Game Theory Dynamic Bayesian Games II

1. Game Theory Dynamic Bayesian Games II Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 3 (January 16,17)

2. Sequential Rationality • Sequential rationality extends and refines PBNE • Rationality condition of PBNE is extended to hold for any info set • Beliefs condition is refined to include info sets that do not arise from asymmetric information

3. Example where sequential rationality is needed • Two NE: (T,U) and (B,D) • SPE and PBNE do not have any bite • Still (T,U) seems unreasonable as it is always best for player 2 to choose D • But SPE and PBNE do not require to specify beliefs in this situation 2,2 T 0,0 U M D 0,1 B 1,0 U D 3,1

4. Sequential equilibrium

5. Reconsider simple example 3,2 accept reject U -1,0 apply S 0,0 1/2 N.A good -1,-3 N accept 1/2 apply reject U -1,0 bad S N.A 0,0 5

6. Is pooling reasonable? • It satisfies PBNE • Specify out-of-eq beliefs P(good/apply) < 3/5 • It satisfies sequential equilibrium • Sequence σn where both types of students choose to apply with probability 1/n. Updating beliefs gives P(good/apply) = ½ so that university has to reject along the sequence σn • But still it seems that the out-of-equilibrium belief should be P(good/apply) = 1. Why? To apply is a dominates strategy for bad student. (he has never incentive to apply; whereas good student may have incentive if she is rejected) • How to formalize this?

7. Domination-based beliefs I • Action a is strictly dominated for type θ if there is another action a’ s.t. Mins’S u(a’,s’,θ) > maxsS u(a,s,θ) • For each action aA define Θ(a) = {θ: there is no a’A that strictly dominates a} • A PBE has reasonable beliefs if for all aA with Θ(a) ≠ empty set, μ(θ / a) > 0 only if θΘ(a) • Apply this definition to simple example

8. Intermediate step to Intuitive Criterion 3,2 accept reject U -1,0 apply exam S 2,1 0,0 1/2 N.A good -1,-3 N accept 1/2 apply reject U -1,0 bad exam S 1,-4 N.A 0,0 8

9. Intermediate step to Intuitive Criterion II • Do reasonable beliefs as defined in sheet 4 rule out the pooling equilibrium? • Is pooling equilibrium then reasonable? • No, still not: apply can only be an optimal action for the bad student if university will take an exam, but he should know that university will never do it.

10. Intermediate step to Intuitive Criterion III • Define S*(Θ,a) as the set of possible equilibrium responses of the Receiver that can arise after action a is observed for some beliefs θ satisfying the property that μ(θ / a) > 0 only if θΘ(a) • For any type θ, action a is strictly dominated in a stricter sense iff there exists action a’ s.t. • Mins’S*(Θ,a’) u(a’,s’,θ) > maxsS *(Θ,a) u(a,s,θ) • For each action aA define Θ’(a) = {θ: there is no a’A that strictly dominates in a stricter sense a} • A PBE has super-reasonable beliefs if for all aA with Θ(a) ≠ empty set, μ(θ / a) > 0 only if θΘ’(a)

11. Apply this definition to new example • Set Θ = {good, bad} • S*(Θ,apply) = {accept, reject} • Bad student does not have an incentive to apply • Mins’S*(Θ,a’) u(not apply,s’,bad) = 0 > • MaxsS *(Θ,a) u(apply,s,bad) = -1 • Good student may have an incentive to apply as • Mins’S*(Θ,a’) u(not apply,s’,good) = 0 < • maxsS *(Θ,a) u(apply,s,good) = 3 • Θ’(apply) = {good} and μ(good / apply) = 1

12. Intuitive Criterion goes yet one step further 3,2 accept reject U -1,0 apply -2,0 S 1/2 N.A 0,0 good -1,-3 N accept 1/2 apply reject U -1,0 bad -2,0 S N.A 0,0 12

13. Intuitive Criterion - preparation • Do super-reasonable beliefs as defined in sheet 7 rule out the pooling equilibrium? No, as • Bad student may now also have an incentive to apply as • Mins’S*(Θ,a’) u(not apply,s’,bad) = -2 < • MaxsS *(Θ,a’) u(apply,s,bad) = -1 • Θ’(apply) = {bad,good} and μ(good / apply) can be anything • Is pooling equilibrium where student never applies then reasonable? • No, still not: we should not look at the lowest pay-off a type can get, but at the equilibrium pay-off and only apply the min or max pay-offs to deviation pay-offs

14. Intuitive Criterion - definition • Define the equilibrium pay-off of type θ as u*(θ) = u(a*(θ),s*(a),θ) • For any type θ, action a’ is equilibrium dominated iff • u*(θ) > maxsS *(Θ,a’) u(a’,s,θ) • For each action a’A define Θ”(a’) = {θ: a’ is not equilibrium dominated} • A PBE satisfies the Intuitive Criterion (IC) if for all a’A with Θ”(a’) ≠ empty set, μ(θ / a’) > 0 only if θΘ”(a’)

15. IC eliminates pooling equilibrium • In a pooling equilibrium where student never applies, university should reject student if not apply • Equilibrium pay-off student is then 0 • Apply is then equilibrium dominated for bad student, but not for good student • Θ”(apply) = {good} and μ(good / apply) = 1 • Given this, the university should accept if student applies. • Given this, good student should apply • Is there another pooling equilibrium here?

16. Exercises I • The truth game is given on the next sheet. A player throws a coin and only player 1 observes the outcome. This player then has to say whether the outcome was heads or tails. Player 2 hears what player 1 says and then also has to guess whether the outcome was head or tails. • What is (are) the sequential equilibria? • What are the PBE satisfying the IC?

17. (3,1) (2,1) „heads“ „heads“ Coin is heads „Heads“ „Tails“ 1 {.8} (1,0) „tails“ (0,0) „tails“ 2 2 (2,0) (3,0) „heads“ „heads“ „Tails“ „Heads“ 1 {.2} (0,1) „tails“ (1,1) „tails“ Coin is tails