Two firms face the following profit matrix: Is it true that,...

Two firms face the following profit matrix:
Is it true that, given this profit matrix, Firm 2 wants to match Firm 1 's price, but Firm 1 does not want to match Firm 2 's price? Does either firm have a dominant strategy? What is the Nash equilibrium in this game? Explain.

Key Concepts

Profit Matrix

A profit matrix is a tabular representation of the outcomes (payoffs) for each combination of strategies chosen by the players in a game. It is a fundamental tool in game theory, allowing one to analyze how each player's decision influences the resulting benefits or costs for themselves and their opponents.

Dominant Strategy

A dominant strategy is one that provides a player with the highest payoff regardless of what the other players do. If a player has a dominant strategy, they will choose it because it guarantees at least as good an outcome as any other strategy no matter how the opponents act.

Nash Equilibrium

A Nash equilibrium occurs when each player's strategy is a best response to the strategies chosen by the others, so no one can unilaterally change their strategy to achieve a better payoff. It represents a state of mutual best responses where every player’s decision is optimal given the decisions of others.

Best Response

The best response is the optimal strategy a player can choose given the strategies selected by the other players. Analyzing best responses helps identify the Nash equilibrium, as it reflects how each player should optimally react to the actions of their opponents.

Transcript

00:01 Looking at this payoff matrix for two firms, we first want to know, would it be true that firm two wants to match firm one's price, but that firm one does not want to match firm two.

00:12 So let's start with firm two matching firm one's price.

00:15 So let's suppose that firm one chooses a low price.

00:18 Well, in that case, if firm two is matching it, they will also choose low.

00:22 In this case, firm two would receive 10 versus the zero they would have had had they not matched it.

00:29 So in that case, yes, if firm one chooses low, firm two also wants to choose low.

00:34 Now let's take a look at this.

00:36 If firm one chooses high, would firm two also want to choose high? well, in this case, they would end up getting 30 versus the five they would get if they chose low.

00:46 So that's true.

00:47 So to answer the first question that this is asking, it would be a yes.

00:52 Firm 2 wants to match the prices.

00:54 But now what we need to do is look at whether or not firm 1 would want to match firm 2's price.

00:58 So to understand that, here we'll get rid of some of this.

01:02 Now let's take a look at firm two choosing first.

01:06 Now, firm two chooses low.

01:08 What would firm one want to do? well, in this case, firm one would want to choose high because they would get 10 versus the zero they would get if they matched it.

01:16 So in this case, it would be a no.

01:19 So it would be true that firm two wants to match, firm one's price, but firm one does not want to match firm two's price...

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