Subgame Perfect Nash Equilibrium Research Articles

Sequential Mergers and Competition Policy under Partial Privatization

This study assesses the conditions under which sequential mergers can emerge in a partially privatized oligopoly with differentiated goods. In particular, it examines: (i) the optimal merger strategies by potential merging firms, (ii) optimal merger policy, and (iii) privatization policy of policymakers. First, under the subgame perfect Nash equilibrium, sequential mergers either completely emerge or do not emerge at all. The parameter range that leads to complete sequential mergers becomes larger as the market is more privatized. Second, policymakers can halt privatization, diminishing the private incentive for further sequential mergers and thus leading to higher welfare. Furthermore, given that some mergers have already taken place, further mergers may actually improve welfare. These welfare-improving mergers may not be privately profitable, however, implying that there may be room for merger-friendly policies. Third, policymakers are better off partially privatizing the public firm unless the goods are perfect substitutes or independent. Our results are applicable to the Japanese life insurance industry and the partial privatization of Japan Posting Insurance.

Sequential mergers and competition policy under partial privatisation

AbstractWe examine optimal merger and privatisation policies in a partially privatised oligopoly with differentiated goods. We first show that under the subgame perfect Nash equilibrium, sequential mergers either emerge completely or do not emerge at all. Given this outcome, we derive the following policy implications. First, the level of social welfare can be U‐shaped with respect to the number of merged firm pairs. That is, given that there are some mergers that have already taken place, further mergers may actually lead to welfare improvement. However, these welfare‐improving mergers may not be privately profitable, implying that merger‐friendly policies are appropriate. Second, policymakers can halt privatisation in order to diminish the private incentive for further sequential (welfare‐deteriorating) mergers and improve welfare. Third, full nationalisation is never optimal unless the goods are homogeneous or independent. Our results are applicable to the Japanese life insurance industry and the partial privatisation of Japan Post Insurance.

Multi-relay selection in energy-harvesting cooperative wireless networks: game-theoretic modeling and analysis

This paper studies distributed multi-relay selection in energy-harvesting cooperative wireless networks and models it as an Indian Buffet Game (IBG). Particularly, the IBG is utilized to model the multi-relay selection decisions of network source nodes, while accounting for negative network externality. Two scenarios are considered: (1) constrained selections (CS), and (2) unconstrained selections (US). In the former scenario, each source is constrained to a maximum number of relay selections; while in the latter scenario, the source nodes can select as many relays as possible. Since the relays are energy-harvesting—and thus intermittently harvest random amounts of energy—the accumulated energy at each relay is unknown to the source nodes, leading to uncertain relays’ energy states. In turn, a non-Bayesian learning (NBL) algorithm is devised for the source nodes to learn the relays’ energy states. After that, two distributed best-response (BR) recursive algorithms, namely BR-CS and BR-US, are proposed to allow the source nodes to make multi-relay selection decisions, while guaranteeing subgame perfect Nash equilibrium. Simulation results are presented to verify the efficacy of the proposed distributed NBL and multi-relay selection algorithms. Specifically, the NBL is shown to efficiently learn the true relays’ energy states. More importantly, the BR-CS algorithm is shown to be comparable to the centralized multi-relay selection—and superior to other relay selection schemes—in terms of network sum-rate improvement (and utility). Lastly, the number of relay selections of the BR-CS algorithm must be constrained to the minimum so as to reduce complexity and fully exploit diversity gains.

On time-inconsistent stopping problems and mixed strategy stopping times

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean–variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

Time Consistent Stopping for the Mean-Standard Deviation Problem --- The Discrete Time Case

Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among stopping times or randomized stopping times may not exist. This motivates us to consider the notion of liquidation strategies, which lets the stopping right to be divisible. We then argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. It turns out that an equilibrium liquidation strategy always exists. We then analyze whether optimal equilibrium liquidation strategies exist and whether they are unique and observe that neither may hold.

Optimal eco-labeling strategy with imperfectly informed consumers

Purpose The purpose of this paper is to find the optimal environmental quality criteria for a strategic eco-labeling authority with three objectives (i.e. maximizing the aggregate environmental quality, maximizing the industry profit and maximizing the social welfare). Particularly, the authors investigate how the existence of imperfectly informed consumers affects labeling criteria determination and competition among firms. Design/methodology/approach A game-theoretic modeling approach was adopted in this paper. A three-stage sequential game was modeled and backward induction was used to solve for a subgame perfect Nash equilibrium. To investigate the impacts of the existence of imperfectly informed consumers, the equilibrium, if all consumers are perfectly informed of the eco-label, was studied as a benchmark. Findings A more strict eco-labeling criterion improves revenues for both the labeled and unlabeled firms. It is interesting to find that the eco-labeling criteria to maximize industry profits are stricter than the criteria to maximize social welfare. Moreover, when the fraction of imperfectly informed consumers increases, the eco-labeling criteria to maximize aggregate environmental quality or industry profits will be more strict, while the criteria to maximize the social welfare will be looser. Originality/value The authors analyze the equilibrium strategies for firms against the eco-labeling criteria certified by authority with different objectives. The obtained optimal labeling strategies could provide insightful guidelines for the certifying authority to select the best suitable labeling criteria to achieve its goals.

Media, fake news, and debunking

We construct a modified Hotelling‐type model of two media providers, each of whom can issue fake and/or real news and each of whom can invest in the debunking of their rival’s fake news. The model assumes that consumers have an innate preference for one provider or the other and value real news. However, that valuation varies according to their bias favouring one provider or the other. We demonstrate a unique subgame perfect Nash equilibrium in which only one firm issues fake news and we show, in this setting, that increased polarisation of consumers (represented by a wider distribution) increases the prevalence of both fake news and debunking expenditures and is welfare‐reducing. We also show, inter alia, that a stronger preference by consumers for their preferred provider lowers both fake news and debunking. Finally, we compare monopoly and duopoly market structures in terms of ‘fake news’ provision and show that a public news provider can be welfare‐improving.

The Folk Theorem for Repeated Games With Time-Dependent Discounting

This paper defines a general framework to study infinitely repeated games with time-dependent discounting, in which we distinguish and discuss both time-consistent and time-inconsistent preferences. To study the long-term properties of repeated games, we introduce an asymptotic condition to characterize the fact that players become more and more patient, that is, the discount factors at all stages uniformly converge to $1$. Two types of folk theorem's are proven under perfect observations of past actions and without the public randomization assumption: the asymptotic one, i.e. the equilibrium payoff set converges to the individual rational set as players become patient, and the uniform one, i.e. any payoff in the individual rational set is sustained by a single strategy profile which is an approximate subgame perfect Nash equilibrium in all games with sufficiently patient discount factors. As corollaries, our results of time-inconsistency imply the corresponding folk theorem's with the quasi-hyperbolic discounting.

Licensing to a competitor and strategic royalty choice in a dynamic duopoly

Network effects encourage original equipment manufacturers (OEMs) to expand their market size by changing their relationships with third-party manufacturers who provide compatible products from competition to coopetition. Moreover, network effects render consumer valuation inherently dynamic. Consumer perceptions of the sales quantity are continuously updated, causing the impact of the network effect to change dynamically over time. To this end, we examine technology licensing and price competition in a dynamic duopoly including an OEM and a third-party manufacturer, considering that consumer utility increases with the dynamic and evolving impact of network effects. However, because of limited product technology, the lower compatibility of third-party products reduces network effects. Thus, the third-party manufacturer licenses technology from the OEM. Because the OEM can strategically choose a static or dynamic royalty under technology licensing in a dynamic pricing game, we derive the firms’ subgame-perfect Nash equilibrium decisions and profits and analyze the effects of licensing mechanisms and market factors on firms’ instantaneous and steady-state equilibrium decisions and profits. Technology licensing enhances firms’ profits when firms’ and consumers’ dynamic behaviors are more significant by exerting stronger network effects. A dynamic royalty is more effective for mitigating price competition intensity and for helping firms maintain higher sales margins. A static royalty induces a lower royalty chosen by the OEM and firms lower prices, which increases the impact of network effects and thus is generally more advantageous for the firms and for social welfare.

COMPARING BEHAVIORAL MODELS USING DATA FROM EXPERIMENTAL CENTIPEDE GAMES

The centipede game posits one of the most well‐known paradoxes of backward induction in the literature of experimental game theory. Given that deviations from the unique subgame perfect Nash equilibrium generates a Pareto improvement, several theoretical models have been employed in order to rationalize this kind of behavior in this social dilemma. The available explanations range from social preferences including fairness, altruism or cooperation motives, errors in playing, inability to perform backward induction or different depths of reasoning. In the present study, we use the Blavatskyy's theoretical contribution, and relax the assumptions of Expected Utility maximization and risk‐neutral attitudes, to test an alternative explanation. We compare various probabilistic decision theory models in terms of their descriptive (in‐sample) and predictive (out‐of‐sample fit) performance, using data from experimental centipede games. We find that introducing non‐Expected Utility preferences to the Quantal Response Equilibrium model, along with a nonlinear utility function, provides a better explanation compared to alternative specifications such as the Level‐k or the Quantal Response Equilibrium model with altruistic motives. (JEL C72, C92, D81, D82)

Perverse Effects of Non-sterilized Interventions on Spot Foreign Exchange Rates

We study the effects of non-sterilized intervention on a spot foreign exchange (forex) rate using a multi-period game-theoretical model which involves an unspecified number of competitive traders, a finite number of strategic traders (forex dealers) with heterogenous initial money balances, and the central bank of the home country. Simulating the subgame-perfect Nash equilibrium of the two-stage game played by the forex dealers in each period, we show that the non-sterilized intervention of the central bank may lead to a perverse effect on the spot forex rate. We call the mechanism underlying this effect strategic trade switching channel that works when an increase in the central bank’s forex currency demand (supply) exerts such a big upward (downward) effect on the forex rate that some sufficiently big dealers, who optimally bought (sold) forex currency in the previous period when the forex rate was sufficiently low, find in the current period selling (buying) it more profitable, thus moving the forex rate in a direction undesired by the central bank. JEL Classification: D43, F31, G20

Dual Channel Management with Strategic Customers

Problem definition: We study how a manufacturer shall manage the retail channel in the presence of a direct channel and strategic customers. We consider two periods, in which the manufacturer can sell through a single retailer or two different retailers in sequence with static or dynamic wholesale pricing. The retailer(s) sets the retail prices that the manufacturer’s direct channel needs to match, while the customers in each channel may strategically time their purchases. Academic/practical relevance: Although direct channel and strategic customer behavior are widely present in practice, how to manage the retail channel in such circumstances has not been investigated. Methodology: We derive complete subgame perfect Nash equilibria of this dynamic game. Results: We find that when the manufacturer can only sell through a single retailer, dynamic wholesale pricing is preferred when the market share of the retail channel is large, as it mitigates double marginalization. On the other hand, static wholesale pricing can alleviate strategic customer waiting, which is the most effective when the customers’ waiting incentive is intermediate. If the manufacturer can sell through two retailers, it can cause intertemporal competition between them, which may not only alleviate double marginalization but also counteract customer waiting. As such, the results change structurally. Selling through a single retailer can be optimal only under static wholesale pricing and only if the market share of the retail channel is small and the customers are moderately strategic. In the other scenarios, selling through two retailers with dynamic (static) wholesale pricing is optimal when the customers’ waiting incentive is weak (strong), and static wholesale pricing may be more preferred when the market share of the retail channel becomes larger. We also find that in the presence of strategic customers, the manufacturer fully controlling the retail prices may not be optimal for itself or the supply chain, particularly when the market share of the direct channel is large. Managerial implications: These results can be practically useful for managing the retail channel in the modern business environment.

Interference Management for Cellular-Connected UAVs: A Deep Reinforcement Learning Approach

In this paper, an interference-aware path planning scheme for a network of cellular-connected unmanned aerial vehicles (UAVs) is proposed. In particular, each UAV aims at achieving a tradeoff between maximizing energy efficiency and minimizing both wireless latency and the interference level caused on the ground network along its path. The problem is cast as a dynamic game among UAVs. To solve this game, a deep reinforcement learning algorithm, based on echo state network (ESN) cells, is proposed. The introduced deep ESN architecture is trained to allow each UAV to map each observation of the network state to an action, with the goal of minimizing a sequence of time-dependent utility functions. Each UAV uses ESN to learn its optimal path, transmission power level, and cell association vector at different locations along its path. The proposed algorithm is shown to reach a subgame perfect Nash equilibrium (SPNE) upon convergence. Moreover, an upper and lower bound for the altitude of the UAVs is derived thus reducing the computational complexity of the proposed algorithm. Simulation results show that the proposed scheme achieves better wireless latency per UAV and rate per ground user (UE) while requiring a number of steps that is comparable to a heuristic baseline that considers moving via the shortest distance towards the corresponding destinations. The results also show that the optimal altitude of the UAVs varies based on the ground network density and the UE data rate requirements and plays a vital role in minimizing the interference level on the ground UEs as well as the wireless transmission delay of the UAV.

A duopoly preemption game with two alternative stochastic investment choices

This paper studies a duopoly investment model with uncertainty. There are two alternative irreversible investments. The first firm to invest gets a monopoly benefit for a specified period of time. The second firm to invest gets information based on what happens with the first investor, as well as cost reduction benefits. We describe the payoff functions for both the leader and follower firm. Then, we present a stochastic control game where the firms can choose when to invest, and hence influence whether they become the leader or the follower. In order to solve this problem, we combine techniques from optimal stopping and game theory. For a specific choice of parametres, we show that no pure symmetric subgame perfect Nash equilibrium exists. However, an asymmetric equilibrium is characterized. In this equilibrium, two disjoint intervals of market demand level give rise to preemptive investment behavior of the firms, while the firms otherwise are more reluctant to be the first mover.

Dynamic Communication Mechanism Design

This paper considers dynamic communication mechanisms in a quasi-linear single-value environment. The mechanism designer gradually identifies agents’ valuations by iteratively offering prices to agents at different stages. Agents pay the maximum price they accepted if their desirable decision is made. We show that sincere reporting is an ex post equilibrium if and only if a communication mechanism is a monotone-price mechanism. English auctions are essentially unique mechanisms that satisfy ex post incentive compatibility and efficiency. When sincere reporting is not an equilibrium, an allocation rule is implemented in a subgame perfect Nash equilibrium under a certain condition. The efficient allocation rule is implemented regardless of details of a communication process.

Time Consistent Stopping for the Mean-Standard Deviation Problem---The Discrete Time Case

Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time-consistent strat...

The Optimal Equilibrium for Time-Inconsistent Stopping Problems---The Discrete-Time Case

We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibria. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.

On the Notions of Equilibria for Time-Inconsistent Stopping Problems in Continuous Time

A new notion of equilibrium, which we call strong equilibrium, is introduced for timeinconsistent stopping problems in continuous time. Compared to the existing notions introduced in Time-Consistent Stopping Under Decreasing Impatience and On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems, which in this paper are called mild equilibrium and weak equilibrium respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.

Driver’s lane selection model based on multi-player dynamic game

With a view to the background of Internet of Things, the vehicle group situation, especially the propensity of different drivers, and the corresponding vehicle types were given a comprehensive consideration on the basis of the factors which influence driving behavior, and the drivers’ lane selection model based on multi-player dynamic game with complete information was built. The drivers’ payoffs under different combination of lane selection strategies were analyzed, and on this basis, the subgame perfect Nash equilibrium solution was solved through backward induction, and the drivers’ optimal lane selection strategy was obtained. Next-Generation Simulation datasets and other data were applied for model calibration and validation. The results showed that the proposed model can objectively reflect the actual operation characteristic of traffic flow on road section. So, it is of great significance to apply the multi-person dynamic game in the field of intelligent transportation. And the theoretical basis of the research on lane selection decision-making can be provided for the command system of intelligent driving especially anthropomorphic driving under the condition of Internet of Things.

A Game Theoretic Framework for Quality of Experience Enhancement in Dense Stadia

In crowded venues, such as sports stadia, maintaining an acceptable network quality of experience (QoE) is hard to achieve. Installing small cells, distributed antenna systems or high-density WiFi in every stadium is too expensive for mobile network operators. Hence, we propose a novel distributed low-cost solution based on user coordination to improve the average QoE when the network capacity cannot be enhanced. Specifically, fans take turns in disabling their cellular connectivity, such that the connected users utilize the relaxed network to obtain then share common match data with the disconnected users via Peer-to-Peer (P2P) connectivity. To eliminate a free-riding behavior, a limited punishment strategy in a large repeated game is proposed and shown to yield an approximate subgame-perfect Nash equilibrium. In addition, we model human irrationality as game noise incorporated into the proposed equilibria. A proposed application-oriented QoE model is first obtained via SimuLTE, an extension of OMNET++, then used in MATLAB simulations to verify the proposed solution. The results show tangible gains realized by the proposed solution under realistic scenarios and parameters.