Stochastic game

About: Stochastic game is a research topic. Over the lifetime, 9493 publications have been published within this topic receiving 202664 citations.

Gaming machine with video mode payoff multiplier

Abstract: A spinning reel slot machine gives a multiplied payoff when certain conditions are fulfilled. Wins including a special symbol on the pay line are multiplied by a multiplier value generated according to the rules of a secondary game. The secondary game is a computer simulation or a video display of a set of reels which simulate the operation of a physical set of reels, and have symbols representing different numerical values. The values on the computer generated reels when they stop are multiplied together to generate the multiplier.

The Competitive Solution for N-Person Games Without Transferable Utility, With an Application to Committee Games

Abstract: This essay defines and experimentally tests a new solution concept for n-person cooperative games—the Competitive Solution. The need for a new solution concept derives from the fact that cooperative game theory focuses for the most part on the special case of games with transferable utility, even though, as we argue here, this assumption excludes the possibility of modelling most interesting political coalition processes. For the more general case, though, standard solution concepts are inadequate either because they are undefined or they fail to exist, and even if they do exist, they focus on predicting payoffs rather than the coalitions that are likely to form.The Competitive Solution seeks to avoid these problems, but it is not unrelated to existent theory in that we can establish some relationships (see Theorems 1 and 2) between its payoff predictions and those of the core, the V-solution and the bargaining set. Additionally, owing to its definition and motivation, nontrivial coalition predictions are made in conjunction with its payoff predictions.The Competitive Solution's definition is entirely general, but a special class of games—majority rule spatial games—are used for illustrations and the experimental test reported here consists of eight plays of a 5-person spatial game that does not possess a main-simple V-solution or a bargaining set. Overall, the data conform closely to the Competitive Solution's predictions.

Theory of hybrid systems and discrete event systems

Abstract: A continuous system has a continuous state space and an evolution law given by a differential or a difference equation. A discrete event system is modeled by an automaton which changes state in response to events. A hybrid system contains both continuous and discrete event sub-systems. In this thesis we study some theoretical problems in the design and analysis of hybrid systems and discrete event systems. We first consider the reachability question for a hybrid system--is a target state reachable from an initial state? We show that for hybrid automata with rectangular inclusions, the reachability question can be answered in a finite number of steps. Hybrid systems with more general dynamics can be reduced to hybrid systems with rectangular inclusions using abstractions. We next consider an Automated Vehicle Highway System (AVHS) design. We consider the safety question: can there be a collision between two vehicles on the AVHS? We show that the AVHS is safe provided the controllers in the vehicles satisfy a set of constraints. The constraints require the reach set $Reach\sb{f}(X\sb0,t$)--the set of states reached after time t starting from an initial set $X\sb0$ for a differential inclusion $\dot x\ \in\ f(x$)--to satisfy a simple criterion. We show that this problem is equivalent to solving an optimal control problem. We then consider some computational questions for differential inclusions. For a Lipschitz differential inclusion $\dot x\ \in\ f(x$), we give a method to compute an arbitrary close approximation of $Reach\sb{f}(X\sb0,t$). For a differential inclusion $\dot x\ \in\ f(x$), and any $\epsilon>$ 0, we define a finite sample graph $A\sp{\epsilon}$. Using graph $A\sp{\epsilon}$, we can compute the $\epsilon$-invariant sets of the differential inclusion--the sets that remain invariant under $\epsilon$-perturbations in f. We also consider some dynamical games played on graphs. The synthesis and the control problem for $\omega$-automata can be formulated as a game between two players. We discuss games on $\omega$-automata and the payoff games. We show that $\omega$-automata games do not necessarily have a value when restricted to positional strategies. We exhibit a bound on the amount of memory required to play these games. We then consider the discounted and mean payoff games. We present the successive approximation and the policy iteration algorithm for solving payoff games. We then show that an $\omega$-automata game with the chain acceptance condition can be solved as a mean payoff game. Solving a chain game is equivalent to solving the model checking problem for propositional $\mu$-calculus. Hence, the policy iteration method can be used to model check $\mu$-calculus formula. This is at present the most efficient algorithm for model checking propositional $\mu$-calculus.

Dynamic Pricing of Perishable Assets Under Competition

Abstract: We study dynamic price competition in an oligopolistic market with a mix of substitutable and complementary perishable assets. Each firm has a fixed initial stock of items and competes in setting prices to sell them over a finite sales horizon. Customers sequentially arrive at the market, make a purchase choice, and then leave immediately with some likelihood of no purchase. The purchase likelihood depends on the time of purchase, product attributes, and current prices. The demand structure includes time-variant linear and multinomial logit demand models as special cases. Assuming deterministic customer arrival rates, we show that any equilibrium strategy has a simple structure, involving a finite set of shadow prices measuring capacity externalities that firms exert on each other: equilibrium prices can be solved from a one-shot price competition game under the current-time demand structure, taking into account capacity externalities through the time-invariant shadow prices. The former reflects the transient demand side at every moment, and the latter captures the aggregate supply constraints over the sales horizon. This simple structure sheds light on dynamic revenue management problems under competition, which helps capture the essence of the problems under demand uncertainty. We show that the equilibrium solutions from the deterministic game provide precommitted and contingent heuristic policies that are asymptotic equilibria for its stochastic counterpart, when demand and supply are sufficiently large. This paper was accepted by Yossi Aviv, operations management.