Identification in Entry Games Using Excluded Shifters

Why Excluded Shifters Matter

In an entry game, a firm’s decision depends not only on its own profitability but also on whether a rival enters. That makes observed entry outcomes jointly determined: a high entry rate for one firm may reflect favorable market conditions, a strong payoff from facing an entrant, or both. An excluded shifter offers a way to separate these explanations. It changes one firm’s profitability while leaving the rival’s payoff unchanged, creating variation in the first firm’s entry probability that can reveal how the rival responds.

The key is not merely to find a variable associated with entry. It must have a defensible exclusion from the rival’s payoff, sufficient variation in the relevant conditional probabilities, and a specified account of equilibrium behavior. The resulting identification is structural: it concerns a payoff parameter under the maintained model, not a causal effect inferred from entry correlations alone.

Model and Excluded Shifter

Consider two firms, \(j=1,2\), in market \(i\). Firm \(j\) chooses \(y_{ij}\in\{0,1\}\), where 1 means enter. Let the difference between its payoff from entering and staying out be

Here \(x_{ij}\) contains observed own-payoff characteristics, \(a_j(x_{ij})\) is the part of the entry payoff determined by them, \(\delta_j\) is the payoff effect of the rival entering, and \(\varepsilon_{ij}\) is a private payoff shock. The sign of \(\delta_j\) describes the strategic effect in firm \(j\)’s payoff: positive values mean entry by the rival raises the payoff from entering; negative values mean it lowers that payoff.

For the identification argument below, assume private information and conditional independence of the firms’ payoff shocks given observed covariates. Normalize each shock to have a standard logistic distribution, so its cumulative distribution function is \(\Lambda(t)=1/(1+\exp(-t))\). This fixes the payoff scale. Assume firms use Bayesian best responses and that the conditional equilibrium entry probabilities are well-defined and stable as the excluded shifter varies. If a model admits multiple equilibria, the equilibrium selection rule must also be specified and maintained.

Exclusion is a substantive assumption. A cost change specific to firm 1 could qualify if it does not also change firm 2’s costs, demand, information, or market composition. A variable that directly changes demand for both firms is not an excluded shifter for firm 1 merely because its association with firm 2’s entry is weaker.

Entry Probabilities and the Identifying Variation

Let \(p_{ij}\) denote firm \(j\)’s conditional equilibrium probability of entry at a given set of market covariates. Under the assumptions above, firm 2 enters when its payoff difference is positive. Averaging over its logistic private shock gives its equilibrium best-response condition:

The term \(p_{i1}\) appears because, conditional on the covariates, firm 2 expects firm 1 to enter with probability \(p_{i1}\). This probability-based expression depends on the stated incomplete-information model; it is not a general formula for every entry game.

If \(z_{i1}\) is excluded from firm 2’s payoff, then changing \(z_{i1}\) can change \(p_{i1}\) without directly changing \(a_2(x_{i2})\). Taking the logit transformation of the equilibrium condition gives

At fixed firm 2 covariates, this is a relationship between firm 2’s conditional log-odds of entry and firm 1’s conditional entry probability. The slope with respect to \(p_{i1}\) is the strategic payoff parameter \(\delta_2\), given the logistic scale normalization. This is the source of identification: the excluded shifter moves firm 1’s entry probability, and the change in firm 2’s conditional entry probability traces out its response.

A Short Identification Derivation

Suppose \(z_{i1}\) is continuous, and consider changes in it at fixed \(x_{i2}\). The logit equation for firm 2 is

Differentiate both sides with respect to \(z_{i1}\). Since firm 2’s payoff index \(a_2(x_{i2})\) does not depend on the excluded shifter, its derivative is zero:

Where \(dp_{i1}/dz_{i1}\ne 0\), the structural response is therefore

This is a relationship between equilibrium probabilities. The change in \(p_{i1}\) may itself reflect strategic feedback from firm 2; that feedback does not invalidate the derivative relation, because the relation follows from firm 2’s equilibrium condition at each value of \(z_{i1}\). It does mean that the excluded shifter’s total effect on firm 2’s entry is not the same thing as \(\delta_2\).

Support, Scale, and What Is Identified

An exclusion does not identify a slope unless it generates variation. If \(z_{i1}\) changes but \(p_{i1}\) remains constant at fixed \(x_{i2}\), then the equation for firm 2 supplies no variation in its \(p_{i1}\) argument. A constant value of \(p_{i1}\) can be absorbed into the intercept \(a_2(x_{i2})+\delta_2p_{i1}\); the data cannot separate that intercept from \(\delta_2\).

With discrete shifters, at least two relevant values of \(p_{i1}\) are needed to identify a linear slope in the log-odds equation, provided the other conditions hold. With richer support, the response can be traced over a wider range. If the observed support covers only a narrow interval, the coefficient may be weakly identified in practice, and conclusions outside that interval rely on the assumed functional form rather than direct variation.

The logistic scale normalization matters as well. If the payoff shock has an unknown scale, the choice probability depends on payoff coefficients divided by that scale. Without a normalization or other scale information, the data do not identify the absolute payoff units of \(\delta_2\). The sign and relative magnitudes may still be informative under additional conditions, but a coefficient should not be reported as a scale-free payoff effect.

The same reasoning applies in the opposite direction if there is an excluded shifter for firm 2. In general, identifying both firms’ strategic effects requires suitable excluded variation for the relevant firm-specific response equations, or other defensible restrictions that supply equivalent information. Symmetry of the firms’ effects should not be imposed merely because it simplifies the analysis.

Why Exclusion Must Be Defended

If \(z_{i1}\) directly affects firm 2’s payoff, the observed response combines two channels: the effect of a change in firm 1’s entry probability and the direct change in firm 2’s entry incentives. For example, if firm 2’s index is \(a_2(x_{i2})+\gamma z_{i1}+\delta_2p_{i1}\), then changes in \(z_{i1}\) cannot isolate \(\delta_2\) unless the direct term \(\gamma z_{i1}\) is modeled and separately identified. Calling \(z_{i1}\) an instrument does not make that direct pathway disappear.

Common Mistake and Practical Interpretation

A common mistake is to interpret a positive association between the firms’ entry outcomes as evidence of strategic complementarity. Markets with favorable demand can encourage both firms to enter even if each firm’s payoff falls when its rival enters. Conversely, strategic complementarity does not guarantee a positive raw association if firms face offsetting cost differences. The excluded-shifter argument instead compares conditional equilibrium probabilities under a specified payoff model and relies on a variable that changes one firm’s profitability but not the other’s.

A second pitfall is to treat any small amount of variation as adequate support. If the excluded shifter barely moves \(p_{i1}\), the identifying ratio can be highly sensitive to probability estimation error. Identification is a population property under assumptions; precision is a separate question about the available data. A credible analysis should explain the economic exclusion, show that the shifter changes the relevant entry probability, state the support being used, and make its equilibrium and scale assumptions explicit.