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In the original text, Figure 5.1 presents a sequential-move game where an entrant considers entering an industry with an incumbent firm. The stages involve the entrant's initial decision to enter, and the incumbent's response to that decision, with payoffs assigned accordingly. The modified scenario introduces a new strategic option for the entrant: the ability to withdraw from the industry after initially deciding to enter, under specific payoff conditions. This essay analyzes the equilibrium outcome of this game with the option to withdraw and assesses whether the entrant benefits from having this withdrawal option.

In the basic sequential game without withdrawal, the entrant faces a straightforward decision: to enter or not, with the payoff structure influencing its choice. If the entrant enters, the incumbent responds, leading to an equilibrium based on backward induction, where both players' optimal strategies are considered. The original payoffs typically reward entry if the anticipated payoff exceeds the cost, and the incumbent's response depends on whether the entrant's entry is profitable or not.

Adding the withdrawal option modifies the strategic landscape significantly. Now, if the entrant enters and anticipates unfavorable payoffs from continued competition, it has the strategic choice to withdraw instead of fighting. This introduces a new set of payoffs: if the entrant fights and loses, it incurs a loss of 1, and the incumbent gains 8; if it stays and fights, both suffer a loss of 5. Conversely, withdrawing results in a loss of 1 for the entrant but avoids the larger conflict costs, and the incumbent's payoff from withdrawal is effectively zero, as they do not engage in fighting.

Analyzing the game through backward induction, the entrant evaluates its options at each decision node. If the entrant considers fighting, it compares the potential payoff of -1 (if it loses) or -5 (if it stays and fights). With the possibility of withdrawal, the entrant will prefer to withdraw if fighting risks a worse payoff. Specifically, the decision hinges on the comparison between "fight" and "withdraw": if the potential outcome of fighting results in a loss of 1, and withdrawing results in a loss of 1, the entrant may be indifferent but likely prefers withdrawal if it wants to avoid conflict. The incumbent, anticipating this, might forego fighting if the entrant is likely to withdraw, which changes the strategic equilibrium.

The equilibrium of this modified game can be characterized as a separating equilibrium, where the entrant chooses to withdraw or fight based on its expectations of the incumbent's response. If the costs of fighting are high and withdrawal offers a lower payoff loss, the entrant will opt to withdraw rather than engage in costly conflict. Consequently, the equilibrium outcome depends on the perceived payoffs and strategic considerations: if the entrant values avoiding conflict more heavily than incurring the initial loss of 1, withdrawal becomes the dominant strategy. The contingent response from the incumbent then depends on whether it prefers to fight or accommodate the entrant's withdrawal, potentially leading to a situation where the entrant chooses to withdraw preemptively, effectively deterring the incumbent from engaging in costly conflict.

Regarding whether the entrant is better off with or without the ability to withdraw, the analysis suggests that the withdrawal option enhances the entrant's strategic position. When able to withdraw, the entrant can avoid costly fights, thereby reducing potential losses associated with fighting, and can threaten to withdraw to deter the incumbent's aggressive actions. This strategic flexibility increases the entrant's bargaining power and potentially results in a more favorable outcome or at least mitigates worst-case scenarios. Conversely, lacking the option to withdraw forces the entrant to face the risk of costly conflict, possibly leading to less optimal payoffs and higher risks of losses.

In conclusion, the inclusion of withdrawal options in strategic games of entry significantly impacts equilibrium outcomes. It allows the entrant to manage potential conflicts proactively, leading to more favorable positionings and strategic stability. This analysis underscores the importance of strategic flexibility in competitive interactions, where the ability to withdraw can serve as a credible threat that influences incumbent behavior and outcomes.

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