Stackelberg competition

The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially (hence, it is sometimes described as the "leader-follower game").

It is named after the German economist Heinrich Freiherr von Stackelberg who published Marktform und Gleichgewicht [Market Structure and Equilibrium] in 1934, which described the model.

Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to move first.

Moving first may be possible if the leader was the incumbent monopoly of the industry and the follower is a new entrant.

The leader then picks a quantity that maximises its payoff, anticipating the predicted response of the follower.

The follower actually observes this and in equilibrium picks the expected quantity as a response.

The follower's actual can now be found by feeding this into its reaction function calculated earlier: The Nash equilibria are all

Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy.

Also referred to as a “decision tree”, the model shows the combination of outputs and payoffs both firms have in the Stackelberg game.

The image on the left depicts in extensive form a Stackelberg game.

Taking the first order derivative and equating it to zero (for maximisation) yields

Suppose marginal costs were equal for the firms (so the leader has no market advantage other than first move) and in particular

Simply by moving first, the leader has accrued twice the profit of the follower.

There may be cases where a Stackelberg leader has huge gains beyond Cournot profit that approach monopoly profits (for example, if the leader also had a large cost structure advantage, perhaps due to a better production function).

There may also be cases where the follower actually enjoys higher profits than the leader, but only because it, say, has much lower costs.

This behaviour consistently work on duopoly markets even if the firms are asymmetrical.

Once the leader has chosen its equilibrium quantity, it would be irrational for the follower to deviate because it too would be hurt.

However, in an (indefinitely) repeated Stackelberg game, the follower might adopt a punishment strategy where it threatens to punish the leader in the next period unless it chooses a non-optimal strategy in the current period.

This threat may be credible because it could be rational for the follower to punish in the next period so that the leader chooses Cournot quantities thereafter.

In Cournot competition, it is the simultaneity of the game (the imperfection of knowledge) that results in neither player (ceteris paribus) being at a disadvantage.

As mentioned, imperfect information in a leadership game reduces to Cournot competition.

However, some Cournot strategy profiles are sustained as Nash equilibria but can be eliminated as incredible threats (as described above) by applying the solution concept of subgame perfection.

Indeed, it is the very thing that makes a Cournot strategy profile a Nash equilibrium in a Stackelberg game that prevents it from being subgame perfect.

Consider a Stackelberg game (i.e. one which fulfills the requirements described above for sustaining a Stackelberg equilibrium) in which, for some reason, the leader believes that whatever action it takes, the follower will choose a Cournot quantity (perhaps the leader believes that the follower is irrational).

Hence, what makes this profile (or rather, these profiles) a Nash equilibrium (or rather, Nash equilibria) is the fact that the follower would play non-Stackelberg if the leader were to play Stackelberg.

[1][2] With the addition of time as a dimension, phenomena not found in static games were discovered, such as violation of the principle of optimality by the leader.

[2] In recent years, Stackelberg games have been applied in the security domain.

[3] In this context, the defender (leader) designs a strategy to protect a resource, such that the resource remains safe irrespective of the strategy adopted by the attacker (follower).

Stackelberg differential games are also used to model supply chains and marketing channels.

[4] Other applications of Stackelberg games include heterogeneous networks,[5] genetic privacy,[6][7] robotics,[8][9] autonomous driving,[10][11] electrical grids,[12][13] and integrated energy systems.