Continuous competition game

Setup

Definition:

Continuous competition game

Players: Two players, Coke and Pepsi.

Strategies: Coke chooses price $p$ ; Pepsi chooses price $q$ .

Rules

Start with Coke and Pepsi facing linear demands; Players simultaneously choose prices $p$ and $q$ .
The player who chooses a profit-maximising price against the rival’s price is optimal.
Demand for Coke: $x = 44 - 2p + q$ ; Demand for Pepsi: $y = 44 - 2q + p$ .
Marginal cost is $8$ for each firm
Profit equals total revenue minus total cost

Derivation (Best Response Analysis)

Coke payoff function

Coke’s profit is

\pi_{\text{coke}} = (p-8)(44-2p+q)

Illustration with fixed rival prices

If $q=10$ , then

\pi_{\text{coke}} = (p-8)(54-2p)=2(p-8)(27-p)

so the first-order condition is

\frac{d\pi_{\text{coke}}}{dp}=2(27-p)-2(p-8)=0

and the best response is $p=17.5$ .

If $q=40$ , then

\pi_{\text{coke}}=(p-8)(84-2p)=2(p-8)(42-p)

so the best response is $p=25$ .

Coke best response

In general,

\frac{\partial \pi_{\text{coke}}}{\partial p}=(44-2p+q)-2(p-8)=60-4p+q

Setting the first-order condition equal to zero gives Coke’s best response:

p=\frac{60+q}{4}

Pepsi best response

Pepsi’s profit is

\pi_{\text{pepsi}} = (q-8)(44-2q+p)

Pepsi’s first-order condition is

\frac{\partial \pi_{\text{pepsi}}}{\partial q}=(44-2q+p)-2(q-8)=60+p-4q

Setting the first-order condition equal to zero gives Pepsi’s best response:

q=\frac{60+p}{4}

Diagram (Best Response Functions)

diagram

Derivation (Nash Equilibrium)

Solve best-response system

Solve the system

p=\frac{60+q}{4}, \qquad q=\frac{60+p}{4}

Rearrange

Rearranging gives

4p-q=60, \qquad -p+4q=60

Equilibrium prices

Solving the two equations yields

p=20, \qquad q=20

Nash Equilibrium

Result:

The Nash equilibrium is
$(p,q) \mapsto (20,20)$

Payoffs at the Nash Equilibrium

Coke’s profit is

\pi_{\text{coke}}=(20-8)(44-40+20)=12 \cdot 24 = 288

Pepsi’s profit is

\pi_{\text{pepsi}}=(20-8)(44-40+20)=288

Collusive Benchmark

If both firms maximise joint profit, they solve

\max_{p,q}\; (p-8)(44-2p+q)+(q-8)(44-2q+p)

The first-order conditions are

52-4p+2q=0, \qquad 52-4q+2p=0

Solving gives

p=q=26

Each firm’s collusive profit is

(26-8)(44-52+26)=18 \cdot 18 = 324

Incentive to Deviate

If Pepsi keeps $q=26$ , Coke’s one-shot best response is

p=\frac{60+26}{4}=21.5

Coke then earns

\pi_{\text{coke}}=(21.5-8)(44-43+26)=13.5 \cdot 27 = 364.5

Since $364.5 > 324$ , unilateral deviation beats the collusive payoff.

Insights

Insight:

Individual optimisation gives lower prices than joint-profit maximisation.

Collusion raises profits but is unstable in a one-shot game.

Best-response analysis and first-order conditions recover the equilibrium cleanly.