Dynamic Game Theory and the Stackelberg Model

Dynamic Game Theory So far we have focused on static games. However, for many important economic applications we need to think about the game as being played over a number of time-periods, making it dynamic. A game can be dynamic for two reasons; interaction and repetation.

Dynamic One-off Games There are two firms (A, B) considering to enter a new market. Unfortunately the market is only big enough to support one of the two firms. If both firms enter the market, they will both make a loss of $10m. If only one firm enters, that firm will earn a profit of $50m and the other firm will just break even. Firm B observes whether firm A has entered the market before it decides what to do.

Since firm B observes what firm A does and decides, it has 4 strategies. In a static game there are only 2 strategies. What are the 4 strategies of firm B? What is the result of this game?

The solution There are 3 pure-strategy Nash equilibria: Firm B threatens always to enter the market irrespective of what firm A does. If firm A believes that threat, it will stay out of the market. Firm B promises always to stay out. If firm A believes the promise, it will always enter. Firm B promises always to do the opposite of what firm A does. If A believes this promise, it will always enter.

In such a game credibility is a key issue. In this game firm B’s threats and promises are not creadible. (Why?) Since we assume that the players are rational, incredible statements will have no effect on other players’ behaviour. Hence the result of this game is that A will always enter and B will always stay out.

Subgame Perfect Nash Equilibrium Game theorists argue that Nash equilibrium concept is too weak. Subgame perfect Nash equilibrium is a stronger concept that does not allow noncredible threats to influence behaviour. Subgame perfection was introduced by Reinhard Selten (1965). A subgame is a smaller game embedded in the complete game. A subgame perfect Nash equilibrium requires that the predicted solution to be a Nash equilibrium in every subgame.

Continuing from the same example There were 3 Nash equilibria. Firm B threatens always to enter the market irrespective of what firm A does. If firm A believes that threat, it will stay out of the market. Firm B promises always to stay out. If firm A believes the promise, it will always enter. Firm B promises always to do the opposite of what firm A does. If A believes this promise, it will always enter.

Let’s see if these strategies are subgame perfect i.This strategy has two subgames. First is; firm A enters and firm Benters. The second is firm A stays out and firm B enters. Considering the subgames only the second is a Nash equilibrium. Hence this strategy is not subgame perfect. ii.Not subgame perfect iii.Subgame perfect

Backward induction This is a convinient method to figure out the subgame perfect Nash equilibria. This principle involves ruling out actions, rather than strategies that players would not play because other actions give higher pay- offs.

Strategic Behaviour Thomas Schelling initiated the formal study of strategic behaviour and introduced many important concepts in his book “The Strategy of Conflict” (1960). Threats: denote a penalty to be imposed on a rival if she takes some action. Promises: involve a reward to be conferred on a rival if she takes some action.

A key issue is whether these threats and promises are credible. The role of a strategic move is to convert a threat or a promise into a commitment. 4 elements are required for a move to be strategic: Sequential moves Communication Affect incentives Rational expectations

Stackelberg Game The Stackelberg game is identical to the Cournut game in that firms compete over quantities. But it differs in the timing of production decisions. In the Stackelberg game output is chosen sequentially. The “leader” moves first and chooses quantitiy. The “follower” firm observe the leader’s move and makes its own quantity choice. The leader takes into account the follower’s optimal response (rational expectations).