# Implications for Firms with Limited Information to Take Advantage of Reference Price Effect in Competitive Settings

Authors: Junhai Ma; Zhanbing Guo
Journal: Complexity
 (2017)

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## Abstract

This paper studies internal reference price effects when competitive firms face reference price effects and make decisions based on partial information, where their decision-making mechanism is modeled by a dynamic adjustment process. It is shown that the evolution of this dynamic adjustment goes to stabilization if both adjustment speeds are small and the complexity of this evolution increases in adjustment speeds. It is proved that the necessary condition for flip bifurcation or Neimark-Sacker bifurcation will occur with the increase of adjustment speed in two special cases. What is more, numerical simulations show that these bifurcations do occur. Then, the impacts of parameters on stability and profits are investigated and some management insights for firms with limited information to take advantage of reference price effects are provided.


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## Body

## 1. Introduction

Reference price plays an important role in consumer purchase decisions. In fact, consumers’ past experiences contribute to the building of reference price. Consumers will compare this reference price with the current price and decide the current demand. The theoretical basis for reference price effect is the adaptation-level theory [1]. Adaptation-level theory holds that the expectation-based reference price is the adaptation level, which the current price is judged against [2]. In particular, if reference price is higher than the current price, the current price will be perceived as “low” and then consumers will be prone to buy more and demand increases. On the contrary, if reference price is lower than the current price, the current price will be perceived as “high” and then demand decreases. Nasiry and Popescu [3] provided an example to illustrate this effect. Apple used to sell digital songs at a low price (99 cents) for a long time. When the company raised these prices to $1.29, the drop in sales was higher than expected. These lost sales were explained by the fact that the new price was perceived as very high when compared to the benchmark (99 cents), so demand dropped significantly.Due to the significant effect of reference price, researchers have provided many construction models for reference price. Kalyanaram and Winer [4] divided reference price into two types: internal reference price and external reference price. Internal reference price of one type of goods is related to its historic prices [5]; customers’ memories of historic prices form the benchmark (internal reference price). External reference price refers to an external standard, which refers to contextual factors, such as prices of competing products and price of the same product in a different location [6]. In addition to purchase frequency, brand loyalty, and involvement levels, Chen [7] demonstrated that self-construal influences the impact of reference cues. Besides the above factors, advertising is often used to influence reference price as an operational management tool [8, 9].Based on this comprehension, scholars have provided various implications for firms to take advantage of reference price effect [10–17]. These studies provided critical implications for firms to maximize their revenues via making use of reference price effect. However, in these models, firms are assumed to be fully rational; that is, firms not only master all necessary information but also choose the best solution determinately. In fact, there are many decision-makers, who face complex environments, and full information is very costly, even impossible, to master for them. Then, some bounded-rationality based decision-making mechanisms are adopted by these decision-makers [18, 19]. However, implications for these firms to take advantage of reference price effect are still rare.In this paper, we try to explore some useful implications for firms with limited information to take advantage of reference price effect in competitive settings. We consider a market with two competitive firms; they sell similar products to customers and compete in price. As the full market information is very difficult and costly to master, they make decisions based on partial information. We assume firms make decisions based on marginal profit, which can be estimated by market experiments. When marginal profit is positive, the firm should increase its product’s price. Conversely, negative marginal profit leads to the decrease of price. We assume that firms estimate the marginal profits by a weighted mean value of past few prices rather than instant price to smooth possible deviations. Although we omit estimation deviation in the model, we also use this mechanism for the following reasons. First, weighted moving average is a widely used estimation method [20, 21]. Second, as will be shown in the following text, adopting an appropriate weight parameter, which belongs to 0.56,0.75, always benefits the stability of the system. Based on the nonlinear dynamic system theory, we study the evolution characteristics of this market and investigate the impacts of relevant factors to explore some managerial implications.This study provides the following implications for firms with limited information to take advantage of reference price effect in competitive settings. First, there is a stability condition which must be satisfied to keep the dynamic system asymptotically stable. And firms get their maximal profits when the system is stable; the period of pricing cycle when the system loses its stability depends on the weight parameter and customer perception coefficient. Second, when customers use the most recent price as reference price, there is a threshold, which depends on customer perception coefficient and belongs to0.56,0.75. The stable region of adjustment speeds increases in weight parameter when the weight parameter is less than this threshold and decreases in weight parameter when the weight parameter is greater than this threshold. What is more, if firms use instant prices to estimate marginal profits, the stable region of adjustment speeds increases in memory parameter. Third, a higher initial reference price relative to the equilibrium price generally benefits both firms when the adjustment process is asymptotically stable. However, this does not hold when the system is unstable, and firms should not improve initial reference prices blindly in this situation.This paper is organized as follows. Section2 introduces the related literature. Section 3 presents the dynamic game model, where firms face reference price effects and make decisions based on partial information. Management insights as well as main results from theoretical analysis and numerical observation are provided in Section 4. Finally, conclusions are given in Section 5.


## 2. Related Literature

This paper is related to the literature on dynamic pricing with reference price effect and implications on taking advantage of reference price effect. The earliest study analyzing the impact of reference prices on pricing strategies was [10]. The author showed that a firm can increase its profit by considering reference price effects. After that, the impacts of reference price effects in various settings have been investigated. Fibich et al. [11] studied a continuous time and asymmetric reference effects model; they showed that the optimal price stabilizes at a steady-state price, which is below the optimal price without reference price effects. Popescu and Wu [12] studied the dynamic pricing problem of a monopolist firm under a discrete time model; they showed that if consumers are loss averse, optimal prices will converge to a constant steady-state price, otherwise, optimal policy cycles. Martín-Herrán and Taboubi [13] investigated whether the results on the efficiency of price coordination in bilateral monopolies still hold when the reference price effect is taken into account. They showed that, for some values of the initial reference price, there is a time interval where channel decentralization performs better than coordination. Zhang et al. [14] studied the dynamic pricing policy for supply chain firms under integrated channels and decentralized channels; they showed that the initial reference price of consumers plays an important role in determining whether price skimming policy or price penetration policy is more profitable. What is more, key parameters effects are also provided. Lin [16] studied the impact of price promotion in a supply chain with reference price effect. It is shown that the reference price effect could mitigate double marginalization effect and improve channel efficiency. This helps us to understand the fact that firms encourage consumers to recall reference price. Hu et al. [22] considered a gain-seeking reference price effect model and identified conditions on parameters such that a high-low pricing strategy is optimal. Besides these studies, some valuable insights are also reached by integrating pricing with other decision variables, such as advertising strategy [8, 23], replenishment policies [24, 25], and preservation technology investment [26]. Our study is significantly different from aforementioned papers in that we consider firms that only master partial information. We model the decision-making mechanism of these firms by a dynamic adjustment based on partial information, and then we explore managerial implications for these firms to take advantage of the reference price effect.Our work is also closely related to the research on the decision-making mechanism of bounded rational players [27–29]. Among this stream of literature, these studies that investigate the dynamic adjustment of players with limited information in competitive settings are most close to our study. Wu et al. [30], Ma and Xie [31], and Elsadany [32] studied the evolution processes when the bounded rational players use naive expectation. Considering that players are heterogeneous, some scholars studied the influence of players’ different adjustment mechanisms on the dynamics of game model (see [33–37]). Some papers extended the decision mechanisms by considering that firms may refer to more than one period when they make decisions, and then delay decision was utilized (see [38–41]). The impacts of players’ exact estimation on the evolutions of the dynamic system were considered in [42, 43]. Besides the above researches, Ma and Guo [44] studied the dynamic competition when one player adopts “one-period look-ahead” behavior. Ahmed et al. [45] and Agliari et al. [46] investigated the dynamic game process when bounded rational firms apply the gradient adjustment mechanism. Although the reference price effect does exist and play an important role in consumer purchase decisions, as far as we know, implication on how to take advantage of reference price effect in competitive settings has never been investigated for decision-makers with partial information, so we build this model to explore some useful implications.


## 3. Model

In this section, we set up a dynamic system model to represent the evolution process of prices and reference prices. We assume there are two firms (two players) in the market, labeled byi=1,2, respectively (we use i to denote both 1 and 2 unless otherwise specified). They offer similar products (firm i offers product i) and form a price game. Firms make decisions in discrete time period t,t=1,2,3,…. Their demand functions in the absence of reference price effect are (1)D^it=ai-bipit+dip3-it,where ai,bi,di>0, and bi>di. Traditionally, ai is viewed as the size of the market base of product i, bi denotes the influence of its own price, and di denotes the influence of the substitute’s price. We consider fixed unit-production cost ci and assume that ai-bici+dic3-i>0.Reference price affects customer demand via the magnitude of perceived “gain” or “loss” relative to the reference point [25]. In this paper, we adopt the linear symmetric reference price effect model: Rr-p,r=kr-p. Then, the demand functions in the presence of reference price effects are modeled as(2)Dit=ai-bipi+dip3-i+kiri-pi,where ki is used to represent the effect of the perceived “gain” or “loss.” We follow the notion given in [47] that the effect of unit perceived “gain” or “loss” on demand is lower than that of unit real “gain” or “loss,” so we define ki=θbi for simplicity, where θ∈0,1. Profit functions are(3)πit=pit-ciai-bipi+dip3-i+θbiri-pi.To model the evolution of reference prices, we adopt an exponential smoothing model. The exponential smoothing model, stemming from the adaptive expectation model, is the most commonly used updated model for reference price (see [12, 13, 25, 48]):(4)rit+1=αrit+1-αpit,where rit is the reference price at time t, pit is the actual price at time t, and α∈0,1 denotes the memory parameter. Memory parameter α captures the strength of past prices which the reference depends on. α is high when customers have a long memory. α is small when customers pay less attention to past prices. If α=0, the reference price becomes the last period’s price; this means customers only have one-period memory.Given the initial reference prices, the long-term profit maximization problem of each firm is(5)Vi=maxpit⁡∑t=1∞ρt-1πirit,pit,pjt,where there is a discount rate of future profit when ρ∈0,1; discount rate is not considered when ρ=1. The updated law of reference prices is (4). pit and pjt are mutually dependent.Solving the optimal strategy for each firm not only is very complicated but also needs complete information about the whole market. This condition is very tough in the real market. Usually, firms only master limited information and have to make decisions based on limited information. To model the decision-making processes of firms with limited information, in this paper, we adopt the widely used gradient mechanism, which provides a good approximation to the practical adjustment when only marginal profit is available. The gradient mechanism assumes that each firm gets its marginal profit with respect to its price via market experiments in each period:(6)∂πip1t,p2t∂pit=ai-2bipit1+θ+dip3-it+θbirit+1+θbici.Based on marginal profit, each makes a myopic decision to maximize its profit in the next period. If this marginal profit is positive (negative), firm i will increase (decrease) price i in the next period (see [31, 32, 49]):(7)pit+1=pit+αipit∂πip1t,p2t∂pit=pit+αipitai-2bipit1+θ+dip3-it+θbirit+1+θbici,where αi>0 is the speed of adjustment.We notice that firms may estimate the marginal profit by a weighted mean value of past few prices rather than instant price to smooth possible deviations (also cited as delay decision; see [38–41]). We include this mechanism here and consider one-period delay decision for simplification:(8)pit+1=pit+αipit∂πip1Dt,p2Dt∂pit=pit+αipitai-2bi1+θwipit+1-wipit-1+diw3-ip3-it+1-w3-ip3-it-1+θbirit+1+θbici,where piDt=wipit+1-wipit-1 and wi(wi∈0,1) is the weight parameter or delay parameter. wi≠1 means firm i adopts one-period delay decision; otherwise, firm i does not adopt delay decision. Then, the dynamic game process of two myopic firms can be modeled as the following dynamic system:(9)p1t+1=p1t+α1p1ta1-2b11+θw1p1t+1-w1x1t+d1w2p2t+1-w2x2t+θb1r1t+1+θb1c1,p2t+1=p2t+α2p2ta2-2b21+θw2p2t+1-w2x2t+d2w1p1t+1-w1x1t+θb2r2t+1+θb2c2,r1t+1=αr1t+1-αp1t,r2t+1=αr2t+1-αp2t,x1t+1=p1t,x2t+1=p2t.


## 4. Main Results

The main goal of this section is to study the evolution characteristics of system (9) under different settings and provide some management insights for firms to make use of the reference price effect.


### 4.1. Equilibrium Point and Stability Condition

By settingpit+1=pit,rit+1=rit, and xit+1=xit, we can get the fixed points of dynamic system (9). As the boundary fixed points (pi=0) are meaningless, we only consider the Nash equilibrium point: E∗=p1∗,p2∗,r1∗,r2∗,x1∗,x2∗, where r1∗=x1∗=p1∗,r2∗=x2∗=p2∗, p1∗=2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2/4b1b2-d1d2+4b1b2θ+b1b2θ2,  and p2∗=2a2b1+2b1b2c2+a1d2+b1c1d2+a2b1θ+3b1b2c2θ+b1c1d2θ+b1b2c2θ2/4b1b2-d1d2+4b1b2θ+b1b2θ2.To investigate the local stability ofE∗, we calculate its linear approximation. The Jacobian matrix of system (9) is(10)J=J11α2d2p2w11-α010α1d1p1w2J2201-α01α1d1p1θ0α0000α2b2p2θ0α00-2α1b1p11-w11+θα2d2p21-w10000α1d1p11-w2-2α2b2p21-w21+θ0000T,where J11=1-2α1b1p1w11+θ+α1a1+d1p2w2+x2-w2x2+b1r1θ+b1c11+θ-2b1p1w1+x1-w1x11+θ, and J22=1-2α2b2p2w21+θ+α2a2+d2p1w1+1-w1x1+b2r2θ+b2c21+θ-2b2p2w2+1-w2x21+θ.Then, the stability ofE∗ can be judged by the eigenvalues of J at point E∗.The characteristic polynomial of (10) takes this form: fλ=λ6+μ1λ5+μ2λ4+μ3λ3+μ4λ2+μ5λ+μ6=0. According to Jury stability criterion, system (9) is asymptotically stable at E∗ if the following condition is satisfied (see [36]):(11)1+μ1+μ2+μ3+μ4+μ5+μ6>01-μ1+μ2-μ3+μ4-μ5+μ6>01-μ6>0,φ0-φ5>0,γ0-γ4>0ν0-ν3>0,ε0-ε2>0,where(12)φ0=μ62-1,φ1=μ5μ6-μ1,φ2=μ4μ6-μ2,φ3=μ3μ6-μ3,φ4=μ2μ6-μ4,φ5=μ1μ6-μ5,γ0=φ02-φ52,γ1=φ0φ1-φ4φ5,γ2=φ0φ2-φ3φ5,γ3=φ0φ3-φ2φ5,γ4=φ0φ4-φ1φ5,ν0=γ02-γ42,ν1=γ0γ1-γ3γ4,ν2=γ0γ2-γ2γ4,ν3=γ0γ3-γ1γ4,ε0=ν02-ν32,ε1=ν0ν1-ν2ν3,ε2=ν0ν2-ν1ν3.


### 4.2. The Dynamic Features with respect to Adjustment Speedαi

Adjustment speed is a parameter reflecting the character of the decision-maker. A radical firm prefers a big adjustment speed with expecting that its profit can increase quickly. A conservative player is more likely to adopt a small adjustment speed to reduce risk. By analyzing the influence of adjustment speeds, we can get the following proposition.Proposition 1.
Nash equilibrium point of system (9) is asymptotically stable when the adjustment speeds of firms are very small.Proof.
First, let us look at criterion (11) when α1=0 and α2=0; then, the subconditions of criterion (11) are (13)1+μ1+μ2+μ3+μ4+μ5+μ6=0,1-μ1+μ2-μ3+μ4-μ5+μ6=41+α2>0,1-μ6=1>0,φ0-φ5=1>0,γ0-γ4=1-α2>0,ν0-ν3=1-α31+α>0,ε0-ε2=0.As all subconditions are continuous in α1 and α2, all subconditions, except the first and the last one, are still positive when both α1 and α2 are very small. Then, we consider whether the first one and last one hold for small adjustment speeds. Define g1=1+μ1+μ2+μ3+μ4+μ5+μ6 and g2=ε0-ε2=ε0-ε2, where ε0=ε2=α2-14>0 when α1=0 and α2=0. We can get their linear approximations:(14)g1α1,α2=g10,0+∂g10,0∂α1α1+∂g10,0∂α2α2+12∂g120,0∂α12α12+12∂g120,0∂α22α22+∂g120,0∂α1∂α2α1α2=1b1b22+θ2-d1d21-α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ∗a1d2+a2b12+θ+b11+θc1d2+b2c22+θα1α2+oα12+α22,g2α1,α2=g20,0+∂g20,0∂α1α1+∂g20,0∂α2α2+oα12+α22=b12+θ1-α51+α+α2+α3a1b22+θ+a2d1+b21+θc2d1+b1c12+θb1b22+θ2-d1d2α1+b22+θ1-α51+α+α2+α3a1d2+a2b12+θ+b11+θc1d2+b2c22+θb1b22+θ2-d1d2α2+oα12+α22.It is easy to see that g1>0 and g2>0 for very small adjustment speeds. So, the Nash equilibrium point is asymptotically stable.Now, we consider two special cases for system (9).Assumption 2.
Consumers only remember the most recent price in the reference price model (α=0).Assumption 3.
Firms only consider the most recent price in the decision-making mechanism; that is, delay decision is not adopted (wi=1).Although these two assumptions seem restrictive, they can provide good approximation to some practical scenarios [22]. We use Fλ=λ4+ξ1λ3+ξ2λ2+ξ3λ+ξ4 to denote the characteristic polynomial of four-dimensional systems. Then, the Jury condition is(15)f1=1+ξ1+ξ2+ξ3+ξ4>0,f2=1-ξ1+ξ2-ξ3+ξ4>0,f3=1-ξ4>0,f4=ψ1-ψ4>0,f5=χ1-χ3>0,where ψ1=1-ξ42, ψ2=ξ1-ξ3ξ4, ψ3=ξ2-ξ2ξ4, ψ4=ξ3-ξ1ξ4, χ1=ψ12-ψ42, χ2=ψ1ψ2-ψ3ψ4, χ3=ψ1ψ3-ψ2ψ4.Under Assumption2, we can get the following proposition.Proposition 4.
When one adjustment speed is very small, Jury stability condition (15) will be violated with the increase of the other adjustment speed. If 8w11+θ-32+θ>0, the violation happens with one eigenvalue being minus one. If 8w11+θ-32+θ<0, the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.Proof.
We assume one adjustment speed is very small (e.g.,α2). Then, the Jury stability condition is (16)f1=1b1b22+θ2-d1d2α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ>0,f2=-2b1b22+θ2-d1d22d1d2+α1b12b2c12+3θ+θ24w11+θ-2-θ+b1a2α1d14w11+θ-2-θ+b2α1c2d11+θ-2-θ+4w11+θ-22+θ2+a1α12+θ4w11+θ-2-θ+o1,∂f2∂α1=-2b14w11+θ-2-θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θb1b22+θ2-d1d2+o1,ξ4=1d1d2-b1b22+θ22α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θd1d2w1-1+w2-w1w2+b1b22w11+θ-2-θ2w21+θ-2-θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ,f3=1-ξ4>0alwaysholds,ψ1=1+o1,ψ4=α1b12w11+θ-2-θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θb1b22+θ2-d1d2+o1.If 2w11+θ-2-θ>0, ψ4 is positive and increases in α1, and if 2w11+θ-2-θ<0, ψ4 is negative and decreases in α1. So, f4 always decreases in α1.
We now investigatef5. χ3 is a quadratic function of α1 and χ3=1 when α1=0. Solving χ3=0, we can get two solutions:(17)α1∗=4b1b2+4b1b2θ+b1b2θ2-d1d2b12+θ-2w11+θ2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2+o1,α1∗∗=4b1b2+4b1b2θ+b1b2θ2-d1d22b1w11+θ2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2+o1.
Case
1 (2+θ-2w11+θ<0). χ3=0 has only one positive solution α1∗∗ and χ3>0 (χ3<0) when 0<α1<α1∗∗  (α1>α1∗∗). Then, solving f5=χ1-χ3=0, we can get α~1=24b1b2+4b1b2θ+b1b2θ2-d1d2/(b14w11+θ-2-θ(2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ + b2c2d1θ+b1b2c1θ2))+o1>α1∗∗. f2 decreases in α1, and f2=o1 when α1=α~1. What is more, f4>0 (0<ψ4=4w11+θ-22+θ/4w11+θ-2+θ+o1<1) when α1=α~1. We can get that all subconditions are satisfied when 0<α1<α~1-ε, where ε is an appropriately small positive constant. When α1=α~1-ε, the characteristic polynomial has one eigenvalue close to but less than +1, one eigenvalue close to but bigger than -1, and two eigenvalues with the product of their moduli less than 1. If the characteristic polynomial has a pair of conjugate complex numbers, the moduli of conjugate complex numbers are less than one. When α1 increased from α1∗-ε to α1∗+ε, f2 changed from positive to negative; then, a single eigenvalue passes through -1 with other eigenvalues still inside the unit circle.
Case
2 (2+θ-2w11+θ>0 and 8w11+θ-32+θ>0). χ3=0 has two positive solutions α1∗ and α1∗∗ and α1∗>α1∗∗. Then, we can get that f5=χ1-χ3=0 has two solutions α1∗ and α~1 and α1∗>α~1>α1∗∗. Then, Case 2 is similar to Case 1.
Case
3
(
4
w
1
1
+
θ
-
2
-
θ
>
0 and 8w11+θ-32+θ<0). χ3=0 has two positive solutions α1∗ and α1∗∗, and α1∗>α1∗∗. Then, we can get that f5=χ1-χ3=0 has only one positive solution α1∗, ∂f5/∂α1α1=α^1+0=-b12+θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ/(b1b22+θ2-d1d2)+o(1)<0. ψ4 decreases in α1 and ψ4=-1+o1(f4=o1) when α1=α1∗. What is more, f2 decreases in α1, f2=-28w11+θ-32+θ/2+θ-2w11+θ+o1>0 when α1=α1∗. Above all, all subconditions are satisfied when α1∈0,α1∗-ε. When α1 increased from α1∗-ε to α1∗+ε, the Jury stability condition was violated with f1>0 and f2>0, so a pair of conjugate complex eigenvalues occur with their moduli equal to one.
Case
4
(
4
w
1
1
+
θ
-
2
-
θ
<
0
). χ3=0 has two positive solutions α1∗ and α1∗∗ and α1∗<α1∗∗. Then, Case 4 is similar to Case 3.
Above all, if8w11+θ-32+θ>0, the violation happens with one eigenvalue being minus one. If 8w11+θ-32+θ<0, the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.Under Assumption3, the following proposition holds.Proposition 5.
For system (9) without delay decision wi=1, if the adjustment speed of one firm is very small, Jury stability condition (15) can be violated by one eigenvalue passing through minus one.Proof.
When delay decision is not considered, system (9) turns to a four-dimensional system. We investigate the case when the adjustment speed of one firm (e.g., firm 2) is a very small positive constant: (18)f1=1b1b22+θ2-d1d2α-12α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ>0,f2=21+α8b1b2θ+2b1b2θ2+2αb1b22+θ2+8b1b2-21+αd1d2b1b22+θ2-d1d2-21+αα1b1b22+θ2-d1d22b1b2c2d1+5b1b2c2d1θ+3b1b2c2d1θ2+αb1b2c2d11+θ2+θ+a1αb1b22+θ2+a1b1b24+8θ+3θ2+a2b1d12+3θ+α2+θ+b12b2c12+3θ+θ22+3θ+α2+θ+o1. We can see that f2 decreases in α1. Solving f2=0 gives (19)α1=1b12+3θ+α2+θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ∗21+αb1b22+θ2-d1d2+o1≜α^1. What is more, when α1=α^1, ξ4=-α2θ+α2+θ+α22+θ/2+3θ+α2+θ+o1, (20)f3=1-ξ4≈1-α2+3θ+2α2+θ+α22+θ2+3θ+α2+θ>0,f4≈1-α31+α2+θ2+3θ+2α2+θ+α22+θ2+3θ+α2+θ2>0.f5=χ1-χ3=χ1+χ3=0,∂f5∂α1α1=α^1<0.
Assume thatε is an appropriately small positive constant. When α1=α^1-ε, the Jury condition is satisfied. Then, the characteristic polynomial has one eigenvalue close to but less than +1, one eigenvalue close to but bigger than -1, and two eigenvalues with the product of their moduli less than 1. If the characteristic polynomial has conjugate complex numbers, the moduli of conjugate complex numbers are less than one. As the moduli of eigenvalues are continuous in α1, when α1 increases from α^1-ε to α^1+ε, f2 changes from positive to negative, and then a single eigenvalue passes through minus one with other eigenvalues still inside the unit circle.Although a single eigenvalue becoming minus one and the modulus of a pair of conjugate complex eigenvalues being equal to one are necessary conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation, respectively, they constitute strong evidence combined with numerical simulations which show that such bifurcations do occur [50]. In the following text, we will resort to some numerical simulations to demonstrate these propositions given above and explore more useful results. To do the simulations, we take the basic parameter values as follows: a1=4.8,a2=4.5,b1=3.5,b2=3.3,c1=0.01,c2=0.01,d1=2.1, and  d2=2.2, and then we can get p1∗=0.734 and p2∗=0.747.Figure1 shows the stable region of adjustment speeds in system (9). Figure 2 gives the bifurcation diagram of prices with respect to α1 when other parameters are fixed, where the blue orbit represents the evolution of p1, the red orbit represents the evolution of p2, and the black orbit shows the largest Lyapunov exponent (LLE) of this system. LLE is widely used to mark chaos; it is positive when chaos occurs [51]. From Figures 1 and 2, we can see that the Nash equilibrium point is locally stable when α1 and α2 are small; this is consistent with Proposition 1. At the same time, if one firm increases its adjustment speed to a certain value, system (9) will become unstable and sink into closed invariant curve via Neimark-Sacker bifurcation. A further increase of its adjustment speed will lead to a series of period-doubling bifurcations. System (9) sinks into chaos eventually via period-doubling bifurcation.Figure 1
The stable region of adjustment speeds whenθ=0.5,α=0.5,andwi=0.5.Figure 2
The bifurcation diagram of prices and LLE with respect toα1 when θ=0.5,α=0.5,wi=0.5, and α2=0.25.We simulate the attractors of system (9) in different states and show them in Figure 3, where Figure 3(a) corresponds to the fixed point attractor, Figure 3(b) corresponds to the closed invariant curve attractor, and Figure 3(c) corresponds to the chaos attractor. We can also see that the complexity of this system increases when adjustment speeds increase.Figure 3
Attractor whenθ=0.5, α=0.5, wi=0.5, and α2=0.25: (a) α1=0.2, (b) α1=0.25, and (c) α1=0.32.
(a)
(b)
(c)Figures4 and 5 show the evolution characteristics of the system given in Propositions 4 and 5, respectively, where the first figure in Figure 4 corresponds to 8w11+θ-32+θ<0 and the second one corresponds to 8w11+θ-32+θ>0. We can see clearly from them that the corresponding bifurcations do occur in this system. What is more important is that although Propositions 4 and 5 give local bifurcation characteristics (bifurcation characteristics when one adjustment speed is very small), they also can shed light on the global bifurcation characteristics (bifurcation characteristics with any adjustment speeds). As is shown in Figures 6–8, the bifurcation type when the system loses its stability is the same for any adjustment speed in each figure, where different colors represent different periods (grey means divergence). What is more, from Figure 9, we can see that firms get their maximal profits when the system is stable.Figure 4
The bifurcation diagram ofp1 with respect to α1 when θ=0.6, α2=0.01, and α=0, where wi=0.5 in the first figure and wi=0.8 in the second figure.Figure 5
The bifurcation diagram of price and LLE with respect toα1 when θ=0.6, α=0.5, wi=1, and α2=0.01.Figure 6
Parameter basin inα1,α2 plane when θ=0.6, α=0, and wi=0.5.Figure 7
Parameter basin inα1,α2 plane when θ=0.6, α=0, and wi=0.8.Figure 8
Parameter basin inα1,α2 plane when θ=0.6, α=0.5, and wi=1.Figure 9
The average profits with respect toα1 when θ=0.5, wi=0.5, and α2=0.25.Above all, we can get that adjustment speeds have obvious impacts on the dynamics of system (9). The system is stable when both adjustment speeds are small. System (9) will lose its stability and bifurcation or even chaos may happen when adjustment speeds increase. Stabilization is a preferable state for firms both from the perspective of dynamic complexity and from the perspective of profit maximization.


### 4.3. The Impacts of Memory Parameterα and Delay Decision Coefficient w

Memory parameterα captures the strength of past prices which the reference price depends on. α is high (low) when customers have a long (short) memory. At the same time, players may adopt delay decision. In this section, the influence of α and w on the dynamics of system (9) under Assumptions 2 and 3 will be studied.Proposition 6.
α
i
∗ increases in wi, α~i decreases in wi, and α^i increases in α.As it is easy to prove Proposition6 by calculating partial derivatives ∂αi∗/∂wi,∂α~i/∂wi, and ∂α^i/∂α, we omit this proof here.Based on Figures10–12, we can get the following observations. (1) The stable region of adjustment speeds is a convex set in α1,α2 plane, so α^i defined in Proposition 5 is also the boundary values of stable region of adjustment speeds. What is more valuable is that (2) the proposition given by Proposition 6 not only applies to boundary values (when one adjustment speed is very small), but also applies to the whole stable region of adjustment speeds. Then, we can get the impacts of parameters on the whole stable region of adjustment speeds. Under Assumption 2, there is w∗=32+θ/81+θ∈0.56,0.75, and the stable region increases in w (wi=w) firstly (w<w∗) and then decreases in it (w>w∗). The stable region of adjustment speeds increases in memory parameter if delay decision is not adopted.Figure 10
The stable region of adjustment speeds whenθ=0.6, α=0, and wi=w.Figure 11
The stable region of adjustment speeds whenθ=0.6, α=0, and wi=w.Figure 12
The stable region of adjustment speeds whenθ=0.6 and wi=1.Above all, the stable region increases in memory parameter when delay decision is not adopted. Firms can expand the stable region by encouraging consumers to recall past prices in this case. When consumers only remember the most recent price, the impacts of delay decision depend on the delay parameter and customer perception coefficient. Firms should be cautious in choosing delay parameter in this situation.


### 4.4. The Influence of Customer Perception Coefficientθ

Perception coefficientθ (θ∈0,1) captures the percentage of perceived “gain” or “loss” effect in response to the real price effect. θ=0 corresponds to the demand in the absence of reference price effect; then demands only depend on current prices. The perceived “gain” or “loss” influences demand when θ≠0, and a bigger value of θ means a bigger influence of perceived “gain” or “loss” effect. We can get the following proposition on θ.Proposition 7.
The equilibrium pricepi∗ and equilibrium profit πi∗ decrease in perception coefficient θ as long as ai-bici-c3-idi>0.Proof.
As the proofs forp2∗ and π2∗ are similar to that for p1∗ and π1∗, we take p1∗ and π1∗ as an example. (21)∂p1∗∂θ=-b2c2d12d2+b1b22+θ2a1-b1c1+a1d1d2+b1d12a22+θ+b2c2θ2+θ+c1d23+2θd1d2-b1b22+θ22.It is easy to get ∂p1∗/∂θ<0 as long as a1-b1c1>0.(22)∂π1∗∂θ=-b1-d1d2+b1b22+θ23∗A∗B,where A=-b22+θa1-b1c1-d1a2+c1d2+b2c21+θ<0 and B=[-4b1b2d1a2-b2c2-4a1b2d1d2-a2d12d2-3b2c2d12d2  −  c1d12d22-4b1b22θa1-b1c1-c2d1-8a2b1b2d1θ-3a1b2d1d2θ  −  5b1b2c1d1d2θ-3b2c2d12d2θ-4b1b22θ2a1-b1c1-3a2b1b2d1θ2-b1b22c2d1θ2  −  3b1b2c1d1d2θ2-b1b22θ3a1-b1c1-b1b22c2d1θ3]<0, so we can get ∂π1∗/∂θ<0.We show the stable region boundary ofα1,α2 in Figure 13. From Figure 13, we can see that the stable region of α1,α2 increases in θ. That is to say, a large perception coefficient of customer is beneficial to the stability of system (9), and the system is more likely to be stable when customers pay more attention to the perceived “gain” or “loss.” Figure 14 shows the bifurcation diagram of p1 and r1 with respect to θ. From Figure 14, we can get that the system turns to a stable state when the value of θ is increased to a certain value. Figure 15 gives firms’ average profits corresponding to Figure 14. We can know that firms’ profits decrease in θ under period 4 status and stabilization status, but profits are not monotonous under the closed invariant curve status. They firstly increase in θ and then decrease in θ after getting their maximal values. What is more, they get their global maximal values in closed invariant curve status with θ≠0.Figure 13
The stable region of adjustment speeds whenα=0.5 and wi=0.5.Figure 14
The bifurcation diagram ofp1 and r1 with respect to θ when α=0.5, wi=0.5, and α1=α2=0.25.Figure 15
The average profits with respect toθ when α=0.5, wi=0.5, and α1=α2=0.25.Above all, equilibrium price and equilibrium profit decrease in customers’ perception coefficient, but the stable region of adjustment speeds increases in it. Compared with the state without reference price effect (θ=0), customer perception coefficient θ≠0 is disadvantageous to firms’ profits when Nash equilibrium is asymptotically stable. But it can be beneficial to profits when Nash equilibrium is unstable. So, when the evolution is stable, firms should try to attenuate customer perception coefficient. But when the system is unstable, firms should try to keep a suitable customer perception coefficient.


### 4.5. The Influence of Initial Reference Price

Reference price is determined by customers but also can be influenced by firms’ activities, such as advertising. Reference price in advertisements (advertised reference price) can draw consumers’ attention and influence consumers’ product evaluations [52]. In this section, we will study the influence of initial reference price on profits and provide some managerial insights about advertising. As the attractor is independent of initial states if initial states belong to the basin of attraction of the same attractor, we only need to consider this influence before system (9) gets its final state (attractor).Figure16 shows players’ average profits of the first 100 periods with respect to initial pricing p11 and initial reference price r11 when Nash equilibrium point is asymptotically stable, where x11=p11, x21=p21=0.7, r21=1, θ=0.5, α=0.5, wi=0.5, and αi=0.2. Figures 17 and 18 show the same content as Figure 16 when system (9) is in closed invariant curve state (Figure 17) and chaos state (Figure 18), respectively, where parameters are the same as those in Figure 16 except for αi=0.25 (in Figure 17) and α1=0.32 and α2=0.25 (in Figure 18). Table 1 shows the maximal profits and their locations for Figures 16–18.Table 1
Maximal profits and their locations.
Figure
Figure16(a)
Figure16(b)
Figure17(a)
Figure17(b)
Figure18(a)
Figure18(b)
r1
3.84
2.98
1.14
1.06
0.81
0.81
p1
1
1
0.73
0.71
0.7
0.7
πi
2.838
2.708
2.762
2.698
2.35
2.434Figure 16
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, and α1=α2=0.2.
(a)
(b)Figure 17
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, and α1=α2=0.25.
(a)
(b)Figure 18
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, α1=0.32, and α2=0.25.
(a)
(b)We can know that when the Nash equilibrium point is asymptotically stable, (1) profits get their maximal values when both initial pricing and initial reference price are obviously bigger than the equilibrium price. In this situation, advertising is generally beneficial to both firms’ profits if the advertising cost is very small compared with the selling profit. But when the system is in closed invariant curve state or chaos state, (2) firms’ profits get their peak values when initial pricing and initial reference price are close to equilibrium point. A higher initial reference price relative to equilibrium price, despite improving the current profit, also enlarges the vibration of the evolution process.


## 4.1. Equilibrium Point and Stability Condition

By settingpit+1=pit,rit+1=rit, and xit+1=xit, we can get the fixed points of dynamic system (9). As the boundary fixed points (pi=0) are meaningless, we only consider the Nash equilibrium point: E∗=p1∗,p2∗,r1∗,r2∗,x1∗,x2∗, where r1∗=x1∗=p1∗,r2∗=x2∗=p2∗, p1∗=2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2/4b1b2-d1d2+4b1b2θ+b1b2θ2,  and p2∗=2a2b1+2b1b2c2+a1d2+b1c1d2+a2b1θ+3b1b2c2θ+b1c1d2θ+b1b2c2θ2/4b1b2-d1d2+4b1b2θ+b1b2θ2.To investigate the local stability ofE∗, we calculate its linear approximation. The Jacobian matrix of system (9) is(10)J=J11α2d2p2w11-α010α1d1p1w2J2201-α01α1d1p1θ0α0000α2b2p2θ0α00-2α1b1p11-w11+θα2d2p21-w10000α1d1p11-w2-2α2b2p21-w21+θ0000T,where J11=1-2α1b1p1w11+θ+α1a1+d1p2w2+x2-w2x2+b1r1θ+b1c11+θ-2b1p1w1+x1-w1x11+θ, and J22=1-2α2b2p2w21+θ+α2a2+d2p1w1+1-w1x1+b2r2θ+b2c21+θ-2b2p2w2+1-w2x21+θ.Then, the stability ofE∗ can be judged by the eigenvalues of J at point E∗.The characteristic polynomial of (10) takes this form: fλ=λ6+μ1λ5+μ2λ4+μ3λ3+μ4λ2+μ5λ+μ6=0. According to Jury stability criterion, system (9) is asymptotically stable at E∗ if the following condition is satisfied (see [36]):(11)1+μ1+μ2+μ3+μ4+μ5+μ6>01-μ1+μ2-μ3+μ4-μ5+μ6>01-μ6>0,φ0-φ5>0,γ0-γ4>0ν0-ν3>0,ε0-ε2>0,where(12)φ0=μ62-1,φ1=μ5μ6-μ1,φ2=μ4μ6-μ2,φ3=μ3μ6-μ3,φ4=μ2μ6-μ4,φ5=μ1μ6-μ5,γ0=φ02-φ52,γ1=φ0φ1-φ4φ5,γ2=φ0φ2-φ3φ5,γ3=φ0φ3-φ2φ5,γ4=φ0φ4-φ1φ5,ν0=γ02-γ42,ν1=γ0γ1-γ3γ4,ν2=γ0γ2-γ2γ4,ν3=γ0γ3-γ1γ4,ε0=ν02-ν32,ε1=ν0ν1-ν2ν3,ε2=ν0ν2-ν1ν3.


## 4.2. The Dynamic Features with respect to Adjustment Speedαi

Adjustment speed is a parameter reflecting the character of the decision-maker. A radical firm prefers a big adjustment speed with expecting that its profit can increase quickly. A conservative player is more likely to adopt a small adjustment speed to reduce risk. By analyzing the influence of adjustment speeds, we can get the following proposition.Proposition 1.
Nash equilibrium point of system (9) is asymptotically stable when the adjustment speeds of firms are very small.Proof.
First, let us look at criterion (11) when α1=0 and α2=0; then, the subconditions of criterion (11) are (13)1+μ1+μ2+μ3+μ4+μ5+μ6=0,1-μ1+μ2-μ3+μ4-μ5+μ6=41+α2>0,1-μ6=1>0,φ0-φ5=1>0,γ0-γ4=1-α2>0,ν0-ν3=1-α31+α>0,ε0-ε2=0.As all subconditions are continuous in α1 and α2, all subconditions, except the first and the last one, are still positive when both α1 and α2 are very small. Then, we consider whether the first one and last one hold for small adjustment speeds. Define g1=1+μ1+μ2+μ3+μ4+μ5+μ6 and g2=ε0-ε2=ε0-ε2, where ε0=ε2=α2-14>0 when α1=0 and α2=0. We can get their linear approximations:(14)g1α1,α2=g10,0+∂g10,0∂α1α1+∂g10,0∂α2α2+12∂g120,0∂α12α12+12∂g120,0∂α22α22+∂g120,0∂α1∂α2α1α2=1b1b22+θ2-d1d21-α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ∗a1d2+a2b12+θ+b11+θc1d2+b2c22+θα1α2+oα12+α22,g2α1,α2=g20,0+∂g20,0∂α1α1+∂g20,0∂α2α2+oα12+α22=b12+θ1-α51+α+α2+α3a1b22+θ+a2d1+b21+θc2d1+b1c12+θb1b22+θ2-d1d2α1+b22+θ1-α51+α+α2+α3a1d2+a2b12+θ+b11+θc1d2+b2c22+θb1b22+θ2-d1d2α2+oα12+α22.It is easy to see that g1>0 and g2>0 for very small adjustment speeds. So, the Nash equilibrium point is asymptotically stable.Now, we consider two special cases for system (9).Assumption 2.
Consumers only remember the most recent price in the reference price model (α=0).Assumption 3.
Firms only consider the most recent price in the decision-making mechanism; that is, delay decision is not adopted (wi=1).Although these two assumptions seem restrictive, they can provide good approximation to some practical scenarios [22]. We use Fλ=λ4+ξ1λ3+ξ2λ2+ξ3λ+ξ4 to denote the characteristic polynomial of four-dimensional systems. Then, the Jury condition is(15)f1=1+ξ1+ξ2+ξ3+ξ4>0,f2=1-ξ1+ξ2-ξ3+ξ4>0,f3=1-ξ4>0,f4=ψ1-ψ4>0,f5=χ1-χ3>0,where ψ1=1-ξ42, ψ2=ξ1-ξ3ξ4, ψ3=ξ2-ξ2ξ4, ψ4=ξ3-ξ1ξ4, χ1=ψ12-ψ42, χ2=ψ1ψ2-ψ3ψ4, χ3=ψ1ψ3-ψ2ψ4.Under Assumption2, we can get the following proposition.Proposition 4.
When one adjustment speed is very small, Jury stability condition (15) will be violated with the increase of the other adjustment speed. If 8w11+θ-32+θ>0, the violation happens with one eigenvalue being minus one. If 8w11+θ-32+θ<0, the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.Proof.
We assume one adjustment speed is very small (e.g.,α2). Then, the Jury stability condition is (16)f1=1b1b22+θ2-d1d2α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ>0,f2=-2b1b22+θ2-d1d22d1d2+α1b12b2c12+3θ+θ24w11+θ-2-θ+b1a2α1d14w11+θ-2-θ+b2α1c2d11+θ-2-θ+4w11+θ-22+θ2+a1α12+θ4w11+θ-2-θ+o1,∂f2∂α1=-2b14w11+θ-2-θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θb1b22+θ2-d1d2+o1,ξ4=1d1d2-b1b22+θ22α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θd1d2w1-1+w2-w1w2+b1b22w11+θ-2-θ2w21+θ-2-θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ,f3=1-ξ4>0alwaysholds,ψ1=1+o1,ψ4=α1b12w11+θ-2-θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θb1b22+θ2-d1d2+o1.If 2w11+θ-2-θ>0, ψ4 is positive and increases in α1, and if 2w11+θ-2-θ<0, ψ4 is negative and decreases in α1. So, f4 always decreases in α1.
We now investigatef5. χ3 is a quadratic function of α1 and χ3=1 when α1=0. Solving χ3=0, we can get two solutions:(17)α1∗=4b1b2+4b1b2θ+b1b2θ2-d1d2b12+θ-2w11+θ2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2+o1,α1∗∗=4b1b2+4b1b2θ+b1b2θ2-d1d22b1w11+θ2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ+b2c2d1θ+b1b2c1θ2+o1.
Case
1 (2+θ-2w11+θ<0). χ3=0 has only one positive solution α1∗∗ and χ3>0 (χ3<0) when 0<α1<α1∗∗  (α1>α1∗∗). Then, solving f5=χ1-χ3=0, we can get α~1=24b1b2+4b1b2θ+b1b2θ2-d1d2/(b14w11+θ-2-θ(2a1b2+2b1b2c1+a2d1+b2c2d1+a1b2θ+3b1b2c1θ + b2c2d1θ+b1b2c1θ2))+o1>α1∗∗. f2 decreases in α1, and f2=o1 when α1=α~1. What is more, f4>0 (0<ψ4=4w11+θ-22+θ/4w11+θ-2+θ+o1<1) when α1=α~1. We can get that all subconditions are satisfied when 0<α1<α~1-ε, where ε is an appropriately small positive constant. When α1=α~1-ε, the characteristic polynomial has one eigenvalue close to but less than +1, one eigenvalue close to but bigger than -1, and two eigenvalues with the product of their moduli less than 1. If the characteristic polynomial has a pair of conjugate complex numbers, the moduli of conjugate complex numbers are less than one. When α1 increased from α1∗-ε to α1∗+ε, f2 changed from positive to negative; then, a single eigenvalue passes through -1 with other eigenvalues still inside the unit circle.
Case
2 (2+θ-2w11+θ>0 and 8w11+θ-32+θ>0). χ3=0 has two positive solutions α1∗ and α1∗∗ and α1∗>α1∗∗. Then, we can get that f5=χ1-χ3=0 has two solutions α1∗ and α~1 and α1∗>α~1>α1∗∗. Then, Case 2 is similar to Case 1.
Case
3
(
4
w
1
1
+
θ
-
2
-
θ
>
0 and 8w11+θ-32+θ<0). χ3=0 has two positive solutions α1∗ and α1∗∗, and α1∗>α1∗∗. Then, we can get that f5=χ1-χ3=0 has only one positive solution α1∗, ∂f5/∂α1α1=α^1+0=-b12+θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ/(b1b22+θ2-d1d2)+o(1)<0. ψ4 decreases in α1 and ψ4=-1+o1(f4=o1) when α1=α1∗. What is more, f2 decreases in α1, f2=-28w11+θ-32+θ/2+θ-2w11+θ+o1>0 when α1=α1∗. Above all, all subconditions are satisfied when α1∈0,α1∗-ε. When α1 increased from α1∗-ε to α1∗+ε, the Jury stability condition was violated with f1>0 and f2>0, so a pair of conjugate complex eigenvalues occur with their moduli equal to one.
Case
4
(
4
w
1
1
+
θ
-
2
-
θ
<
0
). χ3=0 has two positive solutions α1∗ and α1∗∗ and α1∗<α1∗∗. Then, Case 4 is similar to Case 3.
Above all, if8w11+θ-32+θ>0, the violation happens with one eigenvalue being minus one. If 8w11+θ-32+θ<0, the violation occurs with the modulus of a pair of conjugate complex eigenvalues being one.Under Assumption3, the following proposition holds.Proposition 5.
For system (9) without delay decision wi=1, if the adjustment speed of one firm is very small, Jury stability condition (15) can be violated by one eigenvalue passing through minus one.Proof.
When delay decision is not considered, system (9) turns to a four-dimensional system. We investigate the case when the adjustment speed of one firm (e.g., firm 2) is a very small positive constant: (18)f1=1b1b22+θ2-d1d2α-12α1α2a1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θa1d2+a2b12+θ+b11+θc1d2+b2c22+θ>0,f2=21+α8b1b2θ+2b1b2θ2+2αb1b22+θ2+8b1b2-21+αd1d2b1b22+θ2-d1d2-21+αα1b1b22+θ2-d1d22b1b2c2d1+5b1b2c2d1θ+3b1b2c2d1θ2+αb1b2c2d11+θ2+θ+a1αb1b22+θ2+a1b1b24+8θ+3θ2+a2b1d12+3θ+α2+θ+b12b2c12+3θ+θ22+3θ+α2+θ+o1. We can see that f2 decreases in α1. Solving f2=0 gives (19)α1=1b12+3θ+α2+θa1b22+θ+b1b2c12+3θ+θ2+d1a2+b2c21+θ∗21+αb1b22+θ2-d1d2+o1≜α^1. What is more, when α1=α^1, ξ4=-α2θ+α2+θ+α22+θ/2+3θ+α2+θ+o1, (20)f3=1-ξ4≈1-α2+3θ+2α2+θ+α22+θ2+3θ+α2+θ>0,f4≈1-α31+α2+θ2+3θ+2α2+θ+α22+θ2+3θ+α2+θ2>0.f5=χ1-χ3=χ1+χ3=0,∂f5∂α1α1=α^1<0.
Assume thatε is an appropriately small positive constant. When α1=α^1-ε, the Jury condition is satisfied. Then, the characteristic polynomial has one eigenvalue close to but less than +1, one eigenvalue close to but bigger than -1, and two eigenvalues with the product of their moduli less than 1. If the characteristic polynomial has conjugate complex numbers, the moduli of conjugate complex numbers are less than one. As the moduli of eigenvalues are continuous in α1, when α1 increases from α^1-ε to α^1+ε, f2 changes from positive to negative, and then a single eigenvalue passes through minus one with other eigenvalues still inside the unit circle.Although a single eigenvalue becoming minus one and the modulus of a pair of conjugate complex eigenvalues being equal to one are necessary conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation, respectively, they constitute strong evidence combined with numerical simulations which show that such bifurcations do occur [50]. In the following text, we will resort to some numerical simulations to demonstrate these propositions given above and explore more useful results. To do the simulations, we take the basic parameter values as follows: a1=4.8,a2=4.5,b1=3.5,b2=3.3,c1=0.01,c2=0.01,d1=2.1, and  d2=2.2, and then we can get p1∗=0.734 and p2∗=0.747.Figure1 shows the stable region of adjustment speeds in system (9). Figure 2 gives the bifurcation diagram of prices with respect to α1 when other parameters are fixed, where the blue orbit represents the evolution of p1, the red orbit represents the evolution of p2, and the black orbit shows the largest Lyapunov exponent (LLE) of this system. LLE is widely used to mark chaos; it is positive when chaos occurs [51]. From Figures 1 and 2, we can see that the Nash equilibrium point is locally stable when α1 and α2 are small; this is consistent with Proposition 1. At the same time, if one firm increases its adjustment speed to a certain value, system (9) will become unstable and sink into closed invariant curve via Neimark-Sacker bifurcation. A further increase of its adjustment speed will lead to a series of period-doubling bifurcations. System (9) sinks into chaos eventually via period-doubling bifurcation.Figure 1
The stable region of adjustment speeds whenθ=0.5,α=0.5,andwi=0.5.Figure 2
The bifurcation diagram of prices and LLE with respect toα1 when θ=0.5,α=0.5,wi=0.5, and α2=0.25.We simulate the attractors of system (9) in different states and show them in Figure 3, where Figure 3(a) corresponds to the fixed point attractor, Figure 3(b) corresponds to the closed invariant curve attractor, and Figure 3(c) corresponds to the chaos attractor. We can also see that the complexity of this system increases when adjustment speeds increase.Figure 3
Attractor whenθ=0.5, α=0.5, wi=0.5, and α2=0.25: (a) α1=0.2, (b) α1=0.25, and (c) α1=0.32.
(a)
(b)
(c)Figures4 and 5 show the evolution characteristics of the system given in Propositions 4 and 5, respectively, where the first figure in Figure 4 corresponds to 8w11+θ-32+θ<0 and the second one corresponds to 8w11+θ-32+θ>0. We can see clearly from them that the corresponding bifurcations do occur in this system. What is more important is that although Propositions 4 and 5 give local bifurcation characteristics (bifurcation characteristics when one adjustment speed is very small), they also can shed light on the global bifurcation characteristics (bifurcation characteristics with any adjustment speeds). As is shown in Figures 6–8, the bifurcation type when the system loses its stability is the same for any adjustment speed in each figure, where different colors represent different periods (grey means divergence). What is more, from Figure 9, we can see that firms get their maximal profits when the system is stable.Figure 4
The bifurcation diagram ofp1 with respect to α1 when θ=0.6, α2=0.01, and α=0, where wi=0.5 in the first figure and wi=0.8 in the second figure.Figure 5
The bifurcation diagram of price and LLE with respect toα1 when θ=0.6, α=0.5, wi=1, and α2=0.01.Figure 6
Parameter basin inα1,α2 plane when θ=0.6, α=0, and wi=0.5.Figure 7
Parameter basin inα1,α2 plane when θ=0.6, α=0, and wi=0.8.Figure 8
Parameter basin inα1,α2 plane when θ=0.6, α=0.5, and wi=1.Figure 9
The average profits with respect toα1 when θ=0.5, wi=0.5, and α2=0.25.Above all, we can get that adjustment speeds have obvious impacts on the dynamics of system (9). The system is stable when both adjustment speeds are small. System (9) will lose its stability and bifurcation or even chaos may happen when adjustment speeds increase. Stabilization is a preferable state for firms both from the perspective of dynamic complexity and from the perspective of profit maximization.


## 4.3. The Impacts of Memory Parameterα and Delay Decision Coefficient w

Memory parameterα captures the strength of past prices which the reference price depends on. α is high (low) when customers have a long (short) memory. At the same time, players may adopt delay decision. In this section, the influence of α and w on the dynamics of system (9) under Assumptions 2 and 3 will be studied.Proposition 6.
α
i
∗ increases in wi, α~i decreases in wi, and α^i increases in α.As it is easy to prove Proposition6 by calculating partial derivatives ∂αi∗/∂wi,∂α~i/∂wi, and ∂α^i/∂α, we omit this proof here.Based on Figures10–12, we can get the following observations. (1) The stable region of adjustment speeds is a convex set in α1,α2 plane, so α^i defined in Proposition 5 is also the boundary values of stable region of adjustment speeds. What is more valuable is that (2) the proposition given by Proposition 6 not only applies to boundary values (when one adjustment speed is very small), but also applies to the whole stable region of adjustment speeds. Then, we can get the impacts of parameters on the whole stable region of adjustment speeds. Under Assumption 2, there is w∗=32+θ/81+θ∈0.56,0.75, and the stable region increases in w (wi=w) firstly (w<w∗) and then decreases in it (w>w∗). The stable region of adjustment speeds increases in memory parameter if delay decision is not adopted.Figure 10
The stable region of adjustment speeds whenθ=0.6, α=0, and wi=w.Figure 11
The stable region of adjustment speeds whenθ=0.6, α=0, and wi=w.Figure 12
The stable region of adjustment speeds whenθ=0.6 and wi=1.Above all, the stable region increases in memory parameter when delay decision is not adopted. Firms can expand the stable region by encouraging consumers to recall past prices in this case. When consumers only remember the most recent price, the impacts of delay decision depend on the delay parameter and customer perception coefficient. Firms should be cautious in choosing delay parameter in this situation.


## 4.4. The Influence of Customer Perception Coefficientθ

Perception coefficientθ (θ∈0,1) captures the percentage of perceived “gain” or “loss” effect in response to the real price effect. θ=0 corresponds to the demand in the absence of reference price effect; then demands only depend on current prices. The perceived “gain” or “loss” influences demand when θ≠0, and a bigger value of θ means a bigger influence of perceived “gain” or “loss” effect. We can get the following proposition on θ.Proposition 7.
The equilibrium pricepi∗ and equilibrium profit πi∗ decrease in perception coefficient θ as long as ai-bici-c3-idi>0.Proof.
As the proofs forp2∗ and π2∗ are similar to that for p1∗ and π1∗, we take p1∗ and π1∗ as an example. (21)∂p1∗∂θ=-b2c2d12d2+b1b22+θ2a1-b1c1+a1d1d2+b1d12a22+θ+b2c2θ2+θ+c1d23+2θd1d2-b1b22+θ22.It is easy to get ∂p1∗/∂θ<0 as long as a1-b1c1>0.(22)∂π1∗∂θ=-b1-d1d2+b1b22+θ23∗A∗B,where A=-b22+θa1-b1c1-d1a2+c1d2+b2c21+θ<0 and B=[-4b1b2d1a2-b2c2-4a1b2d1d2-a2d12d2-3b2c2d12d2  −  c1d12d22-4b1b22θa1-b1c1-c2d1-8a2b1b2d1θ-3a1b2d1d2θ  −  5b1b2c1d1d2θ-3b2c2d12d2θ-4b1b22θ2a1-b1c1-3a2b1b2d1θ2-b1b22c2d1θ2  −  3b1b2c1d1d2θ2-b1b22θ3a1-b1c1-b1b22c2d1θ3]<0, so we can get ∂π1∗/∂θ<0.We show the stable region boundary ofα1,α2 in Figure 13. From Figure 13, we can see that the stable region of α1,α2 increases in θ. That is to say, a large perception coefficient of customer is beneficial to the stability of system (9), and the system is more likely to be stable when customers pay more attention to the perceived “gain” or “loss.” Figure 14 shows the bifurcation diagram of p1 and r1 with respect to θ. From Figure 14, we can get that the system turns to a stable state when the value of θ is increased to a certain value. Figure 15 gives firms’ average profits corresponding to Figure 14. We can know that firms’ profits decrease in θ under period 4 status and stabilization status, but profits are not monotonous under the closed invariant curve status. They firstly increase in θ and then decrease in θ after getting their maximal values. What is more, they get their global maximal values in closed invariant curve status with θ≠0.Figure 13
The stable region of adjustment speeds whenα=0.5 and wi=0.5.Figure 14
The bifurcation diagram ofp1 and r1 with respect to θ when α=0.5, wi=0.5, and α1=α2=0.25.Figure 15
The average profits with respect toθ when α=0.5, wi=0.5, and α1=α2=0.25.Above all, equilibrium price and equilibrium profit decrease in customers’ perception coefficient, but the stable region of adjustment speeds increases in it. Compared with the state without reference price effect (θ=0), customer perception coefficient θ≠0 is disadvantageous to firms’ profits when Nash equilibrium is asymptotically stable. But it can be beneficial to profits when Nash equilibrium is unstable. So, when the evolution is stable, firms should try to attenuate customer perception coefficient. But when the system is unstable, firms should try to keep a suitable customer perception coefficient.


## 4.5. The Influence of Initial Reference Price

Reference price is determined by customers but also can be influenced by firms’ activities, such as advertising. Reference price in advertisements (advertised reference price) can draw consumers’ attention and influence consumers’ product evaluations [52]. In this section, we will study the influence of initial reference price on profits and provide some managerial insights about advertising. As the attractor is independent of initial states if initial states belong to the basin of attraction of the same attractor, we only need to consider this influence before system (9) gets its final state (attractor).Figure16 shows players’ average profits of the first 100 periods with respect to initial pricing p11 and initial reference price r11 when Nash equilibrium point is asymptotically stable, where x11=p11, x21=p21=0.7, r21=1, θ=0.5, α=0.5, wi=0.5, and αi=0.2. Figures 17 and 18 show the same content as Figure 16 when system (9) is in closed invariant curve state (Figure 17) and chaos state (Figure 18), respectively, where parameters are the same as those in Figure 16 except for αi=0.25 (in Figure 17) and α1=0.32 and α2=0.25 (in Figure 18). Table 1 shows the maximal profits and their locations for Figures 16–18.Table 1
Maximal profits and their locations.
Figure
Figure16(a)
Figure16(b)
Figure17(a)
Figure17(b)
Figure18(a)
Figure18(b)
r1
3.84
2.98
1.14
1.06
0.81
0.81
p1
1
1
0.73
0.71
0.7
0.7
πi
2.838
2.708
2.762
2.698
2.35
2.434Figure 16
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, and α1=α2=0.2.
(a)
(b)Figure 17
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, and α1=α2=0.25.
(a)
(b)Figure 18
The average profits with respect to initialr1 and p1 when θ=0.5, α=0.5, w1=w2=0.5, α1=0.32, and α2=0.25.
(a)
(b)We can know that when the Nash equilibrium point is asymptotically stable, (1) profits get their maximal values when both initial pricing and initial reference price are obviously bigger than the equilibrium price. In this situation, advertising is generally beneficial to both firms’ profits if the advertising cost is very small compared with the selling profit. But when the system is in closed invariant curve state or chaos state, (2) firms’ profits get their peak values when initial pricing and initial reference price are close to equilibrium point. A higher initial reference price relative to equilibrium price, despite improving the current profit, also enlarges the vibration of the evolution process.


## 5. Conclusions

This paper investigates the effect of internal reference price in competitive settings for firms that only master partial information, where reference prices are assumed to evolve according to an exponential smoothing process, and firms’ decision-making mechanism is modeled as a dynamic adjustment based on estimated marginal profits. By investigating the evolution characteristics of this dynamic adjustment and the impacts of key parameters, this study provides the following implications for firms with limited information to take advantage of reference price effect in competitive settings.First, firms get their maximal profits when this evolution converges to the Nash equilibrium point, so firms should let the stability condition be satisfied. Fortunately, it is proved that the stability condition is satisfied when firms’ adjustment speeds are small, so firms should adopt small adjustment speeds when adjusting their decisions. Second, if customers’ reference prices are the last paid prices, there is a threshold belonging to0.56,0.75. The stable region of adjustment speeds increases in delay parameter when the delay parameter is less than this threshold, and the stable region decreases in delay parameter when the delay parameter is greater than this threshold. So, firms can benefit from delay decision by adopting appropriate delay parameters, and this threshold increases when customer perception coefficient decreases. What is more, when delay decision is not adopted, the stable region of adjustment speeds increases in memory parameter, so the implication that firms should encourage consumers to recall reference price given in [16] also applies in limited information settings. A lower customer perception coefficient is beneficial when the system is stable. However, it can be harmful when the system is unstable, so firms should try to keep a suitable customer perception coefficient to maximize their profits when evolution is fluctuant. Third, a higher initial reference price (relative to equilibrium price) of one product generally benefits both firms when the Nash equilibrium point is asymptotically stable. But it also can be harmful when the Nash equilibrium point is unstable. This is an interesting result; if the adjustment process is unstable, too much advertising may be harmful even when the advertising cost is negligible. So, firms should take the potential evolution process into consideration before implementing their advertising strategies.There are several extensions deserving further investigation. First, this paper only considers the internal reference price effect; the results can be more practical if both internal reference price effect and external reference price effect are taken into consideration at the same time. Second, it is assumed that firms are myopic in this paper; reference price effects are not directly reflected in their decision-making mechanism; researchers can investigate the case where firms try to maximize the sum profits of two or more periods. Then, researchers may get some more interesting results. Third, random factors are unavoidable and play important roles in the real market. Incorporating random factors into the study of reference price effect makes implications more profound and robust, so this is another interesting and challenging research direction.



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*Source: 9712626-2017-06-13.xml*