Splitting graphs when calculating Myerson value for pure overhead games

A communication situation consists of a coalitional game and a graph, the nodes of the graph corresponding to the players of the game. To calculate the Myerson value for such situations, we obtain results which extend those well known for trees and cycle-complete graphs. On the other hand, in order to reduce the associated calculus for communication situations with a pure overhead game, the possibility of splitting the graph in several subgraphs is analyzed. For each fixed decomposition of the graph, a subspace of games compatible with this decomposition is given.     

Von Neumann and Newman poker with a flip of hand values

The von Neumann and Newman poker models are simplified two-person poker models in which hands are modeled by real values drawn uniformly at random from the unit interval. We analyze a simple extension of both models that introduces an element of uncertainty about the final strength of each player’s own hand, as is present in real poker games. Whenever a showdown occurs, an unfair coin with fixed bias q is tossed, 0≤q≤1/2. With probability 1−q, the higher hand value wins as usual, but, with the remaining probability q, the lower hand wins. Both models favor the first player for q=0 and are fair for q=1/2. Our somewhat surprising result is that the first player’s expected payoff increases with q as long as q is not too large. That is, the first player can exploit the additional uncertainty introduced by the coin toss and extract even more value from his opponent.     

A new look at the role of players’ weights in the weighted Shapley value

everal new families of semivalues for weighted n-person transferable utility games are axiomatically constructed and discussed under increasing collections of axioms, where the weighted Shapley value arises as the resulting one member family. A more general approach to such weighted games defined in the form of two components, a weight vector λ and a classical TU-game v, is provided. The proposed axiomatizations are done both in terms of λ and v. Several new axioms related to the weight vector λ are discussed, including the so-called “amalgamating payoffs” axiom, which characterizes the value of a weighted game in terms of another game with a smaller number of players. They allow for a new look at the role of players’ weights in the context of the weighted Shapley value for the model of weighted games, giving new properties of it. Besides, another simple formula for the weighted Shapley value is found and examples illustrating some surprising behavior of it in the context of players’ weights are given. The paper contains a wide discussion of the results obtained.     

The Subset Sum game

In this work we address a game theoretic variant of the Subset Sum problem, in which two decision makers (agents/players) compete for the usage of a common resource represented by a knapsack capacity. Each agent owns a set of integer weighted items and wants to maximize the total weight of its own items included in the knapsack. The solution is built as follows: Each agent, in turn, selects one of its items (not previously selected) and includes it in the knapsack if there is enough capacity. The process ends when the remaining capacity is too small for including any item left.     

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley–Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation.     

Introducing competition in healthcare services: The role of private care and increased patient mobility

We study the operational implications from competition in the provision of healthcare services, in the context of national public healthcare systems in Europe. Specifically, we study the potential impact of two alternative ways through which policy makers have introduced such competition: (i) via the introduction of private hospitals to operate alongside public hospitals and (ii) via the introduction of increased patient choice to grant European patients the freedom to choose the country they receive treatment at. We use a game-theoretic framework with a queueing component to capture the interactions among the patients, the hospitals and the healthcare funders. Specifically, we analyze two different sequential games and obtain closed form expressions for the patients’ waiting time and the funders’ reimbursement cost in equilibrium. We show that the presence of a private provider can be beneficial to the public system: the patients’ waiting time will decrease and the funders’ cost can decrease under certain conditions. Also, we show that the cross-border healthcare policy, which increases patient mobility, can also be beneficial to the public systems: when welfare requirements across countries are sufficiently close, all funders can reduce their costs without increasing the patients’ waiting time. Our analysis implies that in border regions, where the cost of crossing the border is low, “outsourcing” the high-cost country’s elective care services to the low-cost country is a viable strategy from which both countries’ systems can benefit.     

A note on “Quality investment and inspection policy in a supplier-manufacturer supply chain”

In a recent paper, Hsieh and Liu (2010) consider quality investment and inspection strategies of one supplier and one manufacturer. They propose solutions of four non-cooperative game models with different degrees of information available about the players’ inspection sampling rates and quality investments. For the two most comprehensive scenarios, we show that these (interior) solutions do not always represent the optimal solutions, as the boundary solutions lead to considerably higher profits. Furthermore, we propose modified algorithms providing the optimal solutions by including the boundary solutions into the consideration.     

Multiple goal non-cooperative conflicts under uncertainty: A fuzzy set approach

This paper investigates strategy selection for a participant in a two-party non-cooperative conflict which involves both uncertainty and multiple goals. Uncertainty arises from the players not knowing the utility functions. Multiple objectives appear as the result of the payoff being a vector of prizes and the players attempt to attain various goals for each prize separately. The main objective is to present a fuzzy set/fuzzy programming solution concept to the conflict situation. In doing so, we compare a Bayesian player to one that employs fuzzy set techniques. We point out some of the advantages of the fuzzy set method. The necessary computations in the fuzzy set method are explained in detail through an example.     

Explicit Stability Zones for Cournot Game with 3 and 4 Competitors

The dynamical system of 3 and 4 competitors in a Cournot game is studied. The stability of its fixed points (Nash-equilibria) are also investigated. The stable and unstable regions are explicitly shown. The bifurcation characteristics are found. Periodic orbits with different periods 7, 25, 18, 13, 17 etc., are detected in both cases. The study of these models is very rich in bifurcation phenomena.     

Distributed computation of Pareto solutions inn-player games

The problem of computing Pareto optimal solutions with distributed algorithms is considered inn-player games. We shall first formulate a new geometric problem for finding Pareto solutions. It involves solving joint tangents for the players' objective functions. This problem can then be solved with distributed iterative methods, and two such methods are presented. The principal results are related to the analysis of the geometric problem. We give conditions under which its solutions are Pareto optimal, characterize the solutions, and prove an existence theorem. There are two important reasons for the interest in distributed algorithms. First, they can carry computational advantages over centralized schemes. Second, they can be used in situations where the players do not know each others' objective functions.