Existence of Truthful Nash Equilibrium

By combining the proposed winner determination and payment determination rules, we now prove that there exists a truthful Nash equilibrium in the proposed auction. We

Algorithm 2 PAYMENT_DETERMINATION Input: N, C,{Qi,c},{Di,c}, {ki, M_i}, and {wi} Output: Payment{pi}

1: for i∈ N do

2: [{Ci}, {c^p_i}, {xi,c,b,t}, ulose] = W IN N ER_DET ERM IN AT ION (N \ {i}, C), {Qi,c}, {Di,c}, {ki, Mi}, {wi};

3: pi= u_l 4: end for

start from a lemma that explains whether a delay-sensitive UE i will report its data re-quirement type untruthfully.

Lemma 3. If a winner UE i reports k^′_i = th instead of k_i = delay as its data requirement type, its experienced delay does not decrease.

Lemma 3 is a direct result from the proposed allocation rule. It shows that a delay-sensitive UE i will never report her type untruthfully. Then, the following lemma provides a necessary condition for the existence of truthful Nash equilibrium.

Lemma 4. If a UE wins the auction by reporting (M, k), then it can wins the action by reporting (M^′, k) with M^′ ≤ M, but may not if reporting (M^′′, k) with M^′′≤ M

Proof. Since M^′ ≤ M, the required number of resource blocks is never greater under M^′ than M . Therefore, the UE's order in the new list is never larger than the original list.

Additionally, there are enough resource blocks to satisfy M^′ since the UE is a winner by reporting M > M^′. So, the UE is still selected as a winner by reporting (M^′, k).

For the case of M^′′ ≤ M, since the required number of resource block is never smaller under M^′′ than M , it is possible that the order of the UE in the new list is smaller than in the original list, according to the winner determination algorithm. This reduces the possibility of the UE to win the auction.

Lemma 4 is important as it suggests that a UE i cannot increase her probability to win in the proposed auction by increasing her requested data amount M_i. Although a UE may choose to decrease her requested data amount M_i in order to increase her probability to win, we will show that reporting M_i^′ ≤ Mi will never be beneficial to the UE i later.

Lemma 5. a UE i's payment piis unrelated to her own bid (M_i, k_i).

Proof. According to the payment determination rule, the UE i's payment is u_i^b(M_i^b, d^b_ib), in which UE i's bid is excluded from the winner determination stage. Since UE i's bid is excluded, it does not have any influence on the winner determination. Therefore, the UE i's bid does not have any influence on her payment.

Finally, we propose the following theorem about the existence of truthful Nash equi-librium in the proposed auction.

Theorem 9 (The existence of truthful Nash equilibrium). A truthful Nash equilibrium exists in the proposed game if the proposed auction is applied.

Proof. Let (M_i^∗, k_i^∗) be the true QoS requirements of UE i,∀i ∈ N. By Lemma 4 and 5, it is guaranteed that reporting M_i ̸= Mi^∗ is never the better response of UE i if k_i^∗ is reported. We now need to check if report k_i^′ instead of k_i^∗ is beneficial to UE i.

Case 1: k_i^∗ = th. In this case, reporting k_i = delay only reduces its weighted valuation and therefore increases its order in the list. This reduces its probability to win the auction and therefore impair its expected utility.

Case 2: k^∗_i = delay. In this case, reporting k_i = th may increase its weighted valua-tion and therefore decrease its order. We should discuss this under two possible cases:

Case 2.a: UE i is not a winner by reporting k_i^∗. In this case, even UE i becomes a winner, its true utility will be negative since he is not a winner in the original list.

Case 2.b: UE i is a winner by reporting k^∗_i. In this case, UE i is still a winner and now follows the allocation rule for throughput-sensitive type. According to Lemma 3, the delay under kiis strictly larger than under k^∗_i if the UE reports the same Mi. In such a case, the valuation of UE i is strictly lower by reporting ki = th instead of k_i^∗ = delay. Given the same payment p_i, the expected utility of UE i is strictly lower by reporting k_i ̸= ki^∗.

Concluding from above, the truthful Nash equilibrium exists if the proposed auction is applied.

4.6 Simulation Results

We evaluate the proposed solutions in Section 4.5 through simulations. We simulate a LTE-Advanced system with 19 cell BSs deployed in a typical 19-cell hexagonal topol-ogy. There are 5 independent carriers, each with 1.4 MHz bandwidth. According to LTE's standard, a frame is composed of 10 subframes, while a subframe is composed of 2 time slot, each with 0.5 ms. A 1.4 MHz carrier is composed of 6 resource blocks at each time slot, while a resource block has a bandwidth of 180kHz and contains 7 symbols. The number of bits contained in a symbol depends on the applied MCS. In the simulations, we apply the MCSs and link-level settings specified by 3GPP [2] according to the signal-to-interference-and-noise ratio (SINR) experienced by UEs [20]. For other settings, we fol-low the IMT-Advanced 4G evaluation guidelines [1]: The downlink transmission power of each cell is 23dBm, while the cell radius is 100 m. The propagation loss model is out-door model, while the shadow fading follows a log-normal distribution of mean 0dB and standard deviation 10dB.

We assume that all cells, except the center one, are full-buffered and are treated as the neighbor cells of the center cell. Each neighbor cell activates all carriers to transmit data to their UEs, while 2 of the activated carriers are assigned as the primary carriers of their serving UEs. For the center cell, we apply the proposed auction design in order to determine the carrier activation and resource block allocations according to the QoS requests from the serving UEs. The auction holds for every 8 frames, which equal to 80 ms. This fits one of the CQI reporting rates defined in LTE-Advanced.

A fixed number of UEs are uniformly and randomly deployed in the system. These UEs request for downlink transmissions. They experience different carrier qualities due to 1) the spatial-diverse and carrier-diverse interference from neighbor cells, and 2) the propagation signal loss due to the distance from the center cell BS to the UE. Each UE supports at most 2 carriers when carrier aggregation is enabled. The QoS requirement of each UE is randomly generated. Given the average request data amount Mavg, the request data amount Miof each UE is randomly generated from 1 kB to 2Mavg kB. Additionally, half of the UEs are throughput-sensitive, while others are delay-sensitive. The valuation

function of throughput-sensitive V^th(m_i) = log(1 + ^{9 min{m}_M^i,Mi}

i ), while the valuation function of delay-sensitive is V^delay(d_i) = log(1 + ⁴⁰_d

i).

The carrier qualities are determined by two factors in the simulations. The first one is the expected resource block data amount (Q_i,c) in each carrier, which represents the data amount a resource block can carry under the currently applied MCS. The value can be derived through simulating the LTE wireless channel model in [20]. The second one is the PDCCH decoding probability (P (D_i,c)). In LTE system, all PDCCH messages are transmitted using the BPSK modulation. Given the transmission power, resource block bandwidth and symbol rates, and the typical length of PDCCH message, we can theoreti-cally calculate the bit-error rate and therefore the message error rate of a PDCCH message (We ignore the coding rate of PDCCH message here for simplicity).

In all simulations, we compare 5 schemes. The first one is the proposed solution, where the proposed auction design is applied. Specifically, the winner determination algorithm in Algorithm 1 is applied here for activating the carriers and allocating the resource blocks to the UEs according to their QoS requirements. We compare the proposed solution with three carrier activation schemes. One is the random scheme, where each UE randomly activates 2 carriers. Another one is the fair scheme, where each carrier is assigned with the same number of UEs (or plus/minus 1 if aliquant). The last one is the Best CQI scheme, where each UE activates two carriers that offer her the highest expected resource block data amount. In all three schemes, round-robin method is applied on the resource block allocation. These schemes represent the traditional carrier activation schemes where no QoS requirements are addressed. In the last scheme, we apply our proposed solution while limiting the number of activated carrier of each UE to 1, which simulate the case that all UEs are using legacy devices with no carrier aggregation support. This helps us to understand how carrier aggregation enhances the performance of the system.

The performance metric we choose in the simulations is the social welfare of the sys-tem, which is defined as follows:

SW = ^∑

i∈N

V^ki(·),

2 4 6 8 10 12 14 16 18 20

Figure 4.3: Simulation Results: Number of UEs

which is the same as the objective function in (4.7). The social welfare represents the total valuation of UEs on the service, which is an important indicator on the system efficiency in game theory. Since our work focus on optimizing the service valuation based on UEs' QoS requirements, we choose this as the performance metric in our evaluations.