Incomplete-Information Entry Games

Why Private Information Changes Entry

In a complete-information entry game, each firm knows the payoffs that determine whether entry is profitable. In many markets, however, profitability also depends on a firm-specific shock observed by that firm but not by its rival. One firm may know that its own costs are unusually low; another cannot observe that information directly. It must instead form a belief about the rival’s entry decision.

The result is not necessarily a single predictable entry decision for each firm. Given a belief about rival entry, a firm has a cutoff rule for its privately observed profitability shock. Across firms, markets, or repeated draws of shocks, that rule generates an entry probability. This tutorial develops that decision rule and shows how beliefs map into probabilistic entry outcomes.

A Binary Entry Game with Private Shocks

Consider two firms, \(i=1,2\), choosing whether to enter a market. Let \(a_i\in\{0,1\}\) denote firm \(i\)’s action, with \(a_i=1\) meaning entry and \(a_i=0\) meaning non-entry. The firms observe public market characteristics \(x\). Each firm also observes its own profitability shock, \(\varepsilon_i\), but not its rival’s shock.

The term \(d_i\) represents strategic substitutability: rival entry makes firm \(i\)’s entry less profitable, as can happen when firms compete for the same customers. The model assumes that \(\pi_i(x)\) and \(d_i\) are known to both firms, while the realizations of the shocks are private. The shock distributions are common knowledge. For the probability calculations below, assume that shocks are continuously distributed and independent across firms conditional on \(x\). Continuity makes exact ties occur with probability zero.

Before observing the rival’s action, firm \(i\) has a belief \(q_i\) about the probability that firm \(j\) enters, conditional on the information available to firm \(i\). Under independent private shocks, the rival’s entry is independent of firm \(i\)’s own shock conditional on public information and the specified belief. This is why a single probability \(q_i\) can be used when calculating expected payoff.

Bayesian Best Response and Entry Probability

Given its belief \(q_i=P(a_j=1\mid\text{firm }i\text{’s information})\), firm \(i\) evaluates entry using its expected payoff. Non-entry yields zero. Entry yields \(\pi_i(x)+\varepsilon_i\) if the rival stays out and \(\pi_i(x)-d_i+\varepsilon_i\) if the rival enters. Therefore the expected payoff difference between entry and non-entry, conditional on the firm’s own shock, is

The firm’s expected payoff from entry is nonnegative exactly when its shock is at or above the cutoff \(d_iq_i-\pi_i(x)\). A higher belief that the rival enters raises the cutoff when \(d_i>0\), because the firm expects tougher competition. The private shock can offset that expected disadvantage.

This expression is a best-response probability conditional on a belief, not a claim that the belief is correct. To turn the expression into a probability of entry, the shock distribution must be specified. For example, if \(\varepsilon_i\) has a logistic distribution with location zero and scale \(\sigma_i>0\), then

The scale \(\sigma_i\) controls how dispersed profitability is across firms or markets. A larger scale spreads out shocks and makes entry less tightly determined by baseline profitability and the rival-entry belief. The logistic formula is one useful specification; the cutoff rule itself applies to any continuous shock distribution.

Worked Logistic Best Response

With logistic shocks, the entry probability has a compact expression in terms of expected profitability after accounting for the rival-entry belief. The next example calculates that probability directly.

From Individual Probabilities to Market Outcomes

If firms’ private shocks are independent and their strategies depend on their own shocks and public information, then their entry decisions are conditionally independent given \(x\). When firm 1 enters with probability \(p_1\) and firm 2 with probability \(p_2\), the probabilities of the four possible market outcomes are products of the corresponding entry and non-entry probabilities.

These probabilities describe the distribution of market outcomes across repeated draws under the specified strategies and shock distributions. Independence matters: if shocks are correlated, knowing that one firm’s private profitability is high can change the probability that the other firm also has a high shock. The product formulas then need not apply.

Beliefs, Information, and Interpretation

A belief is conditional on what a firm knows. If firms observe different public signals, their beliefs about rival entry may differ even when they face the same underlying shock distributions. For instance, a signal that makes market demand look strong may lead a firm to assign a higher probability to rival entry. That belief changes the firm’s expected competitive loss and therefore its entry cutoff.

The independence assumption also clarifies when a probability \(q_i\) is enough to summarize uncertainty about the rival. If firms’ shocks are correlated, firm \(i\)’s own shock may reveal information about firm \(j\)’s shock. Then the probability of rival entry can depend on the realized \(\varepsilon_i\). The expected payoff calculation must use the firm’s conditional belief \(P(a_j=1\mid\varepsilon_i,\text{public information})\), rather than one constant \(q_i\). Likewise, if the rival’s strategy depends on additional private information, the belief must account for that information through the probability distribution known to firm \(i\).

In an equilibrium, beliefs about rivals’ actions must be compatible with how those rivals actually choose. The best-response calculation gives a firm’s decision rule conditional on its belief; equilibrium analysis additionally checks whether the collection of beliefs and decision rules fit together. Keeping these two steps distinct prevents a conditional entry probability from being mistaken for an equilibrium result.

Common Mistakes

• Treating the rival’s entry as certain. A belief \(q_i\) is a probability, not a realized action. The expected competitive loss is \(d_iq_i\), not automatically \(d_i\).
• Using the wrong sign for the cutoff. Entry requires \(\pi_i(x)-d_iq_i+\varepsilon_i\geq 0\), so the shock must satisfy \(\varepsilon_i\geq d_iq_i-\pi_i(x)\).
• Confusing a best response with an equilibrium. A calculated entry probability follows from a given belief. It is not an equilibrium unless beliefs are also consistent with the firms’ strategies.
• Multiplying marginal probabilities without checking independence. The product formulas for joint outcomes require conditionally independent entry decisions. Correlated private information can invalidate them.
• Interpreting an entry probability as a causal effect. A probability describes choices under a specified model and information structure. It does not, by itself, show that changing a market characteristic causes the observed difference.