Subgame Perfect Nash Equilibrium Research Articles

Financial incentives to poor countries promote net emissions reductions in multilateral climate agreements

Financial incentives to poor countries promote net emissions reductions in multilateral climate agreements

A Dynamic Game Approach to Designing Secure Interdependent IoT-Enabled Infrastructure Network

The emerging Internet of Things (IoT) applications that leverage ubiquitous connectivity and big data are facilitating the realization of smart everything initiatives. IoT-enabled infrastructures have naturally a multi-layer system architecture with an overlaid or underlaid device network and its coexisting infrastructure network. The connectivity between different components in these two heterogeneous interdependent networks plays an important role in delivering real-time information and ensuring a high-level situational awareness. However, IoT-enabled infrastructures face cyber threats due to the wireless nature of communications. Therefore, maintaining network connectivity in the presence of adversaries is a critical task for infrastructure network operators. In this paper, we establish a three-player three-stage dynamic game-theoretic framework including two network operators and one attacker to capture the secure design of multi-layer interdependent infrastructure networks by allocating limited resources. We use subgame perfect Nash equilibrium (SPE) to characterize the strategies of players with sequential moves. In addition, we assess the efficiency of the equilibrium network by comparing with its team optimal solution counterparts in which two network operators can coordinate. We further design a scalable algorithm to guide the construction of the equilibrium IoT-enabled infrastructure networks. Finally, we use case studies on the emerging paradigm of the Internet of Battlefield Things (IoBT) to corroborate the obtained results.

Optimal Segmentation over a Generalized Customer Distribution

This paper investigates the impact of consumer preferences on the intensity of competition for companies in a duopoly market. A classical Hotelling’s competition problem will be different if consumers are allowed to distribute non-uniformly. New results in competition intensity are established and conditions for the existence of a subgame perfect Nash equilibrium is identified through a model that considers generic distribution in consumer preferences. A competition strategy is demonstrated to depend on the signs of local change rates of the density function at the endpoints of market segments.

An axiomatic and non-cooperative approach to the multi-step Shapley value

Inspired by the two-step Shapley value, in this paper we introduce and axiomatize the multi-step Shapley value for cooperative games with levels structures. Moreover, we design a multi-step bidding mechanism, which implements the value strategically in subgame perfect Nash equilibrium for superadditve games.

The optimal entry fee-prize ratio in Tullock contests

We analyze a three-stage game where an organizer sets an entry fee for a Tullock contest event, and a finite population of homogeneous agents simultaneously decide whether to participate or not. We show that in the unique symmetric subgame perfect Nash equilibrium, the larger the population size, the lower the probability the agents enter the contest, but the organizer’s optimal entry fee-prize ratio could either increase or decrease with the population size. When the population size approaches infinity, the number of contestants converges to a Poisson distributed random variable.

Virtual Microgrids: Implications of Blockchain Technologies for Peer-to-Peer Trading of Electricity, Renewable Energy Investments, and Resource-Sharing Network Coordination

By transforming how transactions are recorded, verified, and arranged, while eliminating the need for intermediaries, blockchain is generating a shift in focus away from centralized structures (e.g., exchanges, trading platforms, energy companies) towards decentralized systems (e.g., end customers, energy consumers interacting directly). In particular, by ensuring security, authenticity, and execution of financial and energy flows of across the grid, blockchain enables organization of electricity consumers with individual ownership of renewable energy resources (such as solar panels) into {\em virtual microgrids} for the purpose of peer-to-peer trading of locally generated renewable energy. In this paper, we develop an analytical framework to describe equilibrium outcomes in such a virtual microgrid whose each heterogeneous participant first decides, at the beginning of the time horizon, on the level of renewable generation capacity to invest in. Then, throughout the time horizon, he satisfies his (stochastic) demand for electricity either from his own installed generation capacity, or by means of blockchain-mediated purchases of electricity available from his peers in the virtual microgrid. We determine the subgame-perfect Nash equilibrium for this two-stage, simultaneous, non-cooperative, infinite-horizon game with perfect information played by participants in a virtual microgrid. For each such participant, we derive: (i) optimal capacity of his renewable energy investment; (ii) optimal amount of energy bought and sold through peer-to-peer electricity trading at each point in time; and (iii) the marginal value of the energy traded in the microgrid that minimizes his electricity-related costs. We prove that an electricity consumer is always better off with a virtual microgrid than without, with his resulting cost savings averaging 9.7\% in our numerical studies. We establish that the cost performance of a virtual microgrid in equilibrium exactly matches that of a centralized microgrid optimized for total cost. Thus, when participants in a virtual microgrid act optimally to minimize their own individual costs, they also achieve full coordination of the entire system and realize its minimum total cost.

Coordinating a closed loop supply chain with fairness concern by a constant wholesale price contract

The literature on closed loop supply chains (CLSCs) has ignored advantageous inequality aversion while modelling the fairness concern of channel partners and demonstrated that coordinating a decentralised channel requires complex price contracts. In this paper, we show that a constant wholesale price contract can coordinate a decentralised channel in a manufacturer-led CLSC if the retailer's advantageous inequality aversion is sufficiently strong. The result is valid for a range of equitable shares of the channel profit, such that the allocated share of the manufacturer is larger than that of the retailer, and the retailer's share is greater than a minimum threshold. Used product collection rate and channel profit are higher when the retailer is inequality averse compared to when she is a profit maximiser. The results are independent of whether the end-of-use products are collected by the manufacturer or the retailer. We also show that the collection rate is higher, and both channel partners are better-off, under the manufacturer collection model. To obtain these results, we solve multistage sequential move games under the two collection models. We apply Karush–Kuhn–Tucker conditions for constrained optimisation, to determine the boundaries for the existence of the subgame perfect Nash equilibrium.

Rational conjectures and evolutionary beliefs in public goods games

AbstractThis paper applies conjectural variations (CVs) to a model of public good provision and shows that CVs are superior to Nash beliefs. In addition to imposing consistency, as Bresnahan, I show that consistent conjectures (CCs) are obtained from individual payoff maximization. CCs emerge as the unique subgame perfect Nash equilibrium (NE) in a two‐stage game in which beliefs are chosen in Stage 1 and quantities in Stage 2. There is an individual payoff advantage to non‐Nash behavior, generating a Prisoner's Dilemma in conjectures in addition to the usual free‐rider problem associated with public goods. The correct and payoff maximizing conjecture is the unique equilibrium in an evolutionary framework against a player with Nash conjectures. The consistent conjecture equilibrium is the unique evolutionary equilibrium when both players conjectures evolve. Hence, the NE prediction is too optimistic when players have rational conjectures.

A dynamic perspective of government intervention in a competitive closed-loop supply chain

Changes in environmental burden are inherently dynamic because a steady state cannot be instantaneously reached, and the current state is associated with the previous state and continuously evolves. In this study, we examine a dynamic closed-loop supply chain (CLSC) consisting of an original equipment manufacturer (OEM) selling new products and an independent remanufacturer (IR) selling recovered products generated from end-of-life (EOL) products. The government intervenes in the CLSC by imposing taxes or subsidies to reduce the environmental burden. We propose a baseline model without government intervention and six different policies: a fixed tax or subsidy on the firm side, a unit tax or subsidy on the firm side, and a unit tax or subsidy on the consumer side. We derive the firms’ closed-form instantaneous and steady-state decisions at the subgame perfect Nash equilibrium. Finally, we conduct a comparative study to explore the effects of the policies on the environmental burden, firm profits (producer surpluses), consumer surplus, and social welfare. We find that the proposed policies can effectively reduce the environmental burden and mitigate the intensity of price competition, except for the policy with the fixed subsidy to the IR. The policies are all profitable for the IR and, thus, effective in encouraging recovery; however, the policies are not necessarily profitable for the OEM. The policies with a unit tax for consumers and fixed subsidy for the OEM are more conducive to social welfare. Moreover, the degree of dynamics improves the effectiveness of the policies in reducing the environmental burden.

Competition between manufacturers and sharing economy platforms: An owner base and sharing utility perspective

We develop a two-period, five-stage game model with sharing utility to analyze how a manufacturer competes with a sharing economy platform that facilitates sharing of the manufacturer's product from both a long-run and a short-run perspective. We use the subgame perfect Nash equilibrium to reveal how the manufacturer chooses its long-run (period-1) price to foster an owner base (a collection of the period-1 product buyers) that in turn affects its short-run (period-2) competition with the sharing economy platform (hereafter, platform for short). Analytical results show that (1) in the short run, an increase in the owner base has a negative (non-monotonic) impact on the profitability of the manufacturer (platform) while an increase in sharing utility negatively (positively) affects the profitability of the manufacturer (platform); (2) in the long run, the existence of the platform could be irrelevant for, pose a threat to, or benefit the manufacturer contingent upon different levels of sharing utility; (3) in the long run, a win-win scenario is attainable for both the manufacturer and platform when sharing utility is high enough. The managerial implication of this research is that the manufacturer and platform should collaborate to improve sharing utility to a sufficiently high level by enhancing the sense of community belonging, fostering benefits from sharing activities, and addressing sustainability concerns, thereby achieving a win-win result.

On a Competitive Selection Problem

We consider the problem in which n items arrive to a market sequentially over time, where two agents compete to choose the best possible item. When an agent selects an item, he leaves the market and obtains a payoff given by the value of the item, which is represented by a random variable following a known distribution with support contained in [0,1]. We consider two different settings for this problem. In the first one, namely competitive selection problem with no recall, agents observe the value of each item upon its arrival and decide whether to accept or reject it, in which case they will not select it in future. In the second setting, called competitive selection problem with recall, agents are allowed to select any of the available items arrived so far. For each of these problems, we describe the game induced by the selection problem as a sequential game with imperfect information and study the set of subgame-perfect Nash equilibrium payoffs. We also study the efficiency of the game equilibria. More specifically, we address the question of how much better is to have the power of getting any available item against the take-it-or-leave-it fashion. To this end, we define and study the price of anarchy and price of stability of a game instance as the ratio between the maximal sum of payoffs obtained by players under any feasible strategy and the sum of payoffs for the worst and best subgame-perfect Nash equilibrium, respectively. For the no recall case, we prove that if there are two agents and two items arriving sequentially over time, both the price of anarchy and price of stability are upper bounded by the constant 4/3 for any value distribution. Even more, we show that this bound is tight.

Efficient multi-period distribution mechanism for the innovation investment system under uncertainty

In this paper, we study the problem of a venture investor who distributes the budget between several innovation projects under conditions of uncertainty. A common method for solving this problem is through bilateral negotiations with the external evaluation of projects. However, the effectiveness almost entirely depends on the evaluation quality, but external evaluation seldom reduces the knowledge asymmetry for innovation projects. We propose an iterative revelation mechanism for this problem when the investor sequentially offers possible allocations of the limited budget in the form of threshold dividing questions. The binary choices of innovators serve as a signal of internal estimates of the project implementation costs. Under perfect information, such a mechanism, regardless of the method for determining budget allocations, always produces an effective allocation in subgame-perfect Nash equilibrium. Under uncertainty, the method of offering distribution options matters – the optimal solution is found under the English auction class of mechanisms. In an efficient iterative allocation mechanism for innovation investment, the investor proposes a new allocation of the budget each round until an efficient allocation is achieved. The proposed mechanism does not necessarily need to identify the exact minimum budgets for each innovator. Another advantage of the proposed mechanism is the ability to use different processes for organizing rounds.

The evolution of morality and the role of commitment.

A considerable share of the literature on the evolution of human cooperation considers the question why we have not evolved to play the Nash equilibrium in prisoners' dilemmas or public goods games. In order to understand human morality and pro-social behaviour, we suggest that it would actually be more informative to investigate why we have not evolved to play the subgame perfect Nash equilibrium in sequential games, such as the ultimatum game and the trust game. The 'rationally irrational' behaviour that can evolve in such games gives a much better match with actual human behaviour, including elements of morality such as honesty, responsibility and sincerity, as well as the more hostile aspects of human nature, such as anger and vengefulness. The mechanism at work here is commitment, which does not need population structure, nor does it need interactions to be repeated. We argue that this shift in focus can not only help explain why humans have evolved to know wrong from right, but also why other animals, with similar population structures and similar rates of repetition, have not evolved similar moral sentiments. The suggestion that the evolutionary function of morality is to help us commit to otherwise irrational behaviour stems from the work of Robert Frank (American Economic Review, 77(4), 593-604, 1987; Passions within reason: The strategic role of the emotions, WW Norton, 1988), which has played a surprisingly modest role in the scientific debate to date.

Investigating the Adoption of Hybrid Encrypted Cloud Data Deduplication With Game Theory

Encrypted data deduplication, along with different preferences in data access control, brings the birth of hybrid encrypted cloud data deduplication (H-DEDU for short). However, whether H-DEDU can be successfully deployed in practice has not been seriously investigated. Obviously, the adoption of H-DEDU depends on whether it can bring economic benefits to all stakeholders. But existing economic models of cloud storage fail to support H-DEDU due to complicated interactions among stakeholders. In this article, we establish a formal economic model of H-DEDU by formulating the utilities of all involved stakeholders, i.e., data holders, data owners, and Cloud Storage Providers (CSPs). Then, we construct a multi-stage Stackelberg game, which consists of Holder Participation Game, Owner Online Game, and CSP Pricing Game, to capture the interactions among all system stakeholders. We further analyze the conditions of the existence of a sub-game perfect Nash Equilibrium and propose a gradient-based algorithm to help the stakeholders choose near-optimal strategies. Extensive experiments show the feasibility of the proposed algorithm in achieving the Nash Equilibrium of the Stackelberg game. Additionally, we investigate the effects of parameters related to CSP, data owners and data holders on H-DEDU adoption. Our study advises all stakeholders the best strategies to adopt H-DEDU.

Equilibrium concepts for time‐inconsistent stopping problems in continuous time

AbstractA new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, 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 subadditive, 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.

The Interplay of Competition and Cooperation Among Service Providers (Part I)

This paper investigates the incentives of mobile network operators (MNOs) for acquiring additional spectrum to offer mobile virtual network operators (MVNOs) and thereby inviting competition for a common pool of end users (EUs). We consider a base case and two generalizations: (i) one MNO and one MVNO, (ii) one MNO, one MVNO and an outside option, and (iii) two MNOs and one MVNO. In each of these cases, we model the interactions of the service providers (SPs) using a sequential game, identify when the Subgame Perfect Nash Equilibrium (SPNE) exists, when it is unique and characterize the SPNE when it exists. The characterizations are easy to compute, and are in closed form or involve optimizations in only one decision variable. We identify metrics to quantify the interplay between cooperation and competition, and evaluate those as also the SPNEs to show that cooperation between MNO and MVNO can enhance the payoffs of both, while increased competition due to the presence of additional MNOs is beneficial to EUs but reduces the payoffs of the SPs.

A complete folk theorem for finitely repeated games

This paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

On the insufficiency of some conditions for minimal product differentiation

The basic framework is Hotelling’s model of product choice with quadratic transportation cost. Duopolists choose locations in the initial period and compete in prices in subsequent infinite periods. The firms share profits on the profit possibility frontier. Friedman and Thisse (Rand J Econ 24:631–645, 1993) provides a set of sufficient conditions for a unique equilibrium and minimal product differentiation in this setting. This paper reexamines those conditions. In the presence of some mild continuity requirements, there is exactly one profit sharing rule which satisfies those conditions. Furthermore, given any discount factor(s), the corresponding profits cannot be the outcome of a subgame perfect Nash equilibrium at every pair of locations. This brings out an inconsistency in the conditions. A slight weakening of the conditions, to allow for a wider class of profit sharing rules, can result in multiple equilibria and minimal product differentiation need not obtain. Two examples demonstrate this. Thus, neither those conditions nor their weaker variants can be used to characterize a unique equilibrium.

Endogenous vertical structure with network externalities

AbstractThis study examines the endogenous vertical structure in which each manufacturer sells its product to its exclusive retailer who sells network goods to consumers (i.e. a duopoly in the upstream market) under Bertrand competition and Cournot competition with network externalities. We show that with strong (weak) network externalities under Bertrand competition, (a) it is a dominant strategy for each manufacturer to integrate (separate) its retailer; (b) with strong network externalities, the manufacturers’ profits, consumers’ surplus and social welfare are higher under vertical integration than under vertical separation. Under Cournot competition, (a) vertical separation is a unique subgame perfect Nash equilibrium; (b) with strong network externalities, the manufacturers’ profits, consumers’ surplus and social welfare are higher under vertical separation than under vertical integration.

Two-Player Location Game in a Closed-Loop Market with Quantity Competition

This paper considers the two-player location game in a closed-loop market with quantity competition. Based on the Cournot and Hotelling models, a circle model is established for a closed-loop market in which two players (firms) play a location game under quantity competition. Using a two-stage (location-then-quantity) pattern and backward induction method, the existence of subgame-perfect Nash equilibria is proved for the location game in the circle model with a minimum distance transportation cost function. In addition, sales strategies are proposed for the two players for every local market on the circle when the players are in the equilibrium positions. Finally, an algorithm for simulating the competitive dynamics of the closed-loop market is designed, and two numerical simulations are provided to substantiate the effectiveness of the obtained results.