MS&E246:Lecture15PerfectBayesianequilibriumRamesh JohariDynamicgamesIn this lecture, we begin a study of dynamic games ofincomplete information.We will develop an analog of Bayesian equilibrium for this setting, calledperfect Bayesian equilibrium.Whydoweneedbeliefs?Recall in our study of subgame perfection that problems can occur if there are “not enough subgames” to rule out equilibria.Entryexample(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2• Two firms• First firm decides if/how to enter• Second firm can choose to “fight”(0,2)EntryexampleNote that this game only has one subgame.Thus SPNE are any NE of strategic form.(2,1)(-1,-1)Entry2ExitEntry1(0,2)(0,2)(3,0)(-1,-1)RLFirm 2Firm 1EntryexampleTwo pure NE of strategic form:(Entry1, R) and (Exit, L)(2,1)(-1,-1)Entry2ExitEntry1(0,2)(0,2)(3,0)(-1,-1)RLFirm 2Firm 1Entryexample(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2But firm 1 should “know” that ifit chooses to enter,firm 2 will never “fight.”(0,2)Entryexample(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2So in this situation,there are too many SPNE.(0,2)BeliefsA solution to the problem of the entry game is to include beliefs as part of thesolution concept:Firm 2 should never fight, regardless of what it believes firm 1 played.BeliefsIn general, the beliefs of player i are:a conditional distribution overeverything player i does not know,given everything that player i does know.BeliefsIn general, the beliefs of player i are:a conditional distribution overthe nodes of the information set i is in,given player i is at that information set.(When player i is in information set h,denoted by Pi(v | h), for v ∈ h)BeliefsOne example of beliefs:In static Bayesian games, player i’s beliefis P(θ-i| θi) (where θjis type of player j).But types and information sets are in1-to-1 correspondence in Bayesian games,so this matches the new definition.PerfectBayesianequilibriumPerfect Bayesian equilibrium (PBE)strengthens subgame perfection by requiring two elements:- a complete strategy for each player i(mapping from info. sets to mixed actions)- beliefs for each player i(Pi(v | h) for all information sets hof player i)EntryexampleIn our entry example, firm 1 has only one information set, containing one node.His belief just puts probability 1 on this node.(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleSuppose firm 1 plays a mixed action with probabilities (pEntry1, pEntry2, pExit),with pExit< 1.(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleWhat are firm 2’s beliefs in 2.1?Computed using Bayes’ Rule!(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleWhat are firm 2’s beliefs in 2.1?P2(A | 2.1) = pEntry1 /(pEntry1+ pEntry2)(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleWhat are firm 2’s beliefs in 2.1?P2(B | 2.1) = pEntry2 /(pEntry1+ pEntry2)(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABBeliefsIn a perfect Bayesian equilibrium,“wherever possible”,beliefs must be computedusing Bayes’ rule andthe strategies of the players.(At the very least, this ensures information sets that can be reached with positive probability have beliefs assigned using Bayes’ rule.)RationalityHow do player’s choose strategies?As always, they do so tomaximize payoff.Formally:Player i’s strategy si(·) is such thatin any information set h of player i, si(h) maximizes player i’s expected payoff,given his beliefs and others’ strategies.EntryexampleFor any beliefs player 2 has in 2.1,he maximizes expected payoff by playing R.(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleThus, in any PBE, player 2 must play R in 2.1.(3,0)2.1RLRL(-1,-1)(-1,-1) (2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleThus, in any PBE, player 2 must play R in 2.1.(3,0)(2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleSo in a PBE, player 1 will play Entry1in 1.1.(3,0)(2,1)1.1ExitEntry1Entry2(0,2)ABEntryexampleConclusion: unique PBE is (Entry_1, R).We have eliminated the NE (Exit, L).PBEvs.SPNENote that a PBE is equivalent to SPNEfor dynamic games of complete and perfect information:All information sets are singletons,so beliefs are trivial.In general, PBE is stronger than SPNEfor dynamic games of complete andimperfect information.Summary• Beliefs: conditional distribution atevery information set of a player• Perfect Bayesian equilibrium:1. Beliefs computed using Bayes’ rule and strategies (when possible)2. Actions maximize expected payoff,given beliefs and strategiesAnantegame•Let t1, t2be uniform[0,1], independent.•Player i observes ti;each player puts $1 in the pot.• Player 1 can force a “showdown”,or player 1 can “raise” (and add $1 to the pot).• In case of a showdown, both players show ti;the highest tiwins the entire pot.• In case of a raise, Player 2 can “fold” (so player 1 wins) or “match” (and add $1 to the pot).• If Player 2 matches, there is a showdown.AnantegameTo find the perfect Bayesian equilibriaof this game:Must provide strategies s1(·), s2(·); and beliefs P1(· | ·), P2(· | ·).AnantegameInformation sets of player 1:t1: His type.Information sets of player 2:(t2, a1) : type t2,and action a1played by player 1.AnantegameRepresent the beliefs by densities.Beliefs of player 1:p1(t2| t1) = t2(as types are independent)Beliefs of player 2:p2(t1| t2, a1) = density of player 1’s type, conditional on having played a1= p2(t1| a1) (as types are indep.)AnantegameUsing this representation,can you find a perfect Bayesian equilibrium of the