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Let be the number of players, the set of action profiles over the action sets of each player and : be the payoff function for player .. Given a game = (, = …,:), we say that is a potential game with an exact (weighted, ordinal, generalized ordinal, best response) potential function if : is an exact (weighted, ordinal, generalized ordinal, best response, respectively) potential function for .
A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [ citation needed ] The existence of arbitrary numbers of gauge functions ψ ( r , t ) corresponds to the U(1) gauge freedom of this theory.
The "gauge covariant" version of a gauge theory accounts for this effect by introducing a gauge field (in mathematical language, an Ehresmann connection) and formulating all rates of change in terms of the covariant derivative with respect to this connection. The gauge field becomes an essential part of the description of a mathematical ...
A gauge theory is a type of theory in physics. The word gauge means a measurement , a thickness, an in-between distance (as in railroad tracks ), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition ). [ 1 ]
Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
The potential equations can be simplified using a procedure called gauge fixing. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing ...
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.
where v j is the velocity of the point of application of the force F j. In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is = =, =, …,.