Refinements. New notes follow on next page. 1 of 8. Equilibrium path. Belief needed

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1 efiemets Itro from ECO 744 Notes efiemets i Strategic Form: Elimiate Weakly Domiated Strategies - throwig out strictly domiated strategies (eve iteratively ever elimiates a Nash equilibrium (because Nash equilibrium ever has player usig a strictly domiated strategy; goal is to elimiate weakly domiated strategies to get a better defied equilibrium Tremblig Had Perfect - cosider possibility that players make radom errors; look at limit of probabilities of errors to get subset of origial Nash equilibria Theorem - fiite, player game, tremblig had perfect ad elimiatig weakly domiated strategies are the same efiemets i Extesive Form: Tree ules - a little more i depth o the details behid extesive form Successor - odes that ca be reached from a give ode by followig arrows Immediate Successor - ode that's at ed of ay arrow leadig away from a give ode Predecessor - aalogous to successors except we trace backward through the tree; also applies to immediate predecessor Path - sequece of odes that ( starts with the iitial ode, ( eds with a termial ode, ad ( has the property that successive odes i the sequece are immediate successors of each other ule - every ode is a successor of the iitial ode, ad the iitial ode is the oly oe with this property ule - each ode except the iitial ode has exactly oe immediate predecessor; the iitial ode has o predecessors ule - multiple braches extedig from the same ode have differet actio labels ule 4 - each iformatio set cotais decisio odes for oly oe of the players; if this were't the case, at some poit i the game, the players wo't kow who is to make a decisio ule 5 - all odes i a give iformatio set must have the same umber of immediate successors ad they must have the same set of actio labels o the braches leadig to these successors; if this were't the case, the player would be able to distiguish betwee the odes Equilibrium Path - what we observe if we watch the game Subgame - ca't break up a iformatio set ad has to start from a sigle ode Subgame Perfect - if game comes to iitial ode of a subgame, the game will progress as if it were the subgame so we should fid Nash equilibria to all subgames (do't believe people will do irratioal thigs; closely related to throwig out weakly domiated strategies Sequetial atioality - players ought to demostrate ratioality wheever they are called o to make decisios; optimal strategy for a player should maximize his or her expected payoff, coditioal o every iformatio set at which this player has the move; closely related to tremblig had New otes follow o ext page Belief eeded Equilibrium path of 8

2 Sequetial Equilibrium - builds o subgame perfectio Nash Equilibrium - oly addresses the equilibrium path Subgame Perfectio - says all subgames have equilbrium play Kreps & Wilso - apply equilibrium play to all other areas (icludig those where there are o subgames; coditioal o gettig to ay poit i the tree, players procede with optimizig behavior Actios & Beliefs - have to specify actios ad beliefs at each ode; actios are optimizig with respect to beleifs; beleifs must be cosistet with structure of the game Nash eq. says B does't care about play i this subgame B A U A U D D A prefers D, but oly if B plays B No subgames B prefers whe A plays D Fidig Nash Equilibria - easier to do i strategic form (for fiite games; it's easy to overlook a Nash equilibrium i extesive form Player Example - Cosider this simple, -player game: Put it i strategic form ad we see there are (,,, Nash equilibria: ad,, Which is better? (, (, I strategic form, player has weakly domiated by so we toss the equilibrium I extesive form, is ot subgame perfect (if player fids himself at the decisio after player chooses, player would be irratioal to pick over Player Player,,,, Weakly Domiated Commitmet - if player ca figure out a way to commit to, he ca force player to choose (istead of so player would get a payoff of istead of ; views o commitmet: (a some say it's idepedet of the game tree (b commitmet must be built ito the game tree... we'll use this Example - modify example by allowig player to post a bod for commitmet prior to player 's decisio (i.e., player decides whether or ot to post a bod; if he does (C, he pays a small fee (ε ad icurs a large pealty for choosig... effectively, the bod allows player to chage his payoffs for his treat of playig is credible Put this tree i strategic form to fid all the equilibria Note : player 's strategies are listed as XY, where X is the strategy played if player chooses C ad Y is the strategy played if player chooses N Note : we're igorig ε i the payoffs to save space There are 5 of Nash equilibria of 8 (, (, Player (, Player (, (, Subgame Perfectio (,-(,-ε Player N N C C,,, -,,,,,,,, -,,,,, C (,-ε N (, (, (,

3 "Best Equilibrium" (i.e., the most likely or most logical, ot ecessarily the Pareto optimal I strategic form, we ca use iterated weak domiace to fid that (,C is the best Player Extesive form agai uses subgame perfectio to get the same result Complicatio - if ca commit, ca also commit? If so, game tree should show it; we ca also (,-ε (, have multiple levels of commitmet so this (,-(,-ε (, (, method ca get very complicated Commitmet i Game Tree - oce the game tree is writte, do't talk about actios that are't reflected i the tree (i.e., if a player does't have commitmet devices show i the tree, do't talk about the player usig commitmet to get to his preferred equilibrium Problem with Subgame Perfectio - (we're leadig up to sequetial equilibrium, subgame perfect does't apply if there are o proper subgames as i example Example - there are o subgames so every Nash equilibrium is subgame perfect Strategic form - Slutsky did this o the fly ad admitted that his choice was ot the best to work with (i.e., makig player the row player, the colum player ad the matrix player; there are two equilibria: a, b, ad b, b, ( a Player N N C C,,, -,,,,,,,, -,,,,, ( b Player a b a,,,, b,,,, Matrix (plr payoff Colum (plr payoff ow (plr payoff Player Kreps & Wilso - argue that ot all Nash equilibria are equally compellig; look for players doig thigs off the equilibrium path that are irratioal (a, b, a Equilibrium - player is't o the equilibrium path, so check if he's ratioal playig b give his belief that player plays a ad player plays a ; the oly way player gets to make a decisio is if player deviates (i.e., plays b istead of a ; there's o reaso to thik player chages (he does't kow player deviated, so give player gets his choice ad he expects player to play a, player is best off playig a (payoff of 4 istead of of 8 Player Weak Domiace N over N C over C Iterated Weak Domiace over (with N goe over (with C goe a b Player C N Player a b a 4,4,,, b,,,, C (, N (, Subgame Perfectio a b a b a b a b 4 4

4 Tremblig Had - the strategic form versio of what Kreps & Wilso argued (sort of (a, b, a Equilibrium - we should be able to elimiate this; defie: ε = Pr[player deviates] (i.e., plays b istead of a δ = Pr[player deviates] (i.e., plays b istead of a Assume ε ad δ are small Cosider gai to player by playig deviatig (i.e., plays a istead of b : (Note: "deviatio" for player's & are radom errors, but for player it's a choice Coditio Prob Payoff & do't deviate ( ε ( δ = deviates ( does't ε ( δ 4 = deviates ( does't ( ε δ = & deviate εδ = Expected Gai ε ( δ εδ = ε εδ εδ = ε 4εδ = ε ( 4δ player gais if 4δ > δ < / 4 We assumed ε ad δ are small so player will prefer to deviate ad a, b, ( a is ot a tremblig had perfect equilibrium Note: & deviate is "secod order small" (εδ so that loss is't a big deal (b, b, b Equilibrium - defie: ε = Pr[player deviates] (i.e., plays a istead of b δ = Pr[player deviates] (i.e., plays a istead of b Assume ε ad δ are small Cosider gai to player by playig deviatig (i.e., plays a istead of b : Coditio Prob Payoff & do't deviate ( ε ( δ = deviates ( does't ε ( δ = deviates ( does't ( ε δ 4 = & deviate εδ = Expected Gai ( ε ( δ + ( ε δ = ( ε [ + δ + δ ] = ( ε ( + 4δ player gais if + 4δ > δ > / 4 We assumed ε ad δ are small so player will ot prefer to deviate Note: we oly have to fid oe player that wats to deviate so for the a, b, ( a equilibrium we were able to stop; for this equilibrium we still have to check if player or player wat to deviate Check player ; defie: ε = Pr[player deviates] (i.e., plays a istead of b δ = Pr[player deviates] (i.e., plays a istead of b Assume ε ad δ are small Cosider gai to player by playig deviatig (i.e., plays a istead of b : a b a b a b a b of 8

5 Coditio Prob Payoff & do't deviate ( ε ( δ = deviates ( does't ε ( δ = deviates ( does't ( ε δ = & deviate εδ = Expected Gai ε ( δ ( ε δ + εδ = ε εδ δ + εδ + εδ = ε δ + εδ player gais if ε δ > ε > δ / So if player is more tha twice as likely to make a error as player the player will gai by deviatig... ot exactly cocrete; eed to kow player 's beliefs about ε ad δ a b a b a b a b 4 4 Tremblig Had Perfect - basic idea is itroducig a little oise to the game; differet ways to do it (strategic vs. extesive form; players have beliefs i what oppoets will do based o Nash equilibrium, but they also cosider the chace of mistakes i Error - defie error as a mixed strategy E where some positive probability is assiged to every strategy of player i ; by assumptio this error does ot deped o the player's equilibrium strategy; examples where errors are depedet: Football - ca choose strategy to pass or ru; errors are differet (icomplete pass or iterceptio if player passes; fumble if player rus Darts - aim at ad probability of hittig (error is more tha if player aims at 9 Chace of Error - with some small probability ε, player i makes a error (i.e., plays the i i mixed strategy E istead of his Nash equilibrium strategy X Sequece - look at error as a sequece ε with lim ε = Not Serious - ot takig errors seriously; ot modelig it ad do't care what causes it because ε is small Overall Mixed Strategy - pick a ε (or some i the sequece; combiig player's equilibrium strategy ad error results is the mixed strategy i i ( ε X + ε E (ote limit i as is equal to X Example - assume player has three strategies x, x, x ad player has pure strategy equilibrium x ; overall mixed strategy is: E + ε εe ( ε + εe = εe E εe New Equilibrium - this overall mixed strategy game that results from icorporatig errors satisfies the assumptios of the Nash theorem so we ca fid a ew Nash equilibrium usig errors for all the players NE(, E, where: is matrix of all error probability sequeces for each player E is matrix of all mixed strategy vectors for each player Geeral Case - this formulatio allows errors to affect everyoe; Zelto's origial work had player's ow errors affect rivals oly (ot himself 9 5 of 8

6 Tremblig Had Perfect Equilibria - for some i the sequece lim NE(, E NE( set of tremblig had perfect equilibria is lim NE(, E NE( E Subgame Perfect - sice everythig has some positive probability i the error, tremblig had perfect equilibria looks at player decisios at every ode; that meas these equilibria are subgame perfect Sequetial Equilibrium - similar to tremblig had, but ot exactly the same; start with Nash equilibrium ad use equilibrium strategies to defie beliefs at all odes (icludig those ot o the equilibrium path ad those with zero probability of occurrig; there are lots of rules for how to do this Chagig Beliefs - oce deviatio is observed (i.e., ed up o o-equilibrium path cosider how a player's beliefs chage Cosistecy - beliefs must be cosistet with equilibrium strategies, icludig subgame perfect equilibrium strategies (if they get you to that iformatio set; if player is o a ode that's ot part of ay equilibrium path, the beliefs ca be arbitrary Beliefs (,, Beliefs (/,/,/ Beliefs (, Set of cosistet beliefs: lim Beliefs(, E E Mixed Strategies - if a player is usig a mixed strategy i equilibrium, beliefs must be cosistet with that strategy.9. Player must have beliefs cosistet with player 's strategy beliefs are (.9,. Give player 's domiat strategy, player would ever expect to be here durig the game (zero probability evet; still has to be cosistet with player 's iformatio set (add errors for player ad take limit of error goig to zero Beliefs (.9,. Differece - Tremblig had puts i errors, fids NE, the takes limit of equilibria Sequetial puts i errors, puts i beliefs, takes limit of beliefs, the fid equilibrium Their both very similar, but examples ca be costructed where they're differet Tremblig had imposes more restrictios so TH Seq NE Beefit of Seq - emphasizes usig actios ad beliefs (TH does too, but ot explicitly; this is a good bridge to icomplete iformatio Example - shows the differece betwee sequetial ad tremblig had Tremblig Had - effectively geerates a ew payoff Player matrix based o what strategies the players ited to U D play (i this case we have to check (D, ad (U, U,, 6 of 8,,, -, - Player D, -, -

7 Expected Payoff - defie ( x, y as the expected payoff vector whe player plays mixed strategy x ad player plays mixed strategy vector y I this case, we do't really eed to solve all the boxes (it gets very tedious i larger problems; we ca just examie each Nash equilibrium to see if it's still a equilibrium (U, - determie if player really prefers playig : EV = ( δ ( + δ ( = δ EV = ( δ ( + δ ( = δ EV > EV so player actually prefers... (U, is ot tremblig had perfect (D, - determie if player really prefers playig : EV = δ ( + ( δ ( = δ EV = δ ( + ( δ ( = δ EV > EV (recall δ [,] because it's a probability, but it's assumed to be small so player does prefer ; ow check for row player: U EV = ( ε ( + ε ( = EV Player D = ( ε ( + ε ( = ε D U EV > EV (as log as ε < / so player does prefer D (D, is tremblig had perfect imit of Equilibria - effectively took limit (wrt error probabilities of equilibria Gets Hard - with more tha two strategies, eed to look at uio of all distributios of errors Sequetial - takes limit of beliefs (usually easier tha TH Belief - let π (x be sequece of beliefs of colum player about row player choosig strategy x, ad θ ( y be sequece of beliefs of row player about colum player choosig strategy y Cosistet, Not Equilibrium - beliefs to do ot have to be equilibrium beliefs, but have to be cosistet i the limit (i.e., as, the beliefs have to agree with equilibrium For this case, cosider: π ( U = θ ( = / Player Aim at Aim at ( δ U + δd,( ε + ε ( δ U + δd,( ε + ε Aim at U ( ( Aim at D (( δ D + δu,( ε + ε ( ( δ D + δu,( ε + ε / π ( D = / θ ( = / imits as : π ( U =, π ( D = θ ( =, θ ( = This is cosistet because (U, is a Nash equilibrium Sequetial Equilibrium - specifies two thigs: actios ad beliefs: SE = ((U,; π (U =, θ ( =... actio: (U.; beliefs π ( U = & θ ( = atioality - give belief (take at the limit, is actio a Nash equilibrium Cosistecy - limit of beliefs could result from fully mixed errors 7 of 8

8 I this case, sequetial does othig because (D, ad (U, are sequet equilibria Kreps & Wilso Theorem - paraphrased by Slutsky: typically, sequetial ad tremblig had equilibria are the same; for small sets of games ("measure zero" they're ot the same; i those cases, tremblig had elimiates more tha sequetial (TH Seq Nash; theorem is very hard to prove Summary - For players: IWD TH = WD Seq Nash - TH is exactly equivalet to sigle roud of elimiatig weakly domiated strategies - Iteratively elimiatig weakly domiated strategies elimiates more tha TH More tha players: TH WD Nash; TH Seq Nash; ca't compare WD & Seq or aythig with IWD - TH elimiates more equilibria tha a sigle roud of elimiatig weakly domiated strategies Aside - classic game theory movies: Dr. Stragelove - credibility The Maltese Falco - the book by Dashiell Hammett has a sub-plot about husbad disappearig (ot i the movie that deals with tremblig had (error does't mea game is chaged 8 of 8