Undefeatable?
STACKELBERG COMPETITIONS
Stackelberg Model is a strategic game in economics, named after the German economist Heinrich Freiherr Von Stackelberg, which finds its applications in oligopolistic markets. The primary idea behind oligopolistic markets is that few firms supply a sizeable fraction of the market and rule over many in a particular market or industry, offering similar goods and services. Firms compete for the quantity of output, choosing it sequentially.
In Game Theory terms, this game’s players are leaders and followers. The leader chooses the output first, and the follower firm decides the output by observing the former.
Let there be two firms, A (Leader firm) and B (market follower). They produce identical products.
Market output is the sum of the individual output of these firms. Market output can be estimated by backward induction. Backward induction is a process to determine the sequence of initial actions from the end of a problem. That means it starts at the final step by thinking about what a player would do in the initial steps. It helps us to find the solution to a multistage problem.
The leader will use k-level thinking to anticipate the follower’s output and then decide its output to maximize profit. The follower firm has the same objective — to maximize profit. Hence, the output can be estimated by analyzing marginal revenue and cost.
Let us take an example:
Total production, Q=Qₐ + Qᵦ
Where Qₐ : Output of firm A
And Qᵦ : Output of firm B
For Firm B:
Total revenue, TRᵦ = P . Qᵦ
= (C-Qₐ-Qᵦ ) Qᵦ
= C Qᵦ-Qₐ Qᵦ-Qᵦ²
(C is a constant number)
Marginal Revenue for firm B, MRᵦ, is obtained by differentiating TRᵦ wrt Qᵦ
MRᵦ = C-Qₐ-2Qᵦ
Assuming Total cost fn, TCᵦ (Qᵦ)= C₁ + Qᵦ²
Differentiating this, we get,
Marginal cost MCᵦ=2Qᵦ
For maximum profit,
MRᵦ = MCᵦ
C-Qₐ-2Qᵦ = 2Qᵦ
Qᵦ = (C-Qₐ)/4
This is the best response function for firm B.
For Firm A,
Total revenue, TRₐ = P . Qₐ
= (C-Qₐ-Qᵦ ).Qₐ
= C.Qₐ-Qₐ.Qᵦ-Qₐ²
= C.Qₐ-Qₐ .(C-Qₐ)/4-Qₐ²
= 3Qₐ.(C-Qₐ)/4
We thus obtain P[ Qₐ,Qᵦ(Qₐ)]
Marginal Revenue MRₐ= (3C-6 Qₐ)/4
Assuming Total cost fn, TCₐ (Qₐ)= C₁ +Qₐ²
Differentiating it, we get,
Marginal cost MCₐ= 2Qₐ
For maximum profit,
MRₐ = MCₐ
(3C-6.Qₐ)/4 = 2.Qₐ
Qₐ=3C/14
Assuming, C = 26, C₁ = 13
We thus calculate
Total output of Firm A, Qₐ=5.57
Total output of Firm B, Qᵦ=5.11
Since P=26-Qₐ-Qᵦ=15.32, we have:
Total market production Q=10.68
Stackelberg concepts are seen in the security domain, the defender (leader) designs a strategy to protect a resource such that it remains safe irrespective of the strategy of the attacker (follower). Stackelberg games are also used in networks, privacy, robotics, autonomous driving, and supply chains.
The Stackelberg leader gets an advantage of choosing first and earns more profit than followers. The leader knows the follower’s output is dependent on its output level and can maximize its profit by taking into consideration the follower’s thinking process.
