# Analysis of the Proposed Four Stage Game

In this section, we analyze the proposed four-stage Stackelberg game, and obtain its Stackelberg equilibrium. Based on the above analysis, it is known that each stage’s strategy may affect other stages’ strategies. Therefore, we use the backward induction method to analyze the proposed game, as it can capture the sequential dependence of the decisions in each stage of the game.

## Evolution Game Among Users in Stage IV

Communities are formed by groups of users with media service demands. Especially, users in the same community are friends of each other. They can communicate with each other, where the information can be exchanged among them. Therefore, users in the same community can observe others’ decisions on the selection of brokers, and then adjust his strategy to be optimal. We propose an evolutionary game model to solve the broker selection problem. In the evolutionary game, users are the players of the game. The community represents a population in the game.

Replicator dynamic is crucial to analyze the evolution game to obtain the game equilibrium, where the utility of all users in a community are identical. And no player will change his current strategy because the rate of strategy change is zero. For community *j*, the proportion of users who select broker *i* to acquire media service becomes

where *n _{i}, j* is the number of users in community

*j*to connect broker

*i*, and

*Nj*is the number of users in community

*j*. We denote the state of community as the proportions of users to connect brokers. Thus, the state of community

*j*can be obtained by

In the replicator dynamic, the share of a strategy in community grows at a rate which is directly proportional to the difference between the user’s utility and the average utility. It can be denoted as

where *X* is the multiplier of the difference between the user’s utility and the average utility. *Uj (t)* is the average utility of the entire community *j*. It can be calculated by

From the above, it can be obtained X* ^{I}=_{1} X_{i}, j (t) =* 0. Therefore X

*= 1 is satisfied during the broker selection process. Substituting (5.9) into (5.27), we have*

^{I}=_{1}x_{i}, j

We consider the evolutionary equilibrium as the solution to the broker selection game among users. An evolutionary equilibrium is a fixed point of the replicator dynamic. At the fixed point, which can be obtained numerically, the payoff of all users in community *j* are identical. In other words, since the rate of strategy adaptation is zero, the equilibrium can be obtained by solving

To evaluate the stability at the fixed point , which is obtained by solving (5.30), the eigenvalues of the Jacobian matrix which is corresponding to the replicator dynamic needs to be evaluated. The fixed point is stable if each eigenvalue has a negative real part [45]. Here we have the evolutionary equilibrium for any community *j* as follows.