A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.     

Partially Observable Semi-Markov Games with Discounted Payoff

Abstract

We study partially observable semi-Markov game with discounted payoff on a Borel state space. We study both zero sum and nonzero sum games. We establish saddle point equilibrium and Nash equilibrium for zero sum and nonzero sum cases, respectively.     

On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy

We consider a risk minimization problem in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, observable and finite-state Markov chain whose states represent different market regimes. We adopt a particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk. The risk-minimization problem is formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game. One important feature of our model is to allow the flexibility of controlling both the diffusion process representing the financial risk and the Markov chain representing macro-economic risk. This is novel and interesting from both the perspectives of stochastic differential game and stochastic control. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided and some particular cases are discussed.     

Research and Development with Stock-Dependent Spillovers and Price Competition in a Duopoly

This paper investigates the research and development accumulation and pricing strategies of two firms competing for consumer demand in a dynamic framework. A firm’s research and development is production-cost-reducing and can benefit from part of the competitor’s research and development stock without payment. We consider decisions in a game characterized by Nash equilibrium. In this dynamic game, a player’s action depends on whether the competitor’s current research and development stock are observable. If the competitor’s current research and development stock are not observable or observable only after a certain time lag, a player’s action can be solely based on the information on the current period t (open-loop strategy). In the converse case, it can also include the information on the competitor’s reaction to a change in the current value of the state vector (closed-loop strategy), which allows for strategic interaction to take place throughout the game. Given the cumulative nature of research and development activities, a primary goal of this paper is to determine whether, regardless of the observability of the competitor’s current research and development stock, free research and development spillovers generate a lower level of scientific knowledge than research and development appropriability. A second objective of the paper is to determine how the observability of the rival’s current research and development stock affects a firm’s research and development and pricing decisions and payoffs under imperfect research and development appropriability.     

Endogenous timing in duopoly: experimental evidence

In this paper we experimentally investigate Cournot duopolies with an extended timing game. The timing game has observable delay, that is, firms announce a production period (one out of two periods) and then they produce in the announced sequence. Theory predicts simultaneous production in the first period. With random matching we find that, given the actual experimental behavior in the subgames, subjects play a timing game more akin to a coordination game with two symmetric equilibria rather than the predicted game with a dominant strategy to produce early. As a result, a substantial proportion of subjects choose the second period.     

Bayesian optimal knapsack procurement

A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by observable quality and sellers have private reserve prices for their items. The buyer’s problem is to select a subset of maximal quality. Money does not enter the buyer’s objective function, but only his constraints. Sellers quote prices strategically, inducing a knapsack game. We report the Bayesian optimal mechanism for the buyer’s problem. We find that simultaneous take-it-or-leave-it offers are interim optimal.     

Cheap play with no regret

This paper studies a special class of differential information games with pre-play communication —games with “cheap play”. We consider problems in which there are several rounds of payoff-irrelevant publicly observable choice (or discussion) of actions, followed by a final round in which actions are binding and payoff relevant. A natural focal subset of equilibria of such games in one that consists of equilibria involvingno regret. Such games were first studied by Green and Laffont (1987), where a criterion calledposterior implementability is introduced with the intention of identifying regret-free equilibria in games with cheap play. This is simply a restriction on the Bayesian equilibrium of the underlying one-shot game. If indeed such a restriction does characterize regret-freeness, then the analytics of such situations would be enormously simplified since one can ignore the messy extended-form of the cheap play game; merely examining the one-shot game is sufficient. We argue that regret-freeness of an equilibrium has a subtle distinction: regret-freeness in moves and regret-freeness in assessments. We show that the former causes the extended-form to be irrelevant; posterior implementability completely characterizes equilibria with regret-freeness in moves. The latter, on the other hand, does not yield a similar principle: the extended-form cannot be ignored.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

Optimal warranties,reliabilities and prices for durable goods in an oligopoly

This paper addresses the problem of durable goods manufacturers in an oligopoly seeking optimal values for three decision variables: product warranty, reliability and price. Each firm seeks a warranty-reliability-price combination that maximizes expected profit subject to quite general constraints on the firm's decision variables. Warranty serves as a signal of product reliability, which is not observable by consumers. We present a game-theoretic model of warranty-reliability-price competition in such a market and examine Nash equilibria for this game. We show that under fairly general assumptions each firm can optimally set its warranty and reliability independently of price and competitors' actions. In addition, we show that optimal warranties and reliabilities are complementary, and we explore the impact of different market factors on the optimal warranty and reliability. Finally, we show that optimal warranties are longer and products more reliable when consumers are risk averse.     

MODELING: A TOOL FOR ENHANCING THE SURVIVAL PROSPECTS OF AFRICAN WILDLIFE

ABSTRACT. In this article we consider the role of modeling in some aspects of the conservation of African wildlife. The first study is concerned with the endangered black rhino. Animals are being translocated from high density areas to new sites in an attempt to build up the South African population as rapidly as possible. We investigate the efficacy of different translocation strategies. Next we discuss a spatial stochastic metapopulation model used to test management strategies to enhance the survival likelihood of the rare samango monkey. The value of the model is to encourage the use of corridor policies even though there may be little apparent observable benefit in a manager's lifetime. Finally, we look at some commercial aspects of exploiting wildlife on a sustainable basis. By increasing the profitability of wildlife enterprises the incentive to conserve is increased. We look at improving the financial returns from game ranches.     

How characteristic are characteristic functions?

Every game form and effectivity function (EF) generates a family of characteristic functions (CFs) via feasible utility function profiles. We investigate the extent this family characterizes the underlying structure in which agents interact. The strategic structure CFs characterize is found to be limited to that which is implicit in the representative EF. If the dependency between CF and utility function profile is observable and coalitions pursue only pure strategies, then EFs are fully characterizable by their CF progeny. When mixed strategies are viable, CFs are only sufficient for the “effective surface” of an EF. A number of important EF properties are CF characterizable even when dependency between CF and utility function profile is unobservable. Even so, radically different EFs may have the same CF lineage.     

Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

This paper is concerned with Kalman-Bucy filtering problems of a forward and backward stochastic system which is a Hamiltonian system arising from a stochastic optimal control problem. There are two main contributions worthy pointing out. One is that we obtain the Kalman-Bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation. The other is that we develop a backward separation technique, which is different to Wonham's separation theorem, to study a partially observed recursive optimal control problem. This new technique can also cover some more general situation such as a partially observed linear quadratic non-zero sum differential game problem is solved by it. We also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases.     

To wait or not to wait: Strategic behaviors in an observable batch-service queueing system

This paper deals with an observable batch service queueing system in which customers rationally choose whether to form a batch with another customer or not, in addition to deciding whether or not to join the queue. When choosing whether to form a batch, a customer in an incomplete batch decides on an optimal waiting time for the next customer to arrive and share the service fee. When choosing whether to join the queue, customers follow a threshold strategy, which depends on the strategy identified in the former game.     

Optimal inventory threshold for a dynamic service make‐to‐stock system with strategic customers

We consider a make‐to‐stock production system with one product type, dynamic service policy, and delay‐sensitive customers. To balance the waiting cost of customers and holding cost of products, a dynamic production policy is adopted. If there is no customer waiting in the system, instead of shutting down, the system operates at a low production rate until a certain threshold of inventory is reached. If the inventory is empty and a new customer emerges, the system switches to a high production rate where the switching time is assumed to be exponentially distributed. Potential customers arrive according to the Poisson process. They are strategic in the sense that they make decisions on whether to stay for product or leave without purchase on the basis of on their utility value and the system information on whether the number of products is observable to customers or not. The strategic behavior is explored, and a Stackelberg game between production manager and customers is formulated where the former is the game leader. We find that the optimal inventory threshold minimizing the cost function can be obtained by a search algorithm. Numerical results demonstrate that the expected cost function in an observable case is not greater than that in an unobservable case. If a customer's delay sensitivity is relatively small, these two cases are entirely identical. With increasing of delay sensitivity, the optimal inventory threshold might be positive or zero, and hence, a demarcation line is depicted to determine when a make‐to‐stock policy is advantageous to the manager.     

Finding mixed strategies with small supports in extensive form games

The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for thesize of a mixed strategy is the number of strategies in itssupport — the set of pure strategies to which it gives positive probability. This paper investigates the existence of “small” mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set ofrealization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy μ, there exists a realization-equivalent mixed strategy µ′ whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy µ′ (given the realization weights) is NP-hard. On the other hand, if μ is a behavior strategy, µ′ can be constructed from μ in time polynomial in the size of the game tree. In either case, we can use the fact that mixed strategies need never be too large for constructing efficient algorithms that search for equilibria. In particular, we construct the first exponential-time algorithm for finding all equilibria of an arbitrary two-person game in extensive form.     

Infrastructure security games

Infrastructure security against possible attacks involves making decisions under uncertainty. This paper presents game theoretic models of the interaction between an adversary and a first responder in order to study the problem of security within a transportation infrastructure. The risk measure used is based on the consequence of an attack in terms of the number of people affected or the occupancy level of a critical infrastructure, e.g. stations, trains, subway cars, escalators, bridges, etc. The objective of the adversary is to inflict the maximum damage to a transportation network by selecting a set of nodes to attack, while the first responder (emergency management center) allocates resources (emergency personnel or personnel-hours) to the sites of interest in an attempt to find the hidden adversary. This paper considers both static and dynamic, in which the first responder is mobile, games. The unique equilibrium strategy pair is given in closed form for the simple static game. For the dynamic game, the equilibrium for the first responder becomes the best patrol policy within the infrastructure. This model uses partially observable Markov decision processes (POMDPs) in which the payoff functions depend on an exogenous people flow, and thus, are time varying. A numerical example illustrating the algorithm is presented to evaluate an equilibrium strategy pair.     

Cycle games and cycle cut games

Two players play a game on a connected graphG. Each player in his turn occupies an edge ofG. The player who occupies a set of edges that contains a cycle, before the other does it, wins. This game may end in a draw. We call this game the normal cycle game. We define furthermore three similar games, which are called the misère cycle game, the normal cycle cut game and the misère cycle cut game. We characterize the above four games.     

Geometric versions of finite games: Prisoner's dilemma,entry deterrence and a cyclical majority paradox

We provide geometric versions of finite, two-person games in the course of proving the following: if a finite, two-person, symmetric game is constant-sum, it is a location game. If it is not constant-sum, it is a location game with a reservation price. Every finite two-person game is a location game with a reservation price and two location sets, one for each player.We then use location games to resolve a cyclical majority paradox, and to analyze a prisoner's dilemma and an entry deterrence game.     

On linear vector program and vector matrix game equivalence

We define a vector matrix game, which is a vector version of the usual matrix game and consists of more than two skew symmetric matrices and vector ordering. Using vector optimization techniques, we characterize solutions for the vector matrix game. We establish equivalent relations between a linear vector optimization problem and its corresponding vector matrix game.