Why Entry-Game Counterfactuals Need More Than Payoff Estimates
A counterfactual for an entry game asks what firms would do under market conditions different from those observed. Changing a fee, a market-size shifter, or another policy-relevant variable can change firms’ entry incentives. But even with a specified payoff model, the predicted outcome may not be unique: more than one entry pattern can satisfy the equilibrium conditions. A counterfactual therefore depends not only on estimated payoffs, but also on how an equilibrium is selected and whether the new conditions are covered by the data.
The central diagnostic is to separate three questions. Does the proposed market change alter payoffs in a way supported by the model? Does the resulting game have a unique equilibrium, or several? And, if it has several, is the rule used to choose among them justified and held fixed? A prediction can be numerically precise while remaining sensitive to unsupported choices on any of these questions.
For clarity, the examples below use complete-information games with known covariates and a tie convention that is not relevant to the calculations: all entry incentives considered are strictly positive or strictly negative. In an empirical application, unobserved payoff components or uncertain parameter estimates may add uncertainty, but they do not remove the need to address equilibrium multiplicity and selection.
A Two-Firm Game and Its Equilibrium Conditions
Consider a symmetric entry game with strategic substitutes. Each entrant earns a baseline payoff \(a\), while entry by the rival reduces that payoff by \(d>0\). Staying out gives payoff zero. Thus firm \(i\)’s gain from entering is
A firm enters when its gain is positive and stays out when its gain is negative. When \(0<a<d\), entering is profitable if the rival stays out but unprofitable if the rival enters. The resulting multiplicity can be verified directly.
Firm 1 earns \(a>0\) by entering. Firm 2’s gain from deviating to entry is \(a-d<0\). Neither firm wants to switch actions.
By symmetry, firm 2 earns \(a>0\) by entering, and firm 1’s gain from entering is \(a-d<0\). This profile is also an equilibrium.
At \((0,0)\), either firm can gain \(a>0\) by entering. At \((1,1)\), either firm can gain \(d-a>0\) by staying out. Neither profile is an equilibrium.
This gives two equilibria, \((1,0)\) and \((0,1)\). The derivation matters for counterfactual work: a change in \(a\) can move the game into or out of the multiple-equilibrium region, but it does not by itself say which of the two firms enters when both profiles remain equilibria.
Worked Example: A Change in Selection Rule Changes the Prediction
Worked Example: A Change in Selection Rule Changes the Prediction
Suppose a hypothetical market has two symmetric firms, with \(a=0.6\) and \(d=1\). The estimated payoff model is unchanged between the observed and counterfactual conditions. Compare a rule that selects the equilibrium with firm 1 entering to a rule that selects the equilibrium with firm 2 entering.
Check the equilibrium conditions. Here \(0<0.6<1\), so the preceding derivation applies. At \((1,0)\), firm 1’s entry gain is \(0.6\), while firm 2’s gain from entering is \(0.6-1=-0.4\). The profile is an equilibrium. At \((0,1)\), the same calculations with the firm labels reversed show that it is also an equilibrium.
Apply the two rules. A rule selecting \((1,0)\) predicts that firm 1 enters and firm 2 stays out. A rule selecting \((0,1)\) predicts the reverse. Both predictions satisfy the same payoff model and equilibrium conditions; their difference comes entirely from the selection rule.
Interpret the comparison. The number of entrants is one under either rule, but the predicted identity of the entrant changes. A counterfactual report about firm-level market shares would therefore be sensitive to selection even though the aggregate entry count is not. The payoff model alone does not establish that one firm, rather than the other, will enter.
A common error is to use one selection rule to describe the observed game and silently use another for the counterfactual. That change can create an apparent response to policy even when the payoff model and equilibrium set do not change. A defensible analysis should state the rule, explain why it is appropriate, and either apply the same rule consistently or justify why the counterfactual environment changes the selection process.
Worked Example: Reporting Ambiguity When Equilibria Multiply
Worked Example: Reporting Ambiguity When Equilibria Multiply
Now consider a hypothetical game with strategic complements. Firm \(i\)’s gain from entering is \(\Delta_i(x_{-i})=a+\gamma x_{-i}\), where \(a=-0.4\) and \(\gamma=1\). Staying out again gives payoff zero. The counterfactual leaves these payoff values fixed.
Check the no-entry profile. At \((0,0)\), either firm’s gain from entering is \(-0.4\). Both prefer to stay out, so \((0,0)\) is an equilibrium.
Check the both-enter profile. At \((1,1)\), either firm’s gain from entering is \(-0.4+1=0.6\). Both prefer to enter, so \((1,1)\) is also an equilibrium.
Check the asymmetric profiles. At \((1,0)\), the entrant’s gain is \(-0.4\), so that firm would exit. At \((0,1)\), the same argument applies to the other entrant. Neither asymmetric profile is an equilibrium.
The model therefore allows either zero or two entrants. If a specified selection process chooses \((1,1)\) with probability \(0.7\) and \((0,0)\) with probability \(0.3\), the expected number of entrants is
That value is an expectation under the stated selection probabilities; it is not an equilibrium with 1.4 firms entering in a particular market. If the probabilities are not known or justified, the payoff model supports the two outcomes but does not supply this expected value. Reporting the range of equilibrium entry counts, zero to two here, is more transparent than presenting one selected prediction as if it were uniquely implied.
When there are multiple equilibria, a model may still support useful counterfactual conclusions. Some outcomes may be common across every equilibrium; others may vary. For a fixed market, one can report the set of equilibrium predictions or bounds on an outcome across that set. For averages across markets, however, bounds must respect any restrictions on how selection can vary across markets. Pointwise ranges should not automatically be treated as an attainable aggregate range.
Worked Example: Extrapolating Payoffs Beyond Observed Conditions
Worked Example: Extrapolating Payoffs Beyond Observed Conditions
Suppose a hypothetical analysis observes a market shifter \(z\) only between 0 and 2. Over that range, estimated entry incentives follow \(a(z)=0.2+0.2z\), and the strategic-substitutes parameter is \(d=0.8\). A proposed policy sets \(z=6\). Compare the prediction from extending the linear formula with a hypothetical payoff function that levels off after \(z=2\), \(a_{\text{alt}}(z)=0.2+0.2\min(z,2)\).
Extend the fitted line. At \(z=6\), the linear specification gives \(a(6)=0.2+0.2(6)=1.4\). Since \(1.4>0.8\), a firm’s gain from entry remains positive even when the rival enters: \(1.4-0.8=0.6>0\). Under this payoff specification, both firms entering is the unique equilibrium.
Evaluate the alternative function. At \(z=6\), the alternative gives \(a_{\text{alt}}(6)=0.2+0.2(2)=0.6\). Since \(0<0.6<0.8\), the strategic-substitutes derivation applies: \((1,0)\) and \((0,1)\) are both equilibria. The linear extrapolation predicts two entrants; the alternative payoff function permits one entrant.
Interpret the difference. The functions agree over the observed range and differ only in how they behave beyond it. Thus the two counterfactual conclusions cannot be distinguished using those observed values alone. The alternative is illustrative, not evidence that the true payoff function levels off. The example shows why a prediction far outside observed conditions depends on functional-form assumptions that the data may not test.
Extrapolation risk and equilibrium selection are separate issues. Even if a payoff function were trusted at \(z=6\), a multiple-equilibrium game would still require a selection rule. Conversely, a unique equilibrium under an extrapolated payoff function does not make the extrapolation itself well supported. Counterfactual reports should identify both which parts of the prediction rely on extrapolation and whether equilibrium selection affects the result.
A Practical Audit for Counterfactual Predictions
Before interpreting a structural entry-game counterfactual, check the chain from market change to predicted outcome. Each step answers a different question; passing one check does not guarantee that the other steps are sound.
Specify the counterfactual market conditions and distinguish those changes from fixed model features, such as the payoff parameters or other firms’ behavior.
Determine whether the counterfactual lies within the observed support of relevant conditions. If not, describe the functional-form assumptions needed to calculate payoffs there.
Apply the equilibrium conditions to the counterfactual game. Do not stop after finding one equilibrium when others may also satisfy those conditions.
If several equilibria remain, report the selection rule or selection probabilities and explain which conclusions depend on them. If no rule is supported, report the ambiguity rather than hiding it.
Distinguish findings shared by all equilibria and reasonable payoff specifications from findings that change with the selection rule or extrapolation assumptions.
Common Pitfalls
- Changing the selection rule without saying so. Wrong reasoning: “The counterfactual predicts firm 2 enters, although the observed fit selected firm 1.” Correction: check whether the change comes from payoffs, a changed selection assumption, or both. State any change in the selection process explicitly.
- Treating one computed equilibrium as the only equilibrium. Wrong reasoning: “The solver returned this profile, so the model predicts it.” Correction: verify the equilibrium conditions for other profiles. A computational procedure can return one solution without establishing uniqueness.
- Presenting a selected outcome as identified by payoffs alone. Wrong reasoning: “Both firms have the same estimated payoffs, so firm 1 must enter.” Correction: symmetric payoffs can support multiple asymmetric equilibria. A firm-specific prediction needs a selection justification.
- Assuming a good fit near observed markets validates distant predictions. Wrong reasoning: “The line fits the observed range, so its prediction at a much larger \(z\) is reliable.” Correction: a fit within the observed range does not establish how payoffs behave outside it. Show sensitivity to plausible extrapolation assumptions.
- Confusing an expected count with a realized equilibrium. Wrong reasoning: “An expected count of 1.4 means 1.4 firms enter.” Correction: the expectation averages across possible outcomes under stated selection probabilities; each realized two-firm outcome has an integer number of entrants.
Check Your Understanding
Use the equilibrium conditions and distinctions in this tutorial to answer the following questions.
- In the strategic-substitutes game with \(a=0.3\) and \(d=0.7\), identify all equilibria and verify the entry incentives that support them.
- Suppose two selection rules choose different equilibria from the same equilibrium set. What part of the counterfactual prediction changes, and what has not changed?
- In the strategic-complements example, why is an expected entrant count of 1.4 not a realized equilibrium outcome?
- What additional evidence or assumptions would be needed to distinguish two payoff functions that agree throughout the observed range but differ at a counterfactual value of \(z\)?
- A numerical routine returns one equilibrium for a counterfactual game. What must be checked before describing that equilibrium as uniquely predicted?