Asynchronous Choice in Battle of the Sexes Games: Unique Equilibrium Selection for Intermediate Levels of Patience

Transcription

1 Asynchonous Choice in Battle of the Sexes Games: Unique Equilibium Selection fo Intemediate Levels of Patience Attila Ambus Duke Univesity, Depatment of Economics Yuhta Ishii Havad Univesity, Depatment of Economics Septembe 2011 Abstact This pape shows that in infinite-hoizon asynchonous-move battle of the sexes games, fo a full-dimensional set of payoff specifications, thee is an intemediate ange of discount ates fo which evey subgame pefect Nash equilibium induces the same unique limit outcome. The latte is one of the pue Nash equilibia of the stage game, and play is absobed in it in finite time with pobability 1. We fully chaacteize the set of game specifications fo which thee is a unique limit outcome in any Makov pefect equilibium, but show by example that these conditions ae not sufficient fo unique selection in the limit in all subgame pefect Nash equilibia. We also povide sufficient conditions fo the existence of a unique limit outcome in any subgame pefect Nash equilibium. Ou esults complement the findings of Lagunoff and Matsui (1997) and othes, who show that in a class of coodination games asynchonicity of moves leads to unique equilibium selection fo high enough (as opposed to intemediate) levels of patience. Keywods: epeated games, asynchonous moves, battle of the sexes, uniqueness of equilibium, geneicity JEL Classification: C72, C73 aa231@duke.edu ishii2@fas.havad.edu

2 1 Intoduction Asynchonous-move epeated games have been geneating consideable inteest in the game theoy liteatue since the seminal papes of Maskin and Tiole (1988a, 1988b), and moe ecently in empiical industial oganization (Acidiacono et al. (2010), Doaszelski and Judd (2012)). The theoetical inteest stems fom the fact that the pedictions obtained fom these games ae often vey diffeent than those fom standad epeated games with simultaneous moves. 1 In empiical applications, as Acidiacono et al. (2010) points out, the main advantage of asynchonous-move models is that thei computational complexity is geatly educed elative to standad models. This facilitates stuctual estimation and countefactual policy-simulations in complicated dynamic models with a lage numbe of state vaiables. Futhemoe, as pointed out in Schmidt- Dengle (2006) and Einav (2010), in many situations asynchonicity of moves is a moe ealistic modeling assumption. This pape analyzes battle of the sexes games (intoduced by Luce and Raiffa (1957) as the simplest class of games in which playes want to coodinate, but thei inteests diffe in which outcome to coodinate on) in an infinite-hoizon continuous-time famewok, in which playes accumulate payoffs continuously, but they can only change thei actions at andom discete times, govened by independent Poisson pocesses. 2 Despite the stage game has two stict pue 1 Fo example, while Makov pefect equilibia in standad infinitely epeated games ae simply infinite epetitions of Nash equilibia of the stage game, thee can be nontivial stategic dynamics in Makov pefect equilibia of asynchonous-move games - see Maskin and Tiole (1988a, 1988b) and Lagunoff and Matsui (1997). Anothe featue distinguishing asynchonous games fom the standad model is that expected subgame-pefect equilibium payoffs can be stictly below playes minimax values in the stage game - see fo example Takahashi and Wen (2003). 2 This implies that the pobability of two playes changing thei actions exactly at the same time is 0. As Maskin and Tiole (1988a) points out, these games ae closely elated to deteministic altenating-move games, if one esticts attention to Makovian stategies. Thee is a ecent sting of papes using simila continuous-time andom-aival models. Ambus and Lu (2010) investigate multilateal bagaining with a deadline in such a context, while Kamada and Kandoi (2011), and Calgano, Kamada, Lovo and Sugaya (2012) examine situations in which playes can publicly modify thei action plans befoe playing a nomal-fom game. Ambus et al. (2012) and Kamada and Sugaya (2011) use simila famewoks to analyze on-line auctions and policy announcement games, espectively. Simila models wee fist consideed in the macoeconomics liteatue, on sticky pices (Calvo (1983)) and in the seach litea- 1

3 Nash equilibia (each stictly pefeed by one of the playes), we show that thee is an intemediate ange of discount ates fo which fo a full dimensional specification of stage game payoffs asynchonicity of moves leads play in evey subgame pefect Nash equilibium (SPNE) to the same stage game equilibium outcome, no matte what the stating actions ae. Ou esult complements the findings of Lagunoff and Matsui (1997), who show that in an asynchonous-move epeated game famewok the Paeto-dominant equilibium gets selected in a class of pue coodination games fo high enough discount factos. 3 The main diffeence is that in battle of the sexes games unique selection occus fo intemediate levels of patience, not fo high ones. This also implies that thee is no tension between ou esult and the folk theoem, which was shown to hold in geneic asynchonous-move games by Dutta (1995) and Yoon (1999), 4 and that unique selection is a phenomenon that is obust to any small petubations of payoffs and discount ates. Analyzing equilibium possibilities fo intemediate levels of patience in epeated games is a notoiously difficult task, and fo this eason we only povide a set of necessay conditions, and anothe set of sufficient conditions, fo one of the stage game equilibia be the unique limit outcome in any SPNE. We obtain necessay conditions by poviding a complete chaacteization of Makov pefect equilibia (MPE) in asynchonous-move battle of the sexes games. These ae SPNE in which playes stategies only depend on the payoffelevant state, meaning that at any point of time a playe gets the chance to choose an action, he choice only depends on the cuent action choice of tue explaining pice and wage dispesion (Budett and Judd (1983), Budett and Motensen (1998)). 3 See also Faell and Salone (1985) and Dutta (2003) fo simila esults in finite-hoizon asynchonous-move coodination games, and Takahashi (2005) in a class of common inteest games encompassing both finite and infinite hoizons. Moe closely elated to the cuent pape is Calgano et al. (2012), who show that in geneic finite hoizon battle of the sexes games with asynchonous moves thee is a unique SPNE, obtained by backwad induction. The infinite hoizon games we investigate in the cuent pape equie diffeent techniques to analyze, and indeed the esults ae vey diffeent. 4 Lagunoff and Matsui (2001) point out that the geneicity o non-geneicity of anti-folk theoems depends on the ode limits ae taken, as fo any discount facto close to 1, thee is a full-dimensional set of coodination-games fo which all spne payoffs have to be vey close to the payoffs in the Paeto-dominant equilibium of the stage game. 2

4 the othe playe. As Lagunoff and Matsui (1997) show, thee always exists an MPE (on mixed stategies). Hence, a necessay condition fo a unique limit outcome in SPNE is that thee is a unique limit outcome in MPE. Ou complete chaacteization of MPE in these games is also of independent inteest fom the main point of the pape, evealing fo example that one action pofile (when both playes play the action coesponding to thei least pefeed equilibium in the stage game) cannot be pat of an absobing cycle in any SPNE, fo any discount ate. 5 We show by example that uniqueness of the limit outcome in MPE does not imply uniqueness of the limit outcome in SPNE. Hence, the above necessay conditions ae not sufficient fo the esult. Fo this eason, we povide a diffeent set of sufficient conditions fo play to convege to a given playe s favoed Nash equilibium outcome in the stage game, in any SPNE. We also show, by example, that these conditions still do not uniquely pin down equilibium stategies: thee can be multiple SPNE pofiles, with diffeent expected payoffs, with the featue that they all convege to the same stage game outcome. We povide an additional condition that guaantees the uniqueness of SPNE stategies as well (not just limit outcomes). Finally, we investigate that fo which payoff specifications thee exists a ange of discount ates with unique limit outcome. The intuition fo why unique selection occus fo cetain specifications is simple. If the off-equilibium stage game payoff when both playes play the stategies coesponding to thei favoed equilibia is paticulaly bad fo one of the playes, labeled the weak playe, then fo a not too high level of patience the latte playe is bette off giving in when getting the chance, and switch play to the othe playe s favoed outcome, no matte what continuation play she expects. But then if the same out-of-equilibium outcome is not as bad fo the othe playe, labeled the stong playe, and the level of impatience is not too low, the stong playe can foce play to ultimately switch to he favoed equilibium 5 Fo foundations fo MPE in asynchonous-move games see Bhaska and Vega-Redondo (2002) and Bhaska et al. (2012). Fo cetain popeties of MPE of these games see Halle and Lagunoff (2000, 2006). 3

5 outcome, by choosing the coesponding stategy. Actual equilibium dynamics might be moe complicated, and might equie mixing at cetain histoies. The above intuition also eveals why the level of patience has to be fom an intemediate ange fo a unique limit outcome. If playes ae too impatient then eithe of the stage game equilibium outcomes become absobing states. And if playes ae too patient then neithe of them can foce the othe one to play the stategy coesponding to the fist playe s favoed equilibium, and in the limit the folk theoem applies. Which playe becomes stong (fo a ange of discount ates) depends on both on- and off-equilibium payoffs of the stage game. Inceasing a playe s payoff in off-equilibium outcomes, as well as inceasing he payoff in he favoed Nash equilibium makes he stonge. On the othe hand, inceasing he payoff in he dispefeed Nash equilibium makes he weake, as it makes it moe attactive fo he to stay in that outcome. 2 The model Conside the following 2 2 game, efeed to below as the stage game: α 2 β 2 α 1 a, u b, v β 1 c, w d, x Table 1: Stage Game with the paamete estictions a > c, d > b, u > v, x > w, a > d and x > u. The estictions imply that the game has two stict Nash equilibia, (α 1, α 2 ) and (β 1, β 2 ), with playe 1 (P1) pefeing the fist one and playe 2 (P2) pefeing the second. We conside a continuous-time dynamic game with asynchonous moves, in which playes play the above stage game indefinitely, and accumulate payoffs continuously, but they can only change thei actions at andom discete times 4

6 that coespond to ealizations of andom aival pocesses. Moe pecisely, assume that the action pofile at time 0, a 0, is given exogenously. The playes ae associated with independent Poisson aival pocesses with aival ate. A playe can only change he action in the game at the ealizations of he aival pocess, hence in between two aival events she is locked into the action chosen at the fome one. Let a t denote the action pofile played at time t. Playe i s payoff in the dynamic game is then defined as e u i (a t )dt, whee u i () stands fo playe i s payoff in the stage game and is the discount ate (assumed to be the same fo the two playes, fo simplicity). Note that the independence of the aival pocesses implies that it is a 0 pobability event that thee is a point of time when both playes can change thei actions. We assume that the ealizations of the aival pocesses ae egula, and such simultaneous aivals indeed neve happen. Hence playes, when they choose actions, know what action the opponent will be locked in fo a andom amount of time. We assume that aivals ae publicly obseved, hence a publicly obseved histoy of the game at time t, labeled as h t, consists of the path of the action pofile on [0, t) and the set of ealizations of each playes aival pocesses on [0, t). Let H t be the set of such h t. Stategies ae measuable mappings fom H t to {α 1, β 1 } and {α 2, β 2 } espectively, with the value at h t intepeted as the action that the given playe would choose at time t if she had an aival at that time and the histoy at that time was h t. We ae inteested in both all SPNE of the above game, and the set of MPE, in which a playe s stategy when choosing an action only depends on the payoff elevant state, which is simply the action cuently played by the othe playe. Moe pecisely, thee can be 8 diffeent states (types of continuation histoies fom which the continuation game looks the same) at any point of time: (α 1, α 2 ), (α 1, β 2 ), (β 1, α 2 ), (β 1, β 2 ), (., α 2 ), (., β 2 ), (α 1,.) and (β 1,.). The fist fou coespond to points of time when neithe playe has an aival (and one of fou possible action pofiles ae played). The next two coesponding to points of 0 5

7 time when P1 eceives an aival and P2 is locked into playing one of his two actions - these ae the only histoies at which P1 chooses an action. Finally, the last two coespond to points of time when P2 eceives an aival - the only histoies at which P2 chooses an action. Hence, fo each playe thee can only be two states (two types of continuation games) at times when she has to choose an action. Fom now on, wheneve it does not cause confusion, instead of (α 1, α 2 ) and (β 1, β 2 ) we will simply wite (α) and (β). Lastly, we note that we can essentially map the game back to a model in discete time, the following way. Fix a stategy pofile in the dynamic game, and let ω t,j i denote the continuation payoff of playe i in the continuation game stating at t in case the vey fist aival in the game is at time t by playe j. Then we can wite playe i s expected payoff at time 0 as: 0 e 2τ ( (1 e τ )u i (a 0 ) + e τ w τ,1 i This simplifies to: 2 + u i(a 0 ) ) dτ+ 0 e 2τ ( (1 e τ )u i (a 0 ) + e τ w τ,2 i (2 + )e (2+)τ (w τ,1 i + w τ,2 i )dτ. Theefoe effectively, the discount facto is 2/(2 + ). 3 Example of unique limit outcome Conside the following stage game: ) dτ. α 2 β 2 α 1 4, 2 1, 4 β 1 2, 3 2, 4 with discount ate = 1 and aival ate = 1. In this example, we show that α must always be the unique limit outcome egadless of the initial state in any SPNE. Moe stongly, the set of SPNE is unique and Makovian with a unique absobing state at α. 6

8 The eason that the example induces a unique limit outcome is due to the following simple eason. We fist demonstate that P2 must always play α 2 wheneve he obtains an oppotunity to evise his action when P1 is cuently playing α 1 in any SPNE. To show this, note fist that the wost payoff that P1 can obtain in any SPNE with initial state α is u = 2. This is because P2 can guaantee himself a payoff of 2 by always playing α 2 at all histoies, egadless of P1 s stategy. We compae this to the best payoff that P2 could get in any SPNE beginning at the state (α 1, β 2 ). Let us denote this payoff by V (α 1, β 2 ). o Then we can show that this payoff must be bounded above by eithe 2 + v + + v + + x = 1 2 ( 4) = 0 < 2 2 ( v α + 1 ) 2 x = 1 3 ( 4) ( ) 2 4 = 1 12 < 2, whee v α coesponds to the payoff of P2 with initial state α coesponding to the stategy pofile in which P1 plays β 1 egadless of the histoy and P2 plays the action that P2 is cuently playing. The fist expession coesponds to the expected payoff that P2 would get if he chose to play β 2 egadless of histoy and P1 played β 1 wheneve P2 s cuently played action was β 2. Note we ae not stating that such stategy pofiles constitute SPNE. Rathe we ae only poviding an uppe bound on the payoff V (α 1, α 2 ). 6 In the example, both of the expessions above ae stictly less than u, the best payoff that P2 can obtain in any SPNE at state α. Theefoe in any SPNE, P2 must play α 2 at any histoy in which P1 s cuently played action is α 1. This howeve means that P1 should play α 1 at any histoy in which P2 is cuently playing α 2, because by doing so, P1 can obtain his best payoff. Simple calculations eveal that the best ex ante payoff that P1 can obtain in any SPNE beginning at state (β 1, β 2 ) is stictly less than + b + + a. 6 If we indeed examine whethe such a stategy can constitute an SPNE, we can likely shapen the sufficient conditions that we cuently have. 7

9 Howeve this is the payoff that P1 can obtain by playing α 1 at all histoies, povided that P2 always plays α 2 when P1 s cuent action is α 1. Theefoe, this shows that P1 must always play α 1 egadless of the histoy in any SPNE. Theefoe this example shows that stating at any state, the play of the game must eventually absob at (α) almost suely, in any SPNE. The emainde of the pape examines the set of paamete values in battle of the sexes games fo which this is the case. 4 Makov pefect equilibia Fist we chaacteize the set of MPE fo all possible payoff configuations and discount ates. This in paticula yields a necessay and sufficient condition fo all MPE of the game (iespective of the initial action pofile) having the popety that play is eventually absobed in the same action pofile. Fo ease of exposition, we efe to this popety as having a unique limit outcome in MPE. The condition is then also a necessay condition fo having a unique limit outcome in SPNE. If stategies ae Makovian, at times when thee is no aival event, each playe can only have fou continuation values, coesponding to the fou possible action pofiles in the stage game. Let v i (.,.) denote playe i s continuation payoff as a function of the action pofile. We stat the analysis by establishing a lemma on the elationship between continuation payoffs at diffeent states in any MPE, which will be useful to naow down possible stategy pofiles in MPE. Lemma 1 Suppose that v 1 (α 1, β 2 ) v 1 (β). Then in any Makovian equilibium, we must have v 1 (α) > v 1 (β 1, α 2 ). Similaly if v 2 (α 1, β 2 ) v 2 (α) then we must have v 2 (β) > v 2 (β 1, α 2 ). Poof. Suppose that v 1 (β 1, α 2 ) v 1 (α). Then we can wite v 1 (α) 2 + a + 2 ( v 1(α) + 1 ) 2 (pv 1(α) + (1 p)v 1 (α 1, β 2 )) 8

10 v 1 (α 1, β 2 ) = 2 + b + 2 ( v 1(α 1, β 2 ) + 1 ) 2 (pv 1(α) + (1 p)v 1 (α 1, β 2 )) fo some p [0, 1]. Note that the above elies on the condition that P2 conditions his play only on the opponent s cuently played action. The fist is an inequality since if P1 obtains an aival at state α, he has an incentive to play β 1. These expessions then imply that v 1 (α) v 1 (α 1, β 2 ) (a b). + Now next we can wite the value functions when the cuently played action of P1 is β 1 : v 1 (β 1, α 2 ) = v 1 (β) 2 + c d + 2 ( v 1(β 1, α 2 ) + 1 ) 2 (qv 1(β 1, α 2 ) + (1 q)v 1 (β)) ( 1 2 v 1(β) + 1 ) 2 (qv 1(β 1, α 2 ) + (1 q)v 1 (β)). This then implies Then this means that v 1 (β) v 1 (β 1, α 2 ) (d c). + v 1 (α) v 1 (β 1, α 2 ) = (v 1 (α) v 1 (α 1, β 2 )) + (v 1 (α 1, β 2 ) v 1 (β)) + (v 1 (β) v 1 (β 1, α 2 )) = (a b) + (c d) + + (a c + d b) > 0. + This is a contadiction. Theefoe we must have v 1 (α) > v 1 (β 1, α 2 ). Note that the lemma implies that (β 1, α 2 ) cannot be pat of an absobing cycle of the Makovian dynamics in any MPE. Because of the estictions on payoffs (u > v, d > c) it is also impossible that an absobing cycle consists of (α 1, β 2 ) and at most one of the stage game stict Nash equilibia. Hence, fo any specification of the game, we only have the following possibilities fo the Makovian dynamics implied by any MPE: (i) (α) is the unique absobing state; (ii) (β) is the unique absobing state; (iii) (α) and (β) ae both absobing states; (iv) thee is a unique absobing cycle involving (α 1, α 2 ), (α 1, β 2 ) and (β 1, β 2 ). 9

11 Ou cental inteest is chaacteizing the set of paamete values fo which one of the stage game Nash equilibia is the unique absobing state in evey MPE. Without loss of geneality we conduct this fo (α). We do it by chaacteizing all paamete values in which thee is an MPE with an absobing cycle containing (β), and then taking the complement of this set. Lemma 1 implies that the latte is indeed the set of paamete values in which (α) is the unique absobing state. Fist we chaacteize the paamete egion in which thee is an MPE in which (β) is an absobing state, and then the paamete egion in which thee is a nonsingleton absobing cycle. Since this equies a tedious case by case analysis, the poof of the following two lemmas ae in the Appendix. Lemma 2 The necessay and sufficient conditions fo the existence of an equilibium with (β) as a unique absobing state to exist ae: Additionally if + v + + v+ + x u. a + + b c + + d, v + + x u + + w. + x < u and d which both (α) and (β) ae absobing states exist. + b + + a an equilibium in Note that the above implies that fo sufficiently close to zeo, thee always exists a Makovian equilibium with (β) as a unique absobing state (symmetically, this is also tue fo α). Lemma 3 The necessay and sufficient conditions fo the existence of an MPE with a unique absobing cycle involving (α 1, α 2 ), (α 1, β 2 ) and (β 1, β 2 ) 10

12 ae: and + v + + x > u + b + + a > d. We can use the pevious lemmas to chaacteize the set of paamete values fo which (α) is the unique limit outcome in MPE. Symmetic conditions chaacteize the paamete egion in which (β) is the unique limit outcome in MPE. Theoem 1: The necessay and sufficient conditions fo (α) to be the unique absobing state in any MPE ae: and one of the following: 1. + v + + x < u + b + + a > d a+ + b > c+ + d 2. o v + + x < u + + w. Poof. Lemmas 2 and 3 imply that the paamete egion in the statement is exactly the set of paametes fo which thee does not exist an MPE in which (β) is an absobing state and thee does not exist an MPE in which thee is an absobing cycle involving (α 1, α 2 ), (α 1, β 2 ) and (β 1, β 2 ). Lemma 1 then implies that this is exactly the set of paamete values fo which in all MPE the unique absobing state is (α). Intuitively, the fist two conditions equie that it is bette fo P2 if play stays in (α) foeve than tansitioning to (α 1, β 2 ) (even if P1 chooses β 1 with 11

13 pobability 1 at state (., β 2 )), and that if P2 chooses α 2 with pobability 1 at state (α 1,.) then it is bette fo P1 to tansition fom state (α) than staying thee foeve. And eithe of the last two conditions guaantee that thee is no MPE in which P1 tansitions fom (α) because of the fea that othewise P2 would tansition away fom this state. Inceasing a, b, u and w, and deceasing c, d, v and x make the inequalities in the statement of Theoem 1 easie to satisfy. The compaative statics in and ae nonmonotonic, and in fact both fo low enough values of and, and fo high enough values of and one of the conditions ae violated. 5 Geneal subgame pefect equilibia Fist we demonstate by an example that a unique limit outcome in MPE does not imply the uniqueness of limit outcome in SPNE. Then we povide a set of sufficient conditions fo the uniqueness of a limit outcome in SPNE. Lastly, we povide stonge sufficient conditions fo the uniqueness of SPNE stategies as well. 5.1 Example: unique limit outcome in MPE does not imply unique limit outcome in SPNE Conside the following stage game. α 2 β 2 α 1 10, 1 12, 1 β 1 9, 1 1, 2 Let and be such that = 3 4, = 3 5. Note fist that with these paametes, no Makovian equilibium with β as an absobing state exists. Howeve we show that the following non-makovian 12

14 stategy pofile with β as an absobing state is an equilibium. As long as no deviations by P2 have occued, P1 always plays β 1. P2 plays β 2 wheneve play immediately peceding the aival is (β 1, α 2 ), o (β 1, β 2 ). P2 plays β 2 when play immediately peceding the aival is (α 1, α 2 ) if eithe thee have been no aivals in the game o the last playe to aive was P1. Othewise P2 plays α 2. P2 plays α 2 when play immediately peceding the aival is (α 1, β 2 ). If P2 has deviated, then playes evet to the unique Makovian equilibium with an absobing state at (α). Let us illustate how the incentives can be checked. Fist suppose that P2 has deviated. Then incentives to follow the Makovian equilibium is tivial. Theefoe let us suppose that P2 has not deviated. Conside an aival by P1 at state (, α 2 ). Then playing β 1 yields a payoff of = 21 5 = 4.2. Suppose by way of contadiction that playing α 1 is indeed optimal. This then gives a payoff of: whee V = ( 1 4 ( 12) + 3 ( )) 2 v α = = v α = ( ) 5 1 = Thus upon aival when P2 is cuently playing α 2, P1 stictly pefes to play β 1 because of the theat of P2 moving to (α 1, β 2 ). This theat is lage enough to povide incentives fo P1 to play β 1 as soon as she gets the chance. A necessay condition fo this to hold is that fo P1, the path (β 1, α 2 ) (β 1, β 2 ) 13

15 is stictly pefeable to the path (α 1, α 2 ) (α 1, β 2 ) (α 1, α 2 ) (β 1, α 2 ) (β 1, β 2 ). Othewise P1 would have an incentive to play α 1 at states (, α 2 ). Staightfowad calculations establish that this is indeed the case, and that P1 stictly pefes to play β 1 wheneve P2 s cuently played action is β 2. P2 s incentives to play β 2 when P1 s cuently played action is β 1 ae tivial. Futhemoe, incentives to play α 2 when the stategy calls fo it ae also tivial. Thus it emains only to show that it is in he inteest to play β 2 at state (, α 2 ). Note that by staying at α 2, P2 has deviated and thus eceives only a continuation payoff of 1. Howeve if he moves to β 2, then P2 is ewaded by P1 following a stategy in the futue that calls fo the play of β 1 eveywhee on the equilibium path. The game was constucted in such a way to make this stictly geate than 1. The computations yield a payoff of 1 4 ( 1) ( v α ) = 589 = > Thus we have shown that the above stategy pofile is indeed an equilibium. The essential featue of the above example is that it is too costly fo P2 to tansition fom (α 1, α 2 ) to (β 1, β 2 ) via (β 1, α 2 ), even if P1 plays β 1 with pobability 1 wheneve P2 plays β 2. Howeve, P2 is willing to tansition to fom (α 1, α 2 ) to (β 1, α 2 ) if along the equilibium path he can tansition back to (α 1, α 2 ) if he gets the next aival as well, povided that P1 is willing to tansition fom (α 1, α 2 ) to (β 1, α 2 ). The latte is indeed the case because tansitioning via (β 1, α 2 ), is too costly fo P1 as well, and it is in he inteest to pevent it. Lastly, P2 is willing to make the tempoay tansition to (β 1, α 2 ) because othewise he would tigge the MPE as the continuation equilibium. 5.2 Uniqueness of limit outcome in SPNE Fist we povide sufficient conditions fo P2 to choose action α 2 in any histoy at which P1 is cuently playing α 1, in any SPNE. If this is the case then the best 14

16 esponse of P1 is also choosing action α 1 wheneve P2 is cuently playing α 2, establishing that (α 1, α 2 ) is an absobing state in evey SPNE. Subsequently, we will add a futhe condition that guaantees that it is the unique absobing state. To shoten notation define v α = + u + ( + + w + ) + x. We will split the paamete space into two egions (w u and w < u), and fist conside the easie case. Lemma 4 Suppose w u and that: u > u > + v v + x, and ( 1 2 x v α ), then in any SPNE, P2 must play α 2 wheneve P1 s cuently played action is α 1. Poof. Note fist that in any equilibium, at state (α 1, α 2 ), P2 can guaantee himself a payoff of at least u by playing α 1 upon any aival. We now compute an uppe bound on the payoff that P2 can obtain at state (α 1, β 2 ). To do this, denote V (α), V (α 1, β 2 ), and V (β 1, α 2 ) the best possible P2 SPNE payoffs at the espective states. Note that we do not equie that V (α), V (α 1, β 2 ), and V (β 1, α 2 ) be the payoffs at thei espective states in the same SPNE. We use these values to bound the continuation payoffs at the state (α 1, β 2 ) in any SPNE. Note that V (α 1, β 2 ) 2 + v + 2 ( max{v (α 1, β 2 ), V (α)} + 1 ) 2 x. Suppose fist that V (α 1, β 2 ) V (α). Then we have V (α 1, β 2 ) + v + + x < u. 15

17 Theefoe the best possible payoff is less than the payoff that P2 could guaantee at the state (α). Thus we have shown that P2 must play α 2 with pobability one upon aival when the cuently played action of P1 is α 1 in any SPNE. Suppose now that V (α) > V (α 1, β 2 ). Let us bound the payoff V (α): V (α) 2 + u + 2 ( max{v (α), V (α 1, β 2 )} + 1 ) 2 max{v (α), V (β 1, α 2 )} = 2 + u + 2 ( V (α) + 1 ) 2 max{v (α), V (β 1, α 2 )}. Let us assume fist that V (α) V (β 1, α 2 ). Then the above implies that V (α) u. Howeve this means that V (α 1, β 2 ) 2 + v + 2 ( u + 1 ) 2 x < u. Thus again it must be that P2 plays α 2 with pobability one upon aival in any SPNE when the cuently played action of P1 is α 1. Finally assume that V (α) > V (α 1, β 2 ) and V (β 1, α 2 ) > V (α). Then V (β 1, α 2 ) 2 + w + 2 ( V (β 1, α 2 ) + 1 ) 2 x. This implies that V (β 1, α 2 ) + w + + x. This immediately implies that V (α 1, β 2 ) 2 + v + 2 ( v α + 1 ) 2 x < u. Thus in this case again, P2 must play α 2 with pobability one wheneve the cuently played action of P1 is α 1 in any SPNE. The case when w < u is a bit moe subtle. In the pevious case, we used a punishment scheme by which all playes tansition to the Makovian equilibium in which (α 1, α 2 ) is an absobing state. Howeve this only yields a punishment 16

18 payoff of u fo P2. This is an effective punishment against P2 when w u due to the fact that P1 can indeed guaantee himself a payoff of u at the state (α 1, α 2 ). Howeve when w < u this might not be the case anymoe. Theefoe it may be possible to indeed punish P2 using stategies that ae non-makovian that lowe the payoff to P2 to some payoff stictly below u. This leads to an additional poblem that was non-existent in w u. Suppose that the state is (α 1, α 2 ) and if P2 obtains an aival and decides to stay at (α 1, α 2 ) then he gets punished with a continuation payoff that is stictly smalle than u. Howeve note that this may ceate an incentive fo P2 to play α 2 since then he may be ewaded by tansitioning to an MPE with (α) as an absobing state. Thus we must ensue against such a deviation. Lemma 5 Suppose w < u and that + u + + w > + v + x, and + + u + + w > 2 + v + 2 ( x + 1 ) 2 max{v α, u}. Then in any SPNE, P1 must play α 2 in any histoy at which P2 is cuently playing α 1. Poof. The poof follows along the same lines as the pevious lemma. Define as in the pevious lemma the payoffs V (α), V (α 1, β 2 ), and V (β 1, α 2 ). We show that it must always be the case that V (α 1, β 2 ) < Suppose fist that V (α 1, β 2 ) V (α). Then V (α 1, β 2 ) This then implies that V (α 1, β 2 ) 2 + v + + v + Next let V (α) > V (α 1, β 2 ). Thus V (α) 2 + u u + + w x < ( 1 2 V (α 1, β 2 ) + 1 ) 2 x. + u + + w. ( 1 2 V (α) max{v (α), V (β 1, α 2 )} 17 ).

19 Suppose that V (α) V (β 1, α 2 ). Then the above implies that V (α) u. This then means that V (α 1, β 2 ) 2 + v + 2 ( u + 1 ) 2 x < + u + + w. Finally suppose that V (β 1, α 2 ) > V (α) > V (α 1, β 2 ). Then it is easy to see that V (α 1, β 2 ) 2 + v + 2 ( x + 1 ) 2 v α < + u + + w. Thus we have shown that V (α 1, β 2 ) < always guaantee a payoff of + u + + w. Theefoe since P2 can + u + + w at state (α) by always playing α 2 in the futue, P2 must always play α 2 when P1 is cuently playing α 1. Next we investigate the issue of what futhe conditions ae needed fo (α) to be the unique absobing state. Theoem 2 Suppose that { min min u, { u, + b + + a > d, } + w > } + w > + u + + u + + v v + Then in evey subgame pefect equilibium σ, x, and ( 1 2 x max{v α, u} ). P σ ({h : T s.t. h t = α t T } h 0 = a 0 ) = 1 fo any initial state a 0. That is, the histoy must become absobed at state α in finite time almost suely. 18

20 Poof. By the pevious lemma, we know that α must be an absobing state. Suppose by way of contadiction that P σ ({h : T s.t. h t = α t T } h 0 = a 0 ) < 1 fo some a 0. Clealy a 0 cannot be α. Note that a histoy encodes thee pieces of infomation. Fist it specifies the aival times: t 1, t 2,... which specify at what time some playe obtains a evision oppotunity. Secondly it specifies the action chosen at each of these aival times a 1, a 2,.... Finally it specifies the playe who has aived at each of these times. We ignoe this thid piece of infomation fo the puposes of the poof. Thus we denote a histoy h by: h = (a 0, (t k, a k ) k=1) whee a 0 is the initial state. Note fist that we can estict to h such that t k is an infinite sequence since histoies with finitely many aivals occu with pobability zeo unde Poisson aivals. To ease notation, let us define the following set: H k = {h H : t k k s.t. a k α}. Define H k (a) = {h H : t k k s.t. a k = a} and Then the above implies that H(a) = H k (a). k=0 ( ) P σ H k > 0. k=0 But then note that ( ) P σ (H(α 1, β 2 ) H(β 1, β 2 \ (H(β 1, α 2 ) H(α 1, β 2 ))) H(β 1, α 2 )) P σ H k > 0 k=0 19

21 which is less than P σ (H(α 1, β 2 )) + P σ (H(β 1, β 2 ) \ (H(β 1, α 2 ) H(α 1, β 2 ))) + P σ (H(β 1, α 2 )). But note that P σ (H(α 1, β 2 )) = P σ (H(β 1, α 2 )) = 0 since state (β i, α i ), playe i plays α i with pobability one. Thus the above implies that P σ (H(β 1, β 2 ) \ (H(β 1, α 2 ) H(α 1, β 2 ))) > 0. But if h H(β 1, β 2 ) \ (H(β 1, α 2 ) H(α 1, β 2 )), thee exists some T such that h t = (β 1, β 2 ) fo all t T. Hee we ae using the assumption of asynchonous natue of moves. This is because play cannot tansition fom (β 1, β 2 ) to (α 1, α 2 ) without passing though eithe (α 1, β 2 ) o (β 1, α 2 ). But if this is the case, conside the subgame induced at time T. P1 is eceiving a payoff of d. Howeve he can deviate to playing α 1 eceiving a payoff of + b + + a which is stictly bette than d. Theefoe we have aived at a contadiction. We would like to point out that the conditions in Theoem 2 do not guaantee that play is suely absobed in (α) if both playes had a cetain numbe of aivals (even in a specific ode). This is because thee can be mixing in an SPNE at histoies at which play has not eached yet the limit outcome. 5.3 Uniqueness of stategies in SPNE Fist we show that uniqueness of a limit outcome in SPNE does not imply uniqueness of SPNE. Conside the following example. Suppose that the stage game is the following: α 2 β 2 α 1 2, 1 ε, 4 β 1 1, 0 1, 2 20

22 with ε > 0 and let = 1 and = 1 2. We choose ε > 0 so that > ε > 1. Due to the lemma above, we know that in any SPNE both playes play α i wheneve the opponent is cuently playing α i. In the above equilibium, thee exists a Makovian equilibium with the following dynamics. Both playes play α i with pobability one wheneve the opponent s pepaed action is α i. P1 plays α 1 with pobability 9/11 wheneve the opponent s pepaed actions is β 2. P2 plays α 2 with pobability 3ε/(2 ε) wheneve the opponent s pepaed action is β 1. One can easily check that these stategies constitute a SPNE. Futhemoe it is easy to see that the payoffs at each state must be the following: V (α 1, α 2 ) = (2, 1) ( ) 1 V (α 1, β 2 ) = 2 ε + 1, 4 3 ( 1 V (β 1, α 2 ) = 3 ε + 4 3, 1 ) 2 ( 1 V (β 1, β 2 ) = 2 ε + 1, 1 ). 2 In fact the above is the unique MPE fo this set of paametes. Howeve thee also exists the following non-makovian equilibium that makes use of the above Makovian stategy upon deviations. Both playes play α i upon aival as long as eveyone has thus fa played α j upon aival. If at some point, a deviation occued, then playes play accoding to the Makovian equilibium above. 21

23 Note that this non-makovian equilibium aises the payoff of P1 at an initial state (β 1, β 2 ) beyond what he can achieve in the Makovian equilibium. This is because fo P1, going though the path (β 1, β 2 ) (β 1, α 2 ) (α 1, α 2 ) on the way fom (β 1, β 2 ) to (α 1, α 2 ) is bette than moving to α 1 and using the path (α 1, β 2 ) (α 1, α 2 ) as long as P2 is moving to α 2 with pobability one upon aival. In the above stategy pofile when the state is (β 1, β 2 ), eithe P1 aives fist giving him a payoff equal to the Makovian payoff at state (β 1, β 2 ) of V 1 (β 1, β 2 ) o with stictly positive pobability P2 aives fist giving a payoff stictly bette than V 1 (β 1, β 2 ). Note that the punishment via evesion to the Makovian equilibium is necessay since othewise, playes would find it beneficial to simply wait at (β 1, β 2 ) fo the othe playe to aive and move out fo him instead of moving out himself. We conclude this section by poviding sufficient conditions fo uniqueness of SPNE stategies, besides uniqueness of a limit outcome in SPNE. Theoem 3 Suppose that { min min u, { u, + b + + a > d, } + w > } + w > + u + + u + Suppose futhe that o b + w + + v v a > d + + c + u < x + + v. x, and ( 1 2 x max{v α, u} Then thee exists a unique SPNE. Futhemoe the unique SPNE must be the unique MPE. ). 22

24 Poof. We have aleady shown that the fist two inequalities imply that P2 must play α 2 at any histoy when P1 is cuently playing α 1 and P1 must play α 1 at states when P2 is cuently playing α 2. We poceed again as in the pevious poofs. Suppose that the fist inequality holds. The case fo which the second inequality holds poceeds along the same lines and so we omit the poof. We will show that V 1 (β) < + b + + a. Suppose fist that V 1 (β) V 1 (β 1, α 2 ). Then V 1 (β 1, α 2 ) If V 1 (β) V 1 (α 1, β 2 ) then this implies that V 1 (β) + d + + c + + d. + But clealy the above is stictly less than have ( + c + + b + + a. ) + a. Suppose next that V 1 (β) V 1 (β 1, α 2 ) and V 1 (β) < V 1 (α 1, β 2 ). Again we Additionally This then implies that V 1 (β) < V 1 (β 1, α 2 ) V 1 (α 1, β 2 ) 2 + d b + + a. + c + + a. + b + + a. ( 1 2 V 1(α 1, β 2 ) + 1 ) 2 V 1(β 1, α 2 ) Finally let us assume that V 1 (β) > V 1 (β 1, α 2 ). Theefoe V 1 (β) 2 + d ( 1 2 V 1(β) max{v 1(β), V 1 (α 1, β 2 )} 23 ).

25 One can show that this is less than + b + + a. Thus we have shown that in all of these cases, V 1 (β) < + b + + a. Howeve the ight hand side of the above inequality denotes the payoff that P1 can guaantee when she tansitions play to (α 1, β 2 ). Theefoe it must be that in any SPNE, P1 must play α 1 with pobability one at any histoy in which P2 is cuently playing β 2. This establishes the claim. 6 Discussion Payoff specifications compatible with unique limit outcome The conditions we povided fo the uniqueness of limit outcome imply estictions jointly on the payoffs and the discount ate. A natual question to ask is whethe it is tue fo geneic payoff specifications of the battle of the sexes game that thee is an intemediate ange of discount ates such that thee is a unique limit outcome in SPNE. The answe to this question is negative. sexes games: Conside the following battle of α 2 β 2 α 1 1, 0 β 1 2, , 1 0, 1 Ou chaacteization of MPE establishes that thee is no discount ate at which thee is a unique limit outcome in MPE. Moeove, this emains tue fo all small petubations of stage game payoffs, too. Moe geneally, a simple manipulation of ou necessay and sufficient condition fo unique limit outcome eveals that the limit outcome cannot be unique in MPE fo any discount ate wheneve: u 2 (β 1, β 2 ) + u 2 (α 1, β 2 ) < u 2 (β 1, α 2 ) + u 2 (α 1, α 2 ) u 1 (α 1, β 2 ) + u 1 (α 1, α 2 ) > u 1 (β 1, β 2 ) + u 1 (β 1, α 2 ). 24

26 That is, if fo both playes it is tue that the stategy coesponding to the playe s least pefeed stage game Nash equilibium is isk-dominant, then thee is always multiplicity of limit outcomes in equilibium. Expected equilibium payoffs when thee is a unique limit outcome Because uniqueness of the limit outcome occus at an intemediate ange of discount ates, the stong playe (whose favoed stage game Nash equilibium is selected as the limit outcome) is not necessaily bette off. To see this, conside again the example in Section 3: α 2 β 2 α 1 4, 2 1, 4 β 1 2, 3 2, 4 Following the aguments of Section 3, we can show that the unique SPNE is Makovian and follows the following set of dynamics with α as the unique absobing state. Howeve, the ex ante payoffs of P1 at state (β) is: 2 + d + 2 ( ( c + ) + a + 1 ( 2 + b + )) + a = ( ( ( 2) + 1 ) 2 (4) + 1 ( (1) + 1 )) 2 4 = 11 6 < 2. That is, not only P2 but also P1 is wose off at state (β) than if play stayed foeve at state (β). Possible extensions Ou analysis could be easily extended to allowing fo diffeent discount ates and/o aival ates fo the playes. Intuitively, inceasing one of the playe s discount ate makes the playe weake, making it moe likely that the othe playe s favoite stage game equilibium becomes the unique limit outcome in SPNE. Inceasing the aival ate of a playe also makes that playe weake, as 25

27 it makes the othe playe moe like a Stackelbeg leade in the game, since that playe is cedibly locked into playing he cuent action fo a elatively longe time. Chaacteizing MPE and poviding a sufficient condition fo unique limit outcome in SPNE could be done similaly as in ou basic setting, at the cost of moe complicated notation and moe complicated equilibium conditions. As the additional qualitative insights fom this execise ae limited, we do not pusue this diection fomally hee. A much less staightfowad diection would be extending the investigation beyond battle of the sexes games, and chaacteizing moe geneally the set of games fo which thee is an intemediate ange of discount ates fo which the limit outcome is unique in SPNE. As we showed above, such a ange might not exist even fo games within the battle of the sexes class, suggesting that this extension might be difficult. 26

28 A Appendix A.1 Poof of Lemma 2 Below we fist povide necessay conditions fo existence of MPE in which (β 1, β 2 ) is the unique absobing state. Then it is tivial to show that these necessay conditions ae also sufficient. We find it convenient to descibe the type of equilibium gaphically. The aow connecting two states between which one of the playes can tansition play indicates the playe s action conditional on the othe playe playing the action consistent with these states. If the aow points to one state, it implies choosing the coesponding action with pobability 1, othewise mixing between the two actions. The states coloed in blue epesent the elements of the unique absobing cycle. Geneically, note fist that thee ae only thee possible types of dynamics fo any MPE with (β 1, β 2 ) as the unique absobing state: a, u b, v a, u b, v a, u b, v c, w d, x c, w d, x c, w d, x It is easy to see that the fist set of dynamics exist if and only if c + + d a + + b v + + x u + + w. Similaly the second set of dynamics exist if and only if + v + + x u a + + b c + + d. 27

29 The necessay and sufficient conditions fo existence of the thid set of dynamics ae only slightly moe subtle. Fist fo P1 to mix, we need c + + d a + + b. This is because if P2 plays α 2 with pobability one wheneve he is playing α 1, then the unique best esponse fo P1 is to play α 1 wheneve P2 is playing α 2. Fo him to have an incentive to play β 1 when P2 s cuent action is α 2, P1 needs to pefe playing β 1 when P2 plays β 2 with pobability one egadless of the histoy. This is exactly the condition above. Note that this is the fist condition that is necessay and sufficient fo the fist set of dynamics. Since if we also have v + + x u + + w, we showed necessay and sufficient conditions fo existence of equilibia of the fom of the fist kind, let us assume the contay. We have thus fa, c + + d a + + b v + + x < u + + w. With these conditions, simila to the agument we use fo P1, in ode fo P2 to have incentives to mix, he must be willing to play β 2 when P1 plays α 1 with pobability one wheneve his pepaed action is α 2. This means that u < + v + + x. So the set of paametes fo which the thid kind of dynamics can occu as an MPE that ae not coveed by the fist case is exactly c + + d a + + b v + + x < u + + w u < + v + + x. 28

30 Taking the union of these sets of paametes gives us the set of paametes in the statement of Lemma 2. We can pefom a simila analysis to detemine the set of paametes fo which thee exists an MPE in which both (α) and (β) ae the absobing states. a, u b, v c, w d, x The necessay and sufficient conditions fo this to occu ae: u d + v + + x + b + + a. A.2 Poof of Lemma 3 By lemma 1, the only possible set of dynamics suppoting an MPE with (α), (α 1, β 2 ), and (β) as elements of the absobing cycle is the following: a, u b, v c, w d, x Fo P1 to mix, it must be that Othewise P1, would play β 1 played action is β 2 action is α 1. + b + + a > d. with pobability one wheneve P2 s cuently egadless of what P1 plays when P2 s cuently played 29

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