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Concept# Nash equilibrium

Summary

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs.
If each player has chosen a strategy – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium.
If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to

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In this paper we consider a class of hybrid stochastic games with the piecewise open-loop information structure. These games are indexed over a parameter $\varepsilon$ which represents the time scale ratio between the stochastic (jump process) and the deterministic (differential state equation) parts of the dynamical system. We study the limit behavior of Nash equilibrium solutions to the hybrid stochastic games when the time scale ratio tends to 0. We also establish that an approximate equilibrium can be obtained for the hybrid stochastic games using a Nash equilibrium solution of a reduced order sequential discrete state stochastic game and a family of local deterministic infinite horizon open-loop differential games defined in the stretched out time scale. A numerical illustration of this approximation scheme is also developed.

This thesis consists of three chapters on informational frictions in financial markets. The chapters analyze problems related to markets' ability to guide real investment, and what drives liquidity. Both problems are important to ensure efficient resource allocation in the economy.
The first chapter studies the interaction between financial markets and real investments. I develop a model that simultaneously study the equilibrium in financial markets, the choice of investors to produce information, and real decisions by the firm. The chapter provides a new method to overcome non-linearities in the security price, and the equilibrium is surprisingly simple. The results provide insights into when real investments have a substantial impact on market efficiency and when we can analyze equilibrium market efficiency separately. Equilibrium behavior may hide some inefficiencies from standard empirical tests. Some changes in financial markets may increase or have little effect on market efficiency, but reduce real efficiency by increasing the cost of information production.
The second chapter analyzes time-variation in liquidity. I develop a tractable model where conditions among traders vary over time. The resulting equilibrium offers several new predictions on what drives liquidity variation. For example, there may be significant reductions in liquidity from even tiny changes among the traders' conditions. Strategic behavior drives the results, and the model explains how liquidity may suddenly evaporate without a clear cause. Empirical results are in line with the predictions of the model. Surprisingly, everyone may benefit from sometimes restricting some traders from the market. Doing so can reallocate liquidity to periods with more significant liquidity needs.
The third chapter studies the choice of anonymity among traders. All traders end up revealing their identity unless doing so is costly, or the order flow is noisy. The intuition is that there is always at least one trader who prefers to reveal his or her identity. If the order flow is noisy, then there is a threshold type, and more patient traders stay anonymous. The results suggest that a fully anonymous market is most efficient, but the gains from anonymity are distributed unevenly. This result explains why different markets vary significantly in choices related to anonymity.

Due to the increasing number of radio technologies, the available frequency spectrum becomes more and more utilized, hence its clever use becomes a critical issue. Among many proposed solutions, the formulation of the problem as the control of the power of the base stations, also known as the Power Control problem, seems a promising idea. In the present work, we propose to study this problem by first defining a theoretical model. Then, we design a family of non-cooperative games that hopefully stop at a Nash equilibria close to the optimal solution for the network, as well as a few simple tabu search heuristics. We finally developed a java library and a program, in order to experimentally study the behavior of the proposed games and heuristics.

2006