High School

Consider an incumbent and potential entrant game:

1. The entrant decides whether or not to enter the market.
- If the entrant does not enter the market, his payoff is 10, and the incumbent's payoff is 50.

2. If the entrant enters the market, the incumbent must decide whether to attack or accommodate:
- If the incumbent attacks, the payoffs are 8 for the entrant and 12 for the incumbent.
- If the incumbent accommodates, the payoffs are 20 for the entrant and 20 for the incumbent.

What are the Nash equilibria? Explain.

What are the subgame perfect equilibria?

Answer :

The market entry game has two Nash equilibria: (Out, Fight) and (In, Accommodate). The subgame perfect equilibrium refines this to (In, Accommodate). The latter ensures optimal strategies for every subgame.

Nash Equilibria and Subgame Perfect Equilibria in the Market Entry Game

In this market entry game, we can identify the following outcomes based on the decisions of the entrant and the incumbent.

Nash Equilibria

The game has two pure Nash equilibria:

  1. (Out, Fight): In this equilibrium, the entrant decides not to enter the market (Out), and the incumbent plans to fight if the entrant enters. Given the incumbent's strategy to fight, the entrant's best response is not to enter.
  2. (In, Accommodate): In this equilibrium, the entrant decides to enter the market (In), and the incumbent accommodates the entry. Given that the incumbent chooses to accommodate, the entrant's best response is to enter.

Subgame Perfect Equilibria

A subgame perfect equilibrium (SPE) refines the Nash equilibrium by ensuring that players' strategies form a Nash equilibrium in every subgame of the original game. In this scenario, the subgame perfect equilibrium is:

  1. (In, Accommodate): If the entrant enters, the incumbent's best response is to accommodate. Knowing that the incumbent will accommodate, the entrant’s best move is to enter the market.

The Nash equilibrium (Out, Fight) is not a subgame perfect equilibrium because, given the history where the entrant enters, the incumbent's optimal strategy would be to accommodate rather than fight.

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