On the Core of Dynamic Cooperative Games

We consider dynamic cooperative games, where the worth of coalitions varies over time according to the history of allocations. When defining the core of a dynamic game, we allow the possibility for coalitions to deviate at any time and thereby to give rise to a new environment. A coalition that considers a deviation needs to take the consequences into account because from the deviation point on, the game is no longer played with the original set of players. The deviating coalition becomes the new grand coalition which, in turn, induces a new dynamic game. The stage games of the new dynamical game depend on all previous allocation including those that have materialized from the deviating time on. We define three types of core solutions: fair core, stable core and credible core. We characterize the first two in case where the instantaneous game depends on the last allocation (rather than on the whole history of allocations) and the third in the general case. The analysis and the results resembles to a great extent the theory of non-cooperative dynamic games.Comment: 25 page

Core Concepts for Dynamic TU Games

This paper is concerned with the question of how to define the core when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face a finite sequence of exogenously specified TU-games. Three different core concepts are presented: the classical core, the strong sequential core and the weak sequential core. The differences between the concepts arise from different interpretations of profitable deviations by coalitions. Sufficient conditions are given for nonemptiness of the classical core in general and of the weak sequential core for the case of two players. Simplifying characterizations of the weak and strong sequential core are provided. Examples highlight the essential difference between these core concepts.mathematical economics and econometrics ;

Stability and monotonicity in newsvendor situations

Cataloged from PDF version of article.This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, and a supplier. Groups of retailers might increase their expected joint profit by joint ordering and inventory centralization. However, we assume that the retailers impose some level of stock that should be dedicated to them. In this situation, we show that the associated cooperative game has a non-empty core. Afterwards, we concentrate on a dynamic situation, where several model cost parameters and the retailers’ dedicated stock levels can change. We investigate how the profit division might be affected by these changes. We focus on four monotonicity properties. We identify several classes of games with retailers, where some of the monotonicity properties hold. Moreover, we show that pairs of cooperative games associated with newsvendor situations do not necessarily satisfy these properties in general, when changes in dedicated stock levels are in concern

Profit division in newsvendor situations with delivery restrictions

This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, and a supplier. Groups of retailers might increase their expected joint profit by joint ordering and inventory centralization, which means that they give a joint order and allocate this quantity among themselves to maximize the total profit after the demands are realized. Furthermore, we assume that the retailers pose some restrictions on the number of items that should be delivered to them. In this situation, we show that the associated cooperative game has a non-empty core. Afterwards, we concentrate on a dynamic situation, where the retailers change their delivery restrictions. We investigate how the profit division might be affected by these changes. We define four new monotonicity properties, which we think are interesting in general, and we derive necessary and sufficient conditions for pairs of totally balanced TU-games to satisfy these properties. We also show that pairs of cooperative games associated with newsvendor situations do not necessarily satisfy these properties in general. Finally, we define a class of games with retailers having a normally distributed demand where one of the monotonicity properties holds

Equity and power in a cooperative trial-and-error game

General solution concepts in cooperative game theory are static, e.g., the core, the Shapley value and the Nash bargaining solution. Dynamic implementation procedures have been proposed in order to support these static solution concepts. This thesis studies an N-dimensional Markov chain motivated by a dynamic interactive trial-and-error learning model. The state space of the Markov chain is based on a cooperative game (v,N) whose characteristic function v is superadditive and monotone, with conditions on v ensuring non-emptiness of the core. Agents repeatedly bargain over a cooperative surplus by submitting their demand for their share. Each round the payable coalition is chosen, the feasible coalition with the maximum sum of demands. Players in the payable coalition receive their demands as payoffs, the other players receive no payoff. Players adjust their demands according to the following rule: In an efficient state (where the demand sum of all players equals the total surplus, 1) one player is chosen uniformly at random and increases his demand by ε. If demands sum to 1 + ε, one player not in the payable coalition is then chosen to reduce her demand with probability proportional to the size of her demand. An individual’s demand update decision in the learning model is based solely on the observation of his last payoff. Individual updates are in the tradition of reinforcement learning, aspiration adaption, and fictitious play. Selten (1972) found empirical evidence for an inherent equity principle in many outcomes of experimental cooperative bargaining games. By construction, the dynamic learning model presented in this thesis also has an inherent equity principle. The model is a simple modification (and the limit process) of a model introduced by Nax (2010). To our knowledge, this thesis presents the first general results of such a dynamic learning model for general 3-player games and all interesting cases of 4-player games. The transition probabilities of the Markov process studied in this thesis are the transition probabilities between efficient states, obtained by the two steps from an efficient state to a state with demand sum 1+ε and back, of the described trial and error process. The process is a biased random walk on the simplex of efficient states, of which the polytope formed by the grid of core points forms the subset of particular interest. For general N-player games we introduce a coalition structure that exhibits an asymmetry of power between its members: the asymmetric coalition set. We believe the concept of an asymmetric coalition set to be both novel and relevant to the study of dynamic learning models with incremental demand updates for general cooperative games. Along a face of the core polytope generated by an asymmetric coalition set, the asymmetric face, the bias of the process is determined by the interplay between two dynamics: the inherent equity bias, which “drags” the process towards equity, and the asymmetric power, which “drags” the process away from equity. If the core polytope does not contain an asymmetric face, the equity bias of the random walk determines the expected movement along the faces of the polytope. The process can only leave the core polytope from a state on an asymmetric face. We study a special Markov chain in dimension N derived from the N-player bargaining game, where no coalitional constraints are present. Then the bias of the random walk is solely determined by the inherent equity principle: the random walk drifts towards equity, and the equilibrium distribution is concentrated around the equal split, the most equitable allocation. For N = 3, no asymmetric coalition set exists. We show that the set of recurrent states of the Markov chain is the “core polygon”, formed by the grid points in the core. The cooperative outcome co is the unique vector in the core with smallest L2-distance from the equal split. At every state of the core polygon outside a small ball around co, the random walk moves in expectation over one time step towards co. The equilibrium distribution of the Markov chain is concentrated around the vector co. For 3-player games this vector equals the egalitarian allocation, a concept developed by Dutta and Ray (1989). For N ≥ 4, games (v,N) can contain an asymmetric coalition set. For N = 4 the only possible asymmetric coalition set is formed by two distinct two player coalitions. We give three example games (v,4) with combinatorially isomorphic core. Each of the example games has an asymmetric edge in the core. Along the asymmetric edge the inherent equity bias creates a drift dynamic “down” the asymmetric edge, and the asymmetric power creates a drift dynamic “up” the asymmetric edge. In each example game the asymmetric power is extreme, zero or moderate respectively: the equilibrium distribution of the process is concentrated at the “upper” endpoint, the “lower” endpoint (which is co) or around a demand vector in the interior of the asymmetric edge. Furthermore we give simulation results, which indicate that the concept of asymmetric power can be generalized to other dynamic learning processes. Coupling is a powerful and elegant probabilistic tool with which one is often able to calculate tight bounds on the speed of convergence to equilibrium of Markov chains. We believe this technique to be novel to the study of dynamic stochastic learning processes in evolutionary game theory and hence present a general introduction to the technique. We use coupling arguments to show rapid mixing for the cooperative game process for the N-player bargaining game and for general 3-player games

Supply chain collaboration

In the past, research in operations management focused on single-firm analysis. Its goal was to provide managers in practice with suitable tools to improve the performance of their firm by calculating optimal inventory quantities, among others. Nowadays, business decisions are dominated by the globalization of markets and increased competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in operations management has shifted its focus from single-firm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms’ decisions ask for alignment and coordination of actions. Therefore, game theory, the study of situations of cooperation or conflict among heterogenous actors, is very well suited to deal with these interactions. This has been recognized by researchers in the field, since there are an ever increasing number of papers that applies tools, methods and models from game theory to supply chain problems

Study of a Dynamic Cooperative Trading Queue Routing Control Scheme for Freeways and Facilities with Parallel Queues

This article explores the coalitional stability of a new cooperative control policy for freeways and parallel queuing facilities with multiple servers. Based on predicted future delays per queue or lane, a VOT-heterogeneous population of agents can agree to switch lanes or queues and transfer payments to each other in order to minimize the total cost of the incoming platoon. The strategic interaction is captured by an n-level Stackelberg model with coalitions, while the cooperative structure is formulated as a partition function game (PFG). The stability concept explored is the strong-core for PFGs which we found appropiate given the nature of the problem. This concept ensures that the efficient allocation is individually rational and coalitionally stable. We analyze this control mechanism for two settings: a static vertical queue and a dynamic horizontal queue. For the former, we first characterize the properties of the underlying cooperative game. Our simulation results suggest that the setting is always strong-core stable. For the latter, we propose a new relaxation program for the strong-core concept. Our simulation results on a freeway bottleneck with constant outflow using Newell's car-following model show the imputations to be generally strong-core stable and the coalitional instabilities to remain small with regard to users' costs.Comment: 3 figures. Presented at Annual Meeting Transportation Research Board 2018, Washington DC. Proof of conjecture 1 pendin

Robust Dynamic Cooperative Games

Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. Recent works address situations where the values of coalitions are modelled by random variables. In this work we still consider the values of coalitions as uncertain, but model them as unknown but bounded disturbances. We do not focus on solving a specific game, but rather consider a family of games described by a polyhedron: each point in the polyhedron is a vector of coalitions’ values and corresponds to a specific game. We consider a dynamic context where while we know with certainty the average value of each coalition on the long run, at each time such a value is unknown and fluctuates within the bounded polyhedron. Then, it makes sense to define “robust” allocation rules, i.e., allocation rules that bound, within a pre- defined threshold, a so-called complaint vector while guaranteeing a certain average (over time) allocation vector. We also present as motivating example a joint replenishment application

On robustness and dynamics in (un)balanced coalitional games

We build upon control theoretic concepts like robustness and dynamics to better accommodate all the situations where the coalitions’ values are uncertain and subject to changes over time. The proposed robust dynamic framework provides an alternative perspective on the study of sequences of coalitional games or interval valued games. For a sequence of coalitional games, either balanced or unbalanced, we analyze the key roles of instantaneous and average games. Instantaneous games are obtained by freezing the coalitions’ values at a given time and come into play when coalitions’ values are known. On the other hand, average games are derived from averaging the coalitions’ values up to a given time and are key part of our analysis when coalitions’ values are unknown. The main theoretical contribution of our paper is a design method of allocation rules that return solutions in the core and/or $\epsilon$-core of the instantaneous and average games. Theoretical results are then specialized to a simulated example to shed light on the impact of the design method and on the performance of the resulting allocation rules