01282009161043e1rd7UKEDZ - Game Theory Notes: Examples of...

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1 Game Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium Below are two different games. The first game has a dominant strategy equilibrium. The second game has two Nash equilibria. This handout goes through the process of how to find these Nash Equilibria. Example 1 – Dominant Strategy Equilibrium: Consider the following game: o Players: Firm 1, Firm 2 o Strategies: Low Output (Q), High Output (Q) o Payoffs: given by We can think of this as a duopoly game – a game between two firms. These firms can decide to collude – keep output low so that the price is high, in effect they would be acting like a monopolist. However, it is possible for each firm to decide to produce a high output. Each firm will want the other firm to produce a low output in order to keep prices low, while it actually produces a high output in order to take advantage of the low output of its competitor. We want to find a possible solution to this game. Let’s start by looking to see if any player has a dominant strategy. High Output Low Output Firm 2 gets payoff Firm 2 gets payoff Firm 2 gets payoff Firm 2 gets payoff Firm 1 gets payoff Firm 1 gets payoff Firm 1 gets payoff Firm 1 gets payoff 6 10 1 3 61 10 3 Firm 2 Firm 1 Low Output High Output
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2 Def: A strategy is a dominant strategy for a player if it is that player’s best strategy (yields the highest payoff) no matter what the other players do. Let’s look at the strategies of each player. First consider Firm 1 (F1). If Firm 2 (F2) is playing “Low Output” then F1 is better off playing “High Output” (F1 will get 10 if it plays “High Output” versus 6 if it were to play “Low Output”). Now, if F2 were to play “High Output” then F1 is better off playing “High Output” (F1 will get 3 if it plays “High Output” versus 1 if it were to play “Low Output”). We can see that F1 is better off playing “High Output” no matter what F2 is doing. This means that for F1, “High Output” is a dominant strategy . Now consider Firm 2 (F2). If Firm 1 (F1) is playing “Low Output” then F2 is better off playing “High Output” (F2 will get 10 if it plays “High Output” versus 6 if it were to play “Low Output”). Now, if F1 were to play “High Output” then F2 is better off playing “High Output” (F2 will get 3 if it plays “High Output” versus 1 if it were to play “Low Output”).
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01282009161043e1rd7UKEDZ - Game Theory Notes: Examples of...

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