Coalitional Game

In general, game theory can be divided into two branches: noncooperative and cooperative games [16]. The main branch of cooperative games describes the formation of cooperative groups of players, called as coalitions. In the section, we focus on the coalitional game, because we applies a coalitional game to solve the problem in the thesis. A coalitional game focuses on how the players cooperate with each other in the system, in which a coalition is the basic unit. Players in the same coalition have some agreements about forming cooperative group. Hence, the notable issue is how to choose the players to cooperate with. Whereas, the value of a coalition is quantified by the coalition value, which is generated by all the players in the coalition. The players in the system have incentives to join the coalition that increases their own utilities. The assumptions in a coalitional game are different from the basic game described in previous section, so we introduce the coalitional game formulation in the following section.

Game Formulation

A coalitional gameΓ can be formulated as follows

Γ =D

N , {v(S)}S⊆N, {ui}i∈N

E

, (2.4)

whereN ≡ {1, 2, · · ·, N} is the set of players, v(S) is the coalition value of coalition S, and ui is the utility function of playeri. In a coalitional game, the coalition value is the most important element, which is generated by the players in the coalition. Coalition value can be defined in different forms, e.g. rate, power, or payment, according to the game formulation. ui defines the utility of player i and coalition value is generated by the players’ utilities in the coalition. Hence, the coalition valuev(S) is highly related to utility functionui. The player’s utility received in the coalition is called payoff.

In a coalitional game, a coalition is the basic unit and how to divide the players into coalitions is the crucial issue. In this thesis, we consider a special class of coalitional games that all the players would form one coalition, i.e. grand coalition. In the grand coalition, all the players will cooperate with each other with certain agreements, so the whole system is stable. Another assumption of the coalitional game that we are concerned is transferrable utility (TU) [17]. TU property implies that the total utility represented as a real number can be divided in any manner between the coalition members. The utility that a playeri received from the division of v(S) constitutes the player’s payoff denoted asxi. Whether the payoff allocation is stable or not, we can examine it by the solution concept introduced in the following section.

Solution Concept

In the coalitional game theory [16] [18], the most renowned solution concept of a coali-tional game is the core. The relationship between the core and a coalicoali-tional game is similar to Nash Equilibrium and a noncooperative game. The core is directly related to the stability of grand coalition. In other words, the existence of the core implies that the whole system is stable. Due to the superadditivity property, players have incentives to form grand coalitionN , consisting of all players. The definition of superadditivity is as

follows

Definition 4 The two coalitions have the property of superadditivity ifS and Z are dis-joint coalitionsST Z = ∅, then v(S) + v(Z) ≤ v(S S Z). If two disjoint coalitions satisfy the above equation, they are called superadditive.

Hence, if two coalitions are superadditive, they will merge together to form a new coalition. If we discover that all the coalitions in the system are superadditive, all the players will join to form grand coalition. Also, players can increase their payoffs in the grand coalition, so players have no incentive to leave the grand coalition. In other words, the grand coalition is stabilized. The formal definition of the core is given as

Definition 5 A payoff vector x is stable in a coalition S if P

i∈Sxi ≥ v(S), i.e. the playeri has an incentive for the proposed payoff xi. The set of stable payoff allocation, i.e. the core is defined as:

C =

We can see that from the definition of the core, which needs to satisfy two conditions.

The first condition is called as group rational, which the total sum of players’ payoffs is equal to the coalition value of grand coalitionN . The second condition is related to the individually rational. A payoff vector is individually rational if every player can obtain a benefit no less than acting alone, i.e.xi ≥ v(i), ∀i ∈ N . Hence, the second condition can be viewed as the sum payoff at least the same with the coalition valuev(S).

With the definition of the core, we can examine whether the payoff allocation is stable or not. As mentioned above, the core is directly related to the stability of the system.

However, the core is not always guaranteed to exist in a coalitional game. Actually, the core set is empty in many coalitional games, so the grand coalition cannot be guaranteed stabilized. In these situations, we may consider alternative solution concepts, but they are not the main topic in this thesis. Interested readers can refer to [16] [18] for detailed description. In the next chapter, we focus on applying a coalitional game in CCRN.

Chapter 3

Coalitional Game in Cooperative Cognitive Radio Networks

3.1 Problem Setup

We consider a CDMA based cooperative cognitive radio network (CCRN) consisting of Np PUs and Ns SUs, where Np = {1, 2, · · ·, Np} and Ns = {1, 2, · · ·, Ns}. It is an uplink transmission scenario, which PUs aim to transmit signals to primary base station (PBS) whereas SUs want to access the spectrum to transmit data to secondary access point (SAP). There areNb available channels licensed to PUs in the system, whereNb = {1, 2, · · ·, Nb}. There is an example of CCRN with Np = 2,Ns= 2, andNb = 2 as shown in Fig. 3.1. The channels are licensed to PUs, so only PUs have the legal rights to use the channels. If SUs want to access the channels, they need to cooperate with PUs or give payment for channels’ accessing. We apply the idea of cooperative communication in the CCRN. SUs are served as cooperative relays for primary transmission, so PUs’

transmission rates are improved by exploiting cooperative diversity. In return, SUs gain the opportunities to access spare channels. SUs are licensed fractions of time to transmit their own traffic. Hence, both PUs and SUs can benefit from the cooperative scenario. By the assistance of SUs, the transmission rates of PUs increase by exploiting the technique of maximum ratio combining (MRC) at the receiver.

For the time scheduling of channels, we consider a time slot set as t, which is divided

      

into three sub time slots. The first sub time slot used by PUs is set as(1−α)t and the other sub time slots are set asβt and (α−β)t for SU’s relaying and SU’s accessing, respectively.

For notational simplicity, we normalize theses sub time slots by t, so the fractions of time for sub time slots are1 − α, β, α − β, respectively. For PU’s channel allocation, each channel is licensed for one PU, e.g. PU 1 can only uses channel 1 for transmission. In the first sub time slot, PUs transmit their traffic to SUs and PBS simultaneously using the broadcast nature of wireless communication. Then, SUs help to relay the traffic received from PUs to PBS in the second sub time slot, which is for SU’s relaying. In the last sub time slot, each SU can access multiple channels for its own transmission. The reason why each SU can access multiple channels in a sub time slot is explained in the next section.

In the channels allocation of SUs, we exploit a special design to enable that each SU can access multiple channels in a sub time slot. In the relay mode, each SU selects the channel with the best channel quality to assist PUs’ transmission. In other words, each PU’s traffic is relayed by SU’s best channel to increase capacity. Each SU helps to relay the traffic from PUs in the coalition, so it is reasonable that each SU can access available channels in the coalition. On the other hand, the system assigns unique spreading code to each SU in order to differentiate SUs when accessing the same channel. In the next section, we introduce the coalitional game formulation in the CCRN.