Entry Deterrence Game Under Ambiguity

Abstract

In this paper, we introduce ambiguity into an entry deterrence game between a better-informed established firm and a less informed potential entrant, in which the potential entrant has multiple priors on the true state of aggregate demand. In this model, the established firm is also uncertain about the state but is informed of the distribution of the state. We characterize the conditions under which limit pricing emerges in equilibria, and thus ambiguity decreases the probability of entry. Welfare analysis shows that limit pricing is more harmful in a market with higher expected demand than in a market with lower expected demand.

Appendix

A. Proof of Proposition 1

Proof. In this proposition, we intent to prove that at equilibrium, for $\forall x_{1} ,x_{2} \in \left[ {0,1} \right]$, if $x_{1}\, <\, x_{2}$, then $s_{1}^{ *} = s^{ *} \left( {x_{1} } \right) \le\, s^{ *} \left( {x_{2} } \right) = s_{2}^{ *}$. For the case that $\hat{x} \,<\, \tilde{x}$, there are four possibilities that can occur at equilibrium:

• (i). $t^{ *} \left( {s_{1}^{ *} } \right) = t^{ *} \left( {s_{2}^{ *} } \right) = 0$;

• (ii). $t^{ *} \left( {s_{1}^{ *} } \right) = 0,$ and $t^{ *} \left( {s_{2}^{ *} } \right) = 1$;

• (iii). $t^{ *} \left( {s_{1}^{ *} } \right) = 1,$ and $t^{ *} \left( {s_{2}^{ *} } \right) = 1$.

• (iv). $t^{ *} \left( {s_{1}^{ *} } \right) = 1,$ and $t^{ *} \left( {s_{2}^{ *} } \right) = 0$;

In each case, we can show that a violation of the monotonicity leads to a deviation from the equilibrium. Let $A_{1} = \left\{ {x:s^{ *} \left( x \right) = s_{1}^{ *} } \right\}$ and $A_{2} = \left\{ {x:s^{ *} \left( x \right) = s_{2}^{ *} } \right\}$ be the strategies of firm 1 at the particular equilibrium. By contradiction, assume that $s_{1}^{ *} \,>\, s_{2}^{ *}$.

For case (i), let $x_{1}^{ *} = { \inf }\left\{ {x:s^{ *} \left( x \right) = s_{1}^{ *} } \right\}$ and $x_{2}^{ *} = { \inf }\left\{ {x:s^{ *} \left( x \right) = s_{2}^{ *} } \right\}$. Due to the strictly increasing of $\varPi_{2C}^{e} \left( x \right)$ in $x$, firm 2 takes $x_{1}^{ *}$ and $x_{2}^{ *}$ as the worst case by MaxMin utility. If firm 2 doesn’t enter the market at $x_{1}^{ *}$ and $x_{2}^{ *}$, then the optimal response of firm 1 will be

$$ s_{1}^{ *} = \frac{{a^{e} \left( {x_{1}^{ *} } \right) + c_{1} }}{2b},\quad {\text{and}}\quad s_{2}^{ *} = \frac{{a^{e} \left( {x_{2}^{ *} } \right) + c_{1} }}{2b}. $$

If $s_{1}^{ *}\, >\, s_{2}^{ *}$, then $x_{2}^{ *} \,<\, x_{1}^{ *} \,\le\, x_{1}\,<\, x_{2}$. Then the type $x_{2}$ of firm 1 would deviate to $s_{1}^{ *}$. By deviation, $x_{2} \in A_{1}$, and still we have $x_{1}^{ *} = { \inf }A_{1}$, which doesn’t influence the decision of firm 2. The $x_{2}$ type of firm 1 can get higher profit in the first period and the same monopoly profit in the second period.

For case (ii), we have similar argument. If $s_{1}^{ *}\, >\, s_{2}^{ *}$, and firm 2 enters the market observing $s_{2}^{ *}$, then type $x_{2}$ of firm 1 would deviate to get higher profit in the first period and deter the entry by choosing $s_{1}^{ *}$.

For case (iii), since both $s_{1}^{ *}$ and $s_{2}^{ *}$ can’t deter the entry, type $x_{2}$ of firm 1 would deviate to get higher profit in the first period by choosing $s_{1}^{ *}$.

For case (iv), Firstly, we show $x_{1}^{ *} = x_{1}$. If $x_{1}^{ *} \,<\, x_{1}$, then $s_{1}^{ *} \,<\, \frac{{a^{e} \left( {x_{1} } \right) + c_{1} }}{2b}$. Since $s_{1}^{ *}$ induces entry, $x_{1}$ type of firm 1’s profit is $\varPi_{1M}^{e} \left( {x_{1}^{ *} } \right)$. However, the worst case by choosing $s_{1}$ is inducing entry and the profit is $\varPi_{1M}^{e} \left( {x_{1} } \right)$, and $\varPi_{1M}^{e} \left( {x_{1}^{ *} } \right) \,<\, \varPi_{1M}^{e} \left( {x_{1} } \right)$. The $x_{1}$ type of firm 1 would deviate to $s_{1} = \frac{{a^{e} \left( {x_{1} } \right) + c_{1} }}{2b}$. So $x_{1}^{ *} = x_{1}$. Now let’s show that if $s_{1}^{ *} \,>\, s_{2}^{ *}$, one type of firm 1 would deviate. At equilibrium, $x_{2}^{ *} \,<\, x_{1} \,<\, x_{2}$ and the profits of firm 1 given by:

$$ \begin{aligned} & \varPi_{1}^{e} \left( {s_{1}^{ *} ;x_{1} } \right) = \varPi_{M}^{e} \left( {x_{1} } \right) \\ & \varPi_{1}^{e} \left( {s_{2}^{ *} ;x_{2} } \right) = \varPi_{M}^{e} \left( {x_{2}^{ *} } \right) + R^{e} \left( {x_{2} } \right) \\ \end{aligned} $$

If $\varPi_{1}^{e} \left( {s_{1}^{ *} ;x_{1} } \right) \,>\, \varPi_{1}^{e} \left( {s_{2}^{ *} ;x_{2} } \right)$, then $x_{2}$ type of firm 1 would deviate. Choosing a higher price $s_{1}^{ *}$ induces entry but it can get higher profit.

If $\varPi_{1}^{e} \left( {s_{1}^{ *} ;x_{1} } \right) \,<\, \varPi_{1}^{e} \left( {s_{2}^{ *} ;x_{2} } \right)$, The condition for existence of a pooling equilibrium implies that

$$ \varPi_{M}^{e} \left( {x_{2}^{ *} } \right) + R^{e} \left( {x_{2} } \right) - \varPi_{1}^{e} \left( {x_{2} } \right) \,<\, \varPi_{M}^{e} \left( {x_{2}^{ *} } \right) + R^{e} \left( {x_{1} } \right) - \varPi_{1}^{e} \left( {x_{1} } \right). $$

Since $s_{2}^{ *}$ is the optimal choice for $x_{2}$ type of firm 1,

$$ \varPi_{M}^{e} \left( {x_{2}^{ *} } \right) + R^{e} \left( {x_{2} } \right) - \varPi_{1}^{e} \left( {x_{2} } \right) \,\ge \,0. $$

We obtain

$$ \varPi_{M}^{e} \left( {x_{2}^{ *} } \right) + R^{e} \left( {x_{1} } \right) - \varPi_{1}^{e} \left( {x_{1} } \right) \,>\, 0, $$

which implies that $x_{1}$ type of firm 1 would deviate to get higher payoff by choosing a lower price $s_{2}^{ *}$.

B. Proof of Theorem 1

Proof. To prove this theorem, we just follow the definition of Nash Equilibrium (Def. 2.1). Given the strategy of firm 1, observing $s \,<\, \frac{{\left( {a^{e} \left( {\hat{x}} \right) - c_{1} } \right)}}{2}$, firm 2 induces that $x \,<\, \hat{x}$. Firm 2 chooses to stay out of the market with maxmin preference. So $t^{ *}$ is an optimal response to $s^{ *}$. On the other hand, Given the strategy of firm 2, $t^{ *}$, firm 1’s optimal strategy is to maximize its total expected profit. Let $\delta_{1} = 1$, then the total expected profits of firm 1 is:

$$ \varPi_{1}^{e} \left( {s;x} \right) = \left\{ {\begin{array}{*{20}l} {\varPi_{1}^{0e} \left( {s;x} \right) + R^{e} \left( x \right)} \hfill & {{\text{if}}\quad s \,\le\, \frac{{a^{e} \left( {\hat{x}} \right) + c_{1} }}{2b}} \hfill \\ {\varPi_{1}^{0e} \left( {s;x} \right)} \hfill & {{\text{if}}\quad s \,>\, \frac{{a^{e} \left( {\hat{x}} \right) + c_{1} }}{2b},} \hfill \\ \end{array} } \right. $$

We can show that the difference of the profits of firm 1 between deterring and not deterring the entrant is decreasing in $x$ if $\delta_{1} = 1$:

$$ \frac{{d\left( {\varPi_{1}^{e} \left( {\hat{x},x} \right) - \varPi_{1M}^{e} \left( x \right)} \right)}}{dx} = - \frac{{2\left( {a^{H} - a^{L} } \right)\left( {a^{e} \left( x \right) - 2c_{1} + c_{2} } \right)}}{9b} < 0. $$

And we have shown that firm 1 is indifferent at $x = \tilde{x}$. So firm 1 prefers deterring to accommodating the entrant when $x \,<\, \tilde{x}$. When $\tilde{x} \,<\, \hat{x}$, for any $x \in \left[ {0,1} \right]$, firm 1 will choose the monopoly price in the first period because it is too costly to deter the entrant. But when $\tilde{x} \,>\, \hat{x}$, firm 1 has incentive to deter the entrant by pooling strategies for $x \in \left[ {\hat{x},\tilde{x}} \right]$ and what it can do the best is to choose the monoply price at $x = \hat{x}$. Given the strategy of firm 2, for any $x \in \left[ {0,1} \right]$, firm 1 doesn’t want to deviate.