# Dynamic Pricing

**DOI:**https://doi.org/10.1007/978-3-319-32903-1_35-1

## Definitions

Dynamic pricing is a mechanism in which the price is determined dynamically and iteratively through dynamic algorithms. The algorithms can yield optimal or equilibrium pricing solutions that maximize the profit of a single monopolistic seller or multiple oligopolistic sellers, respectively.

## Economic Modeling and Pricing

Economic and pricing approaches have been applied to address many issues in wireless/radio resource management. In the following, major economic and pricing models used in the wireless literature are presented Luong et al. (2017).

**Cost-based pricing**: Cost-based pricing is a common pricing strategy to determine the price of wireless resource and service based on calculating the total cost of acquiring the resource and offering the service as well as adding a percentage of the cost as a desired profit. The goal of cost-based pricing is to ensure that the price yields nonnegative profit to the wireless service provider profitable. The total cost generally consists of a fixed cost and a variable cost. The fixed cost is the cost that does not change when the number of sales of the services changes, for example, hardware costs, for example, servers and network devices, and installation. On the contrary, the variable cost incurs depending to the number of product produced or the amount of services used. For example, the resource costs, for example, energy and bandwidth costs. The advantage of the cost-based pricing is the ease of setting the price. The reason is that the price is a function of the internal cost, i.e., the cost required to generate the service Courcoubetis and Weber (2003). However, this pricing strategy does not consider external market factors, for example, the pricing strategies of other providers and the perceive value and willingness to pay of buyers or users.

**Differential pricing**: The cost-based pricing ignores the requirement and preference of each user. On the other hand, to maximize the profit of providers, differential pricing, also called price discrimination, can be used, utilizing such user’s requirement and preference. In the differential pricing, the provider can charge different prices to different users based on their demand and willingness to pay. By setting higher prices for one type of user, the use of the differential pricing explores the user surplus by the provider. However, it can be unfair to cause one type of users to pay a higher price than that of other types of user.

**Profit maximization**: Profit maximization is the process of determining the output quantity and/or the price that offer the highest profit for a provider Korrapati (2014). The pricing based on profit maximization concept is described in the following. Assume that a service provider needs to determine the amount of radio resource or wireless service units to be sold which is denoted by *q* and the price *p* for their users. The profit of the provider is *π* = *R*(*p*, *q*) − *C*(*q*), where *R*(·,·) is the total revenue and *C*(·) is the total cost. The total cost includes a fixed cost and a variable cost. The revenue is the amount of money that the provider receives from selling *q* resource units to its users. The optimal quantity of radio resource units, i.e., *q*^{∗}, is determined such that the profit is maximized, i.e., *q*^{∗} = max_{q}*π*. The optimal quantity allows to find the optimal price based on the demand curve. The demand curve is typically a linear curve to show the relationship between the price of a resource unit and the quantity of resource units that users are willing to buy. A general demand curve can be expressed as *p* = *a* − *bq*, where *a* and *b* are proper parameters. Thus, at *q* = *q*^{∗}, the optimal price is *p*^{∗} = *a* − *bq*^{∗}.

### Game Theoretic Pricing

*Player*: A player is a participant which makes a decision in the game.*Payoff*: A payoff, i.e., a utility, a profit, or an interest, reflects the desired outcome of the player.*Strategy*: Player’s strategy is a set of actions/instructions that the player can follow to achieve the desired outcome. The payoff is a function of not only the player’s own action, but also the actions of others.*Rationality*: A player is rational if its strategy always aims at maximizing its own payoff.

**Noncooperative game**: In a noncooperative game, each player maximizes only its own payoff without considering the payoff of the other players nor the social welfare of the networks AlSkaif et al. (2015). In this game, the players are self-interested, and they do not form coalitions or make agreements with each other.

*N*players, and

*P*

_{i}is a set of pricing strategies of player

*i*, where

*P*=

*P*

_{1}× … ×

*P*

_{N}. × is the Cartesian product of the sets of strategies of each player. Let

*p*

_{i}∈

*P*

_{i}be the pricing strategy of player

*i*. A vector of strategies of

*N*players is

**p**= (

*p*

_{1}, …,

*p*

_{N}), and a vector of corresponding payoffs is

*= (*

**π***π*1(

**p**), …,

*π*

_{N}(

**p**)), where

*π*

_{i}(

**p**) is the payoff of player

*I*given the player’s chosen strategy and strategies of all the other players. Each player optimizes its best strategy \( {p}_i^{\ast } \) which maximizes its payoff. A set of strategies \( \mathbf{p}\ast =\left({p}_1^{\ast },\dots, {p}_N^{\ast}\right)\in P \) is the Nash equilibrium of the noncooperative game if no player can gain higher payoff by changing its own strategy when the strategies of the others remain the same Friedman (1988), i.e.,

*i*.

**Stackelberg game**: The noncooperative game assumes that players chooses and releases their pricing strategies simultaneously. Also, all the players know each other’s strategies at the same time. However, this may not always hold in real markets. Therefore, sequential games can be used in which players can choose and release their strategies following a certain predefined order. This situation can be modeled as a Stackelberg game Amir and Grilo (1999). In the Stackelberg game, the players are divided into leaders and followers. The follower players decide its own strategic choices after observing the strategies of leader players Kim (2014). It was proved that even if the leader players can choose their strategies first, their payoffs are not less than those at the Nash equilibrium Han (2012), i.e., because of the first-mover advantage. The following provides the definition and properties of the Stackelberg game. Assume that there are two radio resource sellers 1 and 2 in the market. *P*_{1} and *P*_{2} are the sets of pricing strategies of sellers 1 and 2, respectively. Seller 1 chooses its pricing strategy *p*_{1} from set *P*_{1}to maximize its payoff or profit function *π*_{1}(*p*_{1}, *p*_{2}), and seller 2 chooses its pricing strategy *p*_{2} from set *P*_{2} to maximize its payoff function *π*_{2}(*p*_{1}, *p*_{2}). Also, assume that seller 2 selects its strategy before seller 1 decides its selection, and hence seller 1 is able to observe the strategy of seller 2. Seller 2 is then the leader, and seller 1 is the follower. We have the following definition Leitmann (2013):

### Definition 1

If there exists a mapping *F* : *P*_{2} → *P*_{1} such that, for any fixed *p*_{2} ∈ *P*_{2}, *π*_{1}(*F p*_{2}, *p*_{2}) ≥ *π*_{1}(*p*_{1}, *p*_{2}), ∀ *p*_{1} ∈ *P*_{1}, and if there exists *p*_{2s2} ∈ *P*_{2} such that *π*_{2}(*F p*_{2s2}, *p*_{2s2}) ≥ *π*_{2}(*F p*_{2}, *p*_{2}), then the pair (*p*_{1s2}, *p*_{2s2}) ∈ *P*_{1} × *P*_{2}, where *p*_{1s2} = *F*_{p2s2}, is called a Stackelberg strategy pair.

Definition 1 presents that the Stackelberg strategy is optimal for the leader when the follower responds to the leader with the follower’s optimal strategy.

## Case Study of Dynamic Spectrum Pricing

### Pricing Models

**Market-Equilibrium-Based Pricing**: In the market-equilibrium pricing model, the primary service is not aware of other primary services. Thus, at the seller side, the primary service naively sets the price according to the spectrum demand of the secondary service. This price setting is based on the willingness of the primary service to sell spectrum which is generally determined by the *supply function*. For a given price, supply function indicates the size of radio spectrum to be shared by a primary service with the secondary service. At the buyer side, the willingness of a secondary service to buy spectrum is determined by the *demand function*. Again, for a given price, demand function determines the size of radio spectrum required by a secondary service. In this spectrum trading, market-equilibrium price denotes the price for which spectrum supplied by the primary service is equal to the spectrum demand from the secondary service. This market-equilibrium price ensures that there is no excess supply in the market and spectrum supply meets all spectrum demand.

**Competitive Pricing**: In the competitive pricing model, a primary service is aware of the existence of other primary services and all of the primary services compete with each other to achieve the highest individual profit. The competition here occurs in terms of spectrum pricing. In particular, given the spectrum prices offered by other primary services, one primary service chooses the price for its own spectrum so that its individual profit is maximized. This competitive spectrum pricing can be considered as a price war in an oligopoly market. In this market structure, the firms choose their price noncooperatively. The decision of each seller is influenced by other sellers’ actions and action of one seller may be observed by other sellers. A general description for this oligopoly market is as follows. There are *N* sellers, i.e., primary services, in the market offering the same or differentiated product, i.e., spectrum. The seller competes with each other by adjusting the price of product selling to the buyer, i.e., secondary service. In this competitive situation, when one seller varies the price, demand for that product will change and other sellers will observe and adjust their prices to gain higher profit. This competition in oligopoly market can be modeled by noncooperative game.

**Cooperative Pricing**: In the cooperative pricing model, it is assumed that all of the primary services know each other and they fully cooperate (i.e., collude) to obtain the highest total profit by selling spectrum to the secondary service. In an actual environment, to achieve this full cooperation, extensive communication would be required among all primary services. The demand function for spectrum *F*_{i} at the secondary service can be obtained using \( \frac{\partial U\left(\mathbf{b}\right)}{\partial {b}_i}=0 \) as: \( {D}_i\left(\mathbf{p}\right)=\frac{\left({k}_i^{(s)}-{p}_i\right)\left(v\left(N-2\right)+1\right)-v{\sum}_{i\ne j}\left({k}_j^{(s)}-{p}_j\right)}{\left(1-v\right)\left(v\left(N-1\right)+1\right)} \), where **p** denotes a vector of prices offered by all primary services in the market (i.e., **p** = [*p*_{1}…*p*_{i}…*p*_{N}]^{T}).

**Revenue and Cost Functions for a Primary Service**: For a primary service, there are two sources of revenue – from primary users and secondary users. However, a cost is involved which is a function of QoS performance degradation of ongoing primary users due to sharing the radio spectrum with secondary service. We assume that the primary users are charged at a flat rate for a guaranteed amount of bandwidth. However, if the required bandwidth cannot be provided, a primary service offers “discount” to the users, and this is considered as the cost of sharing spectrum with the secondary service. Let \( {R}_i^l \)denote the revenue gained from primary users served by primary service *i*, \( {R}_i^s \) denote the revenue gained from sharing spectrum with secondary users, and *C*_{i} denote the cost due to QoS degradation of primary users. Then, the revenue and cost functions can be defined as: \( {R}_i^s={p}_i{b}_i,{R}_i^l={c}_1{M}_i,{C}_i\left({b}_i\right)={c}_2{M}_i{\left({B}_i^{req}-{k}_i^{(p)}\frac{W_i-{b}_i}{M_i}\right)}^2 \), where *b*_{i} and *p*_{i} denote, respectively, the spectrum size shared with secondary service and the corresponding price, and *c*_{1} and *c*_{2} denote constant weights for the revenue and cost functions at the primary service, respectively. Here, \( {B}_i^{req} \) denotes bandwidth requirement per user, *W*_{i} denotes spectrum size, *M*_{i} denotes the number of ongoing primary users, and \( {k}_i^{(p)} \) denotes spectral efficiency of wireless transmission for primary service *i*. Note that revenue from the primary users is a linear function of the number of concurrent users, while revenue from secondary users is a linear function of the shared spectrum size given the spectrum price. The cost is proportional to the square of the difference between bandwidth requirement and allocated bandwidth to a primary user.

### Solution of Pricing Models

**Solution of Market-Equilibrium Pricing Model**: For each primary service, the spectrum supply function can be derived based on a profit maximization problem. The solution of this optimization formulation is the optimal spectrum size *b*_{i} to be shared with the secondary service for a given price *p*_{i}. Based on revenue (i.e., \( {R}_i^l \) and \( {R}_i^s \)) and cost (i.e., *C*_{i}), profit *P*_{i} of a particular primary service *i* owning spectrum *F*_{i} can be expressed as: \( {P}_i={p}_i{b}_i+{c}_1{M}_i-{c}_2{M}_i{\left({B}_i^{req}-{k}_i^{(p)}\frac{W_i-{b}_i}{M_i}\right)}^2 \). To obtain the optimal spectrum size to be shared, we differentiate the profit function with respect to *b*_{i} (when *p*_{i} is given) and obtain that \( \frac{\partial {P}_i}{\partial {b}_i}=0=-{p}_i+2{c}_2{M}_i\left({B}_i^{req}-{k}_i^{(p)}\frac{W_i-{b}_i}{M_i}\right)\frac{k_i^{(p)}}{M_i} \).

Spectrum supply is given by the optimal value of \( {b}_i^{\ast } \) which is a function of price *p*_{i}. The supply function can be expressed as: \( {S}_i\left({p}_i\right)={W}_i-\frac{M_i}{k_i^{(p)}}\left({B}_i^{req}-\frac{p_i}{2{c}_2{k}_i^{(p)}}\right) \). The market-equilibrium (i.e., solution) is defined as the price \( {p}_i^{\ast } \) at which spectrum supply equals spectrum demand, i.e., \( {S}_i\left({p}_i^{\ast}\right)={D}_i\left({\mathbf{p}}^{\ast}\right),\forall i \), where the vector \( {\mathbf{p}}^{\ast }={\left[\cdots {p}_i^{\ast}\cdots \right]}^T \) denotes the market-equilibrium prices for all primary services.

**Solution of Competitive Pricing Model**: We use a noncooperative game to model the price competition among primary services. The players (i.e., sellers in an oligopoly market) in this game are the primary services. The strategy of each of the players is the price per unit of spectrum. The payoff for each primary service *i* (denoted by *P*_{i}) is the individual profit due to selling spectrum to the secondary service.

Again, based on the demand, revenue, and cost functions, the individual profit of each primary service can be expressed as follows: \( {P}_i\left(\mathbf{p}\right)={R}_i^s+{R}_i^l-{C}_i \), where **p** denotes a vector of prices offered by all of the players (i.e., primary services) in the game. We consider the Nash equilibrium as a solution of this price competition. In this case, the Nash equilibrium is obtained by using the best response function which is the best strategy of one player given others’ strategies. The best response function of primary service *i*, given a vector of prices offered by other primary services **p**_{−i}, is defined as: *B*_{i}(**p**_{−i}) = arg max_{pi}*P*_{i}(*p*_{i}, **p**_{−i}).

The vector \( {\mathbf{p}}^{\ast }={\left[\cdots {p}_i^{\ast}\cdots \right]}^T \) denotes a Nash equilibrium (i.e., solution) of this game on competitive pricing if and only if \( {p}_i^{\ast }={B}_i\left({\mathbf{p}}_{-i}^{\ast}\right)\forall i, \) where \( {\mathbf{p}}_{-i}^{\ast } \) denotes the vector of best responses for player *j* for *j* ≠ *i*. Mathematically, to obtain the Nash equilibrium, we have to solve the following set of equations: \( \frac{\partial {P}_i\left(\mathbf{p}\right)}{\partial {p}_i}=0 \) for all *i*. In this case, the size of the shared bandwidth *b*_{i} in the individual profit function is replaced with spectrum demand *D*_{i}(**p**), and then the profit function can be expressed as: \( {P}_i\left(\mathbf{p}\right)={p}_i{D}_i(\mathbf{p})+{c}_1{M}_i-{c}_2{M}_i{\left({B}_i^{req}-{k}_i^{(p)}\frac{W_i-{D}_i\left(\mathbf{p}\right)}{M_i}\right)}^2 \).

Then, using \( \frac{\partial Pi\left(\mathbf{p}\right)}{\partial {p}_i}=0 \), we obtain that \( 2{c}_2{k}_i^{(p)}{D}_2\left({B}_i^{req}-{k}_i^p\frac{W_i-\left({D}_1\left(\mathbf{p}-i\right)-{D}_2{p}_i\right)}{M_i}\right)+{D}_1\left(\mathbf{p}-i\right)-2{D}_2{p}_i=0 \). Recall that the demand function can be expressed as *D*_{1}(**p**) = *D*_{1}(**p** − *i*) − *D*_{2}*p*_{i}. The solution \( {p}_i^{\ast } \), which is a Nash equilibrium, can be obtained by solving the above set of linear equations by using a numerical method when all the above parameters are available. Then, given a vector of prices **p**^{∗} at the Nash equilibrium, the size of the shared spectrum can be obtained from the spectrum demand function *D*_{i}(**p**^{∗}).

**Solution of Cooperative Pricing Model**: For the cooperative pricing model, an optimization problem is formulated to obtain the optimal price which provides the highest total profit for all primary services. This optimization problem can be expressed as follows:

*W*

_{i}≥

*D*(

**p**) ≥ 0, the Lagrangian can be expressed as: \( L\left(\mathbf{p}\right)={\sum}_{i=1}^N{P}_i\left(\mathbf{p}\right)-{\sum}_{j=1}^N{\lambda}_j\left(-{p}_j\right)-{\sum}_{k=1}^N{\mu}_k\left({D}_k\left(\mathbf{p}\right)-{W}_k\right)-{\sum}_{l=1}^N{\sigma}_l\left(-{D}_l\left(\mathbf{p}\right)\right) \), where

*λ*

_{j},

*µ*

_{k}, and

*σ*

_{l}are Lagrange multipliers for the constraints in Eqs. (3) and (4), respectively. Using Kuhn-Tucker conditions, we can obtain the vector of optimal prices

**p**

^{∗}such that the total profit of all the primary services is maximized.

## Dynamic Pricing Algorithms

In a practical cognitive radio environment, a primary service may not have the complete network information. Therefore, the primary service must learn the behavior of other entities from the history, and a distributed price adjustment algorithm is required which would gradually reach the solution.

**Distributed Implementation of Market-Equilibrium Dynamic Pricing**: The solution of this pricing model is given by market-equilibrium which is defined as the price at which spectrum supply from a primary service is equal to spectrum demand from the secondary service. Therefore, the price offered by each primary service is gradually adjusted in a direction that minimizes the difference between spectrum demand and supply. This process works as follows. The spectrum price is initialized to *p*_{i}[0] and this price is sent to the secondary service. The secondary service replies with the size of spectrum demand which is computed from the demand function *D*_{i}(**p**[*t*]) for spectrum *F*_{i}. Then, the primary service computes the size of the supplied spectrum *S*_{i}(*p*_{i}[*t*]). To obtain the price in the next iteration, the difference between spectrum demand and supply at time *t* is computed, weighted by learning rate *α*_{i}, and added to the price in the current iteration. This process repeats until the difference of prices in current iteration *t* and next iteration *t* + 1 becomes less than the threshold *ε* (e.g., *ε* = 10^{−5}). This price adjustment in each iteration can be expressed as: *p*_{i}[*t* + 1] = *p*_{i}[*t*] + *α*_{i}(*D*_{i}(**p**[*t*]) − *S*_{i}(*p*_{i}[*t*])), where **p**[*t*] is the vector of prices at iteration *t*, i.e., **p**[*t*] = [*p*_{1}[*t*]⋯*p*_{i}[*t*]⋯*p*_{N}[*t*]]^{T}.

**Distributed Implementation of Competitive Dynamic Pricing**: The solution of this dynamic pricing model is given by the Nash equilibrium. It is defined as the prices which satisfies all of the primary services. We assume that a primary service cannot observe the prices of other primary services. Therefore, each primary service can use only local information and spectrum demand information from the secondary service to adjust its strategy. This distributed competitive dynamic pricing works as follows. The spectrum price is initialized to

*p*

_{i}[0] and this price is sent to the secondary service. The secondary service replies with the size of spectrum demand. Also, the primary service estimates marginal individual profit and uses this together with spectrum demand information to compute the spectrum price in the next iteration. The relationship between the prices in the current and the next iterations can be expressed as: \( {p}_i\left[t+1\right]={p}_i\left[t\right]+{\alpha}_i\left(\frac{\partial {P}_i\left(\mathbf{p}\left[t\right]\right)}{\partial {p}_i\left[t\right]}\right) \), where

*α*

_{i}is the learning rate. Let

**p**

_{−i}[

*t*] denote the vector of prices of all primary services except service

*i*at iteration

*t*. To estimate the marginal individual profit, a primary service can observe the marginal spectrum demand for a small variation in price

*ξ*(e.g.,

*ξ*= 10

^{−4}). That is,

**Distributed Implementation of Cooperative Dynamic Pricing**: The solution of this dynamic pricing model yields an optimal price for which the total profit of all primary services is maximized. Again, we assume that a primary service can observe the variation of spectrum demand from the secondary service. In addition, to achieve the highest total profit, primary services can exchange information on current profit among each other. This is possible only if all the primary services are fully cooperative. The distributed cooperative dynamic pricing then works as follows. The spectrum price is initialized to

*p*

_{i}[0] and then it is sent to the secondary service. The secondary service replies with the spectrum demand. Then, the primary service estimates marginal total profit by exchanging information with the other primary services and uses this together with the spectrum demand from secondary service to compute spectrum price in the next iteration. The price update is done as: \( {p}_i\left[t+1\right]={p}_i\left[t\right]+{\alpha}_i\left(\frac{\partial {\sum}_{j=1}^N{P}_j\left(\mathbf{p}\left[t\right]\right)}{\partial {p}_i\left[t\right]}\right) \). Again, similar to that in Eq. (5), to estimate marginal total profit, a primary service can observe the marginal total profit through information exchange among all primary services for a small variation in price

*ξ*as:

Compared with the market-equilibrium and competitive dynamic pricing models, this estimation of marginal total profit incurs the largest communication overhead since the profit information needs to be exchanged among all the primary services.

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