A Worked Likelihood for a Two-Firm Entry Game

From Strategic Payoffs to Data Probabilities

A two-firm entry game produces a pair of observed binary choices in each market: each firm either enters or stays out. Structural estimation asks how likely those observed choices are under a specified payoff model. The likelihood connects the data to payoff parameters by accounting for both firms’ incentives and the private information that makes entry uncertain to the analyst.

The construction requires a model of how private payoff shocks are distributed and a rule describing which equilibrium generates the observed outcomes. In this tutorial, firms have independent private shocks with standard logistic distributions, and the model specifies a unique Bayesian equilibrium for each market. These assumptions let us express the likelihood in terms of equilibrium entry probabilities. If the model instead permits multiple equilibria, a selection rule must be specified before the likelihood is defined.

Model: In market $m$, firm $i$'s payoff difference between entering and staying out is $$ \Delta_{im}=x_{im}'\beta_i-\delta_i y_{jm}+\varepsilon_{im}, \qquad i\ne j. $$ Here $y_{jm}\in\{0,1\}$ indicates whether the rival enters, $x_{im}$ is a vector of observed market and firm covariates, and $\beta_i$ is a vector of payoff coefficients. The parameter $\delta_i\geq 0$ measures how rival entry reduces firm $i$'s payoff (a strategic-substitutes effect). The private shocks $\varepsilon_{im}$ are independent across firms and markets, each with standard logistic cumulative distribution function $\Lambda(t)=1/(1+\exp(-t))$. Firm $i$ enters if and only if $\Delta_{im}\geq 0$.

Because each firm chooses before observing the rival’s action, its Bayesian best response depends on the probability that the rival enters, not on a rival action it has already seen. Write that probability as $q_{jm}$. Under the model, the equilibrium probability $q_{im}$ satisfies

q_{im}=\Lambda\!\left(x_{im}'\beta_i-\delta_i q_{jm}\right).

This equation follows by taking the probability that the firm’s payoff difference is nonnegative, given its belief about the rival: $P(\varepsilon_{im}\geq -[x_{im}'\beta_i-\delta_i q_{jm}])$. The logistic distribution gives the expression above. The entry probabilities are therefore functions of the payoff parameters and the market covariates, evaluated at the model’s specified equilibrium.

Likelihood for the Four Possible Outcomes

Let $Y_m=(y_{1m},y_{2m})$ denote the observed outcome in market $m$. Under the stated assumptions, each firm’s equilibrium decision is a threshold rule in its own private shock. Since the shocks are independent across firms, the two entry indicators are conditionally independent given the covariates and the equilibrium probabilities. Thus, for an outcome $(a,b)$, where $a,b\in\{0,1\}$,

P(Y_m=(a,b)\mid x_{1m},x_{2m}) =q_{1m}^{a}(1-q_{1m})^{1-a} q_{2m}^{b}(1-q_{2m})^{1-b}.

This gives the probabilities of $(1,1)$, $(1,0)$, $(0,1)$, and $(0,0)$ by substituting the corresponding values of $a$ and $b$. The conditional independence here comes from independent private shocks and the specified Bayesian-equilibrium strategies; it is not a general property of all games or all entry data.

Worked Example: Four Outcome Probabilities

Suppose a hypothetical market has symmetric equilibrium entry probability $q_{1m}=q_{2m}=0.5989$. Calculate the probabilities of all four outcomes.

For both firms entering, $P(1,1)=0.5989^2=0.35868121\approx 0.3587$. For exactly one specified firm entering, $P(1,0)=P(0,1)=0.5989(1-0.5989)=0.5989(0.4011)=0.24021879\approx 0.2402$. For neither firm entering, $P(0,0)=0.4011^2=0.16088121\approx 0.1609$.

The rounded probabilities sum to $0.3587+0.2402+0.2402+0.1609=1.0000$, as required. If the observed outcome is $(1,0)$, the market contributes $0.24021879$ to the likelihood. Its log-likelihood contribution is $\log(0.24021879)$, approximately $-1.426$.

Suppose the data contain $M$ independent markets and the covariates are treated as given. Let $\theta$ collect the payoff parameters. For each market, the model and its equilibrium condition determine $q_{1m}(\theta)$ and $q_{2m}(\theta)$. The sample likelihood is the product of the probabilities of the observed outcomes:

L(\theta)=\prod_{m=1}^{M} q_{1m}(\theta)^{y_{1m}}[1-q_{1m}(\theta)]^{1-y_{1m}} q_{2m}(\theta)^{y_{2m}}[1-q_{2m}(\theta)]^{1-y_{2m}}.

The log-likelihood is the sum of the market contributions. Maximizing it gives the maximum-likelihood estimate, provided the maximum exists and the parameters are identified by the model and observed covariate variation. Taking logs is useful computationally because it replaces products of small probabilities with sums; it does not change which parameter values maximize the objective.

Maximum-likelihood estimator: Given observed outcomes and covariates, a maximum-likelihood estimate $\hat{\theta}$ is a parameter value that maximizes $L(\theta)$, or equivalently $\ell(\theta)=\log L(\theta)$, over the parameter space. In a structural entry model, evaluating this objective requires computing the equilibrium probabilities implied by each candidate parameter value.

A Transparent Maximization with Covariates

A small symmetric example shows how observed entry frequencies translate into payoff coefficients. Assume each firm has payoff difference $a+bz_m-\delta q_m+\varepsilon_{im}$, where both firms in a market share the covariate $z_m$, the equilibrium entry probability is $q_m$, and $\delta=0.5$ is fixed and known for this illustration. The common equilibrium condition is

q_m=\Lambda(a+bz_m-0.5q_m).

The numerical example holds the strategic-effect parameter fixed to keep the arithmetic transparent. In a full structural estimation, one could include $\delta$ among the parameters to be estimated, subject to the model’s equilibrium and identification conditions.

Worked Example: Estimating Covariate Payoff Coefficients

Suppose hypothetical data contain 10 independent markets at $z=0$ and 10 at $z=1$, with two firms observed in each market. At $z=0$, the outcome counts are $(1,1):1$, $(1,0):3$, $(0,1):3$, and $(0,0):3$. At $z=1$, the counts are $3,3,3,1$, respectively. Estimate $a$ and $b$ with $\delta=0.5$.

At $z=0$, there are $8$ entries and $12$ nonentries across the $20$ firm-market decisions, so the empirical entry rate is $8/20=0.4$. At $z=1$, the corresponding rate is $12/20=0.6$. Conditional independence makes each group’s likelihood proportional to $q^{E}(1-q)^{20-E}$, where $E$ is the number of entries. Its log is $E\log q+(20-E)\log(1-q)$, which is maximized at $q=E/20$. Therefore the best-fitting group probabilities are $q_0=0.4$ and $q_1=0.6$.

Rearrange the equilibrium equation using the logit function, $\operatorname{logit}(q)=\log(q/(1-q))$:

\operatorname{logit}(q_m)+0.5q_m=a+bz_m.

At $z=0$, this gives $a=\log(0.4/0.6)+0.5(0.4)=-0.405465+0.2=-0.205465$, or $a\approx -0.2055$. At $z=1$, it gives $a+b=\log(0.6/0.4)+0.5(0.6)=0.405465+0.3=0.705465$. Thus $b=0.705465-(-0.205465)=0.910930$, or $b\approx 0.9109$.

These values reproduce the two fitted probabilities in the equilibrium equation. Since the two covariate groups have separate fitted probabilities when $a$ and $b$ vary freely, attaining each group’s binomial maximum also maximizes the likelihood over $a$ and $b$ in this fixed-$\delta$ example. The estimated covariate coefficient is in payoff-index units per one-unit increase in $z$; it is not itself a change in entry probability.

Evaluating and Maximizing the Structural Likelihood

In a more general data set, the equilibrium probabilities need not be available by direct rearrangement. The likelihood is still defined by the same product: for each candidate $\theta$, use the model’s equilibrium probabilities for each market, evaluate the probability of that market’s observed outcome, and sum the log contributions. A numerical optimizer then searches for parameter values with the largest log-likelihood. The exact equilibrium-solving method depends on the game; the essential point here is that the equilibrium probabilities must correspond to the candidate parameters being evaluated.

Specify the model and parameter space.
State payoff differences, shock distributions, parameter restrictions, and the equilibrium or selection rule. These choices determine what probability model is being estimated.

Evaluate equilibrium probabilities.
For each market and candidate parameter vector, obtain the entry probabilities implied by its covariates and the specified equilibrium.

Compute the observed-data log-likelihood.
For each market, use the probability of its actual pair of entry decisions, then add the log contributions across markets.

Optimize and assess the result.
Search over admissible parameter values, check that the solution respects the restrictions, and examine whether the observed covariate variation can distinguish the payoff parameters.

Worked Example: A Market’s Log-Likelihood Contribution

Suppose a hypothetical market has model-implied equilibrium probabilities $q_{1m}=0.5752$ and $q_{2m}=0.5075$, and the observed outcome is $(1,1)$. Calculate its likelihood contribution and log-likelihood contribution.

Conditional independence gives $P(Y_m=(1,1))=q_{1m}q_{2m}=0.5752(0.5075)=0.291914$. Therefore the market’s log-likelihood contribution is $\log(0.291914)\approx -1.23130$, or $-1.231$ to three decimal places. During maximization, this contribution is recalculated whenever the candidate parameters change the equilibrium probabilities.

Why the Assumptions Matter

A likelihood is only as appropriate as the probability model that defines it. In particular, the product of the firms’ entry probabilities assumes independent private shocks. If shocks are correlated, or if an unobserved market factor affects both firms’ payoffs, the joint outcome probabilities generally will not factor into two Bernoulli terms. A likelihood that ignores such dependence can misrepresent how often joint entry or joint nonentry occurs.

The unique-equilibrium assumption also matters. If the same covariates and parameters permit multiple equilibria, the model alone may not give a single probability for an observed outcome. The analyst must specify how an equilibrium is selected or otherwise model the observed outcome probabilities. A selection rule is part of the statistical specification, not a harmless computational detail.

Finally, covariates must provide information that separates payoff effects. If two payoff coefficients always enter the payoff index in the same way across all markets, their separate values cannot generally be learned from the likelihood. Excluded shifters and identification in entry games address this issue; a well-formed likelihood does not by itself guarantee identification.

Common Mistakes

Using independent Bernoulli probabilities without stating why. The factorization follows here from the assumed independent private shocks and equilibrium strategies. It should not be imposed automatically when the model includes correlated shocks or shared unobserved factors.
Treating observed rival entry as known when the decision is made. In this private-information model, a firm’s entry probability depends on its belief about rival entry. The equilibrium probability $q_j$, rather than a rival action observed in advance, enters the payoff calculation.
Maximizing over payoff indices while holding probabilities fixed. The probabilities depend on the candidate payoff parameters through equilibrium. Recompute them when evaluating the likelihood at a new parameter value.
Calling a high likelihood proof of a correct game model. Maximization selects the best-fitting parameter values within the specified model. It does not establish that the payoff specification, shock distribution, or equilibrium-selection assumption is correct.

Key takeaway: A worked entry-game likelihood follows a precise chain: specify payoffs and private information, obtain the model-implied equilibrium probabilities, assign a probability to each observed outcome, and maximize the sum of log probabilities across markets. Every link in that chain depends on the stated assumptions.

Check Your Understanding

Answer each question using the model and likelihood construction in this tutorial.

Under what assumptions does the probability of outcome $(1,0)$ equal $q_{1m}(1-q_{2m})$?
In the symmetric model with $\delta=0.5$, how would you recover the payoff index $a+bz_m$ from an equilibrium probability $q_m$?
Why must the equilibrium probabilities be recalculated as candidate payoff parameters change during likelihood maximization?
What additional modeling ingredient is needed if more than one equilibrium is consistent with a market’s covariates and parameters?
Explain why maximizing a likelihood cannot, by itself, show that the private-shock distribution is correctly specified.