Non-cooperative Game Among Brokers in Stage II and Stage III
Based on the result of the evolutionary game for users, the brokers compete with each other and choose the proper strategies on the price to obtain the maximum utilities. Thus, the non-cooperative game is introduced to model the competition among brokers, and the Nash equilibrium is considered as the solution to the game.
According to the price of the cloud resource determined by media cloud, each broker decides the amount of cloud resource to purchase and then determines the price of cloud resource to charge. Considering the competition, the utility of broker i can be defined as
Here E_,- denotes the vectors of the resource size that brokers have, except broker i. p_; means the price of cloud resource offered by brokers, except broker i.
The Nash equilibrium is considered as the solution of the game, where each broker has an optimal strategy to maximize the utility. In this case, we use the best response function of each broker to find Nash equilibrium, which is the best strategy of a broker based on others’ best strategies. Therefore, when others’ strategies are determined, the best response function of broker i can be defined by
Let p* = (p*, p*,..., p*) and E* = (E*, E|E*) denote Nash equilibrium of the cloud resource price and the cloud resource size obtained from media cloud, respectively. The Nash equilibrium of the game can be obtained by solving
where p_ and E_ is the set of Nash equilibrium of brokers except broker i.
From the above analysis, a broker needs the strategy of other brokers and the equilibrium of the evolutionary game to obtain Nash equilibrium. However, this information may not be available in a practical broker system. Therefore, each broker can only employ the local information and users’ demands to determine the prices and the cloud resource. Then, each broker should adjust its strategy in the direction of utility maximization. Therefore, broker i updates the price of cloud resource and the cloud resource size by
Here pi (т) and Ei (t) are the price of cloud resource to sell and the size of cloud resource purchased from the media cloud. Both of them are determined by broker i at iteration т. mitE and mi,p are used to control the speed of adjustment on the cloud resource size and cloud resource price. The marginal payoff can be used to update the strategy for each broker . It can be calculated by the variation in payoffs with a small variation p (e.g., p = 10-4) as follows: