Competitive Pricing

Definition and motivation

Competitive pricing under incomplete information studies how firms choose prices when they do not know some relevant characteristics of rivals (e.g., marginal costs, capacities, or demand sensitivities) and must form beliefs about those characteristics. Instead of a one-shot Bertrand or Cournot equilibrium with common knowledge (see Oligopoly Models), firms play a Bayesian game (see Bayesian Games) in which each firm's strategy maps its private type to a price and beliefs about rivals are updated via Bayes' rule when observing actions or outcomes.

Core intuition

1) A firm’s optimal price balances margin and the probability of being undercut. Under complete information and Bertrand competition with homogeneous goods, the price is driven down to marginal cost. But with private information (e.g., about costs), a price above marginal cost can be optimal because charging a high price may signal a high cost type or avoid aggressive price undercutting.

2) Competitor response functions capture how rivals change their prices in response to your price and observable market signals. These are often written as reaction functions $r_j(p_i, x)$, where $p_i$ is your price and $x$ are public signals (e.g., market demand shocks). Under incomplete information, firms do not know $r_j$ exactly but have a probabilistic belief over a family of response functions.

3) Bayesian estimation is used both inside the firm (for pricing decisions) and by the researcher (for structural estimation). Firms solve a Bayesian Nash equilibrium when strategies are mutual best responses given beliefs. Researchers invert observed price dynamics to estimate the distribution over competitor reaction functions and to test for tacit collusion (persistent coordinated responses that are not driven by cost shocks).

Why this matters

• •Predictive pricing: Retailers and platforms dynamically set promotions, knowing rivals may match or ignore them. Correctly accounting for uncertainty about matching probabilities alters optimal markdowns.
• •Policy and antitrust: Distinguishing competitive parallel pricing from tacit collusion matters for enforcement. If observed price co-movement results from private cost shocks rather than coordination, enforcement is inappropriate.
• •Empirical methods: Techniques such as structural estimation with Bayesian priors, expectation-maximization (EM), and Markov chain Monte Carlo (MCMC) are necessary to recover latent reaction functions from noisy price data.

Concrete setup (minimal formal model)

Consider two firms i and j. Firm i has private type $\theta_i$ (e.g., marginal cost), drawn from distribution $F_i$ with density $f_i$; similarly for $\theta_j$. Demand for firm i at prices $(p_i,p_j)$ is $D_i(p_i,p_j;\xi)$ where $\xi$ is a public demand shock. In [Price Elasticity], we learned how price elasticities shape demand. Here, a typical parametrization is log-linear (constant elasticity):

Example numeric instance: if $A(\xi)=100$, $\varepsilon_{ii}=2$, $\varepsilon_{ij}=0.5$, and $(p_i,p_j)=(10,12)$, then $D_i=100\cdot 10^{-2}\cdot 12^{-0.5}=100\cdot 0.01\cdot 0.2887\approx0.2887$ units.

Firm i’s profit when setting price $p_i$ and facing $p_j$ is

Under incomplete information, firm i does not observe $\theta_j$ and treats $p_j$ as a random variable generated by firm j's strategy and type. The Bayesian best response solves

This expectation depends on i's belief about firm j's response function. The equilibrium is a profile of strategy functions $p_i^*(\theta_i,\xi)$, one for each firm, forming a Bayesian Nash equilibrium.

Relation to prerequisites

• •In Oligopoly Models we learned how strategy spaces and payoffs determine Cournot/Bertrand equilibria; here we generalize those equilibria to incomplete information using Bayesian Games.
• •In Bayesian Games we learned Harsanyi's transformation; here we treat private types as part of the type space and compute Bayesian Nash equilibria of price-setting games.
• •In Price Elasticity we learned demand forms; here we explicitly use constant elasticity demand to obtain closed-form best responses in many cases.

Preview of next sections

Section 2 will derive the firm’s Bayesian best response and show how to estimate competitor response functions via likelihood and posterior inference. Section 3 will develop equilibrium existence/characterization under incomplete information and show how price wars and tacit collusion arise as equilibria or as empirical artifacts. Section 4 will connect these methods to empirical detection and antitrust practice.

Objective

We want to recover a probabilistic model for how rivals set prices conditional on observable variables. This model — the competitor response function — is often parameterized and estimated by firms or researchers. In the Bayesian approach we place priors on unknown parameters and update them using observed prices.

Parametric reaction function

A useful parametric specification (used widely in empirical IO) is an affine reaction function with random shocks:

where $x$ are public covariates (e.g., demand shifters), $\beta_j=(\beta_{j0},\beta_{j1},\beta_{j2})$, and $\varepsilon_j$ is mean-zero noise. Example numeric instantiation: let $\beta_{j0}=5,\beta_{j1}=0.8,\beta_{j2}=2$ with a single scalar $x=3$, and $\varepsilon_j\sim N(0,1)$. For $p_i=10$, $p_j$ mean = $5+0.8\times10+2\times3=5+8+6=19$, so $p_j\approx19+\varepsilon_j$.

Interpretation: $\beta_{j1}$ measures the sensitivity of competitor j’s price to your price. If $\beta_{j1}\approx1$, they match dollar-for-dollar; if $\beta_{j1}\approx0$, they ignore your price. In [Oligopoly Models] we saw pure best-response functions; here we allow statistical variation around those responses.

Bayesian estimation

Place a prior $\pi(\beta_j,\sigma^2)$, e.g. multivariate normal for $\beta_j$ and inverse-gamma for variance $\sigma^2=\mathrm{Var}(\varepsilon_j)$. Observing a panel of T periods with public covariates $x_t$ and observed prices $(p_{i,t},p_{j,t})$, the likelihood is

Example numeric calculation for one observation: suppose $p_{j,t}=19.5$, $p_{i,t}=10$, $x_t=3$, and candidate $\beta$ as above with $\sigma^2=1$. The residual is $0.5$, so the contribution to log-likelihood is $-\tfrac{1}{2}\ln(2\pi)-\tfrac{0.5^2}{2}= -0.9189 - 0.125 = -1.0439$.

The posterior is

If prior is conjugate (normal-inverse-gamma) the posterior has closed-form updates: posterior mean of $\beta_j$ is the ridge-regression solution blending prior mean and OLS. Conjugate example: take prior $\beta_j\sim N(b_0, V_0)$ and $\sigma^2\sim IG(a_0,b_0)$. With T observations the posterior mean is

where $X$ stacks regressors $(1,p_{i,t},x_t')$ and $y$ stacks $p_{j,t}$. Numeric example: suppose scalar regression with $V_0^{-1}=0.1$, $X'X=10$, $V_0^{-1}b_0=1$, $X'y=50$, then $\hat{\beta}_{post}=(0.1+10)^{-1}(1+50)=10.1^{-1}\cdot51\approx5.0495$.

Nonlinear and latent reaction functions

Competitor response may be nonlinear or depend on latent variables (e.g., unobserved competitor cost). One can specify a parametric nonlinear form, e.g. logistic matching probability times a price response:

where $m_j(\cdot)\in[0,1]$ is the probability of matching behavior (a function with parameters $\gamma$), and $p_j^{(aut)}$ is an autonomous price. Example: if $m_j=0.6$, $\beta_0=2,\beta_1=0.7$, $p_i=10$, $p_j^{(aut)}=12$, expected $p_j=0.6\cdot(2+0.7\cdot10)+0.4\cdot12=0.6\cdot9+4.8=5.4+4.8=10.2$.

Estimation with latent types proceeds via data-augmentation MCMC or EM. For MCMC, treat latent matches or types as additional unknowns and sample them in a Gibbs sampler. For EM, iteratively compute expected sufficient statistics for latent variables given current parameters, then maximize the expected log-likelihood.

Example: MCMC Gibbs step for linear regression with missing match indicator z_t (0/1): sample z_t from Bernoulli with probability proportional to mixture component densities; conditional on z, sample regression coefficients from conjugate posterior.

Firm-side use: posterior predictive response

A firm uses the posterior over $\beta_j$ to form the predictive distribution of $p_j$ given a candidate $p_i$:

If $\pi$ is Gaussian-inverse-gamma, the predictive is a Student-t. Numeric example: using posterior mean $\hat{\beta}=(5,0.8,2)$ and posterior predictive variance $s^2=1.5$, the predictive mean for $p_i=10,x=3$ is $19$ and predictive SD $\sqrt{1.5}=1.225$, so a firm expects $p_j\in[16.55,21.45]$ roughly (95% CI).

Identification and data requirements

• •Variation in $p_i$ exogenous to $\varepsilon_j$ is needed for consistent estimation of $\beta_{j1}$. Instruments may be required if $p_i$ is endogenous (e.g., both prices respond to the same demand shock). In [Bayesian Games] we considered signaling — here the same concern arises: observed covariation may reflect common shocks rather than causal response.
• •Panel data with many repeated interactions helps identify dynamic matching probabilities and persistent heterogeneity.

Empirical diagnostics

• •Posterior predictive checks: simulate $p_j$ from the posterior and compare moments to observed data.
• •Inspect $\beta_{j1}$ posterior: values near 1 across firms suggest high propensity to match prices (important for tacit collusion detection), whereas values near 0 suggest independent pricing.

Summary of Mechanic 1

Bayesian estimation converts noisy panel price data into a full posterior over competitor reaction functions, giving a predictive distribution of rival prices conditional on your candidate price. This posterior is what firms use for optimal pricing and what researchers use as input for structural equilibrium calculations and collusion tests.

Objective

Given a posterior over competitor response functions (from Section 2), compute the firm’s equilibrium pricing strategy and analyze comparative statics: when do we obtain aggressive price-cutting (price wars) versus soft strategies consistent with tacit collusion? We will formalize Bayesian Nash equilibrium in a dynamic or repeated setting and relate static equilibria to dynamic incentives.

Static Bayesian Nash equilibrium (one-shot)

A static Bayesian Nash equilibrium is a profile of strategies $p_i(\theta_i,\xi)$ satisfying, for all types $\theta_i$ and public shocks $\xi$:

The expectation is over rivals’ types and their equilibrium strategies. Existence: under continuity and compactness conditions, a pure-strategy BNE exists. A constructive approach uses best-response operators in a function space and applies Schauder/Tychonoff fixed-point theorems.

Closed-form example with linear demand

Suppose symmetric firms and linear demand for firm i:

with $a,b,c>0$ and $c<b$ to ensure own-price negativity. Let type be marginal cost $\theta_i$. If firm believes rival uses linear strategy $p_j(\theta_j)=\alpha+\gamma\theta_j$, and $\theta_j\sim F$ with mean $\mu$, then the expectation of $p_j$ is $\alpha+\gamma\mu$. The best response solves

First-order condition (FOC):

Hence

Numeric example: let $a=100, b=2, c=0.5, \alpha=10,\gamma=0.5,\mu=20, \theta_i=15$. Then $a+c(\alpha+\gamma\mu)=100+0.5(10+0.5\cdot20)=100+0.5(10+10)=100+0.5\cdot20=110$. Then $p_i(15)=(110+2\cdot15)/(4)= (110+30)/4=140/4=35$.

To find equilibrium $\alpha,\gamma$, we require that the mapping from type to price is consistent with the assumed form. If the equilibrium strategy is linear in $\theta_i$, equate coefficients: for the above FOC-derived $p_i$, coefficient on $\theta_i$ is $b/(2b)=1/2$, so $\gamma=1/2$; intercept $\alpha=(a+c(\alpha+\gamma\mu))/(2b)$ solving for $\alpha$ yields fixed point. This method yields closed-form linear equilibria in many linear-quadratic specifications.

Dynamic repeated interaction: price wars and tacit collusion

When the game is repeated infinitely or with discount factor $\delta$, a firm can use strategies that punish deviations by triggering price wars (grim-trigger or more forgiving strategies). Tacit collusion arises when non-cooperative equilibrium supports higher prices via incentives not to deviate because deviation triggers profitable punishments.

Folk-theorem intuition: with private types, sustaining collusion is harder because detecting deviation is noisy. In [Bayesian Games] we saw how private information complicates equilibrium: imperfect monitoring leads to worse enforcement. If price changes are noisy due to demand shocks or random matching, a firm must weigh false alarms (punishing when no deviation occurred) against real deviations.

Model of imperfect monitoring: public monitoring signal s_t (e.g., aggregated price index) evolves with expected price and noise. Detection relies on likelihood ratios. A strategy: collude at high prices as long as public signal remains in an acceptance region; if signal falls below threshold, switch to a punishment phase (Nash reversion). The incentive constraint for collusion for type $\theta_i$ is

where left-hand side is collusive present value and right-hand side is immediate gain from deviation plus discounted present value of punishment. Numeric example: let $p_C=50,\theta_i=20,D_C=2,p_D=40,D_D=3,\delta=0.95,$ and $V^{pun}=(p_N-\theta_i)D_N/(1-\delta)$ with $p_N=30,D_N=4$. Left PV: $(50-20)2/(1-0.95)=60/0.05=1200$. Immediate gain: $(40-20)3=60$. Punishment PV: $\delta\cdot (30-20)4/(1-0.95)=0.95\cdot40/0.05=0.95\cdot800=760$. RHS = $60 + 760 = 820 < 1200$, so collusion is incentive-compatible here.

Private information undermines this: because $\theta_i$ is private, the firm’s temptation depends on type; low-cost types have more incentive to deviate. Moreover, noisy public signals increase false-punishment probability, reducing the attractiveness of collusion.

Price wars: information cascades and unraveling

Price wars often occur when firms aggressively undercut each other in response to small signals, leading to cascading downward adjustments. Two mechanisms produce price wars:

1) Competitive inference: a small price cut may be interpreted as a signal of low cost, causing rivals to cut prices to avoid losing market share, leading to further inference and cuts. Example: if $\beta_{j1}$ is high (response sensitivity), a small change is amplified.

2) Dynamic retaliation: when punishment for defection is Nash reversion with low future profits (e.g., low $\delta$), firms have weak incentives to maintain high prices and may preemptively cut prices.

Testing whether observed price drops are strategic price wars vs cost shocks can be formalized: if the posterior predictive model from Section 2 places most mass on reaction coefficients $\beta_{j1}$ near 1, then a small cut by one firm should be followed by cuts (consistent with inferred responses). Conversely, if reaction sensitivity is low and contemporaneous demand shocks explain variation, observed co-movement is not strategic.

Empirical equilibrium computation: two-step estimation

1) Estimate competitor reaction $\pi(\beta,\sigma^2\mid\text{data})$ (Section 2).

2) Solve for Bayesian Nash equilibrium in a structural model of pricing that uses the posterior predictive distribution of rival prices. Usually this is done by simulation: draw many samples of $\beta$ from the posterior, for each sample solve the best-response mapping for strategies $p_i(\theta_i,\xi)$ (e.g. via projection on basis functions), and average to get predictive prices.

Inference about collusion

Researchers compare observed prices to model-implied competitive equilibrium prices under no explicit coordination. If observed prices are persistently above competitive model predictions, and alternative explanations (cost shocks, demand shifts, capacity constraints) are ruled out, this is evidence consistent with tacit collusion. Statistical tests involve likelihood-ratio comparisons between models with and without collusive strategies, or computing the posterior probability that behavior arises from coordinated strategies (using mixture models).

Summary of Mechanic 2

Equilibrium pricing under incomplete information requires computing best responses to beliefs over rival strategies. Dynamics and imperfect monitoring determine whether high-price equilibria (tacit collusion) or low-price cycles (price wars) prevail. Empirical implementation couples Bayesian estimation of reaction functions with structural equilibrium solving to make counterfactual and detection analyses feasible.

Overview

This section ties the mechanics to applied problems: antitrust detection of tacit collusion, firm pricing strategy in retail and platforms, and forecasting price wars. We also map to downstream techniques: structural IO estimation, counterfactual simulations, and causal inference with instrumental variables.

Tacit collusion detection in antitrust

Regulators face the challenge of distinguishing lawful parallel pricing from collusion. The framework developed here provides a structured approach:

1) Estimate reaction functions $\pi(\beta\mid\text{data})$ using panel price data for firms in the market (Section 2). For example, in a market with 10 retail chains each observed daily for a year, estimate how each firm responds to rivals’ discounts and to demand covariates.

2) Solve the structural model under two regimes: (a) non-cooperative Bayesian-Nash equilibrium given the estimated distribution over competitor behavior; (b) cooperative or collusive equilibrium where firms coordinate on higher prices subject to incentive constraints. The cooperative equilibrium can be computed by solving a constrained optimization with incentive compatibility constraints as in Section 3.

3) Compute likelihoods or posterior odds: which regime better explains the observed time series of prices after accounting for cost and demand shocks? A large likelihood advantage of the collusive model, robust to alternative shock specifications, is evidence for tacit collusion.

Concrete numeric example (synthetic): suppose competitive model predicts mean daily price 30 with SD 3, while observed mean is 36. The collusive model predicts mean 36 with SD 2. Using Gaussian likelihoods, log-likelihood ratio per day is approx

Over 365 days total LLR = -730, strongly favoring collusion model. (This is a stylized illustration; real tests require controlling for alternative explanations and multiple parameters.)

Retail pricing and dynamic promotions

Retail chains use Bayesian estimates of competitors’ matching probabilities and reaction slopes to design promotion depth and duration. If posterior $\beta_{j1}$ is high, deep temporary markdowns are unattractive because rivals will match and margins evaporate. Conversely, if matching probability is low, targeted steep discounts can steal share profitably.

Numeric policy rule: solve

If predictive rival mean price is 19 with SD 1.2 and demand elasticity implies that lowering price from 20 to 18 increases own demand from 1.5 to 2.0, the expected margin times demand calculation informs whether the promotion increases expected profit net of matching.

Platforms and algorithmic pricing

Algorithmic pricing introduces rapid feedback loops: firms respond in real time using learned reaction functions. Bayesian online updating is natural here: after each observation, update priors for $\beta$ and re-optimize price. But rapid updates can lead to unintentional coordination. Antitrust concerns arise when independent algorithms converge to collusive pricing without explicit communication:

• •Modeling algorithms as agents with Bayesian learning over $\beta$ helps predict whether convergence to high-price equilibria is likely.
• •Regulators may simulate counterfactuals under algorithmic updating rules to assess collusion risk.

Empirical methods and computation

Important practical tools:

• •MCMC (Gibbs, Metropolis-Hastings) to sample posterior over reaction parameters and latent indicators.
• •Simulation-based equilibrium solvers: project strategy functions onto basis (polynomials or splines), iterate best-response mapping to fixed point using Monte Carlo integration over types.
• •Instrumental variables (IV) or control-function methods when price endogeneity (common shocks) confounds reaction estimates. For instance, use cost shifters or lagged rivals’ costs as instruments.

Concrete empirical pipeline

1) Preprocess data: de-seasonalize prices and control for observables $x_t$ (promotions, cost proxies).

2) Estimate reaction function with a Bayesian structural model allowing mixture components for matching vs autonomous pricing.

3) Check predictive fit and posterior of $\beta_{j1}$.

4) Solve structural model for equilibrium prices under competitive and collusive regimes.

5) Perform likelihood ratio or posterior predictive checks to adjudicate between models.

Limitations and robustness

• •Identification: without valid instruments or exogenous variation, estimating causal reaction slopes is challenging. Spurious correlation from common demand shocks can mimic strategic response.
• •Model misspecification: assuming linear reaction functions when actual reactions are state-dependent (e.g., only match small markdowns) biases inference.
• •Endogenous entry/exit and multi-market structure complicate estimation. Hierarchical Bayesian models can pool information across markets while allowing market-specific heterogeneity.

Connections to active research

Recent literature studies algorithmic pricing and collusion, repeated games with private monitoring, and empirical tests distinguishing coordination from parallel adaptation. This lesson equips you with the structural and statistical tools central to that research agenda.

Summary

Bayesian estimation of competitor response functions feeds directly into equilibrium pricing computations that answer managerial and policy questions: will a price cut start a war? Are high prices evidence of tacit collusion? The applied pipeline uses panel estimation, simulation-based equilibrium solving, and careful identification strategies to make these judgments rigorous.