Pooling Problem Research Articles

Ternary Covering Codes

In [5], we studied binary codes with covering radius one via their characteristic functions. This gave us an easy way of obtaining congruence properties and of deriving interesting linear inequalities. In this paper we extend this approach to ternary covering codes. We improve on lower bounds for ternary $1$-covering codes, the so-called football pool problem, when $3$ does not divide $n-1$. We also give new lower bounds for some covering codes with a covering radius greater than one.

Budget Processes and Commitment to Fiscal Discipline

This paper develops a political-economy model of the budget process focusing on the common pool problem of the public budget. We show that the externality arising from the fact that public spending tends to be targeted at individual groups in society while the tax burden is widely dispersed creates a bias towards excessive expenditures and debt. This bias can be reduced by introducing elements of centralization in the budget process, that is, institutional structures that strengthen a comprehensive view of the budget over the particularistic view of the spending ministers and the members of parliament. Using examples from EC countries, we show how budget processes lack or possess such elements. We then present empirical evidence supporting the claim that centralizing elements reduce the deficit bias. The last section concludes with models for reform of the budget process.

A new lower bound for the football pool problem for $7$ matches

A new lower bound for the football pool problem for $7$ matches

A Re-Examination of Near-Bankruptcy Investment Incentives

The traditional justification for American corporate bankruptcy law that bankruptcy's stay of individual creditor collection keeps the wolves at bay. Without bankruptcy, the story goes, creditors might fail to negotiate for united action and in their attempts to claim limited assets dismember an insolvent but viable firm. Anticipation and conduct of this race, moreover, would require creditors to expend resources unnecessarily. Accordingly, the accepted wisdom that bankruptcy's collective proceeding protects the of debtor assets for the aggregate benefit of creditors and consequently saves creditors the expense of strategic positioning. An important component of this exegesis has been that bankruptcy law must protect firms from creditors not only after the onset of bankruptcy but also immediately before it. Thus, in many instances, a firm can recapture transfers to creditors if made while the firm insolvent and within the ninety-day period prior to the firm's bankruptcy. These recapturable transfers are called voidable preferences. In the leading volume on modern bankruptcy theory, Thomas Jackson explains that voidable preference law is essentially a transitional rule designed to prevent individual creditors from opting out of [bankruptcy's] collective proceeding once that event becomes likely. It part of the attempt to ameliorate the effects of a common pool problem that justifies a collective proceeding in the first place.1 On close analysis, however, preference law may not serve the purpose given in this standard justification. Preference law as guardian of the common pool a notion based on a view of the firm as a static pool of assets that exists independently of the

New upper bounds for the football pool problem for 11 and 12 matches

We consider the problem of minimizing the number of words in a code with the property that all words in the space F 3 n are within Hamming distance 1 from some codeword. This problem is called the football pool problem, since the words in such a code can be used in a football pool to guarantee that at least one forecast has at least n − 1 correct results. In this note we show that for 11 and 12 matches, there are 9477 and 27702 words, respectively, having the aforementioned property. Simulated annealing has played an important role in the search for these words.

Why the Principle Over-Predicts Actual Values*

Two articles previously published in this Journal (Watkins 1992 and Adelman 1993) reported that the valuation version of the Hotelling Principle over-predicts in-ground values of oil and gas reserves by a factor of approximately two. This paper shows these results are to be expected once it is understood that: (1) the Principle assumes individual operators have the effective freedom to schedule extraction rates so as to make net prices rise at the rate of discount, regardless of the course of gross (wellhead) prices; and (2) the long-prevailing system of regulating oil well spacing and extraction rates in the United States and Canada, designed to deal with the common pool problem, effectively denies operators that freedom. The discrepancy between actual in-ground values and those predicted by the Hotelling Principle suggests the benefits to be had by substituting compulsory reservoir unitization cum manager freedom for the current system of regulation.

Raid or Trade? An Economic Model of Indian-White Relations

Raid or Trade? An Economic Model of Indian-White Relations

Emerging from the Hobbesian jungle: Might takes and makes rights

The conflict over scarce resources in the Hobbesian jungle may be avoided if rules of obligation delineating property rights develop along with institutions of governance. One possibility is a “duress contract” as the strongest individual threatens others who agree to enslavement. Thus, “might takes rights.” Alternatively, individuals with similar capacities for violence may enter a “consent contract” establishing rules of obligation and then voluntarily participating in governance. They will not agree to a rights assignment that produces less wealth than they expect through violence, however, so “might makes rights.” A might-takes-and-makes-rights analysis is outlined to explain the continuum of legal institutions and property rights allocations that can evolve between these two extremes of duress and consent. Increasingly finely delineated private property rights tend to evolve under institutions produced by consent contracts, while common pool problems tend to arise near the duress contract end of the spectrum.

The “common pool” problem in tourism landscapes

Management of scenic and historic landscapes and other background tourism elements is problematic because they very frequently are what property rights theorists call “common pool” resources. They are characterized by susceptibility to overuse and resource damage and by lack of incentive for productivity-enhancing investment. Property rights theory describes three property rights regimes for managing such resources: privatization, management by government, and common property regimes. All three regimes are widely found in the case of tourism resources, and mixtures of regimes are frequently encountered. Common property regimes, which involve community control or reciprocal actions among individuals, appear to be the least common, yet such arrangements have interesting potential for addressing common pool problems.

Global minimization by reducing the duality gap

We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.

A new upper bound for the football pool problem for nine matches

Consider the set V 3 n of all n-tuples x = ( x 1, …, x n ) with x i ϵ {0, 1, 2}. We are interested in σ n , the minimal size of a subset W of V 3 n , such that for any element x ϵ V 3 n there exists at least one element y ϵ W at a Hamming distance d H ( x, y)⩽ 1, σ n can also be considered as the minimum number of forecasts in a football pool of n matches, such that at least one forecast has at least n − 1 correct results. In this note we present the new upper bound σ 9 ⩽ 53 · 3 3. The bound has been obtained by an approximation algorithm which performs, alternatingly, phases of iterative improvement and randomized backsteps.

Preprocessing Nonlinear Functional Constraints with Applications to the Pooling Problem

Methods to uncover redundancies, infeasibilities and bound structures on variables for nonlinear constraint sets are developed. It is these methods that comprise the preprocessing that is derived here. Bilinear constraint sets ate used to illustrate the methods. Thus, special emphasis is given to bilinearities and a nonlinear programming model in which bilinearities arise naturally, a generic petroleum refinery model and its associated pooling problem. The usefulness of the approaches developed herein include their applications to preoptimization/postoptimization analyses of nonlinear programming problems and the generation of starting values for nonlinear programming optimizers. Numerical examples are given that use existing optimization packages in conjunction with a computer-aided analysis system, and point to the practical implementability of the methods derived. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples

Bounds on packings and coverings by spheres in q-ary and mixed Hamming spaces

In a recent paper by the same author general improvements on the sphere covering bound for binary covering codes were obtained. In the present work it is shown how the main idea can be exploited to obtain “improved sphere bounds” also for nonbinary codes. We concentrate on general q-ary codes and on binary/ternary mixed codes. Special attention is paid to the football pool problem; a few new lower bounds are established. Also, it is shown how a similar method yields upper bounds on packing codes.

Upper bounds for football pool problems and mixed covering codes

Let K( t, b, R) denote the minimum cardinality of any mixed code which has t ternary and b binary coordinates and covering radius R. We collect upper bounds for K( t, b, R) when R ⩽3 and t+ b⩽13 and discuss some simple methods to construct new codes from the known ones.

Supporting individuals in group decision-making

Pooling of different resources is typical among the member countries of the Council of Mutual Economic Assistance participating in joint large-scale construction projects. The problem faced by the members of the Council is to decide, how much of various resources each country should contribute to a construction project. In this paper we present a general approach to supporting individuals involved in such negotiations. We formulate the problem as a multiple criteria/multiple decision-maker model and use our approach to finding a compromise solution for the resource pooling problem within the Council of Mutual Economic Assistance. The approach is implemented on a computer, tested and illustrated using a prototypical example.

Characteristics of Limited Entry Fisheries and the Option Component of Entry Licenses

Entry limitations are frequently proposed as a partial solution to the common pool problem that characterizes most fisheries. Entry limitations have actually been imposed, however, in only a few fisheries.' At first glance, this situation contradicts the self-interest hypothesis of regulation, because most proposals to limit entry would convey entry licenses to many existing fishermen at a nominal price. Johnson and Libecap (1982) attributed the apparent contradiction to fisherman heterogeneity and the costs of devising and enforcing contracts or regulations that would be supported by most fishermen. In a recent paper (Karpoff 1987), I argue that traditional regulations such as capital constraints and season closures are popular because they redistribute the catch toward the politically dominant fishing groups. This paper directly examines conditions under which fishermen are likely to support entry restrictions. Entry restrictions generate positive (quasi-) rents precisely because they exclude or have the potential to exclude some fishermen. The probability of exclusion is positive for most fishermen, so advocacy of entry restrictions is not riskless. For many fishermen the risk of exclusion outweighs the expected gain from restricted competition. When fishermen do advocate entry restrictions, they do so when the expected net gain is positive. One factor that affects the net gain is the variability of fishing income. This point is made by noting that limited entry licenses are similar to call options on future increases in revenue over cost and by developing an exact option-based model to value entry licenses. There are numerous potential applications of this model. It offers the important advantage of requiring information only on factors that can be easily estimated, and it yields comparative static results that can be used to assess the effects on license values of environmental or regulatory changes.

New upper bounds for the football pool problem for 6, 7, and 8 matches

Consider the set V 3 n of all n-tuples x = ( x 1, …, x n ) with x i ∈ {0, 1, 2}. We are interested in σ n , the minimal size of a subset W of V 3 n , such that for any element x ∈ V 3 n there exists at least one element y ∈ W at a Hamming distance d H (x, y) ⩽ 1. σ n can also be considered as the minimal number of forecasts in a football pool of n matches, such that at least one forecast has at least n − 1 correct results. In this note we present new upper bounds on σ 6, σ 7, and σ 8: 73, 186, and 486, respectively. The bounds have been obtained by an approximation algorithm based on simulated annealing. A closer analysis of the result for the 8-matches problem has led to a simple way to construct a large number of subsets W of V 3 8, each consisting of 486 8-tuples and each having the aforementioned property.

Loci and rolling circles

Some mathematics problems, while easy to pose and visualise, can be deceptively difficult to solve. For example, suppose we want to find the area of a three foot wide footpath around the edge of an elliptical shaped swimming pool. The problem is trivial if the pool is circular, in which case we simply subtract the areas of two concentric circles. At first blush, the elliptical pool problem does not appear to be much more difficult, especially if we can recall that the formula for the area of an ellipse is A = πab where a and b are respectively the lengths of the semi-major and semi-minor axes. In any case, we might think that it would be a straightforward calculus problem involving the area between two curves.

Corruption in law enforcement: one consequence of “The tragedy of the commons” arising with public allocation processes

Several corruption scandals involving police, judges, or other law enforcement officials seem to surface every year in the United States. Inevitably, the official explanation is that such corruption is an individualized phenomenon involving a few “rotten apples.” Thomas Wicker, a Louisiana judge, in commenting on the large number of charges brought against judges in the last few years, claimed, for instance, that “most judges are honest and conscientious. The discipline process is taking care of a few bad apples” (Gest 1983, p. 42). Similarly, recent police corruption in Miami produced the following explanation: “Investigators say they believe that only a small percentage of Dade County’s police officers are involved in criminal activity. But authorities agree that even a few bad apples is more than enough” (emphasis added, Anderson and Prout 1985, p. 32). Yet, every detailed study of corruption in law enforcement rejects this “bad apple” theory of corruption. Thus, New York City’s Knapp Commission concluded that “police corruption was found to be an extensive, department wide phenomenon, indulged in to some degree by a sizable majority of those on the force” (1972, p. 61), despite a virtually “official Department doctrine” that “any policeman found to be corrupt must promptly be denounced as a rotten apple in an otherwise clean barrel” (1972, p. 6).’ In all likelihood, the corruption that is exposed each year reveals only a small part of the illegal activities by law enforcement officials. After all, these are only the ones who get caught. Even if a similar number of crimes by public officials were cleared by arrest as crimes in general, the total level of corruption would be substantial. But there are several reasons to believe that the crimes corrupt public officials commit are considerably less likely to be exposed than private sector crime. Indeed, corruption is a direct consequence of the incentives that law enforcement officials face. An alternative to the rotten apple theory of corruption is offered here, one that emphasizes these institutionalized incentives. Corruption by law enforcement officials is a black market for the property rights over which those officials have been given discretionary allocative power (Benson 1981a).2 This discretion arises because of the common pool characteristics of law enforcement resources. It means that rather than assigning rights in reflection of political demands, rights can be sold to the highest bidder. Police, judges, and prosecutors are not monitored closely, again because of incentives arising from common pool problems, so self-interest may dominate and corruption results.