Equilibrium Conditions in Binary Entry Models

Why Entry Outcomes Require Payoff Inequalities

In an entry game, each firm chooses whether to operate in a market, and the payoff to entering can depend on whether the rival also enters. The key equilibrium question is not simply whether entry is profitable in isolation. It is whether either firm would gain by changing its own decision while the rival’s decision is held fixed. Writing those comparisons explicitly gives a direct way to classify possible entry outcomes.

We consider a complete-information game: both firms know the payoff parameters. Each firm $i\in\{1,2\}$ chooses an action $y_i\in\{0,1\}$, where $y_i=1$ means enter and $y_i=0$ means stay out. Staying out yields payoff zero. If firm $i$ enters, its net payoff is $a_i+b_i y_j$, where $j\ne i$. Thus $a_i$ is firm $i$'s net payoff from entering when its rival stays out, and $b_i$ is the change in that payoff when the rival enters. All payoffs are in the same units, such as thousands of dollars of profit.

Definition: In this binary entry game, firm $i$'s payoff is $$ u_i(y_i,y_j)= \begin{cases} 0, & y_i=0,\\ a_i+b_i y_j, & y_i=1. \end{cases} $$ A pure-strategy Nash equilibrium is an entry profile $(y_1,y_2)$ at which each firm’s chosen action maximizes its own payoff given the other firm’s action. The equilibrium conditions use weak inequalities: a firm may be indifferent between its two actions and still be choosing a best response.

The sign of $b_i$ describes how rival entry changes firm $i$'s entry payoff. If $b_i>0$, rival entry raises that payoff; if $b_i<0$, rival entry lowers it. The firms need not have the same values of $a_i$ or $b_i$. In particular, the inequalities below do not assume symmetry.

Deriving the Four Equilibrium Conditions

For each candidate profile, compare a firm's payoff from its prescribed action with the payoff from switching actions, holding the rival's action fixed. The four profiles exhaust the possible pairs of binary decisions.

Both firms stay out.
Firm $i$ must not gain by entering when its rival stays out. Entering would yield $a_i$, so staying out is optimal exactly when $a_i\leq 0$.

Firm 1 enters and firm 2 stays out.
Firm 1's entry payoff against a nonentrant is $a_1$, which must be at least zero. Firm 2's payoff from entering against firm 1 is $a_2+b_2$, which must be at most zero.

Firm 1 stays out and firm 2 enters.
Firm 1's payoff from entering against firm 2 is $a_1+b_1$, which must be at most zero. Firm 2's entry payoff against a nonentrant is $a_2$, which must be at least zero.

Both firms enter.
Firm $i$'s entry payoff is $a_i+b_i$, so each must be at least zero for entry to be optimal.

Theorem: The pure-strategy equilibrium conditions are $$ \begin{aligned} (0,0)\text{ is an equilibrium} &\iff a_1\leq 0,\quad a_2\leq 0,\\ (1,0)\text{ is an equilibrium} &\iff a_1\geq 0,\quad a_2+b_2\leq 0,\\ (0,1)\text{ is an equilibrium} &\iff a_1+b_1\leq 0,\quad a_2\geq 0,\\ (1,1)\text{ is an equilibrium} &\iff a_1+b_1\geq 0,\quad a_2+b_2\geq 0. \end{aligned} $$

For example, at $(1,0)$, it is not enough that firm 1 wants to enter: firm 2 must also prefer staying out when facing an entrant. The condition $a_2+b_2\leq 0$ is precisely that second best-response requirement. Each equilibrium condition is a pair of restrictions, one for each firm's incentive.

Worked Example: A Unique Asymmetric Entry Outcome

Suppose hypothetical payoff parameters, measured in thousands of dollars, are $a_1=1$, $b_1=-2$, $a_2=-1$, and $b_2=-1$. Identify all pure-strategy equilibria.

For $(0,0)$, the conditions are $a_1\leq0$ and $a_2\leq0$. Although $-1\leq0$ for firm 2, $1\leq0$ is false for firm 1, so $(0,0)$ is not an equilibrium.

For $(1,0)$, $a_1=1\geq0$, and $a_2+b_2=-1-1=-2\leq0$. Both conditions hold, so $(1,0)$ is an equilibrium. Firm 1 earns $1$ thousand dollars; firm 2 earns zero and would obtain $-2$ thousand dollars by entering against firm 1.

For $(0,1)$, firm 1's condition holds because $a_1+b_1=1-2=-1\leq0$, but firm 2's condition $a_2\geq0$ fails because $-1<0$. For $(1,1)$, firm 1's condition $a_1+b_1\geq0$ fails because $-1<0$. Therefore $(1,0)$ is the unique pure-strategy equilibrium.

Regions with Multiple or No Pure Equilibria

Each profile's inequalities define a region of the four-dimensional parameter space $(a_1,b_1,a_2,b_2)$. A parameter vector can satisfy the conditions for more than one profile, particularly on boundaries where a firm is indifferent. It can also fail all four pairs of conditions. The next examples illustrate both possibilities.

Worked Example: Two Equilibria Under Complementary Entry Incentives

Suppose both firms have $a_i=-1$ and $b_i=2$. These hypothetical payoffs mean that entering alone yields a loss of $1$ thousand dollars, while entry by the rival raises the net entry payoff by $2$ thousand dollars.

At $(0,0)$, both firms satisfy $a_i=-1\leq0$, so neither wants to enter alone. At $(1,1)$, both satisfy $a_i+b_i=-1+2=1\geq0$, so neither wants to exit when the rival is active.

The two off-diagonal profiles are not equilibria. At $(1,0)$, firm 1's entry condition $a_1\geq0$ fails; at $(0,1)$, firm 2's entry condition $a_2\geq0$ fails. Hence there are exactly two pure-strategy equilibria, $(0,0)$ and $(1,1)$. Whether the firms stay out or both enter depends on which equilibrium is realized; the inequalities alone establish that both are consistent with mutual best responses.

Worked Example: A Parameter Region with No Pure Equilibrium

Now let $a_1=1$, $b_1=-2$, $a_2=-1$, and $b_2=2$, again in thousands of dollars. Firm 1's entry payoff is $1$ when firm 2 stays out and $-1$ when firm 2 enters. Firm 2's entry payoff is $-1$ when firm 1 stays out and $1$ when firm 1 enters.

At $(0,0)$, firm 1 can gain $1$ by entering, so the profile fails. At $(1,0)$, firm 2 can gain $a_2+b_2=-1+2=1$ by entering, so that profile fails. At $(0,1)$, firm 2 prefers staying out because entering would yield $a_2=-1$, while firm 1 also prefers staying out because entering would yield $a_1+b_1=1-2=-1$. Thus $(0,1)$ is not an equilibrium: firm 2 has a profitable deviation from entry to staying out. Finally, at $(1,1)$, firm 1 can gain $1$ by exiting because its entry payoff is $-1$. No profile satisfies both firms' best-response conditions, so there is no pure-strategy equilibrium for these parameters.

This conclusion is specifically about pure strategies, which require each firm to choose one action with certainty. It does not establish that the game has no Nash equilibrium of any kind. In particular, existence of a mixed-strategy equilibrium is a separate question; the classification here is only of the four deterministic entry profiles.

Boundary Cases and Indifference

Weak inequalities matter. If a firm’s entry payoff equals zero given the rival’s action, entering and staying out give the same payoff, so both actions are best responses. As a result, an equality can make two or more profiles equilibria at the same parameter values. This overlap is not a contradiction: Nash equilibrium requires that no firm can improve by deviating, not that every firm has a unique best response.

Worked Example: Indifference Creates Multiple Equilibria

Let $a_1=0$, $b_1=-1$, $a_2=-1$, and $b_2=-1$. At $(0,0)$, firm 1 satisfies $a_1=0\leq0$, and firm 2 satisfies $a_2=-1\leq0$, so $(0,0)$ is an equilibrium. Firm 1 is indifferent between entering and staying out when firm 2 stays out.

At $(1,0)$, firm 1 satisfies $a_1=0\geq0$, while firm 2 satisfies $a_2+b_2=-1-1=-2\leq0$. Thus $(1,0)$ is also an equilibrium. Firm 1 earns zero whether it enters or stays out against firm 2's non-entry. At $(0,1)$, firm 2's condition $a_2\geq0$ fails, and at $(1,1)$ firm 1's condition $a_1+b_1=-1\geq0$ fails. There are exactly two pure equilibria.

A useful check is to translate an inequality back into the relevant deviation. For instance, at $(0,1)$, firm 2 is the entrant and faces a nonentering rival. Its entry payoff is $a_2$, not $a_2+b_2$. If $a_2=-2$, firm 2 strictly prefers staying out, since $0>-2$; it is not indifferent. By contrast, if $a_2=0$, it is indifferent. Keeping track of the rival's action prevents an incorrect equilibrium classification.

Common Mistakes and a Practical Check

Checking only the active firm. At $(1,0)$, verifying that firm 1 wants to enter is insufficient. Firm 2 must also prefer to stay out, requiring $a_2+b_2\leq0$.
Using the wrong entry payoff. Against an entrant, firm $i$'s payoff from entering is $a_i+b_i$; against a nonentrant, it is $a_i$. The rival's action determines which expression applies.
Treating equality as a strict preference. If an entry payoff is zero, the firm is indifferent, not strictly better off entering. Either action can be a best response.
Assuming every game has a pure entry outcome. These four inequality pairs can all fail, as in the no-pure-equilibrium example. This does not rule out a mixed-strategy equilibrium.
Calling multiple equilibria a calculation error. Multiple profiles can each satisfy both firms' best-response conditions, especially when entry by one firm raises the other's payoff or when a firm is indifferent.

Key takeaway: For each candidate entry profile, compare each firm's payoff from its prescribed action with the payoff from switching, holding the rival's action fixed. The resulting four pairs of weak inequalities determine exactly which pure-strategy outcomes are equilibria for the specified payoff parameters.

Check Your Understanding

Use the entry-payoff model above. For each proposed profile, test both firms' incentives rather than checking only one firm's action.

Write the two inequalities required for $(0,1)$ to be a pure-strategy equilibrium, and explain which payoff each inequality compares.
If $a_1=-2$, $b_1=3$, $a_2=-1$, and $b_2=2$, determine whether $(0,0)$ and $(1,1)$ are equilibria.
Why can an equality such as $a_1=0$ permit both $(0,0)$ and $(1,0)$ to be equilibria?
At $(0,1)$, if $a_2=-2$, is firm 2 indifferent between entry and staying out? Explain using its two payoffs.
What does it mean when none of the four profiles satisfies its equilibrium inequalities, and what does that conclusion not establish?