Analysis of Bayesian Nash Equilibrium

It is well known that an equilibrium point may not exist. In the subsection, we are interested in investigating the existence and uniqueness of a BNE in our resource allocation game.

To show the existence of BNE, we need to show that the strategy set of each player is convex, compact and nonempty (79). Moreover, the utility function is concave on the strategy set. The strategy set of each player in our game is nonempty since every admitted connection allows to request bandwidth. Additionally, b_i ⊆ R. Thus, the strategy set is convex and compact. Now we want to show that the utility function is continuous and concave on both bi and b_−i. Our utility function comprises SI and P I. It is easy to show that both SI and P I are continuous on b_i and b_−i. Thus, the utility function is continuous on both bi and b_−i. As shown in Section 4.7, P I for all scheduling classes is modeled based on an exponential function. It is clear to conclude that P I is concave for the payoff function.

We show the concavity of SI by the following Lemma.

Lemma 1: In the proposed game, the SI for all scheduling classes is concave.

proof: As shown in equation (4.14), the SI for nrtPS and BE connections is modeled by a

logarithm function. Thus, it is clear to conclude that the SI for nrtPS and BE connections is concave. We now focus on the SI for rtPS connections. As shown in equation (4.10), the SI for rtPS connections is modeled by a sigmoid function. Additionally, the infection point of the sigmoid function is based on the M SR of the connection. It separates SI for SS i with a rtPS connection as below:

SI_i(b_i) :











convex, 0 ≤ bi≤ MSRi^rt

concave, M SR^rt_i ≤ bⁱ <∞

According to our admission control policy, each SS should receive the guaranteed bandwidth until its M SR is reached. Because of this policy, in our design, the P I of all scheduling classes is zero before the M SR is reached. Moreover, the objective of each SS tries to maximize its payoff which is SI − P I. Therefore, the SS must request bandwidth which is larger or equal to its M SR. Consequently, we can limit our consideration only to the concave part of the sigmoid function and conclude that the SI for rtPS connections is also concave in the proposed game. In summary, the SI for all scheduling classes is concave in the proposed game.

Based on Lemma 1 and description above, the proof for the existence of BNE is com-pleted. Now, we investigate the uniqueness of BNE. We rely on a sufficient condition: a non-cooperative game has a unique equilibrium if the nonnegative weighted sum of the payoff function is diagonally strictly concave (79).

Definition 2. (Diagonally Strictly Concave) A weighted sum function h(x, r) :=Pn

i=1r_iζ(x) is called diagonally strictly concave for all vector x ∈ R^n×1 and fixed vector r ∈ R^n×1, if for any two different vectors x⁰, x¹, we have

Λ(x⁰, x¹, r), (x¹− x⁰)^Tσ(x⁰, r) + (x⁰− x¹)^Tσ(x¹, r) > 0 where σ(x, r) is called pseudo-gradient of f (x, r), defined as:

σ(x^,r),

Definition 1 states that the BNE is obtained when all players obtain their best strategies by giving the strategies of other SSs such that their expected payoffs are maximized. Ad-ditionally, as stated in Definition 2, the payoff is calculated as the expected value of utility function with the corresponding event probability. In this paper, the event probability is considered as the scheduling class of the connection running on the SS. According to IEEE 802.16 standard, the scheduling class has to be determine during admission control proce-dure and it cannot be changed after creation of the connection. Consequently, this makes each SS only have one event with probability of 1. Therefore, the description in Definition 1 and 2 match our utility function shown in equation (4.9).

Lemma 2: The weighted nonnegative sum of average payoff ui in the proposed game is diagonally strictly concave.

proof: We present the weighted nonnegative sum of the average payoff as:

hⁿ(b, r),

n]^T be the pseudo-gradient of hⁿ((b, r)). Each SS i ∈ N serves a connection belonging to one of scheduling classes. Sup-pose b^rt_i , b^nrt_i and b^be_i are the amount of bandwidth for the connection in rtPS, nrtPS and BE, respectively. Note that each connection only has one scheduling class. Thus, only one of b^t_i, t∈ {rt, nrt, be} can be a positive number and the rest of them must be zero. Suppose SS_i serves a rtPS connection, then the average payoff ui can be actually transformed into

a weighted sum function as follows

where γ represents the index for different jointly probability events with corresponding probability wγ. Similarly, the average payoff function can be presented as following if the connection belongs to nrtPS or BE.

u_i(b^s, b^s₋) =X

where s ∈ {nrt, be}. Now, we can write the pseudo-gradient σⁿ as:

σⁿ(b, r) =

To check the diagonally strictly concave, we let b⁰, b¹ be two different vectors and define

Λ(b⁰, b¹, r), (b¹− b⁰)^Tσⁿ(b⁰, r) + (b⁰− b¹)^Tσⁿ(b¹, r) (4.26)

In Lemma 1, we have shown that the SI is concave for both nrtPS and BE connection.

Moreover, for rtPS connection, we also proved that the SS never request bandwidth less than the M SR of the connection. Therefore, only concave part of SI needs to be considered.

Without loss of generality, we assume b¹_i > b⁰_i. It can lead to (^∂SI(b_∂b0⁰ⁱ⁾

i −^∂SI(b_∂b⁰¹ⁱ⁾

i ) ≥ 0. P I for all scheduling classes is modeled by the exponential function which is strictly convex to the amount of reserved bandwidth. Thus, we can have (^{∂P I(b}_∂b1¹ⁱ⁾

i − ^{∂P I(b}_∂b¹⁰ⁱ⁾

i

) > 0. It results λ(b⁰_i, b¹_i, r_i) > 0. Consequently, we can conclud:

Λ(b⁰, b¹, r) =X

∀i

λ(b⁰_i, b¹_i, ri) > 0 (4.28) It shows that the weighted nonnegative sum of average payoff is diagonally strictly concave.

The proof for uniqueness of BNE is completed.