4.2 Numerical Results
4.2.2 Evolutions of Beliefs and Actions
In this section, we simulate and show the evolution of action profile over stage. The simu-lation scenarios are classified into three different cases, they’re 1. all action of all players profile are strictly feasible, which is shown in Fig. 4.6(a), 4.6(b), 4.6(c); 2. some action profile of some players are on the boundary of bandwidth constraints, which is shown in Fig. 4.7(a), 4.7(b), 4.7(c); 3. all action profile of some player are on the boundary of bandwidth constraints, which is shown in Fig. 4.8(a), 4.8(b), 4.8(c).
Fig. 4.6(a), 4.7(a) and 4.8(a) show the equilibrium pricing profile of the actual type corresponding to the one with proposed joint KKT condition and one which uses the same belief as the proposed one at each stage instant but without considering the constraint (unc). Fig. 4.6(b), 4.7(b) and 4.8(b) show the equilibrium demand profile of the actual type corresponding to the one with joint KKT condition and one which uses the same belief as the proposed one at each stage instant but without considering the constraint (unc). Fig. 4.6(c), 4.7(c) and 4.8(c) show the possible minimal equilibrium demand profile corresponding to the one with joint KKT condition and one which uses the same belief as the proposed one at each stage instant but without considering the constraint (unc). The belief update over stage about player of the three cases is also presented in TABLE 4.1, 4.2, 4.3, respectively. Numerical analysis about the action and the belief on the three cases are discussed.
0 1 2 3 4 5 6 7 8
Figure 4.6: The equilibrium strategies over stages of Case 1. (a) Of PS’s. (b) Of SS’s. (c) The possible minimal equilibrium strategies of SS’s.
Case 1: When All Actions Are Strictly Inside Constraints
Next, we study the behavior of the sequence of equilibrium strategies as stage evolves under W1 = 15MHz and W2 = 15MHz, ˆθp1 = 10, ˆθp2 = 10, ˆθs1 = 1, ˆθs2 = 2, γ1p = 15 dB, γ2p = 15 dB, γ11s = 22, γ12s = 18, γ21s = 18, γ22s = 22, Breq1 = 2 Mbps and Breq2 = 2 Mbps. Fig.4.6(a) and Fig.4.6(b) show the equilibrium pricings and the equilibrium demands. Sinceγ11s = 22 is larger than γ12s = 15, SS1 demands more from PS1 than from PS2. Correspondingly, PS1 sets higher price to SS1 than PS2 does. On the other hand, SS2 demands more from PS2 than from PS1 sinceγ22s = 22 is larger than γ21s = 15. Therefore, PS2 sets higher price to SS2 than PS1 does. PS1 asks lower price to SS2 than to SS1 sinceγ11s is larger thanγ21s , while with thatγ22s is larger thanγ12s , PS2 also asks lower price to SS2 than to SS1 since SS2 is with ˆθs = 2 and puts less emphasis on the QoS satisfaction, or equivalently, is more concerned with the monetary expense.
Therefore, both PS’s set lower price to SS2 to stimulate the demand.
We also observe that the difference between utilizing joint KKT condition and without considering the constraints. Although the pricings and demands without considering the constraints evolve into the same value as those considering joint KKT condition after the belief update correctly since the solutions are strictly feasible, they are infeasible at the beginning. It’s because the possible minimal demands of unconstrained case are negative in the beginning stage as Fig.4.6(c) shows, while the minimal demands with joint KKT condition are still feasible, being zero in this case.
The Bayesian game model allows the equilibrium strategies to update according to the beliefs (TABLE 4.1) of all players’ private information. Since these actions are strictly inside feasible region, then the belief and hence the behavior converges in the end.
Table 4.1: Belief Updating versus Stage for Case 1 Belief about PS’s Staget
0 1,2 3,4 t > 4 µ(θp1 = 10|ht) 13 1 1 1 µ(θp1 = 11|ht) 13 0 0 0 µ(θp1 = 12|ht) 13 0 0 0 µ(θp2 = 10|ht) 13 1 1 1 µ(θp2 = 11|ht) 13 0 0 0 µ(θp2 = 12|ht) 13 0 0 0 Belief about SS’s
Staget 1 2,3 4,5 t > 5 µ(θs1 = 1|ht) 13 1 1 1 µ(θs1 = 2|ht) 13 0 0 0 µ(θs1 = 3|ht) 13 0 0 0 µ(θs2 = 1|ht) 13 0 0 0 µ(θs2 = 2|ht) 13 1 1 1 µ(θs2 = 3|ht) 13 0 0 0
Case 2: When Some Actions of Some Players are on the Boundaries of Constraints
Next, we study the behavior of the sequence of equilibrium strategies as stage evolves underW1 = 15MHz and W2 = 15MHz, ˆθp1 = 10, ˆθp2 = 12, ˆθs1 = 1, ˆθs2 = 2, γ1p = 15 dB,γ2p = 15 dB, γ11s = 22 dB, γ12s = 22 dB, γ21s = 22 dB, γ22s = 22 dB, Breq1 = 2 Mbps andBreq2 = 2 Mbps.
Fig. 4.7(a) and Fig. 4.7(b) show that the equilibrium pricings and the equilibrium demands. Each PS asks lower price to SS2 than to SS1 since SS2 is with ˆθs = 2 and so puts less emphasis on the QoS satisfaction, or equivalently, is more concerned with the monetary expense. The penalty, or the cost, of sharing spectrum for PS2 is higher than that for PS1 since PS2 with a higher volume of local connections is more reluctant to share the spectrum, in order to fulfill its primary users’ QoS satisfaction. Consequently, PS2 would set a higher price, that yields lowerb12 andb22. Under this circumstance, SS1
0 1 2 3 4 5 6 7 8
Figure 4.7: The equilibrium strategies over stages of Case 2. (a) Of PS’s. (b) Of SS’s. (c) The possible minimal equilibrium strategies of SS’s.
and SS2 demands moreb11andb21respectively to compensate the insufficiency ofb12and b22.
Under these parameter settings, the unconstrained demand bunc22 would be negative even when the belief is updated to the correct one, while the proposed one is always feasible. Althoughb22 is 0, which is on the boundary of the nonnegative constraint of PS2, the belief about PS2 still converges to the actual one. It is because b12 still isn’t on the boundary, the opponents could still update the belief about PS2. Since the beliefs (TABLE 4.2) could converge, the action profiles converge.
Table 4.2: Belief Updating versus Stage for Case 2 Belief about PS’s Staget
0 1,2 3,4 t > 4 µ(θp1 = 10|ht) 13 1 1 1 µ(θp1 = 11|ht) 13 0 0 0 µ(θp1 = 12|ht) 13 0 0 0 µ(θp2 = 10|ht) 13 0 0 0 µ(θp2 = 11|ht) 13 0 0 0 µ(θp2 = 12|ht) 13 1 1 1 Belief about SS’s Staget
1 2,3 4,5 t > 5 µ(θs1 = 1|ht) 13 1 1 1 µ(θs1 = 2|ht) 13 0 0 0 µ(θs1 = 3|ht) 13 0 0 0 µ(θs2 = 1|ht) 13 0 0 0 µ(θs2 = 2|ht) 13 1 1 1 µ(θs2 = 3|ht) 13 0 0 0
Case 3: When All Actions of Some player are on the Boundaries of Constraints Next, we study the behavior of the sequence of equilibrium strategies as stage evolves underW1 = 15MHz and W2 = 15MHz, ˆθp1 = 10, ˆθp2 = 10, ˆθs1 = 1, ˆθs2 = 1, γ1p = 15
0 1 2 3 4 5 6 7 8
Figure 4.8: The equilibrium strategies over stages of Case 3. (a) Of PS’s. (b) Of SS’s. (c) The possible minimal equilibrium strategies of SS’s.
dB,γ2p = 15 dB, γ11s = 22, γ12s = 22, γ21s = 22, γ22s = 22, Breq1 = 2 Mbps and Breq2 = 4 Mbps. Basically, the parameters of SS1 and SS2 are the same, so PS1 would set the same price to both SS’s, and so does PS2. Likewise, both SS’s would demand the same size of bandwidth from the same PS. Therefore, we letbji denote b1i andb2i andpji denotep1i
andp2i.
As Fig. 4.8(a) and Fig. 4.8(b) show, since PS2 has higher bandwidth requirement for local connection, it asks high price to both SS’s than PS1 does, which makes the demands from both SS’s be 0. The action of PS2 makes the opponents difficult to update the belief about PS2 (TABLE 4.3), but the actions of PS2 converge to those of the actual type of PS2 which is shown in the curves of complete information scenario. It justifies that even thought the belief cannot converge to the actual one, the action profile still converges to the one with correct belief, i.e. complete information, and thus it doesn’t influence the result of the game.
Table 4.3: Belief Updating versus Stage for Case 3 Belief about PS’s Staget
0 2 4 t > 4 µ(θp1 = 10|ht) 13 1 1 1 µ(θp1 = 11|ht) 13 0 0 0 µ(θp1 = 12|ht) 13 0 0 0 µ(θp2 = 10|ht) 13 13 13 13 µ(θp2 = 11|ht) 13 13 13 13 µ(θp2 = 12|ht) 13 13 13 13 Belief about PS’s Staget
1 2,3 4,5 t > 5 µ(θs1 = 1|ht) 13 1 1 1 µ(θs1 = 2|ht) 13 0 0 0 µ(θs1 = 3|ht) 13 0 0 0 µ(θs2 = 1|ht) 13 1 1 1 µ(θs2 = 2|ht) 13 0 0 0 µ(θs2 = 3|ht) 13 0 0 0
Chapter 5
Conclusion and Future Work
5.1 Conclusion
We’ve studied the spectrum trading game with incomplete information in a sequential manner for a cognitive radio network. The incomplete information game is modeled as Bayesian game, which the incomplete information is viewed in Bayesian way. To ensure that the trading is physically practical, we constrain the trading bandwidth. To solve the optimization problem with bandwidth constraints in the multistage game, we’ve proposed using the KKT translation and joint KKT conditions to yield the perfect Bayesian equi-librium at each stage. We’ve demonstrated that the KKT translation technique provides a general rule that can be applied to optimization problems of multistage game theory.
An algorithm for solving joint KKT condition is given, and the complexity of the algo-rithm is analyzed. In addition, we’ve studied the convergence behaviors of belief and action profiles, although belief profiles may not converge to the actual type, the action profiles converge to actual optimal strategy, which means the result is the same as that of complete information. In the simulations, we’ve justified the effectiveness of joint KKT condition, numerically study the convergence of belief and action profiles, and also how the parameters influences the action. Finally, we’ve concluded that the proposed multi-stage Baysian game model with bandwidth constraints is robust and capable of providing more reasonable strategy profiles for players.