Solution of a resource allocation game

Translated from Vychislitel'nye Sistemy i Voprosy Prinyatiya Reshenii, pp. 180–183, 1991.     

On a solution of the discrete resource allocation game

A discrete zero-sum two-person game of breaking attacks at defended posts (a Blotto game) is considered. An algorithm for finding a game solution in mixed strategies is described and generalized to games with budget constraints. In particular cases, explicit formulas for the value of the game and optimal strategies are obtained.     

Dynamic resource allocation in a multi-product make-to-stock production system

We consider optimal policies for a production facility in which several (K) products are made to stock in order to satisfy exogenous demand for each. The single machine version of this problem in which the facility manufactures at most one product at a time to minimise inventory costs has been much studied. We achieve a major generalisation by formulating the production problem as one involving dynamic allocation of a key resource which drives the manufacture of all products under an assumption that each additional unit of resource allocated to a product achieves a diminishing return of increased production rate. A Lagrangian relaxation of the production problem induces a decomposition into K single product problems in which the production rate may be varied but is subject to charge. These reduced problems are of interest in their own right. Under mild conditions of full indexability the Lagrangian relaxation is solved by a production policy with simple index-like structure. This in turn suggests a natural index heuristic for the original production problem which performs strongly in a numerical study. The paper discusses the importance of full indexability and makes proposals for the construction of production policies involving resource idling when it fails.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

Consolidating or non-consolidating queues: A game theoretic queueing model with holding costs

We consider a two-server queueing system in which the servers choose their service rate based on the demand and holding cost allocation scheme offered by the demand generating entity. We provide an optimal holding cost allocation scheme that leads to the maximum possible service rate for each of a pooled and a split system. Our results suggest that careful allocation of holding costs can create incentives that enable minimum turnaround times using a common queue.     

Stability likelihood of coalitions in a two-stage cartel game: An estimation method

Existing formulations of a cartel game aim at finding stable coalitions, i.e. a coalition is labelled stable or not stable. Uncertainty about the underlying structure and/or parameter values gives rise to sensitivity or uncertainty analysis. In this paper we follow a probabilistic robustness concept: What is the probability a product, design or policy really fulfils the requirements or properties it is expected to. Following this idea, we introduce the concept of stability likelihood: What is the probability a coalition can be labelled as stable. Methods are described based on Monte Carlo Simulation and Directional Simulation to estimate such a probability and we illustrate the performance for several cases.     

Dynamic resource allocation: A flexible and tractable modeling framework

This paper presents a binary optimization framework for modeling dynamic resource allocation problems. The framework (a) allows modeling flexibility by incorporating different objective functions, alternative sets of resources and fairness controls; (b) is widely applicable in a variety of problems in transportation, services and engineering; and (c) is tractable, i.e., provides near optimal solutions fast for large-scale instances. To justify these assertions, we model and report encouraging computational results on three widely studied problems – the Air Traffic Flow Management, the Aircraft Maintenance Problems and Job Shop Scheduling. Finally, we provide several polyhedral results that offer insights on its effectiveness.     

The core of a game with a continuum of players and finite coalitions: The model and some results

In this paper we develop a new model of a cooperative game with a continuum of players. In our model, only finite coalitions - ones containing only finite numbers of players - are permitted to form. Outcomes of cooperative behavior are attainable by partitions of the players into finite coalitions: this is appropriate in view of our restrictions on coalition formation. Once feasible outcomes are properly defined, the core concept is standard - no permissible coalition can improve upon its outcome. We provide a sufficient condition for the nonemptiness of the core in the case where the players can be divided into a finite number of types. This result is applied to a market game and the nonemptiness of the core of the market game is stated under considerably weak conditions (but with finite types). In addition, it is illustrated that the framework applies to assignment games with a continuum of players.     

Memra: An equilibrium model for resource allocation in a health care system

The classical approach to resource allocation in Health Care Systems seems inappropriate as it does not take into account the principle of equilibrium between supply and demand, a concept widely used in other areas, e.g. transportation. We propose and validate a model which uses this concept of equilibrium to reproduce medical practice and to predict the quantity of resources consumed and the number of medical services performed given an amount of physician resources available. The formulation reduces to a convex function to be minimized and is very easy to solve. It has been tested on data from the province of Québec with excellent results.     

A resource allocation game on network graphs with two uncertain factors

A solution method is proposed for the problem of allocating a given quantity of resource A on network graphs with two uncertain factors. One of the uncertain factors is the type of the network graph and the other is the capacity of the arcs in the graph.Translated from Programmno-apparatnye Sredstva i Matematicheskoe Obespechenie Vychislitel'nykh Sistem, pp. 163–167, 1994.     

Sharing costs in highways: A game theoretic approach

This paper introduces a new class of games, highway games, which arise from situations where there is a common resource that agents will jointly use. That resource is an ordered set of several indivisible sections, where each section has an associated fixed cost and each agent requires some consecutive sections. We present an easy formula to calculate the Shapley value, and we present an efficient procedure to calculate the nucleolus for this class of games.     

Heterogeneous surface-to-air missile defense battery location: a game theoretic approach

In the context of an air defense missile-and-interceptor engagement, a challenge for the defender is that surface-to-air missile batteries often must be located to protect high-value targets dispersed over a vast area, subject to which an attacker may observe the disposition of batteries and subsequently develop and implement an attack plan. To model this scenario, we formulate a two-player, extensive form, three-stage, perfect information, zero-sum game that accounts for, respectively, a defender’s location of batteries, an attacker’s launch of missiles against targets, and a defender’s assignment of interceptor missiles from batteries to incoming attacker missiles. The resulting trilevel math programming formulation cannot be solved via direct optimization, and it is not suitable to solve via full enumeration for realistically-sized instances. We instead adapt the game tree search technique Double Oracle, within which we embed either of two alternative heuristics to solve an important subproblem for the attacker. We test and compare these solution methods to solve a designed set of 52 instances having parametric variations, from which we derive insights regarding the nature of the underlying problem. Enhancing the solution methods with alternative initialization strategies, our superlative methodology attains the optimal solution for over 75% of the instances tested and solutions within 3% of optimal, on average, for the remaining 25% of the instances, and it is promising for realistically-sized instances, scaling well with regard to computational effort.     

Optimal resource allocation in a two-sector economic model with singular regimes

The article considers the problem of resource allocation in a two-sector economic model with a nonlinear production function of a special type. The main mathematical apparatus is Pontryagin’s maximum principle, i.e., the theorem on necessary conditions of optimality. It is shown that in the given problem the maximum principle provides a necessary and sufficient condition of optimality. A possible singular solution of the problem is found. An extremum solution is constructed in explicit form under various assumptions about the initial values. A “sufficiently long” planning horizon is assumed. An alternative approach is described, which does not use the maximum principle and instead investigates the integral representation of the optimand functional. The detailed theoretical investigation of the problem is accompanied by numerous illustrations.     

Comparing reinforcement learning approaches for solving game theoretic models: a dynamic airline pricing game example

Games can be easy to construct but difficult to solve due to current methods available for finding the Nash Equilibrium. This issue is one of many that face modern game theorists and those analysts that need to model situations with multiple decision-makers. This paper explores the use of reinforcement learning, a standard artificial intelligence technique, as a means to solve a simple dynamic airline pricing game. Three different reinforcement learning approaches are compared: SARSA, Q-learning and Monte Carlo Learning. The pricing game solution is surprisingly sophisticated given the game's simplicity and this sophistication is reflected in the learning results. The paper also discusses extra analytical benefit obtained from applying reinforcement learning to these types of problems.     

Analysis and design of selection committees: a game theoretic secretary problem

Firms often delegate important decisions to committees which are set up specifically for that purpose; for example selection committees. We analyze the equilibrium behavior of a game in which committee members (the players) interview candidates sequentially, either hiring or going on to the next one. The players have differing evaluations of candidates (e.g. one cares about typing skills; the other about IT skills), which become their utilities if the candidate is hired. We then consider the optimal design (rules of the game) of such a committee, from the point of view of the firm. That is, which rules hire candidates which maximize the firm’s utility. Our committee game has a first round in which the members sequentially, by order of player number, say ‘yea’ or ‘nea’ to the candidate. If there are sufficient ‘yeas’ then she is tentatively hired; otherwise she is rejected. In the former case, members who said nea can veto the candidate in the second round. Thus the candidate is either hired, rejected, or vetoed. In the last case, the member casting a veto has one less to use on later candidates. We analyze equilibria where a player may say ‘yea’ to a candidate he would prefer not to hire, in order to force the other player to use up a valuable veto. We show that for the uniform candidate distribution there is a unique equilibrium and better candidates for the firm are hired when there are more vetoes. However we exhibit a candidate distribution where increasing the numbers of vetoes results in hiring worse candidates.     

A game theoretic model for two types of customers competing for service

Two types of customers arrive at a single server station and demand service. If a customer finds the server busy upon arrival (or retrial) he immediately departs and conducts a retrial after an exponential period of time and persists this way until he gets served. Both types of customers face linear costs for waiting and conducting retrials and wish to find optimal retrial rates which will minimize these costs. This problem is analysed as a two-person nonzero sum game. Both noncooperative strategies are studied.     

Dynamic asset allocation with loss aversion in a jump-diffusion model

This paper investigates a dynamic asset allocation problem for loss-averse investors in a jumpdiffusion model where there are a riskless asset and N risky assets. Specifically, the prices of risky assets are governed by jump-diffusion processes driven by an m-dimensional Brownian motion and a(N- m)-dimensional Poisson process. After converting the dynamic optimal portfolio problem to a static optimization problem in the terminal wealth, the optimal terminal wealth is first solved. Then the optimal wealth process and investment strategy are derived by using the martingale representation approach. The closed-form solutions for them are finally given in a special example.     

A generalized DEA model for centralized resource allocation

In this paper, we introduced a new generalized centralized resource allocation model which extends Lozano and Villa’s and Asmild et al.’s models to a more general case. In order to uncover the sources of such total input contraction in the generalized centralized resource allocation model, we applied the structural efficiency to further decompose it into three components: the aggregate technical efficiency, the aggregate allocative efficiency and re-transferable efficiency components. The proposed models are not only flexible enough for the central decision-maker to adjust the inputs and outputs to achieve the total input contraction but also identify the sources of such total input contraction, thereby giving rise to an important interpretation and understanding of the generalized centralized resource allocation model. Finally, an empirical example is used to illustrate the approach.     

A multiobjective planning model for university resource allocation

Application of systematic approaches and dynamic planning for efficient resources allocation in institutions of higher education has been quite prevalent the past fifteen years. However, most models developed for university management have focused upon the analysis of input (budget/resource) requirements. They have generally neglected the existing environment, unique institutional values, and bureaucratic decision structures. In this study, a large scale goal programming model is developed for an efficient allocation of resources for an entire university for a five year planning period. The results of the model have been implemented in the form of administrative actions to achieve a set of multiple objectives.