Stackelberg game: introduction

What is it about?

Adapted from Game Theory 1: Problem set 2

Three oligopolists \(i\) = 1,2,3 operate in a market with inverse demand given by \(P(Q) = a - Q\), where \(Q = q_1 +q_2 +q_3 < a\). Firms \(i\) = 1,2,3 have constant per-unit costs of, respectively, \(c_1\), \(c_2\) and \(c_3\).

Sequence of moves

To determine the subgame-perfect Nash equilibrium, we first consider the sequence of moves, which is as follows:

• At time \(t = 1\): Firm 1 chooses quantity \(q_1 \geq 0\).
• At time \(t = 2\): After observing \(q_1\), firms 2 and 3 simultaneously and independently choose \(q_2 \geq 0\) and \(q_3 \geq 0\), respectively.

Backward induction

This is a sequential game with one proper subgame that starts in \(t = 2\). The subgame-perfect Nash equilibrium (SPNE) consists of a single quantity for firm 1 and functions \(q_2\) = \(q_2(q_1)\) and \(q_3\) = \(q_3(q_1)\) for firms 2 and 3 prescribing subgame perfect Nash equilibrium choices for firms 2 and 3 for every quantity choice by firm 1. We solve the game by backward induction.

How to find the SPNE?

Step 1: \(t = 2\) (1)

Start in \(t = 2\): Maximizing firm 3 profits gives:

\[q_3(q_1,q_2)= \frac{a - c_3 - q_1 - q_2}{2},\]

where \(q_2\) is assumed to be given. For firm 2 analogously:

\[q_2(q_1,q_3)= \frac{a - c_2 - q_1 - q_3}{2}.\]

Step 1: \(t = 2\) (2)

Inserting both into each other yields the Nash equilibrium of the subgame between firms 2 and 3:

\[q_2 = \frac{1}{3}(a - 2c_2 + c_3 - q_1)\ and\ q_3 = \frac{1}{3}(a - 2c_3 + c_2 - q_1)\tag{1}\]

Step 2: \(t = 1\)

Moving to \(t\) = 1: Inserting the expressions in (1) into the objective function of firm 1 gives \(\pi(q_1) = (a - c_1 - q_1 - q_2 - q_3)q_1 = (a - c_1 - q_1 - \frac{1}{3}(a - 2c_2\) \(+\ c_3 - q_1) - \frac{1}{3}(a - 2c_3 + c_2 - q_1))q_1 = \frac{1}{3}(a - 3c_1 + c_2 + c_3 - q_1)q_1.\) The FOC for firm 1 is \(a - 3c_1 + c_2 + c_3 - 2q_1 = 0\), which gives

\[q_1^* = \frac{1}{2}(a - 3c_1 + c_2 + c_3)\tag{2}\]

SPNE and SPNE path

The expression in (1) and (2) together constitute the SPNE of the game. On the SPNE path: \(q_2^* = \frac{1}{6}(a + 3c_1 - 5c_2 + c_3)\), \(q_3^* = \frac{1}{6}(a + 3c_1 + c_2 - 5c_3)\) and so \(Q_a^* = q_1^* + q_2^* + q_3^* = \frac{1}{6}(5a - 3c_1 - c_2 - c_3)\).

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