member of a subset of the total community.17 The second, or postconstitutional, stage of negotiation can commence. At this point, the shift from the two-person to the n-person model introduces other important differences in results. With respect to private-goods trade, exchanges can proceed in the orthodox manner that has been exhaustively

17. Complexities arise in the large-number setting which make any assignment of rights less stable than in the small-number setting. There are two reasons for this difference. In the first place, almost any imputation or assignment that is agreed on will dominate that which the individual would anticipate were he to opt out and to try to exist on his own in pure anarchy. With large-number communities, however, the set of rights assignments that will dominate, for each person, the position that he might expect to secure if any coalition opts out, is much more restricted, and, indeed, this set might be empty in many interactions. These aspects of interactions have been discussed in detail, formally, in modern game theory. To use the appropriate terminology, for large-number groups there may be no imputation or assignment that is in the core, and, if a core does exist, the number of imputations contained may be small. For an introductory discussion of these concepts, see Duncan Luce and Howard Raiffa, Games and Decisions (New York: John Wiley and Sons, 1957).

Second, even if the assignment finally arrived at should qualify for inclusion in the core in the game-theoretic sense, individuals would still find it advantageous to violate the terms if they predict an absence of response on the part of remaining members of the group. This tendency toward instability is present in both the small- and large-number groupings, but the sheer impersonality or anonymity of individuals in large groups makes strategic behavioral calculations much less likely to occur.

treated by economists. Market institutions will be formed; bargaining ranges will be restricted by the presence of multiple alternatives for each buyer and each seller of goods; outcomes will tend to be more determinate than in the small-number setting. The important point to be stressed, for my purposes, is that these exchanges in private goods still take place between separate pairs of traders, between individual buyers and individual sellers. Each exchange remains a two-party transaction, as in the simplified model, despite the addition of numbers. There is no necessity to bring all members of the community into each contract. There is nothing that might be classified as ''social contract'' here.

Over and beyond all such exchanges in private goods, there may exist further potentially realizable surpluses from ''trades'' in public or collective-consumption goods and services, those that can simultaneously meet the demands of all persons in the group. In reference to our simple example, suppose, as before, that Z is such a public good, which may be consumed jointly and simultaneously by all members of, say, a three-man community, A, B, and C. Assume, as before, that all of the available supply of Z is under the disposition of individual B under terms of the basic constitutional contract. ''Trades'' can be made which involve a transfer of privately partition-able goods from A and C to B in exchange for the latter's willingness to make available Z for joint consumption. The point to be emphasized here is that, in such trade, all members of the final consuming group must be brought directly into the contractual negotiations with respect to provision of the public good. It is not possible to factor down a network of economic exchange into separate two-party transactions. This feature categorically distinguishes public-goods trade from private-goods trade when large numbers are involved. With the former, with public-goods trade, something akin to ''social contract'' again comes into being, comparable in numbers of participants to the constitutional contract that delineates individual rights.

Before we can develop further the implications of this similarity, and the confusion that the similarity fosters, certain ambiguities must be clarified. For expository purposes, I have assumed that all scarce goods fall into either one or the other polar extreme—the purely private good or the purely public good.18 Furthermore, I have implicitly assumed that the ''range of public-

18. This is the same assumption that was made by Samuelson in his classic formulation of the welfare norms for public-goods provision. See Paul A. Samuelson, ''The Pure The ness'' for all goods in the public-goods classification extends precisely to the limits of the inclusive community membership. This assumption is perhaps even more restrictive than the polar classification. There may exist many goods that qualify as "public" or ''collective'' over differing numbers of consumers but which, at the same time, become fully rival in consumption as the number of users is increased. The size of the consuming ''club'' that meets efficiency criteria may be much less than the total membership of the community.19 If such ''clubs'' are small, relative to the size of the total community, market institutions can emerge that will internalize the required contractual arrangements, even though the strict pairwise exchange-characteristics of pure markets are violated. For our purposes, having recognized that much of the noncollective sector may embody such small-number jointness, we can treat all ''trades'' that do not involve substantial fractions of the total community membership as taking place in ''private goods.'' In somewhat more practical terms, this amounts to a neglect of local government decision-making, as such.

The analysis suggests that ''social contracting,'' defined as those negotiations which involve all members of the community, may take place conceptually at two levels or tiers: at some initial stage of constitutional contract, in which agreement is reached on an assignment of individual rights, and at some postconstitutional stage in which individuals agree on quantities and cost shares of jointly consumed goods and services. Essentially the same problems emerge at each tier, problems that are created largely by the necessity to bring large numbers of persons into the same contractual arrangements. The sheer cost of getting agreement on any outcome rises sharply as numbers increase. In two-party trading, only the single buyer and the single seller need agree. This difference in numbers alone makes for major differences in costs. But the absence of alternatives exacerbates the attainment of agreement in many-party negotiations. It may not prove overwhelmingly costly for very large numbers of persons to reach agreement if each participant has available to him other avenues for securing comparable objectives. To the extent that ory of Public Expenditure,'' Review of Economics and Statistics 36 (November 1954): 38789.

19. For a more extended treatment, see my ''An Economic Theory of Clubs,'' Economica 32 (February 1965): 1-14.

an individual has other opportunities, he can withdraw from participation in the large-number group if a proposed settlement does not fall within his broadly defined preferences. For the single inclusive community, however, the necessary participation of all members eliminates the prospects of effective alternatives. National states, from which out-migration can take place normally only at significant costs to an individual, rather than local governments, become the real-world organizational units to which the analysis is most directly applicable.

A final qualification must be introduced at this point in order not to give the appearance of inconsistency with the discussion in subsequent chapters. This involves the delineation or assignment of individual rights emerging from constitutional contract. It is essential that the potential variability in the specification of individuals' rights be recognized. The conceptual agreement may range from an assignment that invokes relatively little ''law'' in the formalized sense to an assignment that rigidly constrains individual behavior over many dimensions of adjustment. An assignment of rights is not an all-or-nothing choice. Ceteris paribus, the liberty inherent in anarchistic order without law is the most desirable state of affairs. The extent to which an individual, or the community of individuals, may be willing to trade off the liberty that remains present even in the Hobbesian jungle for the stability promised in regimes with varying degrees of formal restrictiveness will depend on the nastiness of the jungle, the value placed on order, the costs of enforcement, and on many other factors, some of which will be subsequently discussed.

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