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Game Theory
Published in Vincent Knight, Geraint Palmer, Applied Mathematics with Open-Source Software, 2022
Vincent Knight, Geraint Palmer
The definition of a normal form game here and the solution concept of Nash equilibrium are common starting points for the use of game theory as a study of emergent behaviour. Other topics include different forms of games, for example extensive form games, which are represented by trees and more explicitly model asynchronous decision making. Other solution concepts involve the specific study of the emergence mechanisms through models based on natural evolutionary process: Moran processes or replicator dynamics. A good text book to read on these topics is [38].
Game Theory
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Erik G. Larsson, Eduard Jorswiec
In order to compute Nash equilibria for extensive form games more easily, there is a canonical transform to a corresponding game in normal form. All possible strategies in the extensive form game are represented as actions in the normal form game. For the invader game in Figure 19.8, the game matrix is a simple two-by-two matrix which is easily computed in Exercise 19.13.13.
Using game theory for deployment of security forces in response to multiple coordinated attacks
Published in Chongfu Huang, Zoe Nivolianitou, Risk Analysis Based on Data and Crisis Response Beyond Knowledge, 2019
According to Dresher’s air combat model, the proposed 2 × 2 payoff matrix game has no pure strategic N.E (Dresher, 1981). A mixed Nash equilibrium pair (r*, q*) exists in the normal form game if this game has no pure strategy N.E., which is an optimal solution. Player 2’s expected payoff is computed when player 1 and player 2 use mixed strategies r and q, respectively. The mixed N.E. for the probability vector is r* = {r*(u1), r*(u2)} with actions {u1, u2} = {primary attack, diversionary attack} by the terrorist and the vector q* = {q*(d1), q*(d2)} with actions {d1, d2} = {{hide the valued payoffs in primary attack, hide the valued payoffs in diversionary attack} by the security forces commander. The optimal strategies are always mixed, and are the same for the two players: makes a primary attack with a probability ri∗u1=qi∗d1=wi,1−oiri,k−1×oi−wi,2−oi−wi,1,i∈1,…,8;k∈1,…,6.
Reflexion in mathematical models of decision-making
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Dmitry Novikov, Vsevolod Korepanov, Alexander Chkhartishvili
Classic game-theoretic models proceed from the following. In a normal form game, agents choose Nash equilibrium actions. However, investigations in the field of experimental economics indicate this not always the case (e.g. see [29] and the overview [30]). The divergence between actual behavior and theoretical expectations has several explanations:limited cognitive capabilities of agents [31] (evaluation of a Nash equilibrium represents a cumbersome computational problem [32], especially decentralized). It should also be emphasized that the Nash equilibrium does not always adequately describe the real behavior of agents in laboratory experimental one-step games, including because agents do not have time to ‘correct’ their misconceptions about the essential parameters of game. For example, the concept of D. Bernheim’s rationalized strategies requires unlimited rationality of agents [33];agent’s full confidence in that all the opponents would evaluate a Nash equilibrium;incomplete awareness;the presence of several equilibria.Therefore, there exist at least two foundations (‘theoretical’ and ‘experimental’ ones) for considering models of collective behavior of agents with different reflexion ranks.