Competing Over Time — Interactive Pricing Theory

Static pricing models assume firms choose prices once, or at most repeatedly without any link between periods. In reality, firms make investments that change their future competitive position, new competitors enter, and struggling incumbents exit. Dynamic discrete games—pioneered by Maskin and Tirole (1988) and Ericson and Pakes (1995)—provide the framework for understanding these intertemporal strategic interactions.

Introduction

Consider two semiconductor manufacturers deciding how much to invest in fabrication technology. A firm that invests successfully reduces its marginal cost, enabling it to undercut rivals and capture market share. But investment is costly and uncertain—the firm trades off current profits against a stochastic improvement in its future competitive position. This is precisely the tradeoff that static models cannot capture.

Dynamic discrete games model oligopolies where firms make decisions (prices, investment, entry, exit) in every period, and the current state of the industry—captured by a vector of payoff-relevant variables—evolves over time as a consequence of those decisions. The key solution concept is Markov Perfect Equilibrium, in which each firm’s strategy depends only on the current state, not on the full history of play.

Markov Perfect Equilibrium

In a dynamic game, the history of play can be arbitrarily complex. Maskin and Tirole (1988) introduced the concept of Markov Perfect Equilibrium (MPE) to restrict attention to strategies that depend only on payoff-relevant state variables. This yields tractable equilibria while capturing the essential dynamics of competition.

Definition — Markov Perfect Equilibrium

A Markov Perfect Equilibrium of a dynamic game is a profile of strategies

\{\sigma_i(s)\}_{i=1}^N

such that:

Each firm’s strategy $\sigma_i$ depends only on the current state $s \in \mathcal{S}$ , not on the history of play.
Each strategy is a best response to the other firms’ strategies, given the transition probabilities induced by all firms’ actions:
$\sigma_i(s) \in \arg\max_{a_i} \left\{ \pi_i(s, a_i, \sigma_{-i}(s)) + \delta \sum_{s'} P(s' \mid s, a_i, \sigma_{-i}(s)) \, V_i(s') \right\}$

where

V_i(s)

is firm

i

’s value function,

\delta

is the discount factor, and

P(s' \mid s, a)

is the state transition kernel.

The restriction to Markov strategies has both conceptual and computational advantages. Conceptually, it rules out strategies that condition on irrelevant past events (e.g., punishing a rival for an action taken ten years ago that has no bearing on current profitability). Computationally, it reduces the strategy space from the space of all history-dependent policies to a function on the finite state space.

Bellman Equation Formulation

In an MPE, each firm’s value function satisfies a Bellman equation. For a two-firm game where each firm has a quality state $s_i \in \{1, \ldots, K\}$ , the Bellman equation for firm 1 is:

V_1(s_1, s_2) = \max_{a_1} \left\{ \pi_1(s_1, s_2, a_1, \sigma_2(s_1, s_2)) + \delta \, \mathbb{E}[V_1(s_1', s_2') \mid s_1, s_2, a_1, \sigma_2] \right\}

(1)

where the expectation is over the stochastic state transitions. The action $a_1 = (p_1, x_1)$ includes both a pricing decision $p_1$ and an investment decision $x_1 \in \{0, 1\}$ . Investment costs $\kappa$ and succeeds with probability $\alpha$ , moving the firm’s state up by one.

Existence of MPE (Maskin and Tirole, 1988)

In a dynamic oligopoly game with a finite state space $\mathcal{S}$ , compact action spaces, and continuous payoff functions, a Markov Perfect Equilibrium exists. When the state space is finite and actions are discrete, the equilibrium can be computed by value iteration or policy iteration.

The Ericson-Pakes Framework

Ericson and Pakes (1995) developed a general framework for analyzing industry dynamics in which firms invest, enter, and exit. Their model has become the workhorse for empirical industrial organization, with applications to semiconductors, airlines, telecommunications, and many other industries.

The key elements of the Ericson-Pakes framework are:

State vector: Each firm $i$ has a quality or efficiency state $s_i$ . Higher states correspond to lower marginal costs or higher product quality.
Investment: In each period, active firms choose whether to invest. Investment is costly (fixed cost $\kappa$ ) and stochastic: with probability $\alpha$ , the firm’s state improves; otherwise it remains unchanged.
Competition: Given the state vector, firms compete in prices (differentiated Bertrand competition). A firm with a higher state has a cost advantage and earns higher profits.
Entry and exit: Potential entrants pay a sunk entry cost and draw an initial state. Active firms exit if their continuation value falls below a stochastic scrap value.

In our simplified model, we consider two firms competing in a differentiated product market. Firm $i$ with state $s_i$ has marginal cost $c(s_i)$ that decreases linearly from $c_H$ (at state 1) to $c_L$ (at state $K$ ). Demand for firm $i$ ’s product follows differentiated Bertrand:

q_i = a - 2p_i + p_j

(2)

where $a$ is the demand intercept. Equilibrium prices and investment decisions are determined jointly through value iteration.

Interactive: Dynamic Duopoly

The heatmap below shows the equilibrium outcomes across all combinations of firm quality states. Toggle between viewing equilibrium prices (how firms price given their relative positions), value functions (the discounted stream of future profits), and investment policy (whether to invest or extract at each state). The red arrows trace a simulated trajectory from the initial state (1, 1).

The Investment-Extraction Tradeoff

At the heart of dynamic competition is the tradeoff between investing now (to improve future competitive position) and extracting now (to earn high current profits). A firm that invests incurs an immediate cost $\kappa$ but gains a probability $\alpha$ of moving to a higher state with lower costs and higher future profits. A firm that extracts earns maximum current profit but risks falling behind its competitor.

The optimal policy depends on the discount factor $\delta$ . Patient firms (high $\delta$ ) invest aggressively because they value the future cost advantage. Impatient firms (low $\delta$ ) extract because the discounted benefit of investment is too small to justify the current cost.

Semiconductor Fabrication

In the semiconductor industry, moving to a smaller process node (e.g., from 7nm to 5nm) requires billions in fabrication plant investment but yields lower per-chip production costs and higher performance. Intel, TSMC, and Samsung face exactly this invest-vs-extract tradeoff each generation. A firm that falls behind in process technology loses design wins from customers like Apple or Nvidia, which can take years to recover.

The investment incentive is also shaped by the rival’s state. When a firm is far behind its competitor, the marginal value of investment is high because each step closer to the frontier yields a large profit increase. When both firms are at the frontier, the incentive to invest further diminishes because there is less room for improvement.

Interactive: Industry Simulation

The simulation below tracks two firms over 30 periods. The top panel shows their quality states evolving over time (circles mark periods where a firm invests), and the bottom panel shows per-period profits. Adjust the discount factor and investment cost to see how firms shift between invest-heavy and extract-heavy strategies.

Entry Deterrence and Limit Pricing

Beyond investment dynamics among active firms, a fundamental question in industrial organization is whether an incumbent can prevent entry by a potential competitor. Bain (1956) identified three possible outcomes depending on the entrant’s cost structure and the fixed cost of entering the market.

Bain’s Three Regimes

Definition — Entry Deterrence Regimes (Bain, 1956)

Consider an incumbent with marginal cost

c_I

and a potential entrant with marginal cost

c_E

and fixed entry cost

F

. Demand is

q = a - bp

. Three regimes arise:

Blockaded entry: The entrant’s maximum possible profit (even at its best-response price) falls below the fixed cost $F$ . The incumbent charges the monopoly price without concern for entry:
$\frac{(a - bc_E)^2}{4b} \leq F \implies p^* = p^M = \frac{a + bc_I}{2b}$
Deterred entry: Entry would be profitable at the monopoly price, but the incumbent can deter entry by setting a limit price $p_L$ just low enough that the entrant’s residual profit falls below $F$ . The incumbent prefers deterrence to accommodation when the limit-pricing profit exceeds the duopoly profit.
Accommodated entry: The profit from limit pricing is lower than the profit from sharing the market as a duopolist. The incumbent accommodates entry and the market becomes an oligopoly.

The limit price is the highest price the incumbent can set while still making entry unprofitable for the potential rival. Under Stackelberg assumptions, this is approximately:

p_L \approx c_E + 2\sqrt{\frac{F}{b}}

(3)

When the entrant’s cost $c_E$ is high or fixed costs $F$ are large, entry is blockaded and the incumbent enjoys monopoly profits. As $c_E$ falls or $F$ decreases, the incumbent must sacrifice margin to deter entry. Below a critical threshold, deterrence becomes too costly and accommodation is preferred.

Interactive: Entry Regions

The chart below maps the three entry regimes across the (entrant cost, fixed cost) plane. Green regions indicate blockaded entry where the incumbent can charge the monopoly price. Yellow indicates deterred entry where limit pricing is optimal. Red indicates accommodated entry where the market becomes a duopoly. Move the point to explore how the incumbent’s optimal strategy varies.

Practical Implications

Dynamic discrete games provide a rigorous foundation for understanding several phenomena observed in real-world markets:

Technology industries: Firms like Intel, Samsung, and TSMC make enormous capital investments to reduce manufacturing costs. The Ericson-Pakes framework explains why firms at the frontier continue to invest even when current profits are high—the risk of being leapfrogged by a competitor creates a strong investment incentive.
Pharmaceutical generics: When a pharmaceutical patent expires, generic entry depends critically on the fixed cost of regulatory approval and the incumbent’s cost advantage. The three-regime classification predicts when generics enter immediately, when incumbents can deter entry through authorized generics, and when they accommodate competition.
Platform competition: Ride-sharing platforms invest in geographic expansion and driver acquisition. The dynamic investment framework explains why platforms like Uber and Lyft sustained losses for years—they were investing in market position (state improvement) at the expense of current profits.
Pricing strategy: The models show that the optimal price today depends not just on current costs and demand, but on the firm’s investment plans and competitive trajectory. A firm planning to invest aggressively may optimally set lower prices today to fund the investment, accepting temporary margin compression for long-run competitive advantage.

Airline Route Entry

Consider a legacy airline deciding whether to match a low-cost carrier’s entry on a route. The fixed cost of establishing service (crew bases, gate leases) and the entrant’s operating cost determine the regime. If the low-cost carrier has a substantial cost advantage and low fixed costs, entry is accommodated and both carriers coexist. If fixed costs are high (e.g., requiring a new hub), the legacy carrier may deter entry through aggressive limit pricing on the contested route.

References

Bain, J. S. (1956). Barriers to New Competition. Harvard University Press.
Ericson, R. & Pakes, A. (1995). “Markov-Perfect Industry Dynamics: A Framework for Empirical Work.” Review of Economic Studies, 62(1), 53–82.
Maskin, E. & Tirole, J. (1988). “A Theory of Dynamic Oligopoly, I: Overview and Quantity Competition with Large Fixed Costs.” Econometrica, 56(3), 549–569.