Considering repeated interactions that take place in markets, the frequency with which each strategy (i.e., reasoning level) appears in the population is influenced by its success rate (i.e., fitness). In this section, we use evolutionary game theory (Smith and Price 1973; Weibull 1997), to study the evolutionary dynamics of reasoning levels in the population of sellers.
Given the distribution over levels of reasoning x, the frequency change \({\dot{x}}\) is given by the replicator equation (Hofbauer 1985):
$$\begin{aligned} \dot{x}_k = x_k \left[ f_k (x) - \varphi (x) \right] . \end{aligned}$$
(10)
Recall that \(x_k\) is the frequency that strategy Lk appears in the population, \(f_k\) is the fitness of Lk, and \(\varphi (x)\) is the average fitness of the population.
$$\begin{aligned} \varphi (x) = \sum _z x_z f_z(x) \end{aligned}$$
(11)
We revisit the duopoly scenario of the previous section (see Sect. 4.2) to apply the replicator equation. We compute the fitness \(f_k\) for every possible duopoly as follows:
$$\begin{aligned} f_k(x) = \sum _{z=0}^{K} x_z (p_k^* - c) s_k(<p_k^*, p_z^*>), \end{aligned}$$
(12)
where K is the highest reasoning level (here, \(K = 2\)). Figure 2 presents the replicator dynamics for the duopoly model of Sect. 4.2. Arrows at each point of the simplex show the derivative \({\dot{x}}\) (direction and magnitude). We observe that evolution favors the highest reasoning level L2, i.e., L2 has a competitive advantage.
We used the replicator equation to study the evolution over reasoning levels in the duopoly scenario of Sect. 4.2, assuming that a Lk seller believes to be facing a \(L(k-1)\) opponent seller (standard assumption of k-level reasoning). We showed that in such settings the highest reasoning level has always an evolutionary advantage since the belief is not influenced by changes in the distribution x. In addition, this result generalizes to any number of reasoning levels.
Dynamic Belief of Competition
In this section, we alter the standard assumption of k-level reasoning to a dynamic belief model that is influenced by the distribution x.
We generalize our setting to consider an oligopoly market with n sellers, and identical private costs for sellers. We consider that the belief of a Lk seller with regards to opponent levels of reasoning sellers is the real distribution x for all levels lower than k, such that \(\lambda _k = \left\langle x_0, x_1, \ldots , x_{k-1} \right\rangle \). Note that, \(\sum _{z=0}^{k-1} \lambda _k^z < 1\), since \(x_k > 0\), i.e., only lower than k levels of reasoning are included in the belief distribution of Lk. In addition, for \(x_k\) close to one (i.e., Lk dominates the population), \(\sum _{z=0}^{k-1} \lambda _k^z\) is close to zero. We define \(x_{out} = 1 - \sum _{z=0}^{k-1} \lambda _k^z\) as the probability of facing equal or higher levels of reasoning opponents. The probability \(x_{out}\) can only be computed for \(k > 0\), since L0 does not have a belief distribution. Hence, the belief of Lk becomes \(\lambda _k = \left\langle x_0, x_1, \ldots , x_{k-1}, x_{out} \right\rangle \). We interpret the probability \(x_{out}\) as the probability of competing with an unknown opponent, e.g., outside option for buyers. The opponent price associated with the probability \(x_{out}\) is denoted with \(p_{out}\). The price \(p_{out}\) can be set equal to the maximum price buyers are willing to pay to alleviate the risk of extreme prices set by dominant strategies.
Optimal Pricing and Generalized Replicator Equation
We use Eq. (5) to approximate the price of each reasoning level \(p_k^*\). Lk seller draws samples (opponent price vectors \(p_{-i}\) of length \(n-1\)) with regards to its belief \(\lambda _k\). In our experiments, the Lk best response (optimal price for k-level of reasoning) is averaged over 100 sampled opponent price vectors. More samples do not change the behavior of the simulation in experiments presented later in this paper.
Furthermore, to model innovation of strategies in the population, i.e., new sellers that enter competition or sellers that increase/decrease their level of reasoning, we use the generalized replicator equation (Hofbauer and Sigmund 1998):
$$\begin{aligned} \dot{x}_k = \sum _z \left[ x_z f_z (x) Q_{z \rightarrow k} \right] - \varphi (x) x_k, \end{aligned}$$
(13)
where \(Q_{z \rightarrow k}\) is the transition probability of an individual (from the population) from Lz to Lk (i.e., mutation probability). The fitness of Lk, \(f_k (x)\), is computed by:
$$\begin{aligned} f_k(x) = \frac{1}{M} \sum _{\mu =1}^M (p_k^* - c) s_k (\langle p_k^*, p_{z^{(1)}_\mu \sim x}^*, \ldots , p_{z^{(n-1)}_\mu \sim x}^*\rangle ), \end{aligned}$$
(14)
where each \(z^{(j)}_\mu \sim x\) are independent samples (i.e., \(n-1\) opponent prices) from the true distribution over reasoning levels x, and the fitness is averaged out of M sampled opponent price vectors. Considering that the population of sellers is finite, \({\dot{x}}\) is not deterministic for a given x, therefore computing the average fitness improves the approximation (Kemenade et al. 1998). We use \(M = 100\) for experiments presented in the remainder of this paper.
Evolution of Reasoning Levels
Figure 3 illustrates the evolution over levels of reasoning and price with regards to time t for \(c = 0.2\), \(p_0=0.9\), \(p_{out}=1\), and 10 levels of reasoning (from the lowest L0 to the highest L9, here \(K=9\)). The initial distribution \(x_0\) is set to \(\left\langle 1, 0, \ldots , 0 \right\rangle \), only L0 is present at time \(t=0\). The mutation probability is set to 0.01, where transition probabilities are uniformly distributed over all different levels, i.e., \(\sum _{z \ne k} Q_{k \rightarrow z} = 0.01 / (\text {number of levels}-1)\), and \(Q_{k \rightarrow k} = 0.99\). Stack plots placed at the top show the evolution of the distribution x over levels of reasoning, and plots at the bottom show the price evolution for \(\log (\tau ) \in \lbrace -2.7, -0.7, 0 \rbrace \). The bold dashed line shows the average cost for the buyers.
First, we discuss the case of almost perfect rationality, \(\log (\tau ) = -2.7\) (see Fig. 3, top). Given the positive mutation probability in Eq. (13), higher levels (\(L1 - K\)) of reasoning “invade” the population of L0. LK best responds to all lower levels of reasoning, thus it increases its share in x. For \(t > 50\), LK becomes dominant in the population, at the same time the frequency of reasoning levels between L0 and LK diminish in the distribution x. In addition, prices as well as the distribution x are not stable, resulting in price spikes that lead prices higher than the price \(p_0\) (\(p_0 = 0.9\)). Both price spikes and the instability in the evolution of the distribution x are caused due to: (i) the low probability for LK to compete with lower level of reasoning opponents (\(\sum _{j=0}^{K-1} x_j \approx 0.2\)), and (ii) the high probability \(x_{out}\) to face the outside option price \(p_{out}\). The level of price spikes is subject to the outside price \(p_{out}\), higher values for \(p_{out}\) result in higher spikes further away from the price \(p_0\). During price spikes, L0 benefits due to the high prices of (\(L1 - K\)) and increases its share in x. Thereafter, higher levels of reasoning (\(L1 -K\)) decrease their price in face of the increasing share of L0 in x until L0 share decreases again. This results in chaotic evolutionary dynamics while similar behavior is observed for \(\log (\tau ) < -1.7\).
We observe smoother evolutionary dynamics and lower average price for buyers for lower degrees of buyers’ rationality, more specifically, for \(\log (\tau ) > -1.7\). For instance, for \(\log (\tau ) = -0.7\) (see Fig. 3, middle), evolution reaches an equilibrium state at \(t>3\)k, where the distribution x and the prices become stable. On the contrary to the case of almost perfect rationality (see Fig. 3, top), the prices set by higher levels of reasoning (\(L1 - K\)) are lower than \(p_0\) (\(p_0 = 0.9\)), and thus the average cost for the buyers decrease. Note that, the frequency of reasoning levels between L0 and LK is not diminished as in the case of almost perfect rationality. The lower average price for buyers is a result of sustaining competition between different levels of reasoning sellers and the smoother dynamics of the evolution.
Last, we show the evolution of the distribution x and the prices when the buyers’ price selection is almost random (see Fig. 3, bottom). For \(\log (\tau ) = 0\), reasoning levels (\(L1 - K\)) share the distribution x equally, where all reasoning sellers offer prices that exceed the price of L0, \(p_0\), and the price \(p_{out}\), and therefore increase the cost for buyers.
Overall, higher degrees of buyers’ rationality yield higher average cost for buyers than lower degrees of rationality, e.g., \(\log (\tau ) = -0.7\). Furthermore, unstable evolutionary dynamics under almost perfect rationality increase prices further due to price spikes. In our experiments, we additionally used gradual updates to the prices in order to study the possibility more stable states can be reached in the evolution even in the case of perfect rationality. When gradual updates were used, results were consistent to the results presented here, however, the evolution of the distribution x was slower.
Competitive Advantage and Price
We proceed to show how the degree of buyers’ rationality affects the competition in terms of the evolutionary advantage of higher reasoning levels, the resulting prices for buyers, and the stability of the competition.
Figure 4 (left) illustrates the distribution x over levels of reasoning after 10k steps (mean of the last 100 steps) of the evolution averaged over 20 independent runs. LK is the dominant in x for almost all values of \(\tau \), i.e., \(\log (\tau ) < -\,0.25\). For \(\log (\tau ) \approx -\,0.25\), all levels L0 to LK have approximately equal shares in x. This is due to the almost equal prices reasoning levels set (similarly to the duopoly setting examined in earlier sections, see Fig. 1, left). For \(\log (\tau ) > -\,0.25\), the market is shared among levels L1 and LK, since all levels of reasoning but L0 offer very high prices to (almost) random buyers.
We further show the effect of varying degrees of rationality \(\tau \) on buyers’ cost (see Fig. 4, right). The cost is averaged over the last 100 out of 10k steps of evolution and over 20 independent evolution runs. For low \(\tau \), the average cost for buyers is marginally higher than the cost without the presence of higher than L0 reasoning levels, \(p_0 = 0.9\). This is the result of unstable competition dynamics that cause price spikes, during which prices become higher than the price of L0 strategy, \(p_0\). Recall, that \(p_{out} = 1\) alleviates the possibility of extreme prices, and thus the cost for buyers would increase further for higher \(p_{out}\) due to the increasing level of price spikes. In contrast, from \(\log (\tau ) = -\,1.7\) to \(\log (\tau ) = -\,0.2\) buyers’ cost drops below the price \(p_0 = 0.9\), this is mainly caused by the smoother behavior of evolution that converges to stable distributions and alleviate price spikes. In line with our theoretical findings in Sect. 4, we observe that there is a degree of rationality \(\log (\tau ^*) \approx -\,0.7\) that minimizes the average cost for buyers (shown in the figure by the dashed vertical line).
In the presented experiments, we demonstrated that lower degrees than almost perfect buyers’ rationality decrease the prices sellers offer to buyers during the evolution of the competition. For almost perfect buyers’ rationality, the highest reasoning level sellers exploit instances of monopoly situations and increase their prices, while under bounded buyers’ rationality competition is sustained decreasing prices for buyers. In the section that follows, we evaluate the stability of the competition with regards to the degree of buyers’ rationality.
Asymptotic Behavior of the Competition
If the dynamics were known in explicit closed form, one could apply analytical notions of stability (e.g., evolutionary stable strategies, asymptotically stable) to analyze equilibrium strategies (Smith 1972). However, given our implicit dynamics arising from system simulation (see Sect. 5.2), we need to draw on empirical means for characterizing the asymptotic behavior of the evolution. In the remainder of this section we analyze both the first-order derivative and the distribution trajectory x, and examine how the degree of buyers’ rationality influences the stability of the evolution.
First, we use the average magnitude (Euclidean norm) of the derivative of x, \(\overline{|{\dot{x}}|}\), that is shown by the solid line in Fig. 5 (left vertical axis). We compute \(\overline{|{\dot{x}}|}\) over the last 100 out of 10k steps of the evolution while results are averaged over 20 independent runs. The quantity \(\overline{|{\dot{x}}|}\) is maximum for almost perfect buyers’ rationality, specifically, \(\overline{|{\dot{x}}|} > 10^{-3},~\forall \log (\tau ) < -2\). This is in line with our observations in Fig. 3 (top), where we showed chaotic behavior in the evolution of x for a low \(\tau \) value. As \(\tau \) increases, steps in the evolution become smaller and consequently \(\overline{|{\dot{x}}|}\) decreases. For \(\log (\tau ^*) \approx -0.7\), which minimizes the average cost for buyers in Fig. 4 (right), \(\overline{|{\dot{x}}|}\) is very low (\(10^{-5}\)).
Next, we use the Euclidean distance between x and the average distribution \({\bar{x}}\), \(\overline{|x - {\bar{x}}|}\), which is shown by the dashed line in Fig. 5 (right vertical axis). The quantities \({\bar{x}}\) and \(\overline{|x - {\bar{x}}|}\) are computed over the last 100 out of 10k steps of evolution, and averaged over 20 independent runs. Similarly to \(\overline{|{\dot{x}}|}\), \(\overline{|x - {\bar{x}}|}\) decreases as \(\tau \) increases, and hence the distribution x stays closer to the average distribution \({\bar{x}}\) for bounded rational buyers.
Our results suggest that imperfect rationality contributes to smoother competition dynamics, corroborating our observations in Sect. 5.2.1.
Strategy of Zero Reasoning Level
So far we have shown the effects of different degrees of buyers’ rationality on the behavior of retail markets with regards to: the evolution of competition, the resulting prices for buyers, and the stability of evolutionary dynamics. Here, we show that the properties shown in previous sections generalize for different prices of L0 strategy, \(p_0\). Figure 6 illustrates both the degree of rationality \(\log (\tau ^*)\) that minimizes the cost for buyers (left), and the corresponding cost for the values of \(\log (\tau ^*)\) (right). The cost for buyers is minimum if buyers are not perfectly rational for all values of \(p_0\), however as the difference \((p_0 - c)\) becomes larger, \(\log (\tau ^*)\) increases (lower degree of rationality). At the same time, buyers’ cost is relatively lower than \(p_0\) as \(p_0\) increases. Intuitively, the margin between the resulting average cost for buyers (computed for the optimal degree of buyers’ rationality) and the price \(p_0\) increase as the difference (\(p_0 - c\)) increase.