Homogeneous Players Research Articles

Equilibrium analysis in majority-based coalitional bargaining games

This paper introduces majority rule into coalitional bargaining games, adapting traditional models that rely on unanimous consent to more accurately mirror decision-making processes in real-world scenarios. We introduce a majority-based coalitional bargaining game (MBCBG), wherein coalitions pass proposals via majority votes. Our analysis of the stationary subgame perfect equilibrium (SSPE) not only establishes the necessary and sufficient conditions for SSPE strategy profiles but also confirms the existence of no-delay SSPEs in MBCBGs. We further delve into symmetric MBCBGs, delineating conditions that ensure equitable outcomes for homogeneous players. Furthermore, we provide a necessary and sufficient condition for the formation of the grand coalition under SSPEs. Additionally, we briefly explore how asymmetries in coalition values, proposal probabilities, and voting weights may influence both the dynamics of coalition formation and the expected equilibrium payoffs.

Learning Discrete-Time Major-Minor Mean Field Games

Recent techniques based on Mean Field Games (MFGs) allow the scalable analysis of multi-player games with many similar, rational agents. However, standard MFGs remain limited to homogeneous players that weakly influence each other, and cannot model major players that strongly influence other players, severely limiting the class of problems that can be handled. We propose a novel discrete time version of major-minor MFGs (M3FGs), along with a learning algorithm based on fictitious play and partitioning the probability simplex. Importantly, M3FGs generalize MFGs with common noise and can handle not only random exogeneous environment states but also major players. A key challenge is that the mean field is stochastic and not deterministic as in standard MFGs. Our theoretical investigation verifies both the M3FG model and its algorithmic solution, showing firstly the well-posedness of the M3FG model starting from a finite game of interest, and secondly convergence and approximation guarantees of the fictitious play algorithm. Then, we empirically verify the obtained theoretical results, ablating some of the theoretical assumptions made, and show successful equilibrium learning in three example problems. Overall, we establish a learning framework for a novel and broad class of tractable games.

Model-Free Reinforcement Learning for Mean Field Games

In this paper, we have proposed a model-free algorithm, based on sequential decomposition, to obtain optimal policies for. We consider finite horizon with a large population of homogeneous players, sequentially making strategic decisions. Each player observes a private state and a mean-field population state representing the empirical distribution of other players' states. The mean-field state is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">common information</i> among all the players in the game. Authors in [1] provided a sequential decomposition algorithm to compute for such games in linear time than exponential as in prior literature. We extended the idea of sequential decomposition to propose a model-free algorithm for these games using Expected Sarsa in[2]. In this paper, we provide detailed convergence proofs for our algorithm. In addition, we propose an algorithm for with unknown reward functions. The proposed algorithm learns the reward function by studying an expert's behavior and then computes the optimal policy. We illustrate our results using a cyber-physical security example.

Dynamics of a Quantum Common-Pool Resource Game with Homogeneous Players' Expectations.

In this work, we analyse a common-pool resource game with homogeneous players (both have boundedly rational expectations) and entanglement between players' strategies. The quantum model with homogeneous expectations is a differential approach to the game since, to the best of our knowledge, it has hardly been considered in previous works. The game is represented using a Cournot type payoff functions, limited to the maximum capacity of the resource. The behaviour of the dynamics is studied considering how the fixed points (particularly the Nash equilibrium) and the stability of the system vary depending on the different values of the parameters involved in the model. In the analysis of this game, it is especially relevant to consider the extent to which the resource is exploited, since the output of the players is highly affected by this issue. It is studied in which cases the resource can be overexploited, adjusting the parameters of the model to avoid this scenario when it is possible. The results are obtained from an analytical point of view and also graphically using bifurcation diagrams to show the behaviour of the dynamics.

Effect of players’ expectations and memory in a quantum Cournot game

The study of Game Theory is widely developed currently and has many applications in several fields such as economics, psychology, biology, etc. The use of quantum theory is a demonstrated way to improve the results comparing to the classic games due to the entanglement between the players. As well, it is known the positive effect of implementing long term memory in the stability of a dynamical system. In this context, memory is applied to the model of a quantum dynamical Cournot duopoly game considering two different cases, depending on the players’ expectations and the mechanisms they used to maximize their profits. In this work we analyse the game with homogeneous players, considering two boundedly rational players, and the game with heterogeneous expectations, where one of the players is boundedly rational and the other one is a naive player, comparing the results obtained in both cases. Firstly, we come to the conclusion that neither memory nor the type of players (heterogeneous or homogeneous) produces variations in the stable fixed points comparing to the quantum, or even, classic Cournot duopoly game and, therefore, Nash equilibrium is preserved. Secondly, it is observed that the game with homogeneous players, which is not deeply studied previously in quantum games, can improve the local stability of the system versus the game with heterogeneous players, under certain conditions. Finally, it is shown the role of memory as an effective mechanism of chaos control. This achievement is a remarkable economic advantage, since enables the system to reach the stability faster. Throughout this article, these statements are proved analytically and supported widely with several numerical simulations, using different values of the memory factor.

Optimal Relative Performance Criteria in Mean-Field Contribution Games

We consider mean-field contribution games, where players in a team choose some effort levels at each time period and the aggregate reward for the team depends on the aggregate cumulative performance of all the players. Each player aims to maximize the expected reward of her own share subject to her cost of effort. To reduce the free-rider issue, we propose some relative performance criteria (RPC), based on which the reward is redistributed to each player. We are interested in those RPCs that implement the optimal solution for the corresponding centralized problem, and we call such RPC an optimal one. That is, the expected payoff of each player under the equilibrium associated with an optimal RPC is as large as the value induced by the corresponding problem where players completely cooperate. We first analyze a one-period model with homogeneous players and obtain natural RPCs of different forms. Then, we generalize these results to a multiperiod model in discrete time. Next, we investigate a two-layer mean-field game. The top layer is an interteam game (team-wise competition), in which the reward of a team is impacted by the relative achievement of the team with respect to other teams; the bottom layer is an intrateam contribution game where an RPC is implemented for reward redistribution among team members. We establish the existence of equilibria for the two-layer game and characterize the intrateam optimal RPC. Finally, we extend the (one-layer) results of optimal RPCs to the continuous-time setup as well as to the case with heterogenous players.

Mean-Field Game for Collective Decision-Making in Honeybees via Switched Systems

In this article, we study the optimal control problem arising from the mean-field game formulation of the collective decision-making in honeybee swarms. A population of homogeneous players (the honeybees) has to reach consensus on one of two options. We consider three states: the first two represent the available options (or strategies), and the third one represents the uncommitted state. We formulate the continuous-time discrete-state mean-field game model. The contributions of this article are the following: 1) we propose an optimal control model where players have to control their transition rates to minimize a running cost and a terminal cost, in the presence of an adversarial disturbance; 2) we develop a formulation of the micro–macro model in the form of an initial-terminal value problem with switched dynamics; 3) we study the existence of stationary solutions and the mean-field Nash equilibrium for the resulting switched system; 4) we show that under certain assumptions on the parameters, the game may admit periodic solutions; and 5) we analyze the resulting microscopic dynamics in a structured environment where a finite number of players interact through a network topology.

Groundwater Extraction in the South Korea’s Jeju Island: A Real Options Game Approach under Price Uncertainty

This paper uses a standard non-cooperative sequential game with two homogeneous players to analyze investment options of groundwater development project in South Korea’s Jeju island. The model is constructed as an option game taking the uncertainty of water price and the irreversibility of investment into account. The results show that the threshold water price of follower increases with the investment scale of both the leader and the follower while the threshold water price for the leader decreases as the investment scale of the leader increases. This makes the leader choose strategies to maximize the amount of groundwater extraction regardless of the follower’s strategy. Based on the results, it is recommended for policymakers to manage sustainable use of groundwater based on the policy measures such as the groundwater extraction quota system.

Dynamics of a Bertrand Duopoly Game of the Greek Oil Market and Application of the d-Backtest Method

This work presents the complexity that emerges in a Bertrand duopoly between two companies in the Greek oil market, one of which is semi-public and the other is private. The game uses linear demand functions for differentiated products from the existing literature and asymmetric cost functions that arose after approaches using the published financial reports of the two oil companies (Hellenic Petroleum and Motor Oil). The game is based on the assumption of homogeneous players who are characterized by bounded rationality and follow an adjustment mechanism. The players’ decisions for each time period are expressed by two difference equations. A dynamical analysis of the game’s discrete dynamical system is made by finding the equilibrium positions and studying their stability. Numerical simulations include bifurcation diagrams and strange attractors. Lyapunov numbers’ graphs and sensitivity analysis in initial conditions prove the algebraic results and reveal the complexity and chaotic behavior of the system focusing on the two parameters k1 and k2 (speed of adjustment for each player). The d-Backtest method is applied through which an attempt is made to control the chaos that appears outside the stability space in order to return to the locally asymptotically stable Nash equilibrium for the system.

Optimal Relative Performance Criteria in Mean Field Contribution Games

We consider mean-field contribution games, where players in a team chooses some effort level at each time period, and the aggregate reward for the team depends on the aggregate cumulative performance of all the players. Each player aims to maximize the expected reward of her own share subject to her cost of effort. To reduce free-rider issue, we propose some relative performance criteria (RPC), based on which the reward is redistributed to each player. We are interested in those RPCs which implement the optimal solution for the corresponding centralized problem, and we call such RPC an optimal one. That is, the expected payoff of each player under the equilibrium associated with an optimal RPC is as large as the value induced by the corresponding problem where players completely cooperate. We first analyze a one-period model with homogeneous players, and obtain natural RPCs of different forms. Then we generalize these results to a multi-period model in discrete time. Next, we investigate a two-layer mean-field game: The top-layer is an inter-team game (team-wise competition) in which the reward of a team is impacted by the relative achievement of the team with respect to other teams; the bottom layer is an intra-team contribution game where an RPC is implemented for reward redistribution among team members. We establish the existence of equilibria for the two-layer game and characterize the intra-team optimal RPC. Finally, we extend the (one-layer) results of optimal RPCs to the continuous-time setup as well as to the case with heterogenous players.

Improving Selfish Routing for Risk-Averse Players

We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (PoA) wrt. the total latency is improved. The starting point is to introduce some small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases, due to risk-aversion. We adopt the simple model of γ-modifiable routing games, a variant of selfish routing games with restricted tolls. We prove that computing the best γ-enforceable flow is NP-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured γ-enforceable flow whose PoA improves fast as γ increases. We show that the complexity of computing such a γ-enforceable flow is determined by the complexity of computing a Nash flow for the original game. Moreover, we prove that the PoA of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best γ-enforceable flow has the same PoA, and (ii) considering more flexible modifications does not lead to any improvement on the PoA.

Progression in Youth Soccer: Selection and Identification in Youth Soccer Players Aged 13-15 Years.

Bidaurrazaga-Letona, I, Lekue, JA, Amado, M, and Gil, SM. Progression in youth soccer: Selection and identification in youth soccer players aged 13-15 years. J Strength Cond Res 33(9): 2548-2558, 2019-The aim of this study was to identify the factors that are important for the identification and selection of young soccer players. Ninety-four adolescent soccer players from the under-13 (U13; age = 12.3 ± 0.3 years; n = 50) and under-15 (U15; age = 14.0 ± 0.2 years; n = 44) categories belonging to a professional club participated in the study. Anthropometric measurements, physical tests (sprint, agility, endurance and jump), and maturity status (age at peak height velocity) were recorded over 4 seasons. Comparisons were performed among new players joining the club (Enter players, n = 15), players progressing to the next age category (Club players, n = 54), and players leaving the club (Deselected players, n = 25). A 2-way repeated measures analysis of variance (ANOVA) was performed to determine if significant differences existed between groups and testing time. Better physical performance and improvements observed during the season in performance were found to be one of the main factors for U13 players to continue in the club (p < 0.05-0.001). In the U15 group, although body size, maturation and physical performance appeared to be the most important characteristics for being identified to play in the club (p < 0.05), Club players demonstrated better improvements during the season (p < 0.05). Overall, these results indicate that the identification or promotion of players by coaches depends on indicators which are age-dependent. Therefore, this study has shown that the talent identification program was more a selection process than a promotion process, selecting and identifying a posteriori rather than a priori.

Vehicular cloud networking: evolutionary game with reinforcement learning-based access approach

In this paper, we study the vehicular cloud access problem. We model it as an evolutionary game where the vehicles choose to cooperate or to access the conventional cloud through the LTE link. We focus on the centralised case, and we study the equilibrium of both homogeneous and heterogeneous players analytically. We propose an evolutionary game-based vehicular cloud access algorithm (EG-VCA). Moreover, we propose a distributed Q-learning-based vehicular cloud access algorithm (QL-VCA) that allows each vehicle to select the way to access independently to avoid the use of a centralised controller. The simulation results show that QL-VCA and EG-VCA algorithms present almost the same performances. Also, they offer better results compared to the cases of using and accessing only the CC or the VC. Numerical results are also established. They outline the convergence of the two algorithms to the same state of equilibrium.

Non Stationary Markov Perfect Equilibria in Mean-Field Games

In this paper, we consider both finite and infinite horizon discounted dynamic mean-field games where there is a large population of homogeneous players sequentially making strategic decisions and each player is affected by other players through an aggregate population state. Each player has a private type that only she observes. Such games have been studied in the literature under simplifying assumption that population state dynamics are stationary. In this paper, we consider non-stationary population state dynamics and present a novel backward recursive algorithm to compute Markov perfect equilibrium (MPE) that depend on both, a player's private type, and current (dynamic) population state. We present sufficient conditions for existence of equilibria. Using this algorithm, we study a security problem in cyber-physical system where infected nodes put negative externality on the system, and each node makes a decision to get vaccinated. We numerically compute MPE of the game.

Distributed Demand Peak Reduction With Non-Cooperative Players and Minimal Communication

Demand response technology offers the exciting potential to reduce peak energy demand, electricity infrastructure expenditure, and household electricity bills. In this paper, a pricing mechanism that relies on non-cooperative heterogeneous loads knowledgeable of future energy consumption—such as electric vehicles—transferring minimal amounts of information to achieve peak demand response in a distributed fashion, whilst maintaining the privacy of the players. The existence of a Nash equilibrium is proven, as well as convergence conditions proving uniqueness of a Nash equilibrium and the stability of an “Iterated Synchronous Best Response Algorithm.” The price of anarchy (PoA) is proven to approach 1 as the number of homogeneous players approaches infinity, indicating there is no advantage to cooperation for a large number of similar players. Finally, simulation results are presented which suggest that the PoA for a system with heterogeneous players is likely to be proportional to the spread of energy consumption constraints.

Technical Note—Analysis of Scrip Systems: On an Open Question in Johnson et al. (2014)

In a recent paper, Johnson et al. (2014) [Johnson K, Simchi-Levi D, Sun P (2014) Analyzing scrip systems. Oper. Res. 62(3):524–534.]use an infinitely repeated game with discounting, among a set of homogeneous players, to model a scrip system. In each period, a randomly chosen player requests service; all the other players have a choice of whether or not to volunteer to provide service. Among the players who volunteer, the service provider is chosen using the minimum-scrip rule: a player with the minimum number of scrips is chosen as the service provider. The authors study the always-trade strategy for a player, that is, the strategy of a player always willing to provide service, regardless of the distribution of scrips among the players. A key result of their work is that, under the minimum-scrip rule, for any number of players and for every discount factor close enough to one, there exists a Nash equilibrium in which each player plays the always-trade strategy. This result, however, is established under an assumption of severe punishment: If a player selected to provide service refuses to do so even once, then that player will be forever banned from participating in the system, thus losing all potential future benefit. Johnson et al. (2014) explain that this assumption is unsatisfactory, particularly for large scrip systems, because of difficulties in detecting players who refuse to provide service and verifying the reason(s) for their refusal, and the consequent possibility of players being unfairly banned. Motivated by this concern, the authors suggest an important direction of future research: investigate whether or not the always-trade strategy is an equilibrium without the punishment assumption. In this note, we address the question posed in Johnson et al. (2014) for a generalization of their model. Our generalization allows for the possibility that players might genuinely not be available to provide service, say because of sickness. Without using the assumption of punishment, we show that when the number of players is large enough and the discount factor is close enough to one, there exists an ε-Nash equilibrium in which each player plays the always-trade strategy.

Analysis of Price of Total Anarchy in Congestion Games via Smoothness Arguments

Efficiency loss may exist at every stage of repeated play. With the rapid development of our society, how to improve the overall efficiency in repeated games becomes increasingly crucial. This paper studies the performance of a sequence of action profiles generated by repeated play in a multiple origin-destination network. To analyze the overall efficiency of these action profiles, the price of total anarchy (POTA), defined as the worst-case ratio of the average total latency over a period of time and the total latency of any optimal strategy, is adopted. First of all, it is shown that the sequence of action profiles generated by best response principle with inertia possesses almost sure no-regret property. Then, we identify the upper bound of POTA via smoothness arguments for both linear and nonlinear latency cases. After that, we analyze the effect of road pricing on POTA in traffic networks with linear latency function. It is shown that the road price we designed can reduce the upper bound of POTA compared with that without road price. In addition, how the inaccurate parameter information of the latency function affects the POTA is discussed for the linear latency case. Moreover, in a traffic network with heterogeneous players, we show that the upper bound of POTA is the same as that in a network with homogeneous players as time goes to infinity. Finally, the results are applied to a traffic routing problem based on the real traffic data of Singapore.

Recent Developments at the CMA: 2016–2017

We discuss three interesting cases that the Competition and Markets Authority (CMA) has dealt with over the past year. First, we cover the ICE/Trayport vertical merger, which was prohibited by the CMA. Second, we discuss the CMA’s recent market investigation into the UK energy sector. The CMA arrived at the unexpected finding that markets with homogeneous products, multiple players and low entry barriers can still lead to significant competition concerns. Finally, we discuss the CMA’s work in relation to retail most favoured nation clauses, and include econometric results showing the impact of these clauses on platforms’ commission fees .

Strategic formation of homogeneous bargaining networks

We study a model of strategic network formation prior to a Manea (2011a) bargaining game: ex ante homogeneous players form costly undirected links, anticipating expected equilibrium payoffs from the subsequent bargaining game. Assuming patient players, we provide a complete characterization of generically pairwise stable networks: specific unions of separated pairs, odd circles, and isolated players constitute this class. We also show that many other structures, such as larger trees or unbalanced bipartite networks, cannot be pairwise stable at all. The analysis implies that the diversity of possible bargaining outcomes is small in (generically) pairwise stable networks.

Complexity in a Duopoly Game with Homogeneous Players, Convex, Log-linear Demand and Quadratic Cost Functions

Abstract In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have homogeneous expectations. We suppose that the cost function is quadratic and the demand is a convex and log- linear function. The game is modeled with a system of two difference equations. Existence and stability of equilibria of this system are studied. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the speed of adjustment of the players. If this parameter is varied, the stability of Nash equilibrium is lost through period doubling bifurcations. The chaotic features are justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions.