MIT 14 12 - Dynamic Games with Incomplete Information

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1Lectures 15-18Dynamic Games withIncomplete Information14.12 Game Theory Muhamet YildizRoad Map1. Examples2. Sequential Rationality3. Perfect Bayesian Nash Equilibrium4. Economic Applications1. Sequential Bargaining with incomplete information2. Reputation2An ExampleNatureHigh 0.7Low 0.3FirmHireWWorkShirkDo nothire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)What is wrong with this equilibrium?NatureHigh .7Low .3FirmHireWWorkShirkDo nothire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)3What is wrong with this equilibrium? 1BX2 LRRLT(2,6)(0,1) (3,2) (-1,3) (1,5)Beliefs• Beliefs of an agent at a given information set is a probability distribution on the information set.• For each information set, we must specify the beliefs of the agent who moves at that information set.1BX2L R RLT(2,6)(0,1) (3,2) (-1,3) (1,5)µ1−µ4Sequential RationalityA player is said to be sequentially rationaliff, at each information set he is to move, he maximizes his expected utility given his beliefs at the information set (and given that he is at the information set) – even if this information set is precluded by his own strategy. An ExampleNatureHigh 0.7Low 0.3FirmHireWWorkShirkDo nothire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)5Another example1BX2L R RLT(2,6)(0,1) (3,2) (-1,3) (1,5)Example1B2L R RLT(0,10) (3,2) (-1,3) (1,5).9.16“Consistency”Definition: Given any (possibly mixed) strategy profile s, an information set is said to be on the path of play iff the information set is reached with positive probability if players stick to s.Definition: Given any strategy profile s and any information set I on the path of play of s, a player’s beliefs at I is said to be consistent with s iff the beliefs are derived using the Bayes’ rule and s.Example1B2L R RLT(0,10) (3,2) (-1,3) (1,5)7Example2BX3L R RLTE1200121333012011“Consistency”• Given s and an information set I, even if I is off the path of play, the beliefs must be derived using the Bayes’ rule and s “whenever possible,” e.g., if players tremble with very small probability so that I is on the path, the beliefs must be very close to the ones derived using the Bayes’ rule and s.8Example2BX3L R RLTE1200121333012011Sequential RationalityA strategy profile is said to be sequentially rationaliff, at each information set, the player who is to move maximizes his expected utility 1. given his beliefs at the information set, and2. given that the other players play according to the strategy profile in the continuation game (and given that he is at the information set) .9Perfect Bayesian Nash EquilibriumA Perfect Bayesian Nash Equilibrium is a pair (s,b) of strategy profile and a set of beliefs such that1. Strategy profile s is sequentially rational given beliefs b, and2. Beliefs b are consistent with s.NashSubgame-perfectBayesian NashPerfect BayesianExample2BX3L R RLTE120012133301201110Beer – Quiche1101beerbeerquichequichedueldon’tdueldon’tdon’tdueldon’tduel300021201031twts{.1}{.9}Beer – Quiche, An equilibrium1101beerbeerquichequichedueldon’tdueldon’tdon’tdueldon’tduel300021201031twts{.1}{.9}.9.11011Example121(4,4) (5,2)(3,3)(1,-5)121(-1,4) (0,2)(-1,3)(0,-5).9.1Example – solved121(4,4) (5,2)(3,3)(1,-5)121(-1,4) (0,2)(-1,3)(0,-5).9.1µ=7/8α=7/9β=1/212Sequential Bargaining1. 1-period bargaining – 2 types2. 2-period bargaining – 2 types3. 1-period bargaining – continuum4. 2-period bargaining – continuumSequential bargaining 1-p• A seller S with valuation 0• A buyer B with valuation v;– B knows v, S does not– v = 2 with probability π– = 1 with probability 1-π• S sets a price p ≥ 0;• B either – buys, yielding (p,v-p)– or does not, yielding (0,0).SHLppYNYNp2-p00p1-p0013Solution 1. B buys iff v ≥ p;1. If p ≤ 1, both types buy: S gets p.2. If 1 < p ≤ 2, only H-type buys: S gets πp.3. If p > 2, no one buys.2. S offers •1 if π < ½,•2 if π > ½.121pUS(p)Sequential bargaining 2-period• A seller S with valuation 0• A buyer B with valuation v;– B knows v, S does not– v = 2 with probability π– = 1 with probability 1-π1. At t = 0, S sets a price p0≥ 0;2. B either – buys, yielding (p0,v-p0)– or does not, then3. At t = 1, S sets another price p1≥ 0;4. B either – buys, yielding (δp1,δ(v-p1))– or does not, yielding (0,0)14Solution, 2-period1. Let µ = Pr(v = 2|history at t=1).2. At t = 1, buy iff v ≥ p;3. If µ > ½, p1= 2 4. If µ < ½, p1= 1.5. If µ = ½, mix between 1 and 2.6. B with v=1 buys at t=0 if p0≤ 1.7. If p0>1, µ = Pr(v = 2|p0,t=1) ≤π.Solution, cont. π <1/21. µ = Pr(v = 2|p0,t=1) ≤π<1/2. 2. At t = 1, buy iff v ≥ p;3. p1= 1. 4. B with v=2 buys at t=0 if (2-p0) ≥ δ(2−1) = δ Ù p0≤2−δ.5. p0=1:π(2−δ) + (1−π)δ = 2π(1−δ) + δ < 1−δ+δ = 1.15Solution, cont. π >1/2• If v=2 is buying at p0> 2−δ, then– µ = Pr(v = 2|p0 > 2−δ,t=1) = 0;–p1= 1;– v = 2 should not buy at p0> 2−δ.• If v=2 is not buying at 2> p0> 2−δ, then– µ = Pr(v = 2|p0 > 2−δ,t=1) = π > 1/2;–p1= 2;– v = 2 should buy at 2 > p0> 2−δ.• No pure-strategy equilibrium.Mixed-strategy equilibrium, π >1/21. For p0> 2−δ, µ(p0) = ½;2. β(p0) = 1- Pr(v=2 buys at p0)3. v = 2 is indifferent towards buying at p0:2- p0 = δγ(p0) Ù γ(p0) = (2- p0)/δwhere γ(p0) = Pr(p1=1|p0)..1)(1)(21)1()()(0000ππβππβππβπβµ−=⇔−=⇔=−+= pppp16Sequential bargaining, v in [0,1]•1 period:– B buys at p iff v ≥ p;– S gets U(p) = p Pr(v ≥ p);– v in [0,a] => U(p) = p(a-p)/a;–p = a/2.Sequential bargaining, v in [0,1]• 2 periods: (p0,p1)– At t = 0, B buys at p0iff v ≥ a(p0);–p1 = a(p0)/2;– Type a(p0) is indifferent:a(p0)–p0= δ(a(p0)–p1) = δa(p0)/2Ùa(p0) = p0/(1-δ/2)•S gets •FOC:200022/11−+−−δδδppp()()4/3122/10222/1212000δδδδδ−−=⇒=−+−−


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MIT 14 12 - Dynamic Games with Incomplete Information

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