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Analyzing Game Theory in Competitive Markets: Duopoly and Backward Induction Insights

This session covers advanced concepts in game theory, focusing on the Bertrand trap and Cournot duopoly models, highlighting key problems and their solutions. Administrative notes include feedback responses, upcoming deadlines, and contact information for assistance. The agenda emphasizes understanding backward induction, best response strategies, and applying theoretical frameworks in real-world scenarios. Engage in exercises that illustrate strategic interactions in competitive environments, enriching your grasp of economic theories and practices at UC Berkeley's Haas School of Business.

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Analyzing Game Theory in Competitive Markets: Duopoly and Backward Induction Insights

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  1. MBA 299 – Section Notes 4/11/03 Haas School of Business, UC Berkeley Rawley

  2. AGENDA • Administrative • Exercises • Finish off Exercises from Introduction to Game Theory & The Bertrand Trap • Problem 2 (see last weeks section notes) • Problem 5 d. • Problem 6 • Problem 7 (done on the board) • Cournot duopoly • Backwards induction problems

  3. ADMINISTRATIVE • In response to your feedback • Slides in section • More math • CSG entries due Tuesday and Friday at midnight each week • Contact info: • rawley@haas.berkeley.edu • Office hours Room F535 • Monday 1-2pm • Friday 2-3pm

  4. PROOF THAT ALL IDSDS ARE NE (PROBLEM #5D) • Proof by contraction: • 1. Assume not => a NE strategy is eliminated by IDSDS • 2. Suppose in a two player game strategies s1, s2 are a NE • 3. WOLOG Let s1 be the first of the strategies to be eliminated by IDSDS • 4. Then there must exist a strategy si that has not yet been eliminated from the strategy set that strictly dominates s1 • 5. Therefore U(s1,s2) < U(si,s2) • 6. A contradiction of the definition of NE since s1 must be a best response to s2 (Q.E.D.) • Source: Robert Gibbons, “Game Theory for Applied Economists” (1992) p. 13

  5. BERTRAND TRAP PROBLEM 6 (I)Parts a and b • Part a.) K1=K2=50 • 0 if pn>$5 • dn = 50 if pn=$5,n=1,2 • 50 if pn<$5 • profit = 50*(pn-1) • max profit by choosing pn=$5 (no game) • Part b.) K1=K2=100 • 0 if pn> pmin • dn = 50 if pn= pmin • 100 if pn< pmin • profit = X*(pn-1) • max profit by choosing pn=C=$1 . . .

  6. BERTRAND TRAP PROBLEM 6Part b continued and Part c • Part b. • Why does P=C in party b, where K1=K2=100? • Because 50*(P-delta)+50*(P-delta) > 50*P if delta is small • Therefore “defecting” is always the rule until P=C • Part c. • K1=100, K2=50 => there is no pure strategy NE • Why? • If player 2 charges P2=C (and earns zero), player 1 can charge C<P1<=$5 and earn 50*(P1-C) • But if player 1 charges P1>C then player 2 will want to increase his price to P2 = P1 –e earning 50*(P2-C) . . . • But now, if P2>C, player 1 will want to charge P1=P2 –e earning 100*(P1-C) • And so and on . . .

  7. COURNOT DUOPOLYMath • Solution • Profit i (q1,q2) = qi[P(qi+qj)-c] • =qi[a-(qi+qj)-c] • Recall NE => max profit for i given j’s best play • So F.O.C. for qi, assuming qj<a-c • qi*=1/2(a-qj*-c) • Solving the pair of equations • q1=q2=(a-c)/3 • Note that qj < a – c as we assumed • Set-up • P(Q) = a – Q (inverse demand) • Q = q1 + q2 • Ci(qi) = cqi (no fixed costs) • Assume c < a • Firms choose their q simultaneously

  8. COURNOT DUOPOLYIntuition • Observe that the monopoly outcome is • qm=(a-c)/2 • profit m = (a-c)2/4 • The optimal outcome for the two firms would be to divide the market at the monopoly output level (for example qi=qj=qm/2) • But each firm has a strong incentive to deviate at this qm • Check: qm/2 is not firm 2’s best response to qm/2 by firm 1

  9. BACKWARDS INDUCTION (I)Monk’s Cerecloth 1 • What are the BI outcome when a=4? • R = (6,8) L R 2 l r • What is the BI outcome when a = 8? • R = (6,8) a -6 3 2 6 8

  10. BACKWARDS INDUCTION (II)Shoved Environment 1 • What are the BI strategies for each player? • {L1, L2} • {r1, l2, r3} L1 R1 2 1 r1 2 1 2 2 L2 R2 2 l2 r2 3 1 2 1 3 3 1 l3 r3 4 6 2 7 • What is the BI outcome? • L1, r1 = (2,2)

  11. A MAJOR MEDIA COMPANY’S ACQUISITION OF A P2P FILE SHARING COMPANYA Simplified Model of How the Acquisition Was Analyzed 6,6,6,6,6 1=B 2=U 3=T 4=S 5=E Join 5 Join Abstain 4 Join 4,4,4,4,0 Abstain 3 Join 2,2,2,0,0 Abstain 2 Buy 0,0,0,0,0 Abstain 1 -10,0,0,0,0 Don’t buy What do you think happened? What are the limits of BI? 0,0,0,0,0

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