Bridging Nash Equilibrium and Level-K Model

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1 Brdgng Nash Equlbrum and Level-K Model By Dan Levn and Luyao Zhang Oho State Unversty Work n progress; please do not quote wthout permsson May 12, 2015 Abstract: We propose a hybrd model, NLK, that connects Nash Equlbrum and Level k model. A player Level 0 s assumed to randomze over undomnated strateges, and a NLK player Levelk N, k = 1, 2..., n, beleves wth a (pror) probablty λ [0, 1], that ther rval s Levelk 1 N and wth a probablty (1 λ) that ther rval s LevelN k, such as themselves. NLK allows a decson maker n a game to beleve that ther opponent may be as sophstcated as they are, a vew wth strong support n psychology, or as less sophstcated. We show that NLK generates predctons that at tmes are sharply dfferent from those of Nash-equlbrum and/or Level k models. We examne our model n a statc Guessng Game (Arad and Rubnsten, 2012), a dynamc Centpede Game (Palacos- Huerta and Volj, 2009) and a Common Value Aucton (Avery and Kagel, 1997). In all the cases even a smple verson of our model makes much closer predctons than the other two. We dscussed extensons to games wth more than two players, heterogeneous beleves regardng other players, and allowng dstrbutons of lower level players. JEL Classfcaton: D01 C72 C92 Key words: Nash Equlbrum, Level-k model, bounded ratonalty. Acknowledgments: We have benefted from comments of partcpants n the Texas Expermental Economcs Symposum, Mdwest Economc Theory and Trade Conference and from conversatons wth Yaron Azrel, and Phlppe Jehel. 1 Introducton and Lterature Revew There s a mountng robust evdence from laboratory experments of substantal dscrepances between the predctons of Bayes-Nash-Equlbrum, (BNE), and behavor of 1

2 agents (players). 1 In response, several theoretcal models proposed to relax BNE to address such dscrepances. A subset of these models propose to extend BNE by allowng more flexble form of belefs than requred n BNE, but mantanng ndvdual ratonalty by nsstng that players use best response to ther (relaxed) belefs. In ths paper we brdge between BNE and level-k model, probably the leadng model n ths group and one wth several attractve features: 2 Frst, the model put reasonable requrements on the cogntve abltes of such non BNE players, n contrast wth other models. 3 Second, the herarchy of players levels of sophstcaton s constructed teratvely n a natural way by modelng Level 0 as the least sophstcated, and non strategc player,who(typcally) randomly selects a strategy from an allowable set, and then construct hgher levels by assumng that Level k, k = 1, 2,..., best respond to Level k 1. Thrd, the model s qute general, thus applcable and used to predct and analyze data from many laboratory and feld experments. In many games, such as beauty contest and hde and seek, t s ntutve to thnk n the model s terms. Four, and probably the most mportant feature of the model s the clam that t fts data generated n vared expermental envronments by just usng a mxture of players of level 1 and 2. Applcatons to data from laboratory and feld experments can be found n Crawford, Costa-Gomes and Irberr, (2013). Applcatons to statc games wth complete nformaton such as: the p-beauty contest (Bosch-Domenech, Montalvo, Nagel and Satorra, 2002), the two persons guessng game (Costa-Gomes and Crawford, 2006) and the money request game, (Arad and Rubnsten, 2012). For dynamc games, Kawagoe and Takzawa (2012) show that Level k model out predcts the Agent Quantal Response Equlbrum (AQRE) model n a centpede game wth ncreasng pe. Ho and Su (2013) apply dynamc Level k model to both four and sx stages of the centpede game and fnd t fts 1 There are mountng expermental Evdence that (Bayesan) Nash Equlbrum n statc game or Subgame (Perfect) Nash Equlbrum n Dynamc Games fal mserablly. There s a long lst of lterature, for example Mckelvey and Palfrey (1992), Kagel and Levn (2002) etc. 2 The orgn of level k model s attrbuted to Stahl and Wlson (1994, 1995) and was frst appled to analyze expermental data by Nagel (1995). There are many varatons and extensons of the model and we refer the reader to Crawford, Costa-Gomes and Irberr (2013), and references theren. 3 Other models non-bne models who attept to model bounded ratonal, non BNE, decson makers often end up requrng even more cogntve ablty than BNE, e.g. Cursed Equlbrum (CE) by Eyster and Rabn (2005). 2

3 the data well. In Bayesan Games lke auctons, partcularly frst-prce and commonvalue auctons, Crawford and Irberr (2007) clams that Level k model performs better than Cursed Equlbrum (CE) n most cases (and obvously better than NE that fals mserably). The survey by Crawford, Costa-Gomes, and Irberr (2013) documents many success of Level k model (and ts extensons) over other models and BNE. However, as we wll see below, n other cases t predcts poorly. We also know the mportance of BNE for economc analyss. Theoretcally, Level k model has been extended n two ways. Strzaleck (2010) allows belef to vary arbtrarly for players wth a certan cogntve bound. Thus, to construct models for a certan stuaton, we have the choce ether to vary the assumpton of dstrbuton among cogntve levels or belefs for certan type of players. However, the belef s stll restrcted to opponents wth lower cogntve bounds. From another perspectve, Alaou and Penta (2014) show how cogntve bounds, belef for opponents and belef for opponents belef vary accordng to ncentves by a cost-beneft analyss. In Alaou and Penta s model, If agents beleve opponents behave as lower levels than ther cogntve bound, they would behave as one level hgher than opponents. If nstead, agents beleve that the strateges of opponents are reachng ther cogntve bound or even hgher, they would act as ther cogntve bound. So even though Alaou and Penta (2014) consdered the stuaton where opponents are havng the same or hgher cogntve level as ther own, they are treated t beng one level below. Both of those paper have not consdered the strategy dfference when opponents are the same level or even hgher. Ross, Greene and House (1977), clam that people tend to thnk that ther behavor and/or attrbutes are common among others, and snce then there s a rch psychology lterature 4 supportng the fndng of the false consensus effect or self-anchorng argument. One weakness of the Level k model and ts extensons (e.g. allowng heterogenety among players; and respondng to dstrbutons of lower levels) s that t does not allow a Level k player to consder that her rvals are also as sophstcated as they are,.e. 4 Refers to Marks and Mller (1987) and all the lsted references. Ths s also related to nterpersonal percepton n socal psychology. Kenny (1994) shows that self-percepton s closely related to nterpersonal percepton n the way people thnk others are smlar wth themselves. 3

4 also beng a Level k decson maker. Often, n strategc stuatons (or games) a player knows that her opponent s a deeper (more qualfed) player. For example, n chess, t makes lttle sense for an ordnary player to assume that hs opponent, who s ranked as an Internatonal Master, s a lower level player than herself. Our model take a step forward n addressng ths shortcomng and challengng problem. It allows agents Level k players to beleve the possblty that ther opponents mght be of the same level as they are as well as a lower level. For example, n the experment of (Alaou and Penta, 2014), when students from math and scence departments play wth opponents from the feld of humanty, they would put more weght on the later beng Level k 1 than when they play wth fellows from departments as ther own. Furthermore, Palacos-Huerta and Oscar Volj (2009) fnd that n a centpede game, backward nducton explans data well n stuaton where chess player plays wth chess player, but performs badly n the cases where college students plays wth college students. Our hybrd model thus provde a parametrc approach to fnd when equlbrum or Level k model could apply as well as alternatve explanatons when both are not applcable. Our model, as CE and level k, also mantans the assumpton of ndvdual ratonalty but relaxes, relatve to BNE, assumptons on players belefs about other players. For dynamc games, Aumann (1992), and several other papers afterwards 5 have shown that a falure of backward nducton does not mply the falure of ndvdual ratonalty. For example, n the centpede game backward nducton mples that the frst mover stops the game at the frst decson node whch s rarely found n data from experments. However, a relaxaton of common knowledge of ratonalty s suff cent to ratonalze several rounds of contnuaton although all players are ratonal. Whle those paper focus ratonalzng such data, we go one step further and construct a parametrc model for predcton. A related extenson of level k model s Ho and Su (2013). However, ther model s ntended for learnng across repettons whle our model explans strategcal behavor better even 5 Aumann (1992), proves that contnuaton of the game, beyond the frst node as mpled by backward nducton, for several rounds could occur even wth mutual knowledge of hgh degree. Reny (1992), proves that the collapse of backward nducton does not necessarly mply the falure of ratonalty. Smlar arguments are also provded by Ben-Porath (1997), and the refnement by Ashem and Dufwenberg (2003). 4

5 for novel games and wth less parameters. Moreover, our model doesn t restrct the strategy set as Ho and Su (2013) whle we even captures Bayesan updatng across stage wthn a game. For dynamc games wth perfect nformaton and recall, the most related model s analogy-based expectaton equlbrum proposed by Jehel (2005), where agents group the decson nodes of other players nto parttons and form belefs about the average behavor n each analogy class. It concdes wth sub-game perfect, NE for the fnest analogy partton whle t ratonalze, n the centpede game, passng to the last few stages for a large range of parttons n contrast to the mplcaton of backward nducton. Jehel (2008), extends hs anology-based concept to Bayesan games. In such games, our and Jehel s models share advantages but ours allows belefs to be anchored at the begnnng of the game and are updated, by usng Bayes Rule, at each stage. Moreover, whle Jehel s model doesn t provde a specfc way to choose analogy class, our model offers an parametrc estmaton to specfy the model. For statc games wth ncomplete nformaton, Eyster and Rabn (2005) also proposed Cursed Equlbrum (CE) whch extends BNE and ratonalzes behavor (data) from experments where BNE fals mserably. Partcularly, n common-value (CV) auctons where systematc overbddng and losses, the Wnner s Curse, are robust phenomenon, but also n no-trade paradox due to adverse selecton and expermental data from votng and sgnalng models. In fully cursed equlbrum, people correctly predct opponents behavor dstrbuton but gnore ts dependence on opponent s type. Then the belef n χ-ce s a weghted average of fully cursed belef (wth weght χ) and belef n Bayesan Nash Equlbrum (BNE) (wth weght (1 χ)). Heterogenous behavor s thus characterzed by dfferent cursed levels (wth χ = 1 beng fully cursed, and χ = 0 beng BNE). The meanng for 0 < χ < 1 s not obvous, and even fully cursed belef (when χ = 1) needs to be formed after observatons and learnng, whch may explan why level k model outperforms CE n some novel games as n Crawford and Irberr (2007). The level k model also stands out for ts advantage n generalzaton. We thus propose a dfferent hybrd model, NLK, where agent beleves the opponent s one level below wth probablty λ and 5

6 as sophstcated as he s wth probablty (1 λ).in concluson, compared wth Cursed Equlbrum, Our weghts on belef have much clearer meanng as the probablty for an opponents to be dfferent types. Moreover, our model concde wth level k (when λ = 1), whch has better predcton for more general games than fully cursed equlbrum(when χ = 1). Our model brdges equlbrum and the Level k model of bounded ratonalty and t can be appled to many varous games. We construct the basc model and dscuss the exstence of the soluton n the next secton. In secton 3, we compare the predctons of the orgnal Level k model and NE to a smple verson of NLK n a two persons, statc Guessng Game and apply t to data from Arad and Rubnsten (2012). In secton 4, we provde the NLK soluton to the dynamc Centpede Game and compare ts predctons to those of NE and Level k by applyng t to data n Palacos-Huerta and Volj, (2009), Feld Centpedes. In secton 5 we do the same for a Common Value Aucton model by Avery and Kagel (1997). In Secton 6, we dscuss and explore possble extensons to games wth more than two players, heterogeneous beleves regardng other players, and allowng dstrbutons of lower level players. We conclude n secton 7. 2 The Basc Model The smplest Level k model has non BNE players wth a herarchy of sophstcaton levels denoted by k, k = 1, 2,..., n. The model assumes that a player of Level 0 s the least sophstcated and non strategc, who (typcally) selects a strategy randomly from an allowable set of undomnated starteges. 6 Hgher level players are then constructed 6 We do not allow our Level 0 to pck domnated strategy to separate sophstcaton from stupdty. One can fnd other ways of modelng level zero. For example, Crawford and Irrber (2007) consder two ways of modelng level zero n auctons wth ncomplete nformaton: a random Level 0 who bds unformly randomly, ndependent of ts own prvate sgnal, over the range determned by the range of ts sgnal and the value functon; and a truthful Level 0 who bds her value. But why would an unsophstcated Level 0 wth a PV of $30 bd $500 dollars n a sealed bd, frst-prce aucton? Insertng such Level 0 s not nnocent. Wth an alternatve Level 0 players who avod bddng domnated strateges, thus select unformly randomly but only between zero and ther values Level-k model under most expermental assumpton would predct NE and the explanaton for overbddng s lost. We avod such an avenue to ratonalze data. 6

7 recursvely by assumng that a player of Level k s best respondng to the strategy of Level k 1. We further smplfy here by havng only two players and concentratng on a smple symmetrc belefs of the two players to be ntroduced shortly. 7 Our basc model, NLK, starts wth the same assumpton on level 0 players but then t defnes hgher levels, denote by levelk N, k = 1, 2,.., n, n the followng way: Defnton 1 (NLK) In a normal form game wth two players G = (S, u ) =1,2 (where, (S, u ) are, espectvely, the strategy set and the utlty functon of player ) and symmetrc belefs. That s, level N k, k = 1, 2,..., n, beleves her opponent to be leveln k 1 wth probablty λ and levelk N wth probablty (1 λ), λ [0, 1]. Gven the strategy for leveln k 1, ( ) σ Nk 1, and λ. The Startegy of =1,2 leveln k, ( ) σ Nk, satsfes the condton below: =1,2 λu (σ Nk, σ Nk 1 j ) + (1 λ)u (σ Nk, σ Nk j ) λu (s, σ Nk 1 j ) + (1 λ)u (s, σ Nk j ), s S,. (1) Or, we can smplfy equaton 1 by redefnng the utlty functon as, u kλ (x, y) = λu (x, σ Nk 1 ) + (1 λ)u (x, y) : u kλ (σ Nk, σ Nk j ) u kλ (s, σ Nk j ), s S,. (2) Snce we also apply our model to analyze aucton data, we adjust the defnton for a Bayesan Game. Agan to smplfy notaton, we denote u kλ (x, y) = λu (x, σ Nk 1 j (θ j )) + (1 λ)u (x, y). Defnton 2 (BNLK) In a Bayesan Game of ncomplete nformaton of two players, B = (S, u, Θ ) =1,2, where Θ s the set of player s type wth pror dstrbuton p and symmetrc belefs, ths s, levelk N, k = 1, 2,..., n beleves the opponent to be leveln k 1 wth probablty λ and level N k wth probablty, λ [0, 1]. Gven the strategy for leveln k 1, ( σ Nk 1 (θ ) ), and λ. Then, leveln =1,2 k s strategy ( σ Nk (θ ) ) =1,2 satsfes condton below: 7 In secton 6 we ntroduce a natural way to generalze our model. 7

8 p(θ θ )u kλ (σ Nk (θ ), σ Nk j (θ j )) p(θ θ )u kλ (s, σ Nk j (θ j )), θ Θ, s S. = 1, 2. (3) Fnally, n order to apply our model to analyze the Centpede Game, we adjust our defnton for a dynamc game wth perfect nformaton and recall below. Defnton 3 (SPNLK) In a dynamc Game of perfect nformaton of two players P = (A, T, u ) =1,2, where A s an acton set and T s a game tree, wth symmetrc belef,.e. level N k, k = 1, 2,..., n, beleves the pror of opponent to be leveln k 1 wth probablty λ and level N k wth probablty (1 λ), λ [0, 1]. In every decson node wth hstory h, gven the strategy for levelk 1 N, ( σ Nk 1 (h ) ), and λ. Then =1,2 leveln k s strategy restrcted to the subgame ( σ Nk (h ) ) =1,2 and belef pk 1 (h ) for the opponent to be levelk 1 N satsfes the condtons below (we denote u kλ (x, y, h ) = p k 1 (h )u (x, σ Nk 1 (h j )) + (1 p k 1 (h ))u (x, y). j u kλ (σ Nk (h ), σ Nk j, h ) u kλ (a, σ Nk j, h ), a A, where. (4) p k 1 (h ) = λp(h σ Nk for all h that satsfes p(h σ Nk λp(h σ Nk, σ Nk 1, σ Nk 1 j, σ Nk 1 j j ) ) + (1 λ)p(h σ Nk ) > 0 or p(h σ Nk, σ Nk j ) > 0. Note, wth λ {0, 1},there s no Bayesan updatng for all hstores., σ Nk j ), (5) Proposton 1 (Exstence for NLK): Gven the strategy for level k 1, ( ) σ Nk 1 λ, for every fnte strategc-form game G = (S, u ) =1,2, there exst ( ) σ Nk =1,2. =1,2 and Proof. We can check whether the best response correspondence satsfes the condton for Kakutan (1941) s fxed pont theorem. The proof s smlar to the one for Nash Equlbrum (Glksberg, 1952). See detal n the Appendx. Or more ntutvely, gven the strategy for levelk 1 N, ( ) σ Nk 1 and λ, consder the alternatve game =1,2 Gλ = ( ) S, u kλ. =1,2 Frst, there exsts a NE (σ ) =1,2 for G λ. Then t must satsfes equaton (2). So t satsfes 8

9 the condton for ( ) σ Nk for the orgnal game G = (S =1,2, u ) =1,2,that s ( ) σ Nk exsts. =1,2 Remark 1 Obvously, when λ = 1, NLK concdes wth Level k model and when λ = 0, t concdes wth NE. Proposton 2 (Exstence for BNLK and purfcaton): Gven the strategy for levelk 1 N, ( σ Nk 1 (θ ) ) and λ, for every fnte Bayesan game, then there exst ( σ Nk =1,2 (θ ) ). =1,2 Moreover,f the type dstrbuton p s ndependent and twce-dfferentable, fx a set of players, I and strategy space S, for a set of payoffs {u (s)} I,s S of Lebesgue measure 1. Any equlbrum of the payoffs u s the lmt as ε 0 of a sequence of pure-strategy equlbrum of the perturbed payoff u. Proof. Agan, gven the strategy for levelk 1 N, ( σ Nk 1 (θ ) ) =1,2 and λ,we can look at an alternatve Bayesan Game B λ = (S, u kλ, Θ ) =1,2. Then t follows from the fact that whenever B s fnte, B λ s fnte, and snce a fnte game has at least one BNE, the one for B λ s ( σ Nk (θ ) ) for B. By by smlar argument we can also get purfcaton results =1,2 by Harsany (1973). Remark 2 Agan, obvously that when λ = 1, NLK concdes wth Level k model and when λ = 0, t concdes wth BNE. However, snce the aucton game we consder have uncountable nfnte type and acton spaces, ths proposton s of lmted general nterest. But NLK do exsts for any gven λ and k for the model we consder. For ts analog wth BNE and easy style of generalzaton, we suppose that NLK exsts almost all games of economc nterest. Proposton 3 (Exstence for SPNLK):Gven the strategy for level k 1, ( σ Nk 1 (h ) ) =1,2 for every decson node wth hstory h and λ. In any fnte extensve form game, there exst ( σ Nk )=1,2 and pk 1 ( ) whch satsfes condtons for SPNLK. Proof. Consder an alternatve dynamc game of ncomplete nformaton P = (Φ, A k 1, A, T, u ) =1,2, where Φ denotes the possble type for agent whch can be ether θ Nk 1 or θ Nk. A k 1 s the acton set for a player of type θ Nk 1 and A k 1 (h ) = σ Nk 1 (h ) s gven, h.then accordng to Kreps and Wlson (1982), for every fnte extensve form 9

10 game, there exst at least one sequental equlbrum (σ, p ) =1,2. Snce there are only two players, the set of Perfect Bayesan Equlbrum(PBE) concde wth Sequental Equlbrum accordng to Fudenberg and Trole (1991). So (σ, p ) =1,2 s also a PBE. Then ( ( σ (θ Nk ), p )) θ Nk should satsfy equaton (4) and (5) for sequental ratonalty and bayesan updatng. In other words, SPNLK exsts. Remark 3 obvously when λ = 1, NLK concdes wth Level k model and when λ = 0, t concdes wth Subgame Perfect Nash Equlbrum(SPNE) or Backward Inducton. 3 The Arad Rubnsten, Money Request Game. In Arad and Rubnsten (2012), there are two rsk-neutral players; each can request an nteger amount of money from $11 to $20, and receve the amount s/he requests. In addton, a player receves an extra $20 f s/he asks for exactly one nteger less than the other player. In level k model, level 0 randomze unformly on {$11, $12,..., $20}. A level 1, that requests $20 earns $20. Alternatvely, f s/he asks for $19 she would earn $19 for sure and $20 bonus wth probablty 1/10. for a total expected payoffs of $21 It s clear to see that: level 1 pcks $19; level 2 pcks $18;..., level 9 pcks $11. But then level 10 pcks $20; level 11 pcks $19; and so on. The sophstcaton level of player s not well dentfed from the strategy s/he played: A player who requests $11, can be a level 0 or a hghly sophstcated level 9 player. To compare level k, NE and NLK wth dfferent λ, we use maxmum lkelhood method below. We assume that a subject of level k wth belef λ normally follows strategy b k λ but subject to logstc errors of precson ε. 8 Then the probablty of observng strategy b for 8 To ft data, we ntroduced an error structure. Logstc error s wdely used n Industral Organzaton and psychology lteratures. 10

11 level k s P (b k, λ, ε) = belef λ. exp(εu k (b λ) 20 j=11 exp(εu k(j λ), Where u k(b λ) s the payoff for pckng b gven When the error s costler ex-ante, t s less lkely that player would make t. The parameter, ε, represents senstvty to cost of error. When ε 0,players choose any number randomly whle when ε,players make optmal choces gven ther belef. The the model s lkelhood (L) and log-lkelhood (LL) functons are respectvely, N L(ε b, λ, k = 1) = P (b k, λ, ε) and LL(ε b, λ, k = 1) = N =1 log(p (b k, λ, ε). =1 The followng table compares level k, NE and NLK wth dfferent λ: Table 1 Acton d ε LL ( levels 9 ( 8 ) ( 7 ) ( 6 ( 5 ( 4 ( 3 ) ( 2 ) ( 1 ) ( 10 ) 19) ) 15) 14) Level 1 (%) % NE(%) % NLK(%) λ = 1 2 NLK(%) λ = % % DATA * The sum of absolute dfferences between data observaton and modelpredcton. ** The loglkelhood s scared by 108/100 for smplcty whch woundn t make much dffrence. *** The data s from Arad and Rubnsten (2012). For an arbtrary λ = 1, the loglkelhood functon for NLK s lower than that of NE 2 and level 1. However, wth λ = 3, NLK s loglkelhood mproves by about 9% and s hgher 4 than the other two. The followng Table (2) shows that for each λ, there s a unque predcton by NLK. For each λ [0, 19/20), there s a unque mxed strategy for level N 1, (recall that for λ = 0, t concdes wth NE). For each λ [19/20, 1], there s a unque pure strategy to request 19 for level N 1, (recall that for λ = 1, t concdes wth level 1 ). To fnd the optmal λ, our model could do even better. 11

12 Table 2 Acton for level N λ < λ < (5 10λ) 14 λ < (5 2λ) 5(14 20λ) 17 λ < (4 2λ) 15 5(17 20λ) 5(3 2λ) (19 20λ) 5(2 2λ) (1 2λ) 5 (%) In related lteratures such as Crawford and Irberr (2007), the data s usually estmated by a model wth a mxture of level 1, level 2 and level 3 players. So we also want to compare t wth our model. We pcked λ = 3 4 arbtrarly for our model and restrcted to only the type of level N 1. For the orgnal level k model, we allow the mxture of level 1 and level 2. Snce λ = 3 4 s arbtrarly chosen. Now our model has at most the same parameter as the orgnal level k. We also compare our model wth the optmal λ and the orgnal level k model wth the best mxture of level 1, level 2 and level 3, n whch case, our model even has strctly less parameters. Let π k 0, denote the proporton of type k n the populaton, wth k π k = 1. The lkelhood, (L), and loglkelhood (LL) functons are respectvely, N L(ε b, λ, k) = π kp (b k, λ, ε) and LL(ε b, λ, k) = N k =1 log( π kp (b k, λ, ε)). k =1 The estmaton result s summarzed n the followng table: Table 3 π 1 π 2 π 3 LL ε level k restrcted = level N k λ= restrcted = 0 restrcted = level k level N k λ= Restrcted = 0 Restrcted = Our model wth only one level and λ = 3 4 outperforms level k model wth the best mxture of level 1 and level 2. Both the lkelhood and precson s bgger. To maxmze 12

13 loglkelhood and keep a reasonable precson 9, we get the estmaton of precson ε = and λ = 0.85, where the loglkelhood s: 21.61, whch mproves dramatcally than the arbtrarly chosen λ.moreover, our model wth one level and the constrant optmal λ = 0.85 mproves a lot on level k model wth level 1, level 2 and level 3. Both the lkelhood and precson s much bgger. 4 The Centpede Game. Introduced by Rosenthal (1981) the Centpede Game serves as a paradox llustratng devatons from Backward Inducton or from the Subgame Perfect Nash Equlbrum (SPNE) 10. Player A starts frst and player B follows. The ntal pot worth $5. In stage 1, player A has the property rghts to the pot. She can choose ether to end the game by takng 80% of the pot ("and leavng 20% of the pot to player B" ) or allow the pot to double by passng the property rghts to player B. In stage 2, player B has smlar decson. Player A and player B choose alternately. Let s assume the game ends after even perods S = 2N. So the payoff f game ends n odd perod 2k + 1,s ( 2 2k 4, 2 2k), k = 0, 1, 2,..., N 1.The payoff f the game ends n even perod 2k s ( 2 2k 1, 2 2k 1 4 ), k = 1, 2,..., N. Usng backward nducton t s smple to show that the unque SPNE s for player A to take the pot at stage 1, and the game ends. Then, let s look at level k soluton. Frstly, level 1 player B would take at the last stage. For level 1 player A, take at 2N 1 would get 4 x where x s the whole pe, pass would get 5 9 x,so level 5 1 player A would pass at (2N 1). Then for level 2 player A, take at (2N 1) would get 4 5 x where x s the whole pe, pass would get 2 5 x, so level 2 player A would take at (2N 1).Interatvely, for game wth stage S = 2N. Player A would pass whle take at stage s = 2(N h) + 1 for level k, k = 2h, 2h + 1, h N 1. For all player A wth level k = 2N or hgher, t would "take" at the frst stage. Player B would "take" at stage 9 We keep the precson to be ε = 0.346, and then get the optmal λ to maxmze lkelhood. 10 For lterature, refers to Mckelvey and Palfrey (1992), Mckelvey and Palfrey (1996), Fey, McKelvey and Palfrey (1996), Nagel and Tang (1998), Gary Bornsten (1998),Rapoport et al. (2003) and Bornsten, Kugler and Zeglemeyer(2004) whch show that even n the stuaton of hgh stake, consderng altrusm, group decson, backward nducton s stll nadequate to explan players behavor. 13

14 s = 2(N h) + 2 for level k, k = 2h 1, 2h, h N. for all player B wth k = (2N 1) or hgher, t would "take" at the second stage when he frst makes decsons. The bzarre of level k soluton s that t need relatvely hgh level to explan "take" at earler stages especally for game wth more stages. In other words, strateges of dfferent levels are ndepedent of the length of the game. For example, no matter how long the game s, level 1 player would always keep passng to the last node where player B takes. Moreover, no matter what hstory s observed, level k would not change ts belef about opponents. Then, Let s look at NLK. Frstly, t s obvous that the case λ = 0 concdes wth Backward Inducton and λ = 1 concde wth level k model. Let s dscuss the case when 0 < λ < 1.It s obvous that Player B would "take" at the last stage upon reachng. Let s assume player B would take at stage 2n. Then at stage 2n-1, player A s belef for the opponent beng level 0 s p 0 A(2n 1) = ) n 1 λ ( 1 2 λ ( ) 1 n 1 (0, λ) λ If player B take at stage 2n, for stage (2n 1), player A "take" gets 4 x, whle 5 "pass" gets [ p 0 A (2N 1) p0 A (2N 1)] 2 5 x+p0 A (2N 1) x= 2 5 x+p0 A (2N 1) 7 5 x So pass f 2 < 7 p0 A (2n 1) 1 and take otherwse. Because p0 A (2N 1) s decreasng n N, for a longer game, level N 1 s more lkely to take at stage 2N 1. Ths dffers our model from level k where level 1 awalys pass to the end except for the last stage no matter how long the game s. Note that p 0 A (2n 1) s decreasng n n. Then for λ 2, Player A would always 7 take one stage before player B. For λ 2 7, there may be some crtcal value s A,such that p 0 A (2n 1) > 2 7, for n s A, and p 0 A (2n 1) 2 7, for n s A + 1. Let s assume that player A would "take" at stage 2n + 1. Then at stage 2n, player B s belef for the opponents beng level 0 s p 0 B(2n) = ) n λ ( 1 2 λ ( ) 1 n = p 0 A(2n + 1) λ 14

15 Fgure 1: Centpede Game Contnue Contnue Contnue Contnue Contnue Contnue (256,64) stop stop stop stop stop stop (4,1) (2,8) (16,4) (8,32) (64,16) (32,128) Then the strategy for player B s smlar wth that for player A, "pass" f 2 7 < p0 B (2n) 1 and take otherwse. Remark 4 Frstly, for λ 2 7, Both player A and B would "take" 11 at the frst tme they make choces. Game ends at the frst stage. The same results as λ = 0 n SPNE. Secondly, λ > 2, f the game s short enough such that 7 p0 A (2N 1) > 2.Then player A 7 would "pass" to the end and player B would "take" at the last stage. The same result as λ = 1 n level k model. For λ > 2 7 and the game s relatvely long such that p0 A (2N 1) 2 7, game would end earler than level k model. Especally, snce p 0 B (2n) = p0 A (2n + 1) let s be the crtcal value such that p 0 B (2n) > 2 7 for n < s and p 0 B (2n) 2 7 for n s. Then level N 1 Player B would "take" at and after stage 2s whle level N 1 player A would take" at and after stage 2s + 1. Game ends at stage 2s. To compare wth experment data, partcularly, we consder the centpede game n (Palacos-Huerta and Volj, 2009) where N = 3 as Fgure 1, For each λ, there s a unque pure strategy SPNLK as summerzed n table 4 as well as data from Palaco-Huerta and Volj(2009). When 8 13 < λ 1,player A pass to the end and player B take at the last stage. Game ends n stage 6 as for level 1. When λ 2 7, both player would take at the frst tme they make choce. Game ends n stage 1 as n backward nducton. When 2 7 < λ 4 9. Game ends at stage 2. When 4 9 < λ 8 13, game ends at stage 4. When 8 13 λ 1, Game stops at stage We assume each player would take when they are ndfferentce between take and pass. 15

16 Table 4 Data or Predcton f1 f2 f3 f4 f6 f7 level N 1 (0 λ 2 ) ncludes SPNE level N 1 level N 1 ( 2 < λ ) ( level N 4 1 < λ ) ( 8 < λ 1) ncludes 13 level Data 29 (Students vs Students) Students vs Stu mpled stop prob 29 (200,0.03) (194,0.17) (161,0.42) (93,0.65) (6,0.83) Data (Students vs Chess Players) Students vs Chess mpled stop prob. (200,0.30) (140,0.52) (67,0.61) (26,0.69) Data (Chess Player vs Students) Chess vs Stu mpled stop prob. (200,0.375) (125,0.44) (70,0.56) (31,0.61) Data (Chess Player vs Chess Player) Chess vs Chess mpled stop prob. (200,0.725) (55,0.64) (20,0.90) (2,1) From the table, even wthout addng any error structure, you can see there s a trend that when players are more sophstcated, a smaller λ would fts data better. For example, for the treatment when student plays wth students, 0 λ 2 7 fts the data the best. For the treatment when students plays wth chess player 2 7 < λ 4 9 fts data the best. Fnally, 4 9 < λ 8 13 fts data the best for the last two treatments and especally for the one when Chess player plays wth chess player. So, we clam that our model do provde an alternatve explanaton for the scenaros when both the orgnal level k and backward nducton do not apply. Moreover, we tght our hands by only allowng symmetrc belef. Snce when students plays wth Chess players, both sdes are more lkely to have dfferent subjectve λ. We suppose our model would do even better when heterogeous beleves are allowed. More wll be dscussed n secton 6. We also generalze our soluton n the Appendx wth comparson to the result of analogy based equlbrum. 16

17 5 Common Value Aucton In the Common-Value Second-Prce Aucton exprment by Avery and Kagel (1997), there are two bdders, value functon s the sum of two prvate sngal X wth..d U [1, 4]. That s, u (X) = X 1 + X 2 for = 1, 2.There are two mportant functons: v(x, y) = E[u (X) X 1 = x, X 2 = y] = x + y r(x) = E[u (X) X 1 = x] = x = x Then followed from a more general result by Mlgrom and Weber (1982), there s only one symmetrc BNE wth truethful bddng condtonal on just wnnng B 0 (x) = v(x, x) = 2x.Then follows from Crawford and Irberr (2007), strategy for Random level 1 s B 1 1(x) = r(x) = x Denote b k λ ( ) as the NLK strategy for level k wth belef λ.we solve the symmetrc lnear strategy for level N 1. The detal s provded n the Appendx. The data (Avery and Kagel, 1997) are both evaluated by "Cursed Equlbrum", (Eyster and Rabn, 2005) and "Level-K", (Crawford and Irberr, 2007). Eyster and Rabn (2005) shows any other cursed levels (χ (0, 1]) s better than BNE (χ = 0) and gven a cursed level, the model fts better for experenced subjects than for nexperenced subjects n terms of Mean Squared Error (MSE). For only nexperenced bdders, Crawford and Irberr (2007) nstead argue that wth logstc error structure and subject-specfc precsons, a mxture of level 1 and level 2 s better than cursed equlbrum (wth the best mxture of dfferent cursed levels wth restrcton to multples of 0.1 n [0, 1].) accordng to lkelhood and Bayesan nformaton crteron (BIC) BIC penalze models wth more parameter to adjust the lkelhood. 17

18 Table 5 Model b(x) RMSE(nexperenced) RMSE(experenced) NE λ=0 2x level 1 λ=1 level N 1 λ=0.5 Date Inexperenced Data Experenced x x x (0.079) (0.203) 1.313x (0.053) (0.15) We frst constraned λ = 0.5 and compare our model wth NE and level 1. In terms of MSE, level 1 s the best for nexperenced bdders whle our model wth λ = 0.5 s better for experenced bdders. So before we consder the senstvty of error and the mxture of hgher levels, the orgnal level k model dd performs better for nexperenced bdders n the wallet game. However, after 18 perods learnng, NLK wth λ = 0.5 performs better than both level k and NE. We already found that our model can brdge Level k and NE for novel games (the game and the Centpede Game) when players have ntal beleves for opponents to be both less sophstcated and thnk n the same way. Moreover, n the Centpede game, we captures how belef s updated across sectons wthn one game. Now, we clam that our mode also brdges level k and NE n terms of learnng by repettons. To check robustness of the argument, we compare result for more λ and summerze t n the followng table, 18

19 Table 6 Model for level-1 b(x) RMSE (nexperenced) RMSE(experenced) λ = 0 2x λ = 1 x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x λ = x * RMSE= 1 (ˆ b b) n 2 We plot the RMSE for both nexperenced and experenced bdders as n the Fgure 2 and 3. As you can see, The orgnal level 1 dd gve the best predcton for nexperenced 19

20 Fgure 2: Inexperenced Bdders Fgure 3: Experenced Bdders 20

21 bdders. But our model wth λ hgher than 0.3 all out predct the orgnal level 1 for experenced bdders. 6 Dscusson for possble extenson In examples above, we never go to the type of level N 2 for the already excellent performance of level N 1. However, we could generate level N 2 and even hgher levels teratvely as n the orgnal level k model. For example, n the 6 stages Centpede Game of Palaco-Huerta and Volj(2009), for each λ,there exsts a unque pure strategy SPNLK for level N 2 as summerzed n the followng table. Table 7 Predcton for Level N 2 f 1 f 2 f 3 f 4 f 5 f 6 f 7 0 < λ < λ < λ Our model can be extended to games wth more than two players and heterogenous belef for oppoents. For example, f there are N players n the game. The probablty for number g of players to be lower level can be descrbed by Bnomnal dstrbuton p λ (g) = C g N λg (1 λ) N g. It s also possble to allow dstrbutons of all lower type players n the sprts of Cogntve Herarchy Model by Camerer, Ho and Chong (2004). For example, now let λ be the pror for opponents to be all lower levels. Let G(h L), h k, be the probablty that the opponent s level h condtonal on beng lower levels. Then the probablty for the opponent to be level N h s λg(h L), h k and 1 λ for beng leveln k. Moreover, NLK also provde a method to check the robustness of Mult-Equlbrum Issues. Intutvely, when there s mult-equlbrum for BNE, snce the strategy for level ( ) k s usually unque, then there mght exst some threshold λ. ˆ When λ 0, λ ˆ, thre ( ) s multple equlbrums whle there exst unque result for λ ˆλ, 1. For example, 21

22 Rubnsten and Sallant (2015) dscover the exstence of self-smlarty n the chcken gamel as below. Table 8 Dove Hawk Dove 30, 30 20, 70 Hawk 70, 20 0, 0 Let D λ N1 be the probablty of choocng dove for level N1 (n the mxed strategy equlbrum). Results are summerzed n the followng table. You can see, when 2 3 < λ 1, the equlbrum s unque as for level 1 whch s to play hawk for sure. When 0 < λ 2 3, there s always two pure strategy equlbrum where one player chooses dove and the other plays hawk as well as a mx strategy equlbrum. Table 9 Belef D λ N1 Pure strategy equlbrum Multequlbrum? 2 < λ 1 No level 3 N1 always want to play hawk Unque 0 < λ () (Dove, Hawk) (Hawk, Dove) Two Pure, One mx λ = (Dove, Hawk) (Hawk, Dove) Two Pure, One mx 7 Concluson Ths paper has proposed a hybrd model, NLK, that connects Nash Equlbrum and Level k model. It provdes a parametrc way to explore when NE (λ = 0) or level k model (λ = 1) works well and serves as an alternatve approach for estmaton when both fall. It generalzes level k model and ts extenson by allowng level k players to beleve the opponents are possbly as sophstcated as they are. Our analyss shows that n a statc Guessng Game (Arad and Rubnsten, 2012), a dynamc Centpede Game (Palacos-Huerta and Volj, 2009) and a Common Value Aucton (Avery and Kagel, 1997), a smpler verson of our model makes much closer predctons than NE and level k. We clam NLK brdges NE and level k model n three ways. Frst, NLK provdes an alternatve model for statc games when both NE and level k 22

23 do not apply. Second, NLK predcts the behavor of dfferent types of players better n dynamc games and brng Bayesan Updatng back to the level-k argument. Fnally, snce level-k clams they only apples to novel game and all behavor converges to equlbrum n the long run, NLK jumps n for players wth some experence. Moreover, the λ that fts the data best n dfferent treatments concde wth ntuton. For example, n the Centpede Game, when the opponent s chess player rather than students, optmal λ s smaller. And n the Common Value Aucton Model, compared to nexperenced rounds, optmal λ s smaller for experenced rounds. NLK can also be easly generalzed to capture other models of economc nterest Thus NLK not only brdges NE and level k model but also provdes new nsght to explan strategc behavor when players are bounded ratonal. References [1] Ashem, G. B., & Dufwenberg, M. (2003). Deductve Reasonng n Extensve Games*. The Economc Journal, 113(487), [2] Aumann, R. J. (1992). Irratonalty n game theory. Economc analyss of markets and games: Essays n honor of Frank Hahn, [3] Alaou, L., & Penta, A. (2014). Endogenous depth of reasonng. [4] Arad, A., & Rubnsten, A. (2012). The money request game: a level-k reasonng study. The Amercan Economc Revew, 102(7), [5] Ben-Porath, E. (1997). Ratonalty, Nash equlbrum and backwards nducton n perfect-nformaton games. The Revew of Economc Studes, 64(1), [6] Bornsten, G., Kugler, T., & Zegelmeyer, A. (2004). Indvdual and group decsons n the centpede game: Are groups more ratonal players?. Journal of Expermental Socal Psychology, 40(5),

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25 [17] Jehel, P., & Koessler, F. (2008). Revstng games of ncomplete nformaton wth analogy-based expectatons. Games and Economc Behavor, 62(2), [18] Kakutan, S. (1941). A generalzaton of Brouwer s fxed pont theorem. Duke mathematcal journal, 8(3), [19] Kawagoe, T., & Takzawa, H. (2012). Level-< > k</> analyss of expermental centpede games. Journal of Economc Behavor & Organzaton, 82(2), [20] Kenny, D. A. (1994). Interpersonal percepton: A socal relatons analyss. Gulford Press. [21] Kreps, D. M., & Wlson, R. (1982). Sequental equlbra. Econometrca: Journal of the Econometrc Socety, [22] Marks, G., & Mller, N. (1987). Ten years of research on the false-consensus effect: An emprcal and theoretcal revew. Psychologcal Bulletn, 102(1), 72. [23] McKelvey, R. D., & Palfrey, T. R. (1992). An expermental study of the centpede game. Econometrca: Journal of the Econometrc Socety, [24] Mlgrom, P. R., & Weber, R. J. (1982). A theory of auctons and compettve bddng. Econometrca: Journal of the Econometrc Socety, [25] Nagel Rosemare, Unravelng n Guessng Games: An Expermental Study. Amercan Economc Revew, Vol. 85, No. 5, pp [26] Nagel, R., & Tang, F. F. (1998). Expermental results on the centpede game n normal form: an nvestgaton on learnng. Journal of Mathematcal Psychology, 42(2), [27] Reny, P. J. (1992). Ratonalty n extensve-form games. The Journal of Economc Perspectves, [28] Rapoport, A., Sten, W. E., Parco, J. E., & Ncholas, T. E. (2003). Equlbrum play and adaptve learnng n a three-person centpede game. Games and Economc Behavor, 43(2),

26 [29] Rosenthal, R. W. (1981). Games of perfect nformaton, predatory prcng and the chan-store paradox. Journal of Economc Theory, 25(1), [30] Ross, L., Greene, D., & House, P. (1977). The false consensus effect : An egocentrc bas n socal percepton and attrbuton processes. Journal of Expermental Socal Psychology, 13(3), [31] Stahl, Dale and Paul Wlson (1994). Expermental Evdence on Players Models of Other Players. Journal of Economc Behavor and Organzaton, 25, [32] Stahl, Dale and Paul Wlson (1995). On Players Models of Other Players: Theory and Expermental Evdence. Games and Economc Behavor, 10, [33] Strzaleck, T. (2010). Depth of reasonng and hgher order belefs. Harvard Insttute of Economc Research, Harvard Unversty. [34] Palacos-Huerta, I., & Volj, O. (2009). Feld Centpedes. The Amercan Economc Revew, 99(4), Proof for exstence. Gven the strategy for levelk 1 N, ( ) σ Nk 1 and λ. Let f =1,2 : Σ Σ be player s best response correspondence that maxmze u kλ (σ, σ j ) = λu (σ, σ Nk 1 j ) + (1 λ)u (σ, σ j ). Defne the correspondence f : Σ Σ to be the Cartesan product of f. A fxed pont of f s a σ, such that σ f(σ), that s, for each player σ f (σ). Thus a fxed pont of f satsfes equaton (2). Accordng to Katutan s fxed pont theorem, we need to check the followng suff cent condtons for f : Σ Σ have a fxed pont. (1) Σ s a compact, convex, nonempty subset of a fnte dmensonal Eucldean Space. (2) f(σ) s nonempty for all σ. (3) f(σ) s convex for all σ. (4) f( ) has a closed graph: f (σ n, σ n ) (σ, σ), σ n f(σ n ). Then σ f(σ). Let s check all the condtons are satsfed. 26

27 (1) Snce each Σ s a smplex of dmenson #(S ) 1, Σ s the Cartesan product of Σ. (1) follows drectly. (2) Snce each player s utlty u kλ (σ, σ j ) s lnear and thus contnuous n hs own mxed strategy. Then contnuous functons on compact sets attan maxmum. Then condton (2) s also satsfed. (3) If σ,σ both belongs to f(σ), then u kλ (µσ + (1 µ) σ, σ j ) = λu (µσ + (1 µ)σ, σ Nk 1 j ) + (1 λ)u (µσ + (1 µ)σ, σ j ) =λ [ µu (σ, σ Nk 1 j ) + (1 µ)u (σ, σ Nk 1 j ) ] + (1 λ)[µu (σ, σ j ) + (1 µ)u (σ, σ j )] =µ [ λu (σ, σ Nk 1 j ) + (1 λ)u (σ, σ j ) ] + (1 µ) [ λu (σ, σ Nk 1 j ) + (1 λ)u (σ, σ j ) ] =µu kλ (σ, σ j ) + (1 µ)u kλ (σ, σ j ) So the weghted average of σ,σ also belongs to f(σ) snce they then yelds the same expected utlty. So (3) s satsfed. (4) BWOC. Suppose condton (4) s volated, so there s a sequence (σ n, σ n ) (σ, σ) wth σ n f(σ n ), but σ / f(σ) Then σ / f (σ) for some player. Thus there s an ε > 0 and a σ such that u kλ (σ, σ j ) > u kλ ( σ, σ j ) + 3ε (a). Snce u kλ (σ, σ j ) s contnuous n σ and (σ n, σ n ) (σ, σ). Then for n suff cently large we have u kλ (σ, σ n j ) > u kλ (σ, σ j ) ε, u kλ ( σ, σ j ) + ε > u kλ ( σ n, σ n j ) Then substtute (a) n, u kλ (σ, σ n j ) > u kλ ( σ, σ j ) + 2ε > U k ( σ n, σ n j ) + ε. Then σ s strctly better than σ n for σ n j, whch contradcts wth σ n f(σ n ). So (4) s satsfed. 9 More Generalzed result for Centpade agame and Analogy-based Equlbrum Consder the centpede game n (Jehel, 2005). Two player = 1, 2 move n alternatve order startng wth player 2. There are 2K, K 2 Decson nodes as labelled n the fgure. At each node, the player whose turn t s to move, say player, may ether take n 27

28 whch case ths s the end or pass,.e A = {P ass, take}. The game also ends when player 1 Pass at node N (1) 1. The scalars a t and b t defne the payoffs at each termnal node. These scalars are assumed to be non-negatve and satsfy a 2k 1 > a 2k+1 > a 2k and b 2k 2 > b 2k > b 2k 1.Where a 2k+1 > a 2k and b 2k > b 2k 1 make sure tha players always want to take f take happens next perod whle a 2k 1 > a 2k+1 and b 2k 1 > b 2k gurantees that players always want to pass f take happens two perods later. Then the followng remark ndcate that level 1 s strategy strategy concde wth the analogy-based expectaton equlbrum of the coarest analogy groupng under the same condton. Remark 5 Suppose that for all k 1, 1 2 a 2k a 2k > a 2k+1, 1 2 b 2k b 2k 1 > b 2k.Then level 1 player would keep passng throughout the game except n the last node N (1) 1 at whch player 1 takes. Level 2 player 1 play the same strategy as level 1 player 1 whle level 2 player 2 keeps passng whle take at N (1) 2. By nducton player 1 would keep passng and take at stage N (h) 1 for level k = 2h 1, 2h.h K. For player 1 wth level 2K-1 or hgher, t would take at N (K) 1. Player 2 would take at stage N (h) 2 for level k = 2h, 2h + 1, h K 1. For player 1 wth level K or hgher, t would take at N (K) 2. Remark 6 The dsadvanges for level k model stll holds n ths generalzed stuaton. We stll need may levels to explan heterogenety n data, strateges of dfferent levels are ndependent of the length of the game and belefs are not updated throughout the game. In analogy-based equlbrum, we could get dfferent strategy for certan analogy partton. However, belef s stll fxed nsde each analogy group. Remark 7 Suppose that for all k 1, 1 2 a 2k a 2k > a 2k+1, 1 2 b 2k b 2k 1 > b 2k. (The same restrcton could get smlar result as n the smpler model. If λ s small enough, then both player take at the frst tme they make choce, the same as n SPNE. If λ s bg enough or K s small enough, then both player would pass to the end except that player 1 would take at node N (1) 1, whle concde wth the results n level k model. For any other stuaton, there exsts a unque predcton that game ends n some mddle stages. (Whle by proposton 3 of Jehel (2005), game ends n mddle stages could never happen n the coarsest groupng of analogy-based equlbrum.) 28

29 10 Solve the strategy of level N 1 for the Common Value Aucton Model Set up the model for NLK.Let s assume there s a lnear pure strategy NE functon for level N 1, then we could wrte t as b λ (x) = b λ (1) + b λ(4) b λ (1) 3 (x 1) x [1, 4].Denote d λ = b λ (4) b λ (1),the probablty for the opponent to be level 0 condtonal on te gven such proposed NE s q λ =Pr(rval=level 0 te at bd=b)= 6() 6()+λd. Then NE s defned by ndfference n the case of MWP(x)=b(x). MW P (1) = q( ) + (1 q)2 = 1.5q + 2 = b λ (1), MW P (4) = q( ) + (1 q)8 = 8 1.5q = b λ (4). Then d λ = b λ (4) b λ (1) = 6 3q = 6 Then d 2 λ + 3 d λ λ 36() = 0. λ 3λd λ 6()+λd λ. So the bddng strategy s b λ (x) = b λ (1)+ d(λ) 3 (x 1). Where b λ(1) = 1.5q(λ)+2, q(λ) = ()+ ()(1+15λ) 3()+ ()(1+15λ) and d(λ) = 3()+3 ()(1+15λ) 2λ. 29