Nonequilibrium phase transitions in competitive markets caused by network effects

Significance

In the conventional economic theory, competing sellers who maximize profit will all sell goods at the same price, making no profit. Network effects, defined by buyers assigning additional preference to popular sellers, destroy this ideal framework: some sellers spontaneously become more popular than others. Using models based on statistical physics, we predict network effects can also cause persistent dynamics in competitive markets, by driving the spontaneous formation of short-lived fads, with subsequent seller overpricing causing their collapse. This model exhibits both spontaneous price fluctuations and broad distributions of firm sizes, suggesting these empirically observed phenomena might have a common origin.

Abstract

Network effects are the added value derived solely from the popularity of a product in an economic market. Using agent-based models inspired by statistical physics, we propose a minimal theory of a competitive market for (nearly) indistinguishable goods with demand-side network effects, sold by statistically identical sellers. With weak network effects, the model reproduces conventional microeconomics: there is a statistical steady state of (nearly) perfect competition. Increasing network effects, we find a phase transition to a robust nonequilibrium phase driven by the spontaneous formation and collapse of fads in the market. When sellers update prices sufficiently quickly, an emergent monopolist can capture the market and undercut competition, leading to a symmetry- and ergodicity-breaking transition. The nonequilibrium phase simultaneously exhibits three empirically established phenomena not contained in the standard theory of competitive markets: spontaneous price fluctuations, persistent seller profits, and broad distributions of firm market shares.

Economists have an established theory of supply and demand for highly competitive markets in equilibrium. However, our everyday life is full of markets far from static equilibrium, with price fluctuations (1) and fad-driven dynamics (2, 3). A standard assumption is that this is caused by external shocks (4, 5) or technological growth (6) but that the market (if left alone) would equilibrate. Persistent dynamics can arise from market friction or long-term strategizing (7).

To understand the robustness of these assumptions, we devise a theory of a market for a single good, with M competitive sellers and N buyers, using agent-based models inspired by statistical physics (8–12); see ref. 13 for a review. The ingredients in these models which are natural from a physics viewpoint are heterogeneity among N buyers (whose individual preferences among the sellers vary) and network effects (14–17): the preference of buyers to select a product which is already popular. (This does not refer to granularity of a social network in which buyers interact.) In the physics context, we model the market using a time-dependent Potts model, where buyers correspond to spins, with the seller they buy from given by their spin state. Heterogeneous preferences are random fields, and network effects are all-to-all (mean-field) ferromagnetic interactions. The ground state of the Potts model gives a (possibly multivalued) high-dimensional demand curve (18). Sellers individually profit-maximize in this high-dimensional landscape, adjusting prices (corresponding to time-dependent uniform fields in the Potts model) with time. Buyers adjust choices in response, and we numerically simulate the dynamics.

When network effects (i.e., spin interactions) are very weak, there is (nearly) perfect competition (PC): a statistical steady state with negligible seller profits. As network effects increase, there is a nonequilibrium (NE) phase transition, after which fads spontaneously form (18). Sellers exploit this condensation of buyers onto their good by raising their price, after which other sellers undercut them; this cycle causes persistent dynamics, reminiscent of idiosyncratic price fluctuations observed in many markets (1). Sellers make finite time-averaged profit in this NE phase, as we simultaneously observe numerically a broad distribution of sellers’ market shares. As heterogeneity vanishes or buyer dynamics become sufficiently slow, it is also possible for one seller to permanently capture the market and price out possible competitors. Hence, we find two transitions: from an equilibrium symmetric phase to a NE phase which breaks the permutation symmetry (all sellers are just as likely to be preferred) at any fixed time, but not after time-averaging, then to a nonergodic and symmetry-broken (SB) phase where a monopolist captures the market.

Our work complements a large body of recent work investigating how network effects can disrupt the simple model of competitive equilibrium. For example, network effects cause just two competing firms to change pricing strategies (17), even leading to temporal dynamics (19, 20). Work has been more limited on markets with many firms, where the focus has been on classifying multiple equilibria due to network effects (11, 18). However, with nondynamical sellers, a market with many firms will ultimately reach static equilibrium even with network effects (18). The punchline of this paper is that competing profit-maximizing sellers can generically destabilize this static picture. With strong network effects, there is a robust phase in which large and unpredictable temporal fluctuations of market shares persist to infinite time. This phenomenon is endogenous (not caused by external shocks), yet still preferable to the sellers, who make far higher profits. Sellers could therefore prefer to operate in markets far from equilibrium, characterized by persistent boom-and-bust cycles. Our model thus proposes how some highly competitive markets with many interchangeable goods might fail to equilibrate, even in the long run.

Model Setup

We now introduce the details of our model. We do not attempt to capture all possible complex features of economics; rather, we focus on a minimal model which can realize the phenomenology outlined above. Buyers are labeled with $\alpha =1,\dots ,N$ and sellers with $i=1,\dots ,M$. Time is labeled in integer steps: $t=1,2,\dots$. At each time step, seller i sells a good at uniform price p_i to the entire market. Let $q_i(t)$ denote the fraction of buyers who select good i at time t. We assume that a buyer can always purchase from their desired seller and can also choose not to buy from any seller.

When we increment t by 1, first, each of the N buyers updates their decision with probability ρ. Buyer α picks their next decision by maximizing the utility $U_{\alpha ,i}$ of choice i at time t, which we model by (18)

$$U_{\alpha ,i}(t)=u_{\alpha ,i}+J{q}_i(t-1)-p_i(t-1),$$ [1]

where the random fields $u_{\alpha ,i}$ give intrinsic heterogeneity in buyers’ choices, the $+J{q}_i$ term models network effects (with $J\ge 0$ denoting their strength), and $-p_i$ denotes the loss of utility from paying more for a good. We take $u_{\alpha ,i}$ to be independent and identically distributed Gaussian random variables with mean μ and variance ${\sigma }^2$. If buyer α updates, they buy from seller i if $U_{\alpha ,i}\ge U_{\alpha ,j}$ for all j and from no seller (x = 0) if all $U_{\alpha ,i}<0$. We can include this last effect by simply defining $U_{\alpha ,0}=0$ for all buyers.

It is then the sellers’ turn to act. One seller i, chosen uniformly at random, will update their price p_i. To do this, the seller follows textbook economics: they query each buyer α and determine the price point

$$p_{\alpha ,i}^{*}=p_i+U_{\alpha ,i}-\underset{{}_{0\le j\le N,j\ne i}}{\max }{U}_{\alpha ,j}$$ [2]

at which that buyer would be willing to purchase their good (which might be negative). They then set their price by maximizing profit (per total buyer, implicit henceforth)

$${\pi }_i(p_i)=\frac{p_i}{N}{\displaystyle \sum _{\alpha =1}^N\mathrm{\Theta }}(p_{\alpha ,i}^{*}-p_i).$$ [3]

If $p_i^{*}$ is the value at which π_i above is maximized, sellers choose $p_i\to \max (0,p_i^{*})$.

We assume for now that the sellers have no production costs. Inclusion of production costs into the model does not change the nature of the phase diagram discussed below (SI Appendix). The model free of production costs may be more suited as a model of markets such as art, fashion, software, or entertainment. In particular, in both software (21, 22) and fashion (23), there is empirical evidence of strong network effects. Fashion markets are notorious for exhibiting persistent dynamics (24) and thus may be an ideal setting for empirical tests of this theory. Alternatively, a seller selling digitally downloaded software can instantly supply an arbitrary number of goods to interested buyers, analogously to the dynamics in our model.

Note that in this simplified toy model, buyers must rebuy a good in each time step (although wait, on average, ${\rho }^{-1}$ time steps before adjusting their utility maximization calculation). One can imagine that these goods are perishable and/or there is a subscription model for purchasing, such that buyers consistently rebuy goods at each time. Sellers can and must sell to any buyer who wants to purchase from them, suggesting that sellers can and will produce goods on demand, as in the digital download market or perhaps even “fast fashion” markets (25). However, we emphasize that the model’s phase diagram is not sensitive to each precise detail. The core phenomena in this model are likely present in alternative microscopic models with the same key features: incorporating network effects among buyers making discrete (rather than continuous) purchases from competing sellers. With this perspective in mind, we proceed with the analysis.

Phase Diagram of the Model

In a naively rational marketplace, all sellers should perform just as well as all others; there is no objectively better good. In the language of statistical physics, there should be an emergent ${\text{S}}_M$ permutation symmetry in the model in the $N\to \infty$ limit, for each random realization of $u_{\alpha ,i}$. Note that any given realization of randomness ($U_{\alpha ,i}$) microscopically breaks permutation symmetry. However, as is common in statistical physics, we consider this effect to be meaningful only if it has macroscopic effects: namely, ${\lim }_{N\to \infty }{q}_i\ne M^{-1}$. When this identity holds, we say that there is permutation symmetry breaking; when ${\lim }_{N\to \infty }{q}_i=M^{-1}$, we call the phase permutation-symmetric since we could permute the sellers’ labels i without changing the macroscopic observable q_i (at large N).

In a dynamical rational marketplace, we should expect that by the definition above, the market has permutation symmetry among the M sellers: buyers can instantaneously, and without taking on added costs, switch buying from seller i to j. Moreover, sellers can and will make instantaneous and large changes to their prices to undercut other sellers, when possible. In these circumstances, we should find PC, where all sellers have equal market shares and make no profit as $\sigma \to 0$. After all, without network effects, if one seller makes profit at price p₀, another seller can undercut them with price $p_0-\delta$, with $\delta \to 0^{+}$, thus stealing their entire market share. Prices $p_i\to 0$ over time, so in the long run ($t\to \infty$), sellers make no profit. At σ = 0, this effect is not sensitive to whether N is large or small.

However, network effects can drastically modify this picture. It is useful to review the profit maximization problem faced by a single seller (M = 1) in a market with network effects (26). Fig. 1 shows q(p) and $\pi (p)$ for both small and large J. When J is large, the demand curve q(p) is multivalued: this is a hysteresis loop well known from the phase diagram of a ferromagnet in statistical physics, where it is possible for two different collective behaviors to be stable (in this case, many or few buyers purchasing the good). As an extreme limiting case, if $\sigma \to 0$ but J is finite, then buyers are willing to buy at price $p=\mu$ (the mean of $u_{\alpha ,i}$) when $q=0)$ but at $p=\mu +J$ when q = 1; hence, for $\mu <p<\mu +J$, there are two possible (stable) market outcomes. At finite σ, it turns out that the monopolist’s preferred price—namely, the price which maximizes $\pi (p)=q(p)p$—is very close to the critical price p at which the upper branch of q(p) ceases to exist. If p is raised beyond this point, the market will crash (there is a discontinuous phase transition where buyers no longer purchase from this seller). In our simple model, where the seller does not anticipate network effects, they will tend to overprice their good and cause a market crash, as shown in Fig. 1.

Fig. 1.

The supply and demand problem faced by a single seller. We take μ = 3 and σ = 1. At the blue circled point on the true demand curve, the seller (neglecting that network effects allow their higher price point) predicts the demand curve is the black dashed line, and the profit maximizing point is at the black diamond, which will lead to a market crash.

What happens if the monopolist has to compete? We have already noted that in the absence of network effects, $M\ge 2$ sellers will simply undercut each other’s prices for greater market share, pushing $p_i\to 0$ if σ = 0. If $\sigma >0$, sellers stop undercutting each other when $p~\sigma {(\log \frac{M}{2})}^{-1/2}$, which is the price at which a seller can expect to keep a fraction of their buyers even if all other sellers set p = 0 (SI Appendix). The ${(\log M)}^{-1/2}$ scaling comes from Gaussianity of $u_{\alpha ,i}$ and is not universal. The resulting market will then enter a statistically steady state where sellers continue to make small price adjustments to capture a handful of marginal sellers, but they each have average market share $q_i~M^{-1}$, up to subleading corrections in $1/N$. We conclude that this phase is (nearly) PC; crucial features of this phase are low seller profits, statistical stationarity (no macroscopic dynamics in time), and emergent permutation symmetry. An order parameter is

$$\frac{1}{Q}=\frac{1}{{(1-q_0)}^2}{\displaystyle \sum _{i=1}^M{q}_i^2}.$$ [4]

Q is the effective number of sellers if the market share were equally distributed at any time t; here q₀ denotes the fraction of buyers who exited the market, and in the PC phase, Q = M (when $N\to \infty$).

Suppose now that we consider the opposite limit where J is finite while σ = 0. Now an emergent monopolist captures the market and sets $p=J-0^{+}$, crowding out any other seller. This trivial limit spontaneously breaks the permutation symmetry group ${\text{S}}_M$ since buyers collectively and irrationally choose a single seller to buy from, even though that seller is no better than any other. $Q~1$ for the SB phase.

What happens when $J/\sigma$ is neither 0 nor $\infty$? When J is sufficiently small, Eq. 1 will only have one solution for fixed prices (18), and this implies the PC phase is stable. For sufficiently large J, Eq. 1 can have multiple solutions. This occurs when the gain in utility $J{N}^{-1}$ for good i, which arises due to a single buyer switching to that choice, is large enough to cause (on average) $\alpha \ge 1$ buyers to switch to the same choice. This causes an avalanche of decision changes which leads to a condensation of the market onto a single good (i.e., $Q~1$). (At large J, the market can [if pushed in the right direction] condense onto any good—this is why there are multiple solutions to Eq. 1.) We can estimate that (18)

$$\alpha ~J\left|\frac{\partial q}{\partial p_i}\right|~J\frac{M^{-1}}{\sigma {(\log \frac{M}{2})}^{-1/2}}=\frac{J}{\sigma }\frac{\sqrt{\log \frac{M}{2}}}{M}\equiv \overline{J}.$$ [5]

Hence, the PC phase exists for $\overline{J}\ll 1$ but not for $\overline{J}\gg 1$.

To deduce what happens when $\overline{J}\gg 1$, we must think about dynamics. At early times, an emergent monopolist will capture the market. As in Fig. 1, if they capture the entire market, they will overprice their good (SI Appendix) by not anticipating how much of their good’s value derives from network effects and thus precipitate a market crash, leading to their market share decaying as ${(1-\rho )}^t\approx {\text{e}}^{-\rho t}$ (for $\rho \ll 1$). However, if the buyers are sufficiently slow, the sellers will ramp up their price more slowly, which allows them to more accurately estimate the demand curve. This monopolist will maintain market share when $\overline{\rho }\ll 1$, where (SI Appendix)

$$\overline{\rho }=\frac{\rho M}{\log M}.$$ [6]

In contrast, when $\overline{\rho }\gg 1$, the market crashes due to overaggressive pricing. Once the monopolist has lost market share, network effects will again drive an instability wherein a different seller will capture the market share and become an emergent monopolist. This cycle of market condensation and crashes forms an NE phase, in which the permutation symmetry is broken at any fixed time but is restored on long times, in the sense that each seller will have the same time averaged behavior: e.g.,

$$\underset{T\to \infty }{\lim }\frac{1}{T}{\displaystyle \sum _{t=1}^T{q}_i}(t)=\frac{1}{M}.$$ [7]

A useful dynamical order parameter for the NE–SB phase transition is the rate γ at which the seller with the largest market share changes, which is finite in NE and 0 in SB.

In SI Appendix, we detail two finite size effects, quite visible in simulations, which can appear to modify the parameters where phase transitions occur. Let ${\overline{J}}_{\text{c}}$ denote the critical value at which the buyers would (in the absence of seller dynamics) condense into a single seller’s good. In our model, ${\overline{J}}_{\text{c}}\approx 0.7$. First, suppose that M, N are very large and $\overline{J}-{\overline{J}}_{\text{c}}\ll {\overline{J}}_{\text{c}}$. If

$$\frac{1}{\overline{J}-{\overline{J}}_{\text{c}}}\frac{1}{\overline{\rho }}\mathrm{\gtrsim }\frac{\log M}{\log N},$$ [8]

a seller raises prices fast enough during the decision change avalanche described above that they significantly slow down the avalanche. The market then mostly appears to be in PC but exhibits extreme and short-lived bursts in single seller market share when (by random chance) a seller does not update prices for a long time. We associate such dynamics with the NE phase, but the transition from PC is continuous and can appear quite slow (Fig. 2). In contrast, when N is small and M > 2, finite-size fluctuations in prices p_i can push the market out of the metastable (18) equilibrium of PC; we find this occurs if

$${\overline{J}}_{\text{c}}-\overline{J}\mathrm{\lesssim }{M}^{1/3}{N}^{-1/3}.$$ [9]

Fig. 2.

Dynamics in a competitive market with M = 10 sellers (each different color) and $N=5,000$ buyers. (Top) Seller i’s market share $q_i(t)$ and (Bottom) price $p_i(t)$. In the NE phase, the mechanism driving oscillations is clearly visible: the seller with high market share raises price and will be undercut unless they can catch the effect in time to lower their price and stop the cascade of buyers to another seller.

Here the transition to NE or SB appears nearly discontinuous.

As a semantic point, therefore, any thermodynamic limit of $N,M\to \infty$ in which the phases described above are well defined must be taken carefully, with the limit $N~M^a\to \infty$ taken together in a suitable fashion. This technicality aside, agent-based simulations agree well with our expectations. Fig. 2 shows the behavior of $q_i(t)$ and $p_i(t)$ in one model realization in various phases of the model. Sellers overpricing goods and causing market crashes is easily visible as the mechanism behind persistent dynamics in NE. Fig. 3 demonstrates that the advertised order parameters behave as expected near the PC–NE transition as well as the NE–SB transition, along with a phase diagram of our model in the $(\overline{J},\overline{\rho })$ plane. Finite size corrections to the critical ${\overline{J}}_{\text{c}}$ where PC ceases to exist are significant but are consistent in magnitude with basic estimates (SI Appendix).

Fig. 3.

(Left) The NE–PC transition at fixed $\overline{\rho }=0.8$, visible in both average seller profit $\langle {\pi }_i\rangle$ and in M / Q. Corrections to the critical point due to finite N / M are clearly visible. (Middle) The NE–SB transition at fixed $\overline{J}=0.9$, visible in γM. We also show that the industry-averaged profit is finite in both phases. (Right) Numerical determination of the phase diagram of the model in the $(\overline{J},\overline{\rho })$ plane at M = 30. To classify each point to a phase, we demanded $Q>M/3$ for the PC phase and a median flip rate γ = 0 for the SB phase between nine random realizations. All simulations used $N=100M$ and N time steps.

If one incorporates production costs into the model, the qualitative phase diagram does not change. In fact, since a standard assumption is that the production costs w(q)—which modify the firm’s profit to ${\pi }_i=p_i{q}_i-w(q_i)$—obey ${\text{d}}^{2}w/\text{d}{q}^2<0$, we might expect that production costs destabilize PC because sellers who (by some random fluctuation) get a few extra buyers can lower their cost relative to other sellers even further, while simultaneously, the network effects also drive more buyers to this same seller. This enhances the tendency of buyers to condense into purchasing from a single seller. This effect is realized in our simulations: details and results from numerical simulations are presented in SI Appendix.

A second extension to the model is to include noise in buyer/seller decisions. There are three types of noise that we consider in SI Appendix: 1) sellers can only sample a fraction of buyer preferences when setting their profit-maximizing price; 2) sellers set their price at $p_i^{*}+\eta$, with η a random variable; and 3) buyers do not always pick the highest utility choice. Types 1 and 2 are found empirically to destabilize PC, in the former case because small N effects, described by Eq. 9, are amplified in seller behavior and in the latter case because for moderately large $\overline{J}$, PC is a metastable phase, and the larger seller price swings may push the market out of the permutation-symmetric stability basin for the buyer dynamics. In contrast, type 3 stabilizes PC, analogous to how finite temperature stabilizes a disordered permutation-symmetric phase in a random-field Potts model.

Distribution of Market Shares

Having established the phase diagram in our model, we now predict heavy-tailed distributions in the distribution of sellers’ market shares, q_i, in the NE phase. While part of this tail simply arises from the emergent monopolist, we predict heavy tails in the distribution even among less popular sellers. When a monopolist loses market share, the newly free buyers will select their next seller i at a rate proportional to $u_{\alpha ,i}+J{q}_i-p_i$. p_i values and $u_{\alpha ,i}$ may be similar for all sellers, but the Jq_i term suggests a preferential attachment (“rich get richer”) mechanism, whereby a seller who just happens to have a large market share will gain an even larger one with time (and so fluctuations in q_i get amplified with time). Numerically, we confirm that the probability density $P(q_i)$ of market shares has heavy tails. If one desires to fit to a power law, the best fit appears to be roughly $P(q_i)~q_i^{-2}$. Curiously, this exponent is known to appear in the preferential attachment model of ref. 27, but its applicability to our model is unclear: in particular, growing systems obtain power-law distributions, yet the number of sellers in our model is fixed. Regardless of the microscopic origin, in our simulations, heavy tails are present over a broad range of scales and (SI Appendix) should be expected over at least the range $M^{-2}\mathrm{\lesssim }{q}_i\mathrm{\lesssim }{M}^{-1}$. The heavy-tailed distribution in NE sharply contrasts with PC, where $P(q_i)$ is concentrated around $q_i\approx M^{-1}$. In SB (besides an obvious spike for $q_i~1$) we find a more rapidly decaying tail at small q_i (Fig. 4). Numerically, the sharpest $P(q_i)~q_i^{-\nu }$ (with $\nu \approx 2$) scaling occurs near the NE–SB transition (SI Appendix).

Fig. 4.

$P(q_i)$ measured in 10 simulations with $N=10,000$, M = 100, μ = 5, and σ = 1. PC, $\overline{J}=0,\hspace{0.17em}\overline{\rho }=0.6$. NE, $\overline{J}=0.9,\hspace{0.17em}\overline{\rho }=0.6$. SB, $\overline{J}=0.9,\hspace{0.17em}\overline{\rho }=0.05$. We observe heavy-tailed $P(q_i)$ in NE, sharply concentrated $P(q_i)$ in PC, and a highly bimodal distribution near $q_i=0,1$ in SB. Note that P(0), not displayed, is substantial in SB for this N, M.

It is empirically established (28, 29) that there is a heavy tail in the market shares of firms; this is known as Gibrat’s law. Economists commonly deduce this using, e.g., firm employee counts; what is important is the broad distribution of firm sizes that arises in real markets. Our model, which accounts for very few of the complications of supply-side economics, already includes one microscopic mechanism for these heavy-tailed distributions.

Universality of NE Phase

Let us now argue that the qualitative conclusions of our model can be robust; namely, even if seller pricing strategies are rather different, profit-maximizing sellers will generically drive the market out of the PC phase into either the NE or SB phases. First, as noted above, the qualitative nature of the phase diagram with PC at small $\overline{J}$ and NE or SB at large $\overline{J}$ are unchanged in the presence of noise or production costs. More broadly, in the spirit of effective theories in physics, consider each seller maximizing

$${\mathrm{\Pi }}_i={\displaystyle \underset{0}{\overset{T}{\int }}\text{d}}t\left({\pi }_i(p_i;p_1,\dots ,p_N)-\zeta {\left(\frac{\text{d}{p}_i}{\text{d}t}\right)}^2+\cdots \right)\text{},$$ [10]

where the phenomenological parameter ζ captures stiffness or locality in price dynamics. We can think of Π_i as a (negative) action (a la Lagrangian mechanics) which seller i wishes to maximize. Let us schematically carry out this maximization. In the PC phase, $q_i~M^{-1}$ and $p~\sigma {(\log \frac{M}{2})}^{-1/2}$ are constant in time, thus leading to total profit $\mathrm{\Pi }~\sigma M^{-1}{(\log \frac{M}{2})}^{-1/2}T$ in time T, if seller i prices similarly. However, if the seller can drive the market to a symmetry broken point, then (if $\eta \to 0$) in the NE phase they can make profit $\mathrm{\Pi }\mathrm{\gtrsim }{M}^{-1}JT$ by setting price J whenever they capture the market (on average for a time T / M). Thus, whenever $\overline{J}\mathrm{\gtrsim }1$, forward-thinking profit-maximizing sellers drive the market into the NE phase. Of course, a sufficiently wise seller may subsequently stabilize the market in an SB phase once they capture the market (17, 19, 30): the emergent monopolist could set a lower price to avoid market crash. However, if sellers need to learn the value of J, there may be a long period of oscillatory dynamics before a permanent monopolist arises.

Conclusions

There are many extensions of this model which will be important to analyze in order to further verify the robustness of our conclusions against features of a real economy we have not yet accounted for. 1) Supply-side constraints—notably, that sellers may need to produce their goods in advance of buyers choosing them (thus necessitating estimates of future demand)—are not included yet. The model described here may best model markets for software downloads and/or online entertainment, where production costs or infrastructure are less critical and sellers can instantaneously provide a good to a buyer who pays for it. Alternatively, the persistence of bursty fad-driven dynamics is consistent with empirical observations in fashion markets (24). 2) A game-theoretic analysis of optimal seller strategies in this model, accounting for both network effects and the strategies of other sellers (19), is needed. It is unclear if the NE phase could still exist as $t\to \infty$ without technological growth (6) or buyer/seller entry/exit into the market. It is tempting to speculate that analogously to repeated prisoner’s dilemma games where cooperation can be desirable (31, 32), in a competitive market with large network effects, firms may learn to accept oscillatory dynamics in exchange for being assured a larger time-averaged profit than in a competitive phase. 3) Heterogeneous NE phases that arise due to locality on a buyer/seller interaction graph are likely to exist, as in other evolutionary games (33). 4) Including firm heterogeneity may lead to broad distributions in firm growth/decay rates, as well as in $P(q_i)$ (34). 5) Including macroeconomic market dynamics coupling the markets for many goods together (35–37) is important.

Although our cartoon model is certainly incomplete as a model of an actual competitive market, the essential point of this paper is that network effects could generically lead to an unexpected failure of economic lore: that a market of memoryless utility-maximizing buyers and profit-maximizing sellers may never reach any static equilibrium, exhibiting unpredictable dynamics for all time. This conclusion holds even without any exogenous shocks. We emphasize that this phenomenon is not equivalent to chaos that can occur when buyers maximize a utility function $u(x(t-1),x(t))$ that depends on choices at two (or more) times (38): in our model, buyers only gain utility from their current decision and, without feedback from sellers, reach equilibrium even with network effects. The long run analysis of even highly competitive markets will be misleading if equilibrium is unstable. If the macroeconomy is already fragile even in the absence of network effects (4, 37), it is all the more crucial to understand whether network effects play an important role in persistent price dynamics observed in real economies.

We found that the NE phase will exhibit both the strongest temporal fluctuations in prices and the broadest distributions of firm’s market shares; looking for the presence or absence of this correlation in empirical data may be a simple test of our model. Following refs. 39–41, a more quantitative check for whether fluctuations are endogenous (intrinsic to the market) or exogenous (driven by external shocks) is to study the time correlations of market properties such as Q(t). In particular, in our model, exogenous shocks in the PC phase cause rapid drops in Q(t) followed by slow increases as the market returns to equilibrium. In contrast, in and near the NE phase, the largest jumps in Q(t) are when it increases abruptly as the dominant seller abruptly loses market share. In SI Appendix, we confirm this expectation quantitatively in numerical simulations using time correlations in Q(t). Hence, with neither the ability to control network effects nor knowledge of the times or strengths of exogenous shocks, it may be possible to confirm our theory for a competitive market, driven far from equilibrium by endogenous fluctuations, by studying dynamics of prices and market shares in real world markets.

Data, Materials, and Software Availability

MATLAB code and data files for the figures has been deposited in GitHub (https://github.com/ajlucas90/220405314) (42).