1. Some examples in Cournot Competition

2. Cournot Duopoly A duopoly with inverse demand
P = a-bQ (P/Q:market price/quantity)
ci: the constant marginal cost of Firm i
Each firm chooses qi to max. its profit
Firm 1 chooses q1 to
Max (a-bQ-c1)q1
Firm 2 chooses q2 to
Max (a-bQ-c2)q2

3. A simultaneous-move game
N.E.(q1*, q2*) solves the simultaneous eqs. (2 first-order conditions)
a-2bq1-bq2-c1 = (A)
a-bq1-2bq2-c2 = (B)
q1*=(a-2c1+c2)/3b
q2*=(a+c1-2c2)/3b

4. Stackelberg Competition
A similar game where Firm 1 acts first with his actions observed by Firm 2. (A leader and a follower in the same industry. A sequential-move game)
From Firm 2’s F.O.C, equation (B), Firm 2’s best response to every q1 is
q2=(a-bq1-c2)/(2b) →q2(q1)
q2(q1) captures the equilibrium in every subgame of q1 (there’s a subgame after Firm 1 announces q1 )

5. Firm 1 takes this into account (foreseeing that q2=q2(q1)) and chooses q1 to max its profit
Max [a-b(q1+q2)-c1]q1
s.t. q2=q2(q1) →simply replace q2 with q2(q1)
SPNE
q1*=(a-2c1+c2)/(2b)
q2*=q2(q1*)=(a+2c1-3c2)/(4b)

6. Static Cournot with asymmetric info.
The problem with asymmetric info. →the game is no longer common knowledge
Static (Simultaneous-move) Bayesian Games (Harsanyi)
Assume with probability t, c1=cH, and (1-t) c1=cL. (Firm 1 also knows this is how Firm 2 expects Firm 1’s costs though Firm 1 knows exactly its own cost.
First consider a slightly different game where even Firm 1 doesn’t know its own cost before the game is played but soon it will realize after the nature has made a choice. So that we can interpret the original game in this way.

7. Firm 2 has only 1 information set because it is a simultaneous-move game
q1H t cH
Nature q2 cL
1-t 1
q1L

8. A N.E. will specify what Firm 1 will do when cH and when cL, and what Firm 2 will do. It’s like now a 3-player game.
Indeed Firm 1 with high cost will (it competes with q2, not q1L)
max [a-b(q1H+q2)-cH]q1H
→q1H=(a-bq2-cH)/(2b)………….(I1)
Similary Firm 1 with low cost will
max [a-b(q1L+q2)-cL]q1L
→q1L=(a-bq2-cL)/(2b)…………..(I2)

9. Firm 2 will maximize (with prob
Firm 2 will maximize (with prob. t it’s competing with q1H, and 1-t with q1L)
t[a-bq1H-bq2-c2]q2+(1-t)[a-bq1L-bq2-c2]q2
→q2=[a-c2-tbq1H-(1-t)bq1L]/(2b)…(I3)
The N.E. is (q1H, q1L, q2) that solves (I1),(I2) and (I3) simultaneously.

10. q2*=[a-2c2+tcH+(1-t)cL]/(3b]
q1H*=[2a+2c2-(3+t)cH-(1-t)cL]/(6b)
q1L*=[2a+2c2-tcH-(4-t)cL]/(6b)
One can compare the result to the deterministic cases with low-cost/high-cost Firm 1 to see the differences in price/quantity.

11. We’ll see a similar game when we introduce auction where all players have private information regarding its own valuation toward to item on auction and they have to bid simultaneously.