Imperfect vs Incomplete Information Games In a ggame of imperfect p information players p y are simply unaware of the actions chosen by other players However they know who the other players are what are hat their possible strategies actions are are and the preferences payoffs of these other players Hence information about the other players p y in imperfect information is complete In incomplete information games players may or may nott know k some iinformation f ti about b t the th other th players e g their type their strategies payoffs or their o t e ppreferences e e e ces Example 1a of an Incomplete Information Game Prisoner s Dilemma Game Player y 1 has the standard selfish preferences but Player 2 has either selfish preferences or nice preferences Player 2 knows her type but Player 1 does not know 22 ss type 2 2 1 C D C D 4 4 0 6 6 0 2 2 y 2 selfish Player 1 C D C D 4 6 0 4 6 2 2 0 Player y 2 nice Recall that C cooperate D defect If player 2 is selfish then player 1 will want to choose D but if player 2 is nice player 1 best 1 s b t response is i still till to t choose h D D since i D is i a dominant d i t strategy for player 1 in this incomplete information game Example 1b of an Incomplete Information Game Prisoner s Dilemma Game Player y 1 has the standard selfish preferences but Player 2 has either selfish preferences or nice preferences Suppose player 1 s preferences now depend on whether player 2 is nice or selfish or vice versa versa 2 2 1 C D C D 4 4 0 6 6 0 2 2 y 2 selfish Player 1 C D C D 6 6 2 4 4 2 0 0 Player y 2 nice If 2 is selfish then player 1 will want to be selfish and choose D but if player 2 is nice player 1 s best response is to play C Be B nicer i to t those th who h play l nice i mean to t those th who h play l mean Example p 1b in Extensive Form Where Player 2 s Type is Due to Nature Information Set Prevents 1 From Knowing 2 s Type and 2 s Move Example 1b Again But With a Higher Probability that Type 2 is Selfish Analysis y of Example p 1b Player 2 knows his type and plays his dominant strategy D if selfish C if nice Player 1 s choice depends on her expectation concerning the unknown type of player 2 If player 2 is selfish selfish player 1 s 1 s best response is to play D D If player 2 is nice player 1 s best response is to play C Suppose player 1 attaches probability p to Player 2 being selfish lfi h so 1 p 1 is i the th probability b bilit th thatt Pl Player 2 iis nice i Player 1 s expected payoff from C is 0p 6 1 p Player y 1 s expected p ppayoff y from D is 2p 4 1 p p p 0p 6 1 p 2p 4 1 p 6 6p 4 2p 2 4p p 1 2 Player 1 s best response is to play C if p 1 2 D otherwise In I first fi version i p 1 3 1 3 play l C C iin second d p 2 3 2 3 play l D D The Nature of Nature Nature What does it mean to add nature as a player It is simply a proxy for saying there is some randomness in the type of player with whom you play a game The probabilities associated with nature s move are the subjective probabilities of the player facing the uncertainty about the other player s type When thinking about player types two stories can be told The identity of a player is known but his preferences are unknown I know I am playing against Tom but I do not know whether he is selfish or nice Nature whispers to Tom his type and I the other player have to figure it out Nature selects from a population of potential player types I am going to play against another player but I do not know if she is smart or dumb forgiving or unforgiving rich or poor etc Nature decides Example 2 Michelle and the Two Faces of Jerry Dancing Dancing M Frat Party 2 1 0 0 J Frat Party 0 0 1 2 Jerry likes company Dancing M Dancing Frat Party 2 0 0 1 J Frat Party 0 2 1 0 Jerry is a loner Assume that Jerry knows his true type and therefore which of the two games are being played Assume Michelle attaches probability p to Jerry liking company andd 1 p 1 to Jerry J bbeing i a lloner Big assumption Assume Jerry knows Michelle s estimate of p assumption of a common prior Bayes Nash Equilibria Bayes Nash equilibria is generalization of Nash equilibrium for an incomplete information game 1 First 1 First convert the game into a game of imperfect information 2 Second use the Nash equilibria of this imperfect i f information ti game as the th solution l ti concept t Apply this technique to the Michelle and Jerry Game Michelle s pure strategy choices are Dancing D or Party P She can also play a mixed strategy D with probability m Jerry s strategy is a pair one for each type of Jerry the first component is for the Jerry who likes company Jerry type 1 and 2 Pure the second component is for Jerry the loner Jerry type 2 strategies for Jerry are thus D D D P P D and P P Jerry also has a pair of mixed strategies g1 and g2 indicating the probability Jerry plays D if type 1or if type 2 Focus on pure strategies Pure Strategy Equilibria gy Bayes Nash y q Suppose Michelle plays D for certain Type 1 Jerry plays D Type 2 Jerry plays P Jerry D P Does Michelle maximize her payoffs by playing D against the Jerrys pure strategy of D P With probability p p she gets the D D D D payoff 22 and with probability 1 p she gets the D P payoff 0 So expected payoff from D against Jerry D P is 2p If instead instead she played P against Jerry D P D P she would get with probability p the P D payoff 0 and with probability 1 p she gets the P P payoff 1 So expected payoff from P against Jerry D P is 1 p Thus playing D against Jerry D D P P is a best response if 2p 1 p or if 3p 1 or if p 1 3 If p 1 3 it is a Bayes Nash equilibrium for Michelle to play D while hil the th Jerrys J …