Lloyd Shapley Research Articles

Revisiting Zakat with a distribution of weighted Shapley value

PurposeThis paper aims to extend the Shapley value (SV) into a discussion of Zakat, a Pillar of Islam. Lloyd Shapley was awarded the Nobel Prize in Economics in 2012. This study shows that their relationship is significant for all nations, that of levelling up. An important but neglected paper by Datta (1939) showed insights provided by the Power Law, or as it is sometimes called, the Pareto distribution, into the role of Zakat in raising the income of all above the subsistence level. The Pareto distribution describes the prevailing tendency. The SV illustrates the interdependence perspective of Zakat with the Pareto distribution, wealth, income and poverty. Payoffs apply equally to both givers and receivers. For this study’s purposes, payoffs are considered as transferable utilities. They are formed by individuals who willingly cooperate in society rather than atomistic individuals who act independently. Zakat represents the recognition that society needs to be cooperative rather than individualistic; people cooperate in groups or societies to create value. SV implications and axioms are evaluated with an illustration.Design/methodology/approachThis study extends Datta’s approach by introducing distribution weights into the SV. The authors set out the concept of weighted Shapley values that retain the elements of randomness and marginal contribution to a coalition contained in pure/true SVs and weights that follow a ley-Pareto distribution. This paper is a viewpoint work that relies primarily on the author’s qualitative interpretation.FindingsThe findings indicate that individual members of a coalition make multiple contributions that are often unrewarded. The contribution of one member of a coalition is dependent upon the contribution of others. The measure of contributions is payoffs, which have both monetary and non-monetary aspects; transferable payoffs or utilities are usually assumed. Furthermore, the significant agents in society or an organisation are stakeholders rather than the usual categories: managers, staff, shareholders, etc.Practical implicationsContextualising these concepts within the Islamic values and principles that guide Zakat administration is crucial to ensure that the distribution of Zakat funds is fair, equitable and meets the needs of all eligible recipients. By applying these concepts appropriately, Zakat administrators can ensure that the Zakat system functions effectively and fulfils its religious obligation.Originality/valueThe novelty of this paper is that it blends the SV and the idea behind Zakat by introducing the idea of alternatives of Shapley weights. The link between the institution of Zakat and SV in terms of equality, poverty elimination and wealth distribution should be at the top of the research agenda.

Stochastic integral representation of solutions to Hodge theoretic Poisson's equations on graphs, and cooperative value allocation of Shapley and Nash

The fundamental connection between stochastic differential equations (SDEs) and partial differential equations (PDEs) has found numerous applications in diverse fields. We explore a similar link between stochastic calculus and combinatorial PDEs on graphs with Hodge structure, by showing that the solution to the Hodge-theoretic Poisson's equation on graphs allows for a stochastic integral representation driven by a canonical time-reversible Markov chain. When the underlying graph has a hypercube structure, we further show that the solution to the Poisson's equation can be fully characterized by five properties, which can be thought of as a completion of the Lloyd Shapley's four axioms.

A formula for the value of a stochastic game

In 1953, Lloyd Shapley defined the model of stochastic games, which were the first general dynamic model of a game to be defined, and proved that competitive stochastic games have a discounted value. In 1982, Jean-François Mertens and Abraham Neyman proved that competitive stochastic games admit a robust solution concept, the value, which is equal to the limit of the discounted values as the discount rate goes to 0. Both contributions were published in PNAS. In the present paper, we provide a tractable formula for the value of competitive stochastic games.

An Account of 'The Core' in Economic Theory

The concept of ‘the core’ originates in cooperative game theory and its introduction to economics in the 1960s as a basis for proofs of existence of general equilibrium is one of the earliest attempts to use game theory to address big questions in economics. Discovery of the core was met with enthusiasm among the community of economic theorists at the time. However, use of the core eventually waned and the concept faded into the backdrop of economic theory. This paper makes use of unpublished correspondence between Herbert Scarf, Lloyd Shapley, and Martin Shubik, as well as other archival and secondary resources, to provide an account of the development of the core and the trajectory of this concept, including those who developed it, after its initial appearance. It is found that the core’s eventual decline is explained by the combined effect of the slowing general equilibrium research program in the 1970s, the increasing prominence of non-cooperative game theory, and subtle issues with the concept that shaped Scarf and Shubik’s research programs after the 1960s.

Competitive pricing and the core: With reference to matching

Lloyd Shapley's contributions with respect to the core are interpreted as subdifferentiability characterizations of the pricing of individuals that is similar to the pricing of commodities in economic models of exchange with transferable utility. Differentiability of the core is interpreted as perfect substitutability with respect to the pricing of individuals. Differentiability implies, but is not implied by, equivalence of the core and Walrasian equilibria. Differentiability eliminates opportunities for strategic misrepresentation of utilities. The assignment model with transferable utility is framed in the setting of exchange economies and its individual and commodity pricing is extended to non-transferable utility.

Lloyd Shapley and Chess with Imperfect Information

Abstract Anyone who has ever studied game theory knows the name Lloyd Shapley. Just recall Matching, Deferred-Acceptance Algorithm, Core, Market Games, Stochastic Games, Shapley value, and Shapley vector. 1 But Professor Shapley was also a great lover of chess with imperfect information. Upon our first encounter at Stony Brook in 1998, I was fortunate to investigate the chess problems he set before me. In this essay I analyze some of those problems, in commemoration of Lloyd Shapley's contributions to the study of chess and chess with imperfect information.

Lloyd Shapley and chess with imperfect information

Anyone who has ever studied game theory knows the name Lloyd Shapley. Just recall Matching, Deferred-Acceptance Algorithm, Core, Market Games, Stochastic Games, Shapley value, and Shapley vector.1 But Professor Shapley was also a great lover of chess with imperfect information. Upon our first encounter at Stony Brook in 1998, I was fortunate to investigate the chess problems he set before me. In this essay I analyze some of those problems, in commemoration of Lloyd Shapley's contributions to the study of chess and chess with imperfect information.

Modification of Shapley Value and its Implementation in Decision Making

Abstract The article presents a solution of a problem that is critical from a practical point of view: how to share a higher than usual discount of $10 million among 5 importers. The discount is a result of forming a coalition by 5 current, formerly competing, importers. The use of Shapley value as a concept for co-operative games yielded a solution that was satisfactory for 4 lesser importers and not satisfactory for the biggest importer. Appropriate modification of Shapley value presented in this article allowed to identify appropriate distribution of the saved purchase amount, which according to each player accurately reflects their actual strength and position on the importer market. A computer program was used in order to make appropriate calculations for 325 permutations of all possible coalitions. In the last chapter of this paper, we recognize the lasting contributions of Lloyd Shapley to the cooperative game theory, commemorating his recent (March 12, 2016) descent from this world.

Three little-known and yet still significant contributions of Lloyd Shapley

This paper makes three corrections in the literature by restating three little-known and yet still significant contributions of Lloyd Shapley: 1) Shapley (1955), not Gillies (1953, 1959), first defined the core solution; 2) Shapley (1956) first defined and studied the noncooperative solution in multiobjective games; and 3) Shapley (1987) revealed a fundamental error in existing noncooperative game theory.

An annotated bibliography of Lloyd Shapley's contributions

An annotated bibliography of Lloyd Shapley's contributions

An ordinal minimax theorem

In the early 1950s Lloyd Shapley proposed an ordinal and set-valued solution concept for zero-sum games called weak saddle. We show that all weak saddles of a given zero-sum game are interchangeable and equivalent. As a consequence, every such game possesses a unique set-based value.

Stochastic games

In 1953, Lloyd Shapley contributed his paper “Stochastic games” to PNAS. In this paper, he defined the model of stochastic games, which were the first general dynamic model of a game to be defined, and proved that it admits a stationary equilibrium. In this Perspective, we summarize the historical context and the impact of Shapley’s contribution.

A note on uncertainty and perception concerning measurable utility

A linkage to reconcile measurable utility derived from intensity comparisons or from probability mixtures is provided in this note. This brief note is in honor of Lloyd Shapley whose relatively unknown seminal paper on measurable utility from axioms involving the fineness of perception offered a different view on utility measurement.

Course Allocation via Stable Matching

The allocation of students to courses is a wide-spread and repeated task in higher education, often accomplished by a simple first-come first-served (FCFS) procedure. FCFS is neither stable nor strategy-proof, however. The Nobel Prize in Economic Sciences was awarded to Al Roth and Lloyd Shapley for their work on the theory of stable allocations. This theory was influential in many areas, but found surprisingly little application in course allocation as of yet. In this paper, different approaches for course allocation with a focus on appropriate stable matching mechanisms are surveyed. Two such mechanisms are discussed in more detail, the Gale-Shapley student optimal stable mechanism (SOSM) and the efficiency adjusted deferred acceptance mechanism (EADAM). EADAM can be seen as a fundamental recent contribution which recovers efficiency losses from SOSM at the expense of strategy-proofness. In addition to these two important mechanisms, a survey of recent extensions with respect to the assignment of schedules of courses rather than individual courses is provided. The survey of the theoretical literature is complemented with results of a field experiment, which help understand the benefits of stable matching mechanisms in course allocation applications.

Theory of stable allocations

The Swedish Royal Academy awarded the 2012 Nobel Prize in Economics to Lloyd Shapley and Alvin Roth, for the theory of stable allocations and the practice of market design. These two American researchers worked independently from each other, combining basic theory and empirical investigations. Through their experiments and practical design they generated a flourishing field of research and improved the performance of many markets. Shapley provided the fundamental theoretical contribution to this field of research, whereas Roth, a professor at the Harvard University in Boston, developed and upgraded these theoretical investigations by applying them to the American market of medical doctors. Namely, their research helps explain the market processes at work, for instance, when doctors are assigned to hospitals, students to schools and human organs for transplant to recipients.

Is Policy Too Important to be Left to Empiricists? Lessons of the 2012 Nobel Prize in Economics

Fifty years ago, a paper entitled 'College Admissions and the Stability of Marriage' was published in a somewhat obscure journal, the American Mathematical Monthly (currently a 'B' journal, according to the Australian Business Deans Council). The research program and policy developments that have flowed from that abstract and apparently slight seven-page paper recently led to the award of the 2012 Nobel Prize for Economics to one of its authors, Lloyd Shapley. (Shapley's co-author, David Gale, died in 2008.) Shapley shared the Nobel Prize 'for the theory of stable allocations and the practice of market design' with US economist Alvin Roth, who has been responsible for much of the applied work that has built on Gale and Shapley's insights. The history of the path leading from the abstract Gale/Shapley insights to the design of resource allocation mechanisms in 2012 is a fascinating and instructive one for many reasons. This article tries to give the reader an idea of what this literature is about, and of the many ways in which Matching Theory has led to real improvements in the design of operational resource-allocation mechanisms.

Matching theory: kidney allocation

Lloyd Shapley and Alvin E. Roth have recently been awarded the Nobel Prize in Economics for their work in matching theory. Although branching from the field of economics, matching theory has had many implications in the world of medicine. For example, the National Residency Matching Program in the United States is an application of matching theory. The focus of this article is the application of matching theory to kidney transplant allocation. Kidney transplantation is the best treatment for end stage renal failure. Unfortunately, the demand for kidneys exceeds supply. Kidney paired exchange programs, which have begun to garner great success in increasing the number of kidney transplants worldwide, base their foundations on matching theory. Overviewed in this paper will be how these programs were created and work, their successes, and some of the unique challenges and logistical obstacles they face.

Lloyd Shapley's Matching and Game Theory

AbstractThis is a survey of Lloyd Shapley's contributions to matching theory and game theory in general, starting with the work that inspired the Swedish Academy to award the 2012 Nobel Memorial Prize in Economic Sciences to Lloyd Shapley and Alvin Roth.

KOLEJNY SUKCES TEORII GIER: NOBLIŚCI Z EKONOMII 2012

Game theory has earned recognition in economics along with the development of mathematical economics. It was reflected in Nobel Prizes, especially in 1994, 2005 and 2007. Lloyd Shapley and Alvin Roth award honours the achievements of a distinguished, elderly champion in the paradigm of game theory and a younger scholar who is also an effective organiser of matching pairs in the allocation of markets which fit the framework of game theory. Successes in marriage matchmaking, school recruitment and regulations of the exchange of human organs for the purpose of transplantation have already become an inspiration for further research and new markets improvement.

Two-Sided Markets: Theory and Applications <i> (Nobel Memorial Prize in Economics 2012)</i>

The paper provides an overview of major results in the theory of two-sided markets that brought the 2012 Nobel Memorial Prize in economics to Lloyd Shapley and Alvin Roth. We describe the celebrated Gale & Shapley deferred acceptance algorithm and Shapley & Scarf top trading cycles algorithm in applications to marriage and college admissions markets, and provide simple examples to illustrate the basic mechanics of these algorithms. We also explain how Gale & Shapley algorithm can be used to improve efficiency of college admission in Russia.