Some examples in Cournot Competition

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

A simultaneous-move game N.E.(q1*, q2*) solves the simultaneous eqs. (2 first-order conditions) a-2bq1-bq2-c1 = 0-------------(A) a-bq1-2bq2-c2 = 0-------------(B) q1*=(a-2c1+c2)/3b q2*=(a+c1-2c2)/3b

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 )

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)

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.

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

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)

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.

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.

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.