🎲 Oligopoly Models & Strategic Game Theory

In oligopolistic markets, few firms compete, making mutual strategic interdependence the central driver of economic outcomes. Game theory provides the mathematical toolkit to model strategic interactions where each agent's optimal payoff depends on rivals' actions.

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1. ♟️ Strategic Normal-Form Games & Nash Equilibrium

A normal-form game $\mathrm{\Gamma }=(N,(S_i{)}_{i\in N},(u_i{)}_{i\in N})$ comprises:

1. Set of players $N=\{1,2,\dots ,n\}$.
2. Pure strategy spaces $S_i$ for each player $i$.
3. Payoff functions $u_i:S_1\times S_2\times \cdots \times S_n\to \mathbb{R}$.

1.1 Nash Equilibrium ($NE$)

A strategy profile $s^{\ast }=(s_1^{\ast },s_2^{\ast },\dots ,s_n^{\ast })$ is a Nash Equilibrium if no player has an incentive to unilaterally deviate:

$$\mathrm{\forall }i\in N,u_i(s_i^{\ast },s_{-i}^{\ast })\ge u_i(s_i,s_{-i}^{\ast })\mathrm{\forall }{s}_i\in S_i$$

1.2 Classic Normal-Form Archetypes

Game Archetype	Player 2: C	Player 2: D	Key Strategic Property
Prisoner's Dilemma	$(3,3)$	$(0,5)$	Defect is strictly dominant strategy; unique $NE=(D,D)$ is Pareto inefficient!
(P1: C, P1: D)	$(5,0)$	$(\mathbf{1}\mathbf{,}\mathbf{1})$
Battle of the Sexes	$(\mathbf{3}\mathbf{,}\mathbf{2})$	$(0,0)$	Two pure-strategy $NE$ with conflicting coordination preferences.
(P1: Ballet, P1: Fight)	$(0,0)$	$(\mathbf{2}\mathbf{,}\mathbf{3})$
Matching Pennies	$(1,-1)$	$(-1,1)$	Strictly zero-sum; zero pure-strategy $NE$; unique mixed strategy $NE=(1/2,1/2)$.
(P1: Heads, P1: Tails)	$(-1,1)$	$(1,-1)$

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2. ⚔️ Oligopoly Competition Models

Consider a linear inverse market demand $P(Q)=a-bQ$ where $Q=q_1+q_2$ and constant marginal cost $c<a$.

Market Structures Output Comparison:
  Monopoly/Cartel       Cournot Duopoly       Stackelberg (Leader+Follower)       Bertrand/PC
      Q_m = (a-c)/2b        Q_c = 2(a-c)/3b                 Q_s = 3(a-c)/4b             Q_pc = (a-c)/b
  ◄────────────────────────────────────────────────────────────────────────────────────────►
  Low Output                                                                      High Output
  High Price                                                                      Low Price = MC

2.1 Cournot Quantity Duopoly (Simultaneous Choice)

Each firm chooses output $q_i$ taking rival output $q_j$ as given:

$$\underset{q_1}{max}{\pi }_1=(a-b(q_1+q_2)-c)q_1⟹\frac{\partial {\pi }_1}{\partial q_1}=a-2b{q}_1-b{q}_2-c=0$$

Reaction (Best-Response) Functions:

$$R_1(q_2)=\frac{a-c-b{q}_2}{2b},R_2(q_1)=\frac{a-c-b{q}_1}{2b}$$

Solving simultaneously yields the symmetric Cournot-Nash Equilibrium:

$$q_1^{\ast }=q_2^{\ast }=\frac{a-c}{3b},Q^{\ast }=\frac{2(a-c)}{3b},P^{\ast }=\frac{a+2c}{3},{\pi }_i^{\ast }=\frac{(a-c{)}^2}{9b}$$

For an $n$-firm symmetric Cournot oligopoly:

$$Q_n^{\ast }=\frac{n}{n+1}\frac{a-c}{b},P_n^{\ast }=\frac{a+nc}{n+1}\stackrel{n\to \mathrm{\infty }}{\to }c\text{(Converges to Perfect Competition)}$$

2.2 Bertrand Price Duopoly (Simultaneous Price Choice)

Firms produce homogeneous goods and set prices $p_1,p_2$. Consumers buy exclusively from the lower-priced firm.

• If $p_1<p_2$, Firm 1 gets entire market demand.
• If $p_1=p_2$, firms split demand $50/50$.
• If $p_1>p_2$, Firm 1 sells zero.

Bertrand Paradox: In equilibrium, price competition drives price down to marginal cost with just $N\ge 2$ firms:

$$p_1^{\ast }=p_2^{\ast }=c,{\pi }_1^{\ast }={\pi }_2^{\ast }=0$$

2.3 Stackelberg Leadership (Sequential Quantity Competition)

Firm 1 (Leader) commits to $q_1$ first; Firm 2 (Follower) observes $q_1$ and chooses $q_2=R_2(q_1)$. Leader solves by backward induction:

$$\underset{q_1}{max}{\pi }_1=(a-b(q_1+\frac{a-c-b{q}_1}{2b})-c)q_1=\left(\frac{a-c-b{q}_1}{2}\right)q_1$$$$\frac{d{\pi }_1}{d{q}_1}=\frac{a-c-2b{q}_1}{2}=0⟹q_1^{\ast }=\frac{a-c}{2b}(\text{Leader Output})$$$$q_2^{\ast }=R_2(q_1^{\ast })=\frac{a-c-b\frac{a-c}{2b}}{2b}=\frac{a-c}{4b}(\text{Follower Output})$$$$Q_{Stackelberg}^{\ast }=\frac{3(a-c)}{4b},{\pi }_{Leader}=\frac{(a-c{)}^2}{8b}>{\pi }_{Cournot}>{\pi }_{Follower}=\frac{(a-c{)}^2}{16b}$$

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3. 🔄 Dynamic Games & Subgame Perfect Nash Equilibrium (SPNE)

In extensive-form games with perfect information:

• Subgame: A subset of the game starting at a single decision node that contains all subsequent nodes.
• Subgame Perfect Nash Equilibrium (SPNE): A strategy profile that induces a Nash equilibrium in every subgame. Solved via backward induction.

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Collusion in Infinitely Repeated Bertrand Duopoly

Problem: Two identical firms engage in Bertrand price competition repeated indefinitely over discrete periods $t=0,1,2,\dots$ with common discount factor $\delta \in (0,1)$. Total monopoly profit per period is ${\mathrm{\Pi }}_M$.

1. If both collude to charge monopoly price $P_M$, each earns ${\mathrm{\Pi }}_M/2$ per period.
2. If a firm deviates by slightly undercutting $P_M-ϵ$, it captures the entire monopoly profit ${\mathrm{\Pi }}_M$ in that period, triggering Grim Trigger punishment (eternal pricing at $P=MC$ with $\pi =0$). Find the minimum discount factor ${\delta }^{\ast }$ required to sustain tacit collusion.

Step-by-Step Rigorous Derivation:

1. Payoff from Collusion (${\mathrm{\Pi }}_{\text{collude}}$):

   $${\mathrm{\Pi }}_{\text{collude}}=\frac{{\mathrm{\Pi }}_M}{2}+\delta \frac{{\mathrm{\Pi }}_M}{2}+{\delta }^2\frac{{\mathrm{\Pi }}_M}{2}+\cdots =\frac{{\mathrm{\Pi }}_M}{2}\sum _{t=0}^{\mathrm{\infty }}{\delta }^t=\frac{{\mathrm{\Pi }}_M}{2(1-\delta )}$$

2. Payoff from Defection (${\mathrm{\Pi }}_{\text{defect}}$):

   $${\mathrm{\Pi }}_{\text{defect}}={\mathrm{\Pi }}_M+\delta (0)+{\delta }^2(0)+\cdots ={\mathrm{\Pi }}_M$$

3. Incentive Compatibility Constraint for Collusion:

   $${\mathrm{\Pi }}_{\text{collude}}\ge {\mathrm{\Pi }}_{\text{defect}}⟹\frac{{\mathrm{\Pi }}_M}{2(1-\delta )}\ge {\mathrm{\Pi }}_M⟹\frac{1}{2(1-\delta )}\ge 1$$$$1\ge 2(1-\delta )=2-2\delta ⟹2\delta \ge 1⟹{\delta }^{\ast }\ge \frac{1}{2}$$

   Economic Result: If firms value future profits sufficiently ($\delta \ge 0.50$), monopoly collusion is an SPNE under the grim trigger strategy!