Randomized Strategy Equilibrium and Bertrand Equilibrium in the

Randomized Strategy Equilibrium and Bertrand Equilibrium in the Action Commitment Game with a Small Cost of Leading Joint work with Takeshi Murooka and Akira Ogawa OT 2010 1

Plan of the Presentation (1) Stackelberg or Cournot (Bertrand) (2) Endogenous Timing (3) Observable Delay (4) Action Commitment (5) Gains for Waiting and Equilibrium in Action Commitment (6) Mixed Strategy Equilibrium OT 2010 2

Stackelberg or Cournot (Bertrand) ModelとStackelberg Modelのど ちらを使うべきか？ simultaneous move modelを使うべきかsequential move modelを使うべきか？ 既存企業(incumbent)と新規参入者(new entrant)の 競争→sequential-move model こういう外生的な非対称性のない普通の状況 →simultaneous-move model でも現実には企業は生産量だけでなくタイミング も選ぶことができる OT 2010 3

代表的なTiming game (1) Observable delay game (2) Action commitment game (3) Infinitely earlier period model (4) Seal or disclose (5) Two production period model OT 2010 7

Observable Delay Game Duopoly First stage: Two firm choose period 1 or period 2. Second Stage: After observing the timing, the firm choosing period 1 chooses its action. Third Stage: After observing the actions taking at the second stage, the firm choosing period 2 chooses its action. Payoff depends only on its action (not period). OT 2010 8

Possible Outcomes Both firms choose period 1 ⇒Cournot (Bertrand) Both firms choose period 2 ⇒Cournot (Bertrand) Only firm 1 chooses period 1 ⇒Stackelberg Only firm 2 chooses period 1 ⇒Stackelberg OT 2010 9

Equilibrium in Observable Delay Game: Symmetric Cases Strategic Substitutes ⇒Both firms choose period 1 (Cournot, Bertrand) Leader ≫ Cournot ≫ Followerだから Strategic Complements ⇒Only firm 1 chooses period 1 (Stackelberg) or Only firm 2 chooses period 1 (Stackelberg) Leader ≫ Cournot (Bertrand)かつ Follower ≫ Cournot (Bertrand)だから OT 2010 10

Asymmetric Case Consider a price-setting competition under product differentiation. (Not always but ) usually it yields the cases of strategic complements ⇒Two Stackelbergs Suppose that one firm is more efficient than the other. Which firm does more likely becomes the leader? van Damme and Hurkens (2004), Amir and Stepanova (2006) ⇒The more efficient firm more likely becomes the leader. OT 2010 11

Payoff Dominance ２ １ OT 2010 L F L (0, 0) (4, 3) F (1, 2) (0, 0) 12

Payoff Dominantな均衡が実現？ ２ １ L F L (-100, 0) (4, 3) F (1, 2) (0, -100) Payoff Dominance →(L, F) Risk Dominance →(F, L) OT 2010 13

Risk Dominance ２ １ L F L (4, 4) (0, 0) F (0, 0) (1, 1) (L, L) risk dominates (F, F). OT 2010 15

Risk Dominance ２ １ L F L (4, 4) (-100, 0) F (0, -100) (1, 1) (F, F) risk dominates (L, L). OT 2010 16

Payoff dominance and risk dominance in the observable delay game Matsumura and Ogawa (2009) If (L, F) payoff dominates (F, L) and (L, L), either (i) (L, F) is the unique equilibrium or (ii) both (L, F) and (F, L) are equilibria and (L, F) risk dominates (F, L). OT 2010 17

Action Commitment Game (1) Duopoly First stage: Two firms choose period 1 or period 2. Second Stage: Without observing the timing, the firm choosing period 1 chooses its action. Third Stage: After observing the actions taking at the second stage, the firm choosing period 2 chooses its action. Payoff depends only on its and the rival's actions (not period). 第１期の相手の行動を見てはじめて相手が選んだ タイミングが判る OT 2010 18

Action Commitment Game (2) Duopoly First stage: Each firm chooses whether it takes actions in period 1 or not. Firms chooses period 1 chooses its action. Second Stage: After observing the actions taking at the second stage, the firm choosing period 2 chooses its action. Payoff depends only on its and the rival's actions (not period). 第１期の相手の行動を見てはじめて相手が選んだ タイミングが判る OT 2010 20

Equilibrium in the Action Commitment Game (1) Both firms choose period 1 (Cournot) (2) Only firm 1 chooses period 1 (Stackelberg) (3) Only firm 2 chooses period 1 (Stackelberg) 戦略的代替・補完によらず一つのoutcome（両企業 とも第２期を選ぶ）以外は均衡になる。 Asymmetricなケースも同じ。企業１の反応曲線が 右下がり、企業２のそれが右上がりでも同じ。 OT 2010 22

Instability of Cournot Outcome in the Action Commitment Game (1) Both firms choose period 1 (Cournot) 企業１がdeviateして第２期を選ぶ 企業２は第１期にCournot outcomeを選ぶ→企業１ は第２期にCournot outcomeを選ぶ⇒結局利得は deviation前と同じ off pathでは？ OT 2010 25

Instability of Cournot Outcome in the Action Commitment Game off path: 企業２が第２期を選んでいたら？ ⇒deviationの前後で利得変わらない 企業１が第１期でCournot outcome以外を選んでい たら？ ⇒deviationによって利得増える 第１期を選んでCournot outcomeを生産するのは第 ２期を選ぶ戦略にweaklyにdominateされている。 Cournotはrobustではない。 OT 2010 26

The Set of Pure Strategy Equilibria 均衡におけるP２ Equilibrium Outcomes P 2 L P 2 F P 2 B 0 OT 2010 ε 29

The Set of Equilibria 均衡におけるP２ Equilibrium Outcomes P 2 L P 2 F P 2 B 均衡の集合にはある種の連続性がある 0 OT 2010 ε 31

Notations pi: Firm i's price Vi(pi, pj): Firm i's profit when it chooses period 2 εi: Cost of leading Ri(pi): Firm i's reaction function Superscript L: outcome when it is the Stackelberg leader Superscript F: outcome when it is the follower Superscript B: Equilibrium value in Bertrand εi: = Vi(pi. L, pj. F)- Vi(pi. B, pj. B) pi 1 : Firm i's first period price at the mixed strategy equilibria qi : Probability that firm 1 choose first period price at the mixed strategy equilibria OT 2010 33

Model Action commitment game with price-setting competition Firm i's payoff is Vi(pi, pj) if it chooses period 1 and is Vi(pi, pj)-εi if it chooses period 2. Strategic complements, stability condition is satisfied, and V is twice differentiable. OT 2010 34

Mixed Strategy Equilibrium First order condition for choosing pi 1 (See equation 1 on page 5) Indifference between choosing periods 1 and 2 (See equation (3) on the same page) 4 equations →four endogenous variables OT 2010 35