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Static Games of Complete Information: Subgame Perfection

Static Games of Complete Information: Subgame Perfection. APEC 8205: Applied Game Theory. Objectives. Why is Nash not enough in dynamic games of complete information? Subgame Perfect Refinement to Nash Equilibrium What is a subgame? Definition of Subgame Perfect Equilibrium

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Static Games of Complete Information: Subgame Perfection

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  1. Static Games of Complete Information: Subgame Perfection APEC 8205: Applied Game Theory

  2. Objectives • Why is Nash not enough in dynamic games of complete information? • Subgame Perfect Refinement to Nash Equilibrium • What is a subgame? • Definition of Subgame Perfect Equilibrium • Application of Subgame Perfect Equilibrium • Stackelberg Duopoly • Stackelberg Rent Seeking • Empirical Weaknesses of Subgame Perfection

  3. Why is Nash not enough in dynamic games of complete information? Player 1 Player 1’s Strategies: Do Not Enter {Do Not Enter, Enter} Enter Player 2 Player 2’s Strategies: High Output Low Output (70,60) {If Enter, High Output; If Enter, Low Output} (60,10) (90,50) (Player 1’s Payoff, Player 2’s Payoff) How will/should this game be played?

  4. Normal Form Game & Nash Equilibria • Player 1 • 1(L| E=1) = 90L + 60 (1 - L) = 30L + 60 • 1(L| E=0) = 70L + 70 (1 - L) = 70 • 1(L| E=1) >/=/< 1(L| E=0) for L >/=/< 1/3 • Player 2 • 2(E| L=1) = 50E + 60 (1 - E) = 60 - 10E • 2(E| L=0) = 10E + 60 (1 - E) = 60 - 50E • 2(E| L=1) ≥ 2(E| L=0) • Nash Equilibria: • {(1,0),(1,0)}, {(0,1),(0,1)}, & {(0,1),(1/3 ≥ L> 0,1- L)} There are an infinite number of Nash equilibria!

  5. Can we do better than this? Consider the Nash equilibrium {(0,1),(0,1)}! Player 1 If Player 1 unilaterally deviates by Entering, Player 2 will choose a High Output according its equilibrium strategy. Do Not Enter Enter However, executing such a strategy means that Player 2 will receive 10 instead of 50 from choosing a Low Output. Player 2 High Output Low Output (70,60) Is this sensible? (60,10) (90,50) If not, why should Player 1 ever choose Do Not Enter? (Player 1’s Payoff, Player 2’s Payoff) This equilibrium is sustained by what is called an incredible threat! If we rule out such threats as unreasonable, we are left with a unique equilibrium: {(1,0),(1,0)}!

  6. How can we rule out incredible threats? We can rule out incredible threats by solving the game backward! Player 1 Do Not Enter Since Player 2 moves last, we will start by asking the question, what is Player 2’s best response assuming Player 1 Enters? Enter Player 2 Low Output High Output Low Output (70,60) Since Low Output is a best response to Enter, lets eliminate High Output as an option for Player 2 given Enter? (60,10) (90,50) Now that High Output is out of the picture, what is Player 1’s best response? (Player 1’s Payoff, Player 2’s Payoff) Enter Therefore, we are left with the unique equilibrium: {(1,0),(1,0)}!

  7. Subgame Definition • A subgame in an extensive form game: • begins at a decision node n that is a singleton information set, • includes all the decision and terminal nodes following n in the game tree (but no nodes that do not follow n), and • does not cut any information sets (i.e. if a decision node n’ follows n in the game tree, then all other nodes in the information set that contains n’ must also follow n and must be included in the subgame). • A subgame is a piece of a larger game that can be solved without considering the rest of the game!

  8. Example Subgames Player 1 Player 2 Player 3 Player 4 Player 4 How many subgames are there in this game? 1 + 1 + 1 = 3

  9. Player 1 Player 2 Player 3 Player 4 Player 4 Example Subgames How many subgames are there in this game? 1 = 1

  10. Subgame Perfect Equilibrium • A Nash equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium in every subgame (Selten, 1965).

  11. Application: Stackelberg Duopoly • Who are the players? • Two firms denoted by i = 1, 2. • Who can do what when? • Firm 1 chooses output. • After Firm 1 chooses output, Firm 2 chooses output. • Who knows what when? • Firm 1 does not know Firm 2’s output when choosing. • Firm 2 knows Firm 1’s output when choosing. . • How are firms rewarded based on what they do? • gi(qi,qj) = (a – qi– qj)qi – cqi for i≠ j. • Question: What is a strategy for each firm? • Firm 1: q1≥ 0 • Firm 2: q2(q1) ≥ 0 for all possible q1.

  12. What is the subgame perfect equilibrium? • Firm 2 has the last move knowing Firm 1’s output, so lets start here! • Firm 2’s optimization problem is then: • FOC for interior: a – 2q2 – q1– c = 0 • SOC: –2 < 0 is satisfied • Solve for q2: • This is Firm 2’s Nash equilibrium strategy for the subgames starting after Firm 1 has chosen its output. • Note that there are an infinite number of these subgames.

  13. Now we know Firm 2’s best response, lets solve for Firms 1 taking into account this information? • Firm 1’s optimization problem is: • FOC for interior: • SOC: • But: , and . • So, the SOC is satisfied and .

  14. Stackelberg: Strategies Outputs Total Output Profits Cournot: Strategies Outputs Total Output Profits And the subgame perfect equilibrium is?

  15. Implications Regarding the Impacts of Better Information for Firm 2 • Output • Firm 1’s Increases • Firm 2’s Decreases • Total Increases • Profit • Firm 1’s Increases • Firm 2’s Decreases • Total Profit Decreases Even though Firm 2 has better information to make its choice, it is worse off!

  16. Is this the only Nash equilibrium strategies? • No! • How about ? • This is a Nash equilibrium strategy! • It is not subgame perfect, because q2 = a – c is not a best response to any q1≥ 0. • There are in fact an infinite number a Nash equilibria for this game.

  17. Application: Stackelberg Rent Seeking • Who are the players? • Two firms denoted by i = 1, 2 competing for a lucrative contract worth Vi. • Who can do what when? • Firm 1 chooses effort (x1) for preparing its proposal. • After Firm 1 chooses effort, Firm 2 chooses effort (x2). • Who knows what when? • Firm 1 does not know Firm 2’s effort when choosing. • Firm 2 knows Firm 1’s effort when choosing. . • How are firms rewarded based on what they do? • gi(xi,xj) = Vi xi / (xj + xj) – xi for i≠ j. • Question: What is a strategy for each firm? • Firm 1: x1≥ 0 • Firm 2: x2(x1) ≥ 0 for all possible x1.

  18. What is the subgame perfect equilibrium? • Firm 2 has the last move knowing Firm 1’s effort, so again lets start here! • Firm 2’s optimization problem is: • FOC for interior: • SOC: • Solve for x2:

  19. Now we know Firm 2’s best response, lets solve for Firms 1 taking into account this information? • Firm 1’s optimization problem is: • FOC for interior: • SOC satisfied: • Solve for x1: • Which implies:

  20. Stackelberg Strategies Rent Dissipation Payoffs Cournot Strategies Rent Dissipation Payoffs Is Firm 2 better or worse off from knowing Firm 1’s effort?

  21. Implications • Firm 1 better off with Firm 2 knowing its effort. • Firm 2 may be better or worse off knowing Firm 1’s effort: • Better off if V2 > V1 > 0 • Worse off if 2V2 > V1 > V2 • So we can get results contrary to the duopoly model. Having more information is not always bad! • Why? • Information Effect: Beneficial to Second Mover • Timing Effect: Detrimental to Second Mover • In the Duopoly model with linear demand the timing effect always dominates. • In the Rent Seeking model, the information effect can dominate.

  22. Predictive Weaknesses of Subgame Perfection • Incredible Threats That Are Actually Credible • Pareto Improving Non-Equilibrium Play

  23. Player 1 Do Not Enter Player 2 High Low (70,60) (60,48) (90,50) 32%/12.5% 32%/12.5% 36%/75% Incredible Threats That Are Actually Credible Player 1 Do Not Enter Player 2 High Low (70,60) 12%/0% (60,10) (90,50) 0%/25% 88%/75% Subgame Perfection Does Pretty Well! Subgame Perfection Does Pretty Poorly! (Player 1’s Payoff, Player 2’s Payoff) Percentage of Observed Play

  24. Player 1 2 1 2 (6.40,1.60) Pass Pass Pass Pass Take Take Take Take (0.40,0.10) (0.20,0.80) (1.60,0.40) (0.80,3.20) Player 1 2 1 2 1 2 (25.60,6.40) Pass Pass Pass Pass Pass Pass 4% Take Take Take Take Take Take (0.40,0.10) (0.20,0.80) (1.60,0.40) (0.80,3.20) (6.40,1.60) (3.20,12.80) 1% 6% 20% 38% 25% 5% Pareto Improving Non-Equilibrium Play 5%/0% 7%/50% 36%/25% 37%/0% 15%/25% (Player 1’s Payoff, Player 2’s Payoff) Percentage of Observed Play

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