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Simulation of a Random Variable and its Application to Game Theory

Valizadeh Gurttapeh, Mehrdad | 2020

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 53168 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Aminzadeh Gohari, Amin
  7. Abstract:
  8. We study the optimization of the payoff of a decision-maker (a player or a team) in the presence of an adversary who wishes to minimize the payoff of the decision-maker. In this situation, the decision-maker can puzzle the adversary by playing random actions and hence, gain more payoff. We assume that the decision-maker has limited access to randomness sources and is only allowed to randomize his actions by conditioning them to the limited randomness sourse to which he has access. We consider two models: in the first model, in each stage, the decision maker observes a random source independent of past, and distributed according to a time invariant probability distribution. The adversary observes a noisy version of the random source. At each stage, the decision-maker and the adversary must choose their action as a deterministic function of their observations (from the randomness source and played actions) up to that stage. In the second model, the decision-maker is a team of players who can randomize their respective actions privately, but don't have access to any explicit source of shared randomness to coordinate their actions. Here, we assume that the adversary monitors the actions of the team players through a noisy channel. Therefore, the team players can extract shared randomness from their actions in past stages.In the above models, at each stage, the decision-maker needs to simulate his action in such a way that it's distribution is almost as desired, and it is independent of the observations of the adversary. In previous works, the accuracy of the simulation was quantified with Absolute Kullback Distance, and the method of types was utilized to generate the sequence of random actions. However, here, we take the total variation distance as the measure of accuracy, and use random hashing for generation of the desired simulator. Then, we provide a new probablestic tool for simulation of target random variable from a given randomness source with side information. The tool gives a deterministic mapping that when applied to the randomness source outputs a random variable whose distribution is close to the distribution of the target random varible, and it is almost independent of the side information. The accuracy of the simulation vanishes exponentialy in the difference of the Renyi entropies of the randomness source and the target random variable. Using the simulation tool, we not only simplify the prior results but also extend them
  9. Keywords:
  10. Random Variables ; Rényi Entropy ; Payoff Optimization ; Limited Randomness Source ; Random Variable Simulation ; Repeated Zero-Sum Game ; Game Theory

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