Endogenous timing game with non monotonic reaction functions.

Transcription

1 CERDI, Etudes et Documents, E Document de traval de la sére Etudes et Documents E Endogenous tmng game wth non monotonc reacton functons. Magnus Hoffmann* and Grégore Rota Grazos *: Unversty of Magdeburg, Department of Economcs, Insttute of Publc Economcs, Unverstaetsplatz 2, Magdeburg, Germany. Emal: magnus.hoffmann@ovgu.de : CERDI CRS, Unversté d'auvergne, 65 boulevard Franços Mtterrand, Clermont Ferrand, France Emal: gregore.rota_grazos@u clermont1.fr 27 Avrl 2010

2 Aprl 27, 2010 Endogenous tmng game wth non-monotonc reacton functons. Abstract The am of ths paper s to generalze the endogenous tmng game proposed by Hamlton and Slutsky (1990) to cases where the reacton functons are non-motononc, as for nstance n the lterature on contest. Followng the taonomy of socal dlemma provded by Eaton (2004) we consder several possble stuatons dependng on the nature of nteractons (plan complementarty or plan substtuablty and strategc complementarty or strategc substtutablty). Under the assumptons of the estence and the unqueness of the ash and Stackelberg equlbra, we hghlght the presence of a frst-mover advantage or a second-mover ncentve only dependng on the nature of cross-effects n players payoff functons and the slopes of ther reacton functons at the ash equlbrum of the statc game. These propertes allow us to determne rgorously the Subgame Perfect ash Equlbrum (SPE) n the ten studed stuatons. We establsh under whch condtons on the nature of nteractons a leader emerges at the SPE. Keywords: endogenous tmng game; frst-mover advantage; second-mover ncentve; Subgame Perfect ash Equlbrum. JEL classfcaton: C72; D43; L13. Magnus Hoffmann* and Grégore Rota Grazos *: Unversty of Magdeburg, Department of Economcs, Insttute of Publc Economcs, Unverstaetsplatz 2, Magdeburg, Germany. Emal: magnus.hoffmann@ovgu.de : CERDI-CRS, Unversté d Auvergne, Mal address: 65 boulevard Franços Mtterrand, Clermont-Ferrand, France Emal: gregore.rota_grazos@u-clermont1.fr

3 1 Introducton The semnal artcle of Hamlton and Slutsky (1990) endogenezes the Stackelberg leadershp n a duopoly game. Ths model s mportant snce t provdes a smple formalzaton of the tradeoff between pre-commtment and fleblty. The consdered commtment conssts n movng before the others and corresponds to a pure uncondtonal commtment n Schellng s termnology. Amr (1995) completes Hamlton and Slutsky (1990) by establshng the necessty of an addtonal condton, that s the monotoncty of each payoff n the other player s actons. Several developments of Hamlton and Slutsky (1990) have been proposed n the lterature. For nstance, Sadanand and Sadanand (1996) consder demand uncertanty n ths model. In order to deal wth the multple equlbra ssue, whch may appear van Damme and Hurkens (1999) ntroduce the noton of rsk-domnance as defned by Harsany and Selten (1988). Amr and Stepanova (2006) use the tool of supermodularty to make the mnmum of assumptons. They dstngush three possble cases dependng on the slope of the reacton functons (postve or negatve for both players or postve for one, negatve for the other). They establsh general condtons on the demand functons to hghlght frst- or second-mover advantage and then to solve the endogenous tmng game. However, these developments and others restrct themselves to stuatons where the reacton functons are monotonc. Even though the assumpton of monotonc reacton functons encompasses often the man specfcatons proposed n the IO lterature, there are mportant eceptons. For eample Bulow, Geanakoplos, and Klemperer (1985a) show that for the case of the constant elastcty of demand n Cournot duopoly, the reacton functons are non-monotonc. The same holds for the best responses of duopolsts n the endogenous-tmng duopoly model eamned by Maskn and Trole (1988). In an nfnte horzon framework the dynamc reacton functons n a prce competton model also ehbt non-monotoncty. Adner and Zemsky (2005) eamne a model of technology nnovaton n dfferent markets n a Cournot-framework. In ther model the emergence of competton between technologes alters the shape of the demand functons for products n two dfferent markets n a manner whch nduces non-monotonc best responses. The monotoncty of reacton functons assumpton appears also to be too restrctve n 1

4 other felds, as for nstance nternatonal trade or publc economcs, where commtments are an mportant ssue too. For eample, Syropoulos (1994) shows the (non) equvalence of polcy nstruments (quotas and tarrffs) n the contet of non-cooperatve polcy games wth endogenous tmng. In ths framework the tarff reacton functons of a country ehbts strategc substtutablty and complementarty, contngent on the other country s decson on t s tarff. Dt (1987) and Bak and Shogren (1992) analyze a contest for an eogenous rent between two players whom reacton functons are ncreasng and then decreasng. Ths game s nether supermodular nor submodular snce the players actons are strategc complements for the favorte, whle they are strategc substtutes for the underdog 1 at the ash Equlbrum. Fnally, we can also menton the recent work of Stengel (2010), who compares the leader and follower payoff n a symmetrc duopoly game wthout the assumpton of monotonc reacton functons. Ths author concludes that the follower payoff s ether hgher than the leader payoff, orlowerthan n the smultaneous game. In ths paper, we propose a generalzaton of Hamlton and Slutsky (1990) by consderng non-monotonc reacton functons. Followng Eaton (2004) who provdes a useful taonomy of socal dlemma, we consder ten possble stuatons dependng on the nature of nteractons (plan complementarty or plan substtutablty and strategc complementarty or strategc substtutablty). Admttng the estence and the unqueness of the ash and Stackelberg equlbra, we hghlght the presence of a frst-mover advantage or a second-mover ncentve only dependng on the nature of cross-effects n players payoff functons and the slopes of the reacton functons at the ash equlbrum of the statc game. These propertes are then suffcent to determne the Subgame Perfect ash Equlbrum(s) (SPE) of the endogenous tmng game proposed by Hamlton and Slutsky (1990). By dong ths, we then etend the taonomy of Eaton (2004) n determnng the SPE(s) n each studed case. The rest of ths paper s organzed as follows: secton 2 precses the assumptons of Hamlton and Slutsky (1990) and derves some propertes of the payoff functons; secton 3 establshes the SPE(s) of the endogenous tmng game; secton 4 concludes. 1 Dt (1987) defnes the favorte (respectvely the underdog ) as the player who has a probablty to wn the contet above (respectvely below) one half at the ash equlbrum of the statc game. 2

5 2 Model Frst, we present our assumptons. Secondly, we consder the three basc games (the statc game and the two Stackelberg ones) and gve a suffcent partal rankng of the equlbrum values. Thrd, we dentfy the presence of a frst-mover advantage or a second-mover ncentve. 2.1 Assumptons The (contnuous) payoff functons are gven by (, j ),where X (respectvely j X j ) s the strategy of player (j) defned over a nonempty compact nterval of the real lne. 2 We make the assumpton that player s payoff functons s concave n her own acton and monotone n the other player s acton. Thus, estence of a ash equlbrum (E) s assured. Followng EatonandEswaran(2002), wedefne plan nteractons, more precsely plan complements (PC) and plan substtutes (PS). Defnton 1 The game ehbts plan complements (plan substtutes) for player f (, j ) X X j, (, j ) j j (, j ) > 0 (< 0). The noton of plan nteractons s very close to ths of spllovers, but t appears more precse than the latter. The necessary condton gven by Amr (1995) s then equvalent to the assumpton that the property of PC or PS holds for any values of and j. Followng Bulow, Geanakoplos, and Klemperer (1985b), we wll also use the noton of strategc complementarty (SC) or strategc substtuably (SS). Let denote by 2 (, j ) j j (, j ) the second cross dervatve of the payoff functon for player. We have SC (respectvely SS) for player f j (, j ) > 0 (respectvely j (, j ) < 0). Snce we consder non-monotonc best responses the property of SC and SS may vary for a player, contngent on the strateges chosen by both players. 3 We therefore wll defne the property of SC or SS at the ash Equlbrum (E) of the statc game. 2 The acton may be for nstance quantty n Cournot duopoly, prce n Bertrand duopoly, effort n a rent-seekng game or ta rate n a model of ta competton. 3 Ths contrast wth the the property of PC or PS whch holds for each player n any studed games. 3

6 Defnton 2 At the ash Equlbrum of the statc game, defned by, j, player regards strategc complements (strategc substtutes) f 2 (, j ) (, j ) X X j, j, j > 0 (< 0). j These defntons allow us to encompass several dfferent settngs, whch are consdered by Eaton (2004). For nstance, the classc verson of the Cournot duopoly s a game of PS and SS, whle the Bertrand duopoly wth dfferentated goods s a game of PC and SC. The model of prvate provson of a publc good proposed by Bergstrom, Blume, and Varan (1986) ehbts PC and SS for players and a formalzaton of defense ependture among enemy countres would be a game of PS and SC (see Eaton (2004)). However, we wll also consder some med stuatons gong beyond the taonomy proposed by Eaton (2004) and gettng closer to the semnal work of Hamlton and Slutsky (1990). For nstance, our results may be appled to some partcular and nfluental frameworks, as the model of Sngh and Vves (1984), or to the contest games usually used n the rent seekng lterature. Sngh and Vves (1984), whch s consdered n Hamlton and Slutsky (1990) dscuss an asymmetrc case, where two frms n a dfferentated duopoly market can only make two types of bndng contracts wth consumers, a prce contract and a quantty contract. In the case where both frmsdecdetomakedfferent decsons regardng the nature of the contract, we obtan a game of PC and PS for one frm (whch chooses the prce contract) and a game of PS and SS for the other frm (whch chooses a quantty contract). An mportant applcaton of our results would be the lterature, whch uses the Contest Succes Functon. 4 Indeed, snce we allow for non-monotonc best responses the property of SC and SS may vary for a player, contngent on the strateges chosen by both players. In the case of rent seekng, as provded by Dt (1987), player s best response functon ncreases n j for lower values of j, and decreases n j forhghervaluesof j.inotherterms, each contestant regards actons as SC or SS dependng on her effort and the opponent s effort. Estence and unqueness of the the ash and Stackelberg equlbrums usually nvolve several restrctons on the payoff functons. In order to reman as general as possble, we make the followng assumptons 4 The CSF s the functon that maps players efforts nto probablty of wnnng the prze. 4

7 Assumpton 1 () The ash equlbrum of the smultaneous game s unque. () The equlbrum of the Stackelberg games ests and s unque. The frst assumptons guarantees that our defnton of SC, or SS respectvely, s unque foreachplayernthestatcgame.asuffcent condton for the unqueness of the Stackelberg equlbrum s the concavty of the Stackelberg leader s payoff functon, F j ( ),where F j ( ) s the reacton functon of player j. More formally, we assume for the rest of the paper that d 2, F j ( ) d 2 < 0. (1) Gven Assumpton (1), n partcular the assumpton of unqueness regardng the Stackelbergequlbrum, we know that the sgn of the slope of players best response functons at the E are dentcal to the sgn of the slope of the follower s best response functon n the Stackelberg equlbrum. 2.2 A suffcent partal rankng of the equlbrum values at the three basc games We consder three basc games, denoted by Γ, Γ S 1 and Γ S 2, whch respectvely correspond to the statc game and to the two Stackelberg games. Let denote by, j the ash equlbrum values of the statc game, and L, F j L the equlbrum values of the Stakelberg equlbrum, where player leads. We have arg ma X (, j ) j arg ma j X j j ( j, ). (2) The Stackelberg equlbrum s determned by backward nducton. We have: F j ( ) arg ma j X j j ( j, ), (3) 5

8 and L arg ma X, F j ( ). (4) Gven the optmzng behavor n the basc games (epressons 2, 3 and 4), we are now n the poston to establsh some partal rankngs of the levels of agents actons at the dfferent equlbrum, whch would be suffcent for our prncpal result. We then have: 5 Lemma 1 Under Assumpton (1), wehave: (, j) {1, 2} 2 and 6= j, ½ j > L j j, j > 0 j ( j, ) < 0 or ½ j, j < 0 j ( j, ) > 0, (5) and ½ j < L j j, j > 0 j ( j, ) > 0 or ½ j, j < 0 j ( j, ) < 0. (6) Proof. See Append A.1. Ths Lemma compares the E s level wth ths choosen by the leader. The obtaned rankngs result from the concavty of the objectve functon of the leader gven n (1). Let consder for nstance the case of SC for player at the E and PS for player j. By defnton, the leader antcpates how the follower would react to a change n her acton wth respect to the E values. When player j leads, t s n her own nterest to nduce a reducton of the acton of the otherplayer(),sncethsactonsapsforplayerj. Due to the property of SC for player, the leader (player j) knows that by reducng her acton she ntates the other player to do the same. That s the reason why we obtan n ths case: L j < j. Smlar reasonngs apply for the other stuatons. 2.3 Frst-mover advantage and second-mover ncentve We now compare the payoffs nthethreebascgames(γ, Γ S 1 and Γ S 2 ), whch wll gve us the opportunty of detectng potental frst-mover (second-mover) advantages or frst-mover (second-mover) ncentves. We defne these notons as follows Defnton 3 () Player has a frst-mover advantage (a second-mover advantage) f her equlbrum payoff n the Stackelberg game n whch she leads, denoted by Γ S, s hgher (lower) than 5 For the rest of the paper, we pose F j F j (L ). 6

9 n the Stackelberg game n whch she follows Γ j S. () Player has a frst-mover ncentve (a second-mover ncentve) f her equlbrum payoff n the Stackelberg game n whch she leads (she follows), denoted by Γ S Γ S j, s hgher than n the statc game Γ. From Lemma (1) and Defnton 3, we can state that: Lemma 2 Under Assumpton (1), wehave Player has a frst-mover advantage f Actons are SC at the E for player and they nduce PC for one player and PS for the other, Or f actons are SS at the E for player and they are PC or PS for both players. More formally, we have L, ½ ½ F j > F, L j j, j > 0 j (, j ) j j ( or, j < 0 j, ) < 0 j (, j ) j ( j, ) > 0. Player has a second-mover ncentve f Actons are SC at the E for player and they nduce PC for one player and PS for the other Or f actons are SS at the E for player and they are PC or PS for both players. Equvalently,, ½ ½ j > F, L j j, j > 0 j (, j ) j j ( or, j < 0 j, ) > 0 j (, j ) j ( j, ) < 0. Proof. see Append A.2. The precedng Lemma takes nto account the non-monotoncty of the reacton functons, snce t reles only on the sgn of the second cross dervatves at the E of the statc game. The ntuton s the followng. If, for nstance, we have SC for player at the E and a game of PC for player j, then, gven Lemma 1, we know that the Stackelberg leader wll ncrease j compared to the E-level. The effect of ths on player s payoff s twofold. There s a drect effect, the payoff effect, whch s postve (negatve), gven that s a game of PC (PS) for player. And there s an ndrect effect, the strategc effect, whch causes to ncrease (decrease) gven that player regards j as SC (SS). However, the ndrect effect, caused by the ncrease of j on player s reacton s unambguous, snce player wll mamze her payoff by movng towards her best response functon. The net effect on player s payoff s postve f t s a game of PC for 7

10 player, and negatve f t s a game of PS for player. Therefore, n ths settng, player has a second-mover ncentve n the former case and a second-mover dsncentve, 6 or equvalently, a frst-mover advantage, n the latter one. ote, that once we know whether plan nteractons (PS or PC) have a smlar effect for both players, we only need to know whether a player s best response functon s n- or decreasng to dentfy a frst-mover advantage or a second-mover ncentve. For nstance, n the classcal Cournot Duopoly game (wth PS) as well as n the prvate provson of publc goods framework (wth PC) we establsh a frst mover advantage for both players. Moreover, n the case where the cross-effects n the payoff functons are of dfferent sgns, a game wth SS for player and SC for player j at the E, player j wll always have a frst-mover advantage and player j a second-mover ncentve. The reason for ths s that player, whether she ncreases or decreases as a Stackelberg leader compared to the E, wll always choose a pont n the strategy space that les nsde the Pareto-superor set, defned as P S 1, 2 (, j ) > (, j ), j =1, 2 and 6= j ª. Player j, on the other hand, always chooses a pont n the strategy space outsde P S. 3 Resolvng the endogenous tmng game The ssue of endogenous tmng s eamned accordng to the etended game wth observable delay proposed by Hamlton and Slutsky (1990). Ths game, denoted by Γ, allows players to choose non-cooperatvely and smultaneously ther tmng decson n a preplay stage ether as soon as (early)oraslateaspossble(late). Ther decson s announced by the players subsequently. The players choose then ther acton accordng to ther tmng decson to whch they are commtted. If both players decde to play at the same tme (whether early or late), the statc game Γ s played. If player chooses to move early and player j chooses to move late, the Stackelberg game Γ S where player leadssplayed. Theetendedgame ³ Γ has 6 See epresson (15) n Append A.2. 8

11 the followng reduced normal form: Table 1: ormal form of the etended game Player 2 early Player 1 early 1 1, 2, 2 2, 1 late 1 F 1, L 2, 2 L 2, F 1 late 1 L 1, F 2, 2 F 2, L 1 1 1, 2, 2 2, 1 The soluton of the game ³ Γ s equvalent to the soluton of the leadershp problem. There s no leader f both players choose the same tmng decson; a leader emerges when they choose complementary roles. Under our assumptons, the soluton of the tmng game can be drectly eplaned by the nature of the nteractons among the two players at the ash equlbrum of the statc game only. We obtan the followng Theorem: Theorem 1 Under Assumpton (1), we have: If payoff functons ehbt smlar plan nteractons j (, j ) j ( j, ) > 0,then 1. The SPEs are the two Stackelberg outcomes, f players strateges are SC at the E. 2. The SPE s the outcome of the statc game, f players strateges are SS at the E. 3. The SPE s the outcome of the Stackelberg game Γ S where player for whom actons are SS at the E wll act as a Stackelberg leader and player j for whom actons are SC at the E wll act as a Stackelberg follower. If payoff functons ehbt opposte plan nteractons j (, j ) j ( j, ) < 0,then 1. The SPE s the outcome of the statc game, f players strateges are SC at the E. 2. The SPEs are the two Stackelberg outcomes, f players strateges are SS at the E. 3. The SPE s the outcome of the Stackelberg game Γ S where player for whom actons are SS at the E wll act as a Stackelberg follower and player j for whom actons are SC at the E wll act as a Stackelberg leader. Proof. See Append A.3. Theorem (1) shows that the SPE(s) s contngent on the nature of plan and strategc nteractons. For nstance f plan nteractons are of the same sgn for both players, t s suffcent for a sequental move game to emerge as an SPE, f one of the players regards 9

12 actons as SC. The suffcent condton f the sgns of plan nteractons dffer s that one of the players regards actons as SS. The reason for ths s that a leader, thus the outcome of a Stackelberg game, would only emerge at the SPE f at least one best response functon les n the Pareto-superor set. Only n ths case we have a second-mover ncentve for at least one player. Fgures 1 and 2 provde a graphcal llustraton of our result. In both fgures player 1 ehbts SC and player 2 SS. Fgure 1 represents the cases when plan nteractons are smlar. The bold lnes represent the non-monotonc reacton functons of player 1 and 2. The dashed (dotted) lnes represent the so-payoff-curves of the players when the game ehbts PS (PC) for both players. In the case of PS (PC) the Pareto-superor set P S, represented by the grey surface, les to the south-west (north-east) of the E. Thus, a sequental move game wll emerge as a SPE f at least one of the players regards actons as a SC, here player 1. If the sgns of plan nteractons dffers the opposte holds. In fgure 2 the dashed (dotted) lnes represent the so-payoff curves of the players when the game ehbts PS (PC) for player 1 and PC (PS) for player 2 and the grey surface to the south-east (north-west) represents P S. Hence, at least one player has to regard actons as a SS n order to guarantee a Stackelberg outcomeasthespeoftheetendedgame(here,player2). Fgure 1: Smlar plan nteractons. 10

13 Fgure 2: Opposte plan nteractons. Gven Theorem (1), we can also state the followng Corollary. Corollary 2 Under Assumpton (1), we have: () If the unque SPE of the etended game s the outcome of a Stackelberg game, then the SPE Pareto-domnates the smultaneous game and the other Stackelberg game. () If the SPEs of the etended game are the outcomes of the two Stackelberg games, then both SPEs Pareto-domnate the statc game, but can not be ranked n a Pareto sense among each other. () If the unque SPE of the etended game s the smultaneous move game, then the outcomes of the three basc games can not be ranked n a Pareto sense. Proof. Immedate It s worth mentonng that n the case of multplcty of SPE (PC and SS or PS and SC), several authors as van Damme and Hurkens (1999), van Damme and Hurkens (2004) or Amr and Stepanova (2006) use the noton of rsk-domnance. However, ths noton may not be appled wthout specfyng the payoff functons. For nstance, van Damme and Hurkens (1999) or Amr and Stepanova (2006) use lnear demand functons n a duopoly contet. An mmedate llustraton of our results s the analyss of contest provded by Dt (1987) and Bak and Shogren (1992). In a two-players contest for an eogenous rent, these authors defne the favorte and the underdog through the probablty of wnnng the contest at the ash equlbrum of the statc game: the favorte (respectvely the underdog ) has a probablty superor (respectvely nferor) to one half. We can easly establsh that ths contest 11

14 corresponds to a game of PS for both players where for one player, n fact the underdog, appropraton efforts are SS, whle for the other, the favorte, efforts are SC. For nstance, let consder a logt Contest Success Functon, denoted by p (, j ). We have p (, j )= f ( ) f ( )+f j ( j ), where f 0 ( ) > 0 >f 00 ( ) by assumpton. Ths s also the probablty of wnnng the eogenous rent R for player dependng on her effort and the effort of the other player ( j ).Thepayoff of players are (, j )=p (, j ) R and j ( j, )=(1 p (, j )) R j. We note that j (, j )=p j (, j ) R = j j ( j, ), Thus, the strategc nteractons among players can never be of the same sgn. Moreover, we have p j (, j )= [f ( ) f j ( j )] f 0 ( ) f 0 j ( j ) [f ( )+f j ( j )] 3 > 0 f ( ) >f j ( j ) p (, j ) > 1 2. Thus, the Stackelberg outcome where the underdog leads s the SPE of the commtment game as Bak and Shogren (1992) establshed t, snce players actons are SS at the E for the underdog and plan nteractons are of the same sgn for both players (PS). 4 Concluson We generalze the Hamlton and Slutsky game of commtment by takng nto account the nonmonotoncty of the reacton functons. We consder several stuatons dependng on the nature of nteractons among two players. The underlyng games may then be nether supermodular nor submodular. By sgnng the slopes of the reacton functons at the ash equlbrum of 12

15 the statc game, we are able to provde a partal rankng of the equlbrum values of players actons. Ths rankng allows us to establsh the presence of a frst-mover advantage or a secondmover ncentve, whch n turn yeld to determne the SPE. We conclude that a leader and obvously a follower emerge at the SPE(s) f for at least one of the players actons are strategc complements (strategc substtutes) at the ash Equlbrum of the statc game and the crosseffects n the payoff functons are of the same (dfferent) sgn(s) for both players. 13

16 References Adner, R., and P. Zemsky (2005): Dsruptve technologes and the emergence of competton, The Rand Journal of Economcs, 36(2), Amr, R. (1995): Endogenous tmng n two-player games: a countereample, Games and Economc Behavor, 9(2), Amr, R., and A. Stepanova (2006): Second-mover advantage and prce leadershp n Bertrand duopoly, Games and Economc Behavor, 55(1), Bak, K. H., and J. F. Shogren (1992): Strategc behavor n contests: comment, Amercan Economc Revew, 82(1), Bergstrom, T., L. Blume, and H. Varan (1986): On the prvate provson of publc goods, Journal of Publc Economcs, 29(1), Bulow, J. I., J. D. Geanakoplos, and P. D. Klemperer (1985a): Holdng dle capacty to deter entry, TheEconomcJournal, 95(377), (1985b): Multmarket olgopoly: strategc substtutes and complements, Journal of Poltcal Economy, 93(3), Dt, A. (1987): Strategc behavour n contests, Amercan Economc Revew, 77(5), Eaton, B. C. (2004): The elementary economcs of socal dlemmas, Canadan Journal of Economcs, 37(4), Eaton, B. C., and M. Eswaran (2002): Appled Mcroeconomc Theory. EdwardElgarPublshng. Hamlton, J. H., and S. M. Slutsky (1990): Endogenous tmng n duopoly games: Stackelberg or cournot equlbra, Games and Economc Behavor, 2(1), Harsany, J. C., and R. Selten (1988): A general theory of equlbrum selecton n games. MIT Press, Cambrdge, MA. Maskn, E., and J. Trole (1988): A Theory of Dynamc Olgopoly, II: Prce competton, knked demand curves, and Edgeworth cycles, Econometrca, 56(3), Sadanand, A., and V. Sadanand (1996): Frm scale and the endogenous tmng of entry: a choce between commtment and fleblty, Journal of Economc Theory, 70(2), Sngh,., and X. Vves (1984): Prce and quantty competton n a dfferentated duopoly, RAD Journal of Economcs, 15(4), Stengel, B. v. (2010): Follower payoffs n symmetrc duopoly games, Games and Economc Behavor, Forthcomng. Syropoulos, C. (1994): Endogenous tmng n games of commercal polcy, Canadan Journal of Economcs, 27(4),

17 van Damme, E., and S. Hurkens (1999): Endogenous Stackelberg leadershp, Games and Economc Behavor, 28(1), (2004): Endogenous prce leadershp, Games and Economc Behavor, 47(2),

18 A Append A.1 Proof of Lemma 1 (a partal rankng) Let defne the functon Ψ ( ) as Ψ ( ), F j ( ) + j, F j ( ) df j ( ). (7) d Ths functon corresponds to the frst dervatve of the leader payoff functon: Ψ ( )= (, F ()) j.for L, we obtan the FOC of the leader, that s Ψ L =0. Under Assumpton (1) we have Ψ 0 ( ) < 0. If j j j, d > 0, actons are SC for the player j at the ash equlbrum, or equvalently F j () d > 0 at = (the reacton functon of player j s ncreasng n at the ash equlbrum). From (7), we have Ψ =, F j ( ) + j, F j ( ) df j ( ) d = j, F j ( ) df j ( ), d snce, F j ( ) =, j of PC or PS. =0. We now consder two stuatons dependng on the presence If j (, j ) > 0, from precedng we obtan snce df j () d Ψ > 0=Ψ L, > 0. The decreasng of Ψ (.) n nvolves In constrast, PS for player j (, j ) < 0 nvolves Ψ > Ψ L < L. (8) Ψ < Ψ L > L. (9) If j j j, < 0, thenwehave d F j ( ) d < 0 at the ash equlbrum. The two cases are then If j (, j ) > 0, we deduce that Ψ < Ψ L > L. (10) If j (, j ) < 0, wehave Ψ > Ψ L < L. (11) A.2 Proof of Lemma 2 (Frst-mover advantage and second-mover ncentve) By defntons of the Stackelberg and the ash equlbra, we have L,F j L >, F j ( ) >,F j = (12), j, 16

19 where the frst nequalty results from the leader s mamzaton program, and the second from the specfcaton of to. The leader of the Stackelberg game always has a payoff level superor or equal to ths obtaned at the ash equlbrum. In other terms, players always have a frst-mover ncentve. We determne under whch condtons n terms of PC, PS, SC or SS player wll have a frst-mover advantage or a second-mover ncentve. A.2.1 Frst-mover advantage From the mamzaton s program gven n (4), we know that player s payoff as a Stackelberg leader s L, F j >, j =ma, j > F, j, (13) where the frst nequalty results from (12) and the second from the defnton of the ash mamzaton program. The estence of a frst-mover advantage may be reduced to determne the condtons such that the followng nequalty holds: F, j > F, L j. (14) Ths nequalty only emerges f The cross-effect n the player s payoff functon s postve ( j (.) > 0) and j Lemma (1), sonlyconsstentwth > L j,whch,gven ½ j, j > 0 j (, j ) < 0 or ½ j, j < 0 j (, j ) > 0. The cross-effect n the player s payoff functon s negatve ( j (.) < 0) and j Lemma (1), sequvalentto < L j,whch,gven ½ j, j > 0 j (, j ) > 0 or ½ j, j < 0 j (, j ) < 0. We remark that a player who has a frst-mover advantage has also a second-mover ds-ncentve, formally defned as, j > F, L j. (15) A.2.2 Second-mover ncentve From (3), we know that F, L j =ma, L j >, L j. (16) To hghlght a second-mover ncentve, we have to establsh condtons for the followng nequalty:, L j >, j. (17) Ths nequalty only emerges f The cross-effect n the player s payoff functon s postve ( j (.) > 0) and j Lemma (1), sonlyconsstentwth < L j,whch,gven ½ j, j > 0 j (, j ) > 0 or ½ j, j < 0 j (, j ) < 0. The cross-effect n the player s payoff functon s negatve ( j (.) < 0) and j Lemma (1), sonlyconsstentwth > L j,whch,gven ½ j, j > 0 j (, j ) < 0 or ½ j, j < 0 j (, j ) > 0. 17

20 A.2.3 Payoffs rankng From (12), (13), (14), (16), (17) and (15) we deduce the followng rankngs of the player s payoff functon at the equlbrum of the three basc games dependng on the estence of a frst-mover advantage or a second-mover ncentve: Player has a frst-mover advantage L, F j >, j > F, L j, (18) and ½ F Player has a second-mover ncentve,l j >, j L,F j > (19), j. A.3 Proof of Theorem 1 (Subgame Perfect ash Equlbra) A player always has a frst-mover ncentve: L,F j >, j, {1, 2} (see epresson 12). In order to determne the SPE, we use the results n Lemma (2) and may compare the payoff levels when the country follows and when t plays smultaneously ( F,L j, j ). If payoff functons ehbt smlar plan nteractons ³ j (, j ) j ( j, ) > 0, 1. SC at the E for both players: j, j > 0 for (, j) {1, 2}. Both players have a secondmover ncentve (see Lemma 2). Sncebothhavealsoafrst-mover ncentve, we deduce that the SPE are the two Stackelberg outcomes: f a player chooses to play early, the other prefers to move late; f a player chooses to play late, the other prefers to play early. 2. SS at the E for both players: j, j < 0 for (, j) {1, 2}. Both players have a frstmover advantage (see Lemma 2) and a second-mover dsncentve:, j > F,L j (see epresson 18). The SPE are then the ash outcome: movng early s a strctly domnant strategy for both players. 3. SC for player 2 and SS for player 1 (at the E): , 1 > 0 > , 2. Lemma (2) establshes that player 1 has a frst-mover advantage, whle player 2 has a second-mover ncentve. Combnng wth the frst-mover ncentve for both players, we have: ½ 1 L 1, F 2 > ma 1 1, ª 2, 1 F 1, L 2 2 F 2, L 1 > 2 2, (20) 1. The SPE s the outcome of the Stackelberg game where player 1 leads and player 2 follows. If payoff functons ehbt opposte plan nteractons ³ j (, j ) j ( j, ) < 0, 1. SC at the E for both players. From Lemma (2) both players have a frst-mover advantage and a second-mover dsncentve (see epresson 18). Thus, the domnant strategy for both players s to play early. The unque SPE s then the outcome of the smultaneous move game. 2. SS at the E for both players. Both players have a second-mover ncentve (see Lemma 2). Snce both have also a frst-mover ncentve, we deduce that the SPE are the two Stackelberg outcomes. 3. SC for player 2 andssforplayer1 (at the E): , 1 > 0 > , 2.FromLemma (2), we know that player 1 has a second-mover ncentve and player 2 has a frst-mover advantage. The SPE s then the outcome of the Stackelberg game where player 1 follows and player 2 leads. 18