Complete-Information Entry Games

Entry as a Complete-Information Game

A firm deciding whether to enter a market must consider not only whether entry is profitable, but also which rivals will enter. If several firms’ profits depend on the same market conditions and on one another’s participation, their choices form a strategic game. In a complete-information entry game, the firms know the payoff functions and market conditions relevant to the game before choosing. In a simultaneous-move game, each firm chooses whether to enter without first observing the other firms’ choices.

The equilibrium question is not simply whether each firm would like to operate in the market in isolation. A pure-strategy equilibrium is a profile of entry decisions in which no firm can improve its payoff by changing its own decision alone, holding all rivals’ decisions fixed. This makes it essential to check two kinds of incentives: whether an entrant wants to exit, and whether a nonentrant wants to join.

Write \(x\) for observed market conditions, such as market size or demand, and let \(a_{-i}\) denote the choices of all firms other than firm \(i\). Let \(E_i(a_{-i};x)\) be firm \(i\)’s payoff if it enters when rivals choose \(a_{-i}\), and let \(O_i(a_{-i};x)\) be its payoff if it stays out. These payoffs can include revenues, operating costs, fixed entry costs, and any value associated with remaining outside the market. Normalizing the non-entry payoff to zero is convenient in some examples, but it is an assumption about payoffs, not part of the definition of an entry game.

Equilibrium Conditions: Check Each Firm’s Deviation

A profile can be an equilibrium even if some firms earn low profits, provided that no individual firm can do better by changing its own action. Conversely, a profile is not an equilibrium merely because its entrants have positive profits: a nonentrant may still find that joining would raise its payoff.

The first condition compares an entrant’s current payoff with its payoff after unilaterally exiting. If \(i\) is an entrant, the other entrants are \(S\setminus\{i\}\), and those firms remain in the market under the deviation. The second condition compares a nonentrant’s current payoff with its payoff after unilaterally entering. In this case, all firms in \(S\) remain entrants and become \(i\)’s rivals. These comparisons exhaust the possible unilateral deviations because each firm has just two actions.

A Symmetric Entry-Payoff Benchmark

A useful special case has identical firms and a payoff that depends on the number of other entrants, rather than on their identities. Suppose an entering firm earns \(a-bk\), where \(k\) is the number of its rivals that enter, \(a\) summarizes market conditions, and \(b>0\) measures the reduction in entry payoff from each additional entrant. Suppose also that non-entry pays zero. For a profile with \(m\) entrants, each entrant faces \(m-1\) rivals, while each nonentrant faces \(m\) rivals if it joins.

The equilibrium conditions therefore depend on whether the firm is currently in or out. For \(m\) between 1 and \(n-1\), an entrant must prefer its current payoff to zero, and a nonentrant must prefer zero to the payoff from joining:

Rearranging gives \(b(m-1)\leq a\leq bm\). The boundary cases have only one type of firm to check. If \(m=0\), every firm is out, so the profile is an equilibrium exactly when joining does not pay: \(a\leq 0\). If \(m=n\), every firm is in, so the profile is an equilibrium exactly when each entrant prefers to stay: \(a-b(n-1)\geq 0\). There is no nonentrant in the all-enter profile whose incentive to join needs checking.

These are weak inequalities because a firm that is indifferent between its two actions has no strictly profitable deviation. At a boundary value, more than one entry count can satisfy the conditions. Moreover, if an entry count is an equilibrium, any particular set of \(m\) identical firms entering gives the same payoffs and also satisfies the equilibrium conditions.

Why the Conditions Characterize Equilibrium

The conditions above are not an approximation. They follow directly from the definition of a pure-strategy Nash equilibrium. Fix a candidate profile with entrant set \(S\). Each entrant has a current payoff given by entry against the firms in \(S\setminus\{i\}\); its only alternative is non-entry against those same rivals. Each nonentrant has a current payoff given by non-entry against the firms in \(S\); its only alternative is entry against those same rivals.

This reasoning also explains why checking only entrants is incomplete. An entrant’s incentive to remain says nothing about whether an outsider would benefit from joining. Similarly, checking only outsiders does not establish that current entrants prefer to stay. Both sets of inequalities are needed whenever both entrants and nonentrants are present.

Market Conditions, Strategic Effects, and Equilibrium

Market conditions affect equilibrium by shifting the payoffs used in the deviation comparisons. For example, a larger market may raise an entrant’s payoff both when it is the only firm and when rivals also enter. A higher entry cost may lower it. The equilibrium analysis itself remains the same: evaluate payoffs at the proposed market condition, then check every firm’s incentive to change its own action.

The way rival participation changes payoff also matters. In the symmetric benchmark, \(b>0\) means that each additional entrant reduces a firm’s entry payoff. In other settings, participation can instead raise payoffs—for example, if firms benefit from a shared network or a market that becomes more attractive as participation grows. The pure-strategy equilibrium conditions do not require either direction of effect; they require only that the relevant entry and non-entry payoffs be compared correctly.

Common Mistakes and Practical Checks

• Checking only current entrants. Positive profits for every entrant do not rule out a profitable entry by an outsider. Check each nonentrant’s payoff after joining.
• Using the wrong number of rivals. An entrant who exits leaves the other current entrants in the market. A nonentrant who joins faces all current entrants. Count rivals after the contemplated deviation, not before it.
• Inventing a deviation that is not available. In an \(n\)-firm game, an all-enter profile has no nonentrant who can join. To check that profile, consider each entrant’s exit deviation; do not evaluate the payoff of a hypothetical additional firm.
• Treating indifference as a profitable deviation. Equilibrium requires that no firm can strictly improve by deviating. If the two payoffs are equal, the firm is still choosing a best response.
• Assuming a unique equilibrium. A complete-information game can have more than one pure-strategy equilibrium, as the heterogeneous-cost and complementary-entry examples show. A payoff calculation alone does not select among equilibria.