This paper considers the problem of how to allocate power among competing users sharing a frequencyselective interference channel. We model the interaction between selfish users as a non-cooperative game. As opposed to the existing iterative water-filling algorithm that studies the myopic users, this paper studies how a foresighted user, who knows the channel state information and response strategies of its competing users, should optimize its transmission strategy. To characterize this multi-user interaction, the Stackelberg equilibrium is introduced, and the existence of this equilibrium for the investigated non-cooperative game is shown. We analyze this interaction in more detail using a simple two-user example, where the foresighted user determines its transmission strategy by solving as a bi-level program which allows him to account for the myopic user's response. It is analytically shown that a foresighted user can improve its performance, if it has the necessary information about its competitors. Since the optimal solution of Stackelberg equilibrium is computationally prohibitive, we propose a practical low-complexity approach based on Lagrangian duality theory. Numerical simulations verify the performance improvements. Possible ways to acquire the required information and to extend the formulation to more than two users are also discussed.Index Terms-interference channel, power control, non-cooperative game, Stackelberg equilibrium First, the formulation of the multi-user wideband power control problem as a non-cooperative game has appeared in several recent works [1] [2]. An iterative water-filling (IW) algorithm was proposed to mitigate the mutual interference and optimize the performance without the need for a central controller [1]. At every decision stage, selfish users deploying this algorithm try to maximize their achievable rates by water-filling across the whole frequency band until a Nash equilibrium is reached. Alternatively, self-enforcing protocols are studied in the non-cooperative scenario, in which incentive compatible allocations are guaranteed [2]. By imposing punishments in the case of misbehavior and enforcing users to cooperate, efficient, fair, and incentive compatible spectrum sharing is shown to be possible.Second, there also have been a number of related works studying dynamic spectrum management (DSM) in the setting of cooperative games [3]- [6]. Two (near-) optimal but centralized DSM algorithms, the Optimal Spectrum Balancing (OSB) algorithm and the Iterative Spectrum Balancing (ISB) algorithm, were proposed to solve the problem of maximization of a weighted rate-sum across all users [4] [5]. OSB has an exponential complexity in the number of users. ISB only has a quadratic complexity in the number of users because it implements the optimization in an iterative fashion. An autonomous spectrum balancing (ASB) technique is proposed to achieve near-optimal performance autonomously, without real-time explicit information exchanges [6]. These works focus on cooperative games, becaus...

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To reduce variance in estimating security prices via Monte Carlo simulation, we formulate a parametric minimization problem for the optimal importance sampling measure, which is solved using infinitesimal perturbation analysis (IPA) and stochastic approximation (SA). Compared with existing methods, the IPA estimator we derive is more universally applicable and more computationally efficient. Under suitable conditions, we show that the objective function is a convex function, the IPA estimator is unbiased, and the stochastic approximation algorithm converges to the optimum. Lastly, we demonstrate how combining importance sampling with indirect estimation using put-call parity can lead to further substantial variance reduction.

The present study was designed to examine the developmental changes in the character-complexity and word-length effects when reading Chinese script. Character complexity was defined in two ways: (1) the number of constituent strokes for characters (Experiment 1), and (2) the number of constituent radicals for characters (Experiment 2). The word length was defined as the number of characters in words (Experiment 3). The three experiments involved a lexical recognition task, and the participants consisted of 25 second graders, 24 fourth graders, 24 sixth graders, and 25 university students. In Experiment 1, it was found that the response latencies increased with the number of strokes in characters for second graders, whereas no effects were evident for fourth graders, sixth graders, or university students. In contrast, in Experiment 2 no character-complexity effect was found for the number of radicals. In Experiment 3, only a partial word-length effect was found for the number of characters for second graders. These results suggest that beginning Chinese readers process characters in an analytical way, but that the decoding process changes gradually from analytic to holistic as their reading skills develop. The educational implication of this result is discussed.

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