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Linking of Repeated Games. When Does It Lead to More Cooperation

and Pareto Improvements? Henk Folmer and Pierre von Mouche

NOTA DI LAVORO 60.2007

MAY 2007 ETA – Economic Theory and Applications

Henk Folmer, Wageningen Universiteit and Rijksuniversiteit Groning Pierre von Mouche, Wageningen Universiteit and Universiteit Utrecht

Linking of repeated games. When does it lead to more cooperation and Pareto improvements? Summary Linking of repeated games and exchange of concessions in fields of relative strength may lead to more cooperation and to Pareto improvements relative to the situation where each game is played separately. In this paper we formalize these statements, provide some general results concerning the conditions for more cooperation and Pareto improvements to materialize or not and analyze the relation between both. Special attention is paid to the role of asymmetries

Keywords: Environmental Policy, Linking, Folk Theorem, Tensor Game, Prsioners' Dilemma, Full Cooperation, Pareto Efficiency, Minkowski Sum, Vector Maximum, Convex Analysis

JEL Classification: C72

Address for correspondence: Pierre von Mouche Wageningen Universiteit Algemene Economia Hollandseweng 1 Postbus 8130 6700 EW Wageningen The Netherlands E-mail: pvmouche@gmx.net

1 Introdu tionThere has developed an interest in the theory and appli ations of linking, also alled `inter onne -tion'. The basi idea is the following. Consider a group of de ision makers who are simultaneouslyinvolved in several di�erent real world problems (issues). The standard approa h is to onsider thede ision making pro ess for ea h problem in isolation. In pra ti e, however, the de ision makingpro ess with respe t to one problem is usually in�uen ed by the de ision making pro esses withrespe t to the other problems (spill-over e�e ts or links). Dis arding the links among the issuesand analyzing the de ision pro ess on ea h issue separately rather than in a multi-issue de isionmaking ontext is likely to lead to biased out omes. Parti ularly, a single issue approa h ignoresthe possibility that if the issues have ompensating asymmetries of similar magnitudes, an ex- hange of on essions may allow and enhan e ooperation whi h extends beyond ooperation inthe single issue ontext. Some well-known real world examples of linking are the negotiations `onland for pea e' between Israel and Palestina and the deal on WTO membership and parti ipationin the Kyoto agreement between the EU and Russia.In the e onomi s literature the notion of linking has been applied in the ontext of multimarketbehavior in oligopolisti markets (see e.g. Bernheim and Whinston, 1990; Spagnolo, 1999) and ofinternational environmental problems (see e.g. Folmer et al., 1993; Botteon and Carraro, 1998;Carraro and Sinis al o, 1999; Finus, 2001).A game theoreti al framework for the linking of repeated games was developed by Folmer et al.(1993) and by Folmer and von Mou he (1994). In Folmer and von Mou he (2000) the followingthemes for linking of repeated games were suggested: linking may sustain more ooperation,1 mayeliminate so ial welfare losses, may bring Pareto improvements and may fa ilitate ooperation.We observe that `may' is used here to indi ate that the hara teristi s of linking of repeatedgames mentioned do no hold un onditionally but depend on the parti ular nature of the problemat hand. However, to our best knowledge, the onditions under whi h these hara teristi s holdhave not yet been thoroughly analyzed whi h is a major omission in the light of the pra ti al andtheoreti al relevan e of linking. Admittedly, some results about the onditions under whi h the hara teristi s of more ooperation and Pareto improvements hold an be found in Ragland (1995)and Just and Netanyahu (2000). However, these results are limited in s ope be ause the settingsin these publi ations on ern the spe ial ase of linking of two repeated 2 × 2-bimatrix games.The main purpose of this paper is to identify lasses of isolated stages games for whi h thethemes 'linking may sustain more ooperation' and 'linking may bring Pareto improvements' mate-rialize or not. For that purpose we formalize the themes 'linking may sustain more ooperation'and'linking may bring Pareto improvements'. Our results apply to the linking of an arbitrary numberof repeated games with an arbitrary number of (the same) players. In se tion 2 we present pre-liminaries and introdu e on epts. In se tion 3 we present �gures that illustrate these on eptsand that will be referred to in the next se tions. In se tion 4 we dis uss 'more ooperation' and inse tion 5 Pareto improvements. Se tion 6 on ludes. Various proofs will be given in the appendix.2 PreliminariesNegotiation sets. Consider a game in strategi form among N players. That is, for ea h playeri ∈ N := {1, . . . , N} we have a non-empty (a tion) set X i and a real-valued (payo�) fun tion f ion the set of multi-a tions X := X1 × · · · × XN . In order to avoid some te hni alities we willrestri t ourselves here often to what we all regular games in strategi form, whi h are games instrategi form that satisfy the following three assumptions. First, ea h payo� fun tion is bounded.This assumption assures that the minimax payo� vj of ea h player j is a well-de�ned real number.Se ond, without any loss of generality, we assume that vj = 0 for ea h player j. This assumptionimplies that a payo� ve tor (i.e. an element of RN ) is individually rational if and only if it belongsto RN

+ , i.e. the losed positive o tant of RN . Third, denoting f(x) := (f1(x), . . . , fN (x)), the1This is the ounterpart of the theme `repetition enables ooperation' for repeated games. 'More' is relative tothe single issue ase. 2

feasible set, i.e. the onvex hull co(U) of the set U := {f(x) | x ∈ X} of basi payo� ve tors, isassumed to be losed. This ondition is always satis�ed in the ase ea h a tion set is �nite.2For a regular game in strategi form Γ, the interse tion of its set of individually rational payo�ve tors and its feasible set is an important obje t. We all it here simply the negotiation set of Γand denote it by H :3H := o(U) ∩ RN

+ .The three assumptions presented above ensure that H is a ompa t set.4Be ause ea h Nash equilibrium payo� ve tor of Γ is individually rational, H ontains the set ofNash equilibrium payo� ve tors. By PB(H) we denote the Pareto boundary of H and by PBw(H)its weak Pareto boundary.5 Be ause H is ompa t, PB(H) 6= ∅ if H is non-empty. Also we have(see Appendix A.4)PB(H) = PB(co(U)) ∩ RN

+ . (1)Given a game in strategi form Γ we all a maximizer x of the total payo� fun tion ∑Nj=1 f ja full- ooperative multi-a tion. The set of su h multi-a tions will be denoted by Y . It is easy tosee that (see Appendix A.4) for a regular game in strategi form we have

Y 6= ∅. (2)Dire t sum games and anoni al mapping. Consider M games in strategi form 1Γ, . . . , MΓamong (the same) N players. We refer to them as isolated stage games and use pre-subs ripts torefer to obje ts related to them. Let M := {1, . . . , M}, the set of issues. Let kXj be the a tionset of player j in kΓ. De�ne for ea h k ∈ M

kX := kX1 × · · · × kXNand for ea h player j

∗Xj := 1X

j × · · · × MXj .Moreover, de�ne the mapping Ψ : 1X× · · · × MX → ∗X1 × · · · × ∗X

N byΨ(

1x...Mx

) := (∗x

1, . . . , ∗xN ).

Ψ is alled the anoni al mapping. Note that the anoni al mapping is a bije tion.For M games in strategi form 1Γ, . . . , MΓ among N players, the trade-o� dire t sum game(⊕Γ)α is de�ned as the game in strategi form where player j has a tion set ∗X

j and his payo�fun tion is given by6f j(∗x

1, . . . , ∗xN ) :=

M∑

k=1

kf j(1x1, . . . , 1x

N ).(In the ase of two bimatrix games (⊕Γ)α is the tensor sum of the individual bimatrix games.)The set of possible payo�s ve tors Uα of (⊕Γ)α equals ∑k∈M kU := 1U + · · ·MU .72 Note that for a regular game in strategi form it is possible that its feasible set does not ontain 0. Indeed,this for example holds for the regular bimatrix game „

−2; 2 0; −41; −3 −2; 0

«.3The negotiation set plays an important role in Folk theorems whi h relate to the geometri stru ture of the setof (average) subgame perfe t Nash equilibrium payo� ve tors for repeated games < Γ > with Γ as stage game. Inthis ontext it is ustomary to assume that repeated games are with dis ounting and that ea h player has the samedis ount fa tor δ ∈ (0, 1). Finally, if we onsider several repeated games below (with the same players) together,then it is assumed that in ea h of them the periods are the same and the dis ount fa tors are the same. For thepurpose of this paper it is not ne essary to go into the details of (te hni ally ompli ated) Folk theorems. For this,we refer to, for example, Benoît and Krishna (1996).4This set may be empty, as for example is the ase for the bimatrix game in footnote 2.5See appendix A.3 for Pareto boundaries.6The α refers to the fa t that in this formula the payo�s of the isolated games are added (with weights 1).7For two subsets A, B of RN its Minkowski sum A + B is de�ned by A + B := {a + b | a ∈ A, b ∈ B}.3

Let kE be the set of Nash equilibria of kΓ, kY the set of full- ooperative multi-a tions of kΓ,Eα the set of Nash equilibria of (⊕Γ)α and Yα the set of full- ooperative multi-a tions of (⊕Γ)α.It an be shown that (see Folmer et al., 1993; Folmer and von Mou he, 1994)

Ψ(1E × · · · × ME) = Eα, (3)Ψ(1Y × · · · × MY ) = Yα. (4)Suppose ea h kΓ is regular. Then (⊕Γ)α also is regular. The negotiation set of kΓ is

kH := RN+ ∩ co(kU).Using the fa t that a onvex hull of a sum is the sum of the onvex hulls, the negotiation set of

(⊕Γ)α isHα = RN

+ ∩∑

k∈M

co(kU).Linking. Again, let 1Γ, . . . , kΓ be M regular games in strategi form and onsider the repeatedgames < kΓ >. Linking of the (isolated) repeated games < kΓ > is done by ombining them intoa repeated game (⊗Γ)α, a so- alled trade-o� tensor game. This trade-o� tensor game has as stagegame the trade-o� dire t sum game (⊕Γ)α.In order to analyse the e�e ts of linking, we de�ne the aggregated negotiation set asHag :=

∑

k∈M

kH.

Hag may be onsidered as the negotiation set when the M repeated games are not linked butmerely aggregated. We remark that Hag = ∅ when some kH is empty. Be ause∑

k∈M

(RN+ ∩ co(kU)) ⊆

∑

k∈M

RN+ ∩

∑

k∈M

co(kU) = RN+ ∩

∑

k∈M

co(kU) (5)it follows thatHag ⊆ Hα. (6)We observe that equality holds in (6) if and only if the ⊆-symbol is a =-symbol in (5).More ooperation and Pareto improvements. In Folmer et al. (1993) it is shown that Nashequilibria for ea h repeated game < kΓ > lead in a anoni al way to a Nash equilibrium for thetrade-o� tensor game (⊗Γ)α.8 In general, the trade-o� tensor game also has other (subgameperfe t) Nash equilibria. Folk theorems are useful in order to investigate the question how manymore subgame perfe t Nash equilibria there are, parti ularly by fo ussing on the set Hα \ Hag.This leads to the following de�nition:De�nition 1 There is an enri hment of the aggregated negotiation set if the stri t in lusion Hag

⊂ Hα holds. ⋄Hen e, enri hment of the aggregated negotiation set an be interpreted as `Linking sustains more ooperation'.We all u ∈ PB(Hag) a (strong) expansion point of PB(Hag) if there exists w ∈ Hα su h that9w ≫ u and a weak expansion point of PB(Hag) if there exists w ∈ Hα su h that w > u. By EXPwe denote the set of expansion points and by EXPw the set of weak expansion points. Of ourse,EXP ⊆ EXPw and EXP ⊆ PB(Hag). Moreover, (see Appendix A.4)

EXP = PB(Hag) \ PBw(Hα). (7)Below we shall only deal with strong expansion points.8It is straightforward to show that this statement remains valid if one repla es `Nash equilibrium' by `subgameperfe t Nash equilibrium'.9For a = (a1, . . . , aN ), b = (b1, . . . , bN ) ∈ RN we write a ≥ b if ai ≥ bi for all i. We write a > b if a ≥ b and

a 6= b. And we write a ≫ b if ai > bi for all i. 4

De�nition 2 We speak of partial expansion (of the Pareto boundary of the aggregated negotia-tion set) if ∅ ⊂ EXP ⊂ PB(Hag). In the ase EXP = ∅ we say that there is expansion nowhere.Finally, in the ase ∅ ⊂ EXP = PB(Hag) there is expansion everywhere. ⋄We observe that by virtue of Folk theorems the existen e of an expansion point of PB(Hag)is related to possible Pareto improvements. This may be interpreted as `Linking brings Paretoimprovements'.Finally, we observe that if there is no enri hment of the aggregated negotiation set, i.e. ifHag = Hα, then Hag and Hα have the same Pareto boundaries and thus, by virtue of (7), EXP = ∅.3 FiguresIn this se tion we present �ve �gures that illustrate the on epts de�ned above. Moreover, we wilrefer to these �gures in se tions 4 and 5. The �gures present the linking of two repeated games,where the isolated stage games are (regular) 2 × 2- bimatrix games.Figure 1 relates to the games

1Γ :=

(2; 1 −3; 2

5; −1 0; 0

), 2Γ :=

(1; 2 −1; 5

2; −3 0; 0

).

–4

–2

0

2

4

6

–4 –2 2 4 6Figure 1: Expansion everywhere.Figure 1, and also Figures 2 � 5, are to be interpreted as follows. Four polygons are drawn:the feasible sets co(1U), co(2U), the sum of these two sets and the aggregated negotiation setHag = 1H + 2H . Be ause the minimax payo� ve tors for 1Γ and 2Γ are 0, the sets 1H and 2H anbe distinguished. Hag = 1H + 2H is the boldfa ed polygon. Be ause the minimax payo� ve torfor (⊕Γ)α is 0, the set Hα an also be distinguished. For reasons of onvenien e these four setsfor Figure 1 are drawn below.

–2

2

4

–2 2 4

–4

–2

0

2

4

6

–4 –2 2 4 6

0

1

2

3

4

1 2 3 4The sets in the above three �gures respe tively on ern co(1U) and co(2U), co(1U) + co(2U)and Hag = 1H + 2H . 5

We note that in the ase of Figure 1(⊕Γ)α =

3; 3 1; 6 −2; 4 −4; 74;−2 2; 1 −1;−1 −3; 26; 1 4; 4 1; 2 −1; 5

7;−4 5;−1 2;−3 0; 0

.Figure 2 relates to the two games

1Γ :=

(0; 2 3; 1−3; 0 0; 0

), 2Γ :=

(0; 1 1; 0.5−2; 0 0; 0

).

0

0.5

1

1.5

2

2.5

3

–4 –2 2 4Figure 2: No enri hment of the aggregated negotiation set.Figure 3 relates to the two games1Γ :=

(7; 1 −3; 3

10; −2 0; 0

), 2Γ :=

(1; 7 −2; 10

3; −3 0; 0

).

–4

–20

2

4

6

8

10

12

–4 –2 2 4 6 8 10 12Figure 3: Partial expansion (non-symmetri isolated stage games).Figure 4 relates to the two games1Γ :=

(2; 2 −2; 4

4; −2 0; 0

), 2Γ :=

(2; 2 −1; 1

1; −1 0; 0

)Finally, Figure 5 relates to the two games1Γ :=

(2; 2 −2; 10

10; −2 0; 0

), 2Γ :=

(3; 3 −3; 4

4; −3 0; 0

).6

–2

2

4

6

–2 2 4 6Figure 4: Enri hment of the aggregated negotiation set and expansion nowhere.–4

–20

2

4

6

8

10

12

14

–4 –2 2 4 6 8 10 12 14Figure 5: Partial expansion (symmetri isolated stage games).4 Linking sustains more ooperationThe next theorem, proven in Appendix A.4, identi�es three ases where linking does not lead toan enri hment of the aggregated negotiation set.Theorem 1 Ea h of the following onditions is su� ient for that there is no enri hment of theaggregated negotiation set.1. For ea h k the payo� fun tion of ea h player in kΓ is a positive multiple kr of its payo�fun tion in 1Γ; this result holds in parti ular if all isolated stage games are identi al.2. In ea h isolated stage game ea h basi payo� ve tor is individually rational.103. Hα = ∅. ⋄Theorem 1 is a negative result and learly shows that the stru ture of the isolated stage gamematters to a hieve more ooperation. Figure 2 shows that there are situations of no enri hmentof the aggregated negotiation set that are not overed by Theorem 1. In all other �gures there isan enri hment.Now we turn to the onditions under whi h a positive general result holds, i.e. linking leadsto an enri hment of the aggregated negotiation set. For that purpose we present Theorem 2 asa �rst general result. This theorem deals with isolated stage games that have ` ompensatingasymmetries of exa tly the same magnitude'. This notion is de�ned as follows. Given isolatedstage games 1Γ, . . . , NΓ (so M = N) we say that they have ` ompensating asymmetries of ex-a tly the same magnitude' if there are N permutations π1, . . . , πN of N with π1 := Id (i.e. theidenti al permutation) su h that for ea h j ∈ N one has {π1(j), . . . , πN (j)} = N and su h thatkΓ := πk(1Γ) (k ∈ M). So ea h kΓ is a permutation of 1Γ (see Appendix A.1 for permuted games),but not all N ! permuted games of 1Γ are allowed.1110Note that this is equivalent with `in ea h isolated stage game ea h point of its feasible set is individuallyrational'.11It should be noted that regularity of 1Γ implies regularity of ea h kΓ and that if one of then is symmetri , allare su h. 7

Another ondition in Theorem 2 is that Γ has a defe t (Folmer and von Mou he, 2000): a gamein strategi form with bounded payo� fun tions has a j-defe t (where j ∈ N ) if for player j nofull- ooperative payo� ve tor is individually rational. The game has a defe t if it has a j-defe t forsome j. Of ourse, a defe t ex ludes the possibility that a Nash equilibrium is full- ooperative.12It also ex ludes the possibility that the game is symmetri and regular.13Theorem 2 Consider isolated regular stage games that have ompensating asymmetries of exa tlythe same magnitude. If Γ := 1Γ has a Nash equilibrium and a defe t, then there is an enri hmentof the aggregated negotiation set. Moreover, the game (⊕Γ)α has a Nash equilibrium for whi hthere exists a full- ooperative unanimous Pareto improvement. ⋄The proof of Theorem 2 is given in Appendix A.4. Note that in Theorem 2 all the isolatedstage games have a defe t, but (⊕Γ)α does not have. Theorem 2 explains the enri hment of theaggregated negotiation set in Figure 1 (where Γ has a 2-defe t). Figures 3�5 show that there aresituations of enri hment of the aggregated negotiation set that are not overed by Theorem 2. Weobserve that Theorem 2 does not ex lude the possibility that in the ase the isolated stage gamesare symmetri (without having ompensating asymmetries of exa tly the same magnitude), there ould be an enri hment of the aggregated negotiation set (Figures 4 and 5).We note that in Figures 1, 3 and 5 the isolated stage games are prisoners' dilemma games,14but that this is not the ase for Figure 4. Con erning this aspe t:Corollary 1 Consider isolated regular stage games that are 2 × 2-bimatrix prisoners' dilemmagames, with a unique full- ooperative multi-a tion that have ompensating asymmetries of exa tlythe same magnitude, Then there is an enri hment of the aggregated negotiation set. Moreover,(⊕Γ)α has a Nash equilibrium for whi h there exists a full- ooperative unanimous Pareto improve-ment. ⋄Indeed, for this situation 1Γ automati ally has a Nash equilibrium and a j-defe t for some j.155 Linking brings Pareto improvementsWe have already seen that if there is no enri hment of the aggregated negotiation set, then there isexpansion nowhere. A natural question now is whether enri hment of the aggregated negotiationset implies that there is an expansion point. The answer is `no' as Figure 4 shows. Note that inthis �gure the Pareto boundary PB(2H) is the singleton {(2, 2)}.Theorem 1(2) implies that if in ea h isolated stage game ea h point of its feasible set is indi-vidually rational, then there is expansion nowhere. Also in Figure 2 there is expansion nowhere,but this an not be explained in this way. Individual rationality of ea h point of the feasible setsis a strong ondition. In Theorem 4 there is a weaker ondition that also guarantees expansionnowhere and explains expansion nowhere in Figure 2. The proof of Theorem 4 uses the te hniqueof normal ones16 and is a little bit ompli ated. Therefore, before we turn to this theorem, westate a spe ial ase of it, Theorem 3, for whi h we an provide a simple proof.12In this sense one may say that a defe t implies that ea h Nash equilibrium has a welfare loss. For su h a gamethe welfare loss remains when we repeat the game. See Folmer and von Mou he (1994, Proposition 4.2.) for apre ise statement.13Here is a proof of this statement, by ontradi tion. Suppose Γ is symmetri , regular and has a j-defe t. Thenfor ea h permutation π of N the game π(Γ) has a π−1(j)-defe t. But π(Γ) = Γ, so Γ has an i-defe t for ea hi ∈ N . By (2) there exists a full- ooperative multi-a tion y. Let n be a Nash equilibrium. Then one has (usingthe fa t that ea h Nash equilibrium payo� ve tor is individually rational) PN

j=1fj(n) ≥

PNj=1

0 >PN

j=1fj(y), a ontradi tion.14We all a game in strategi form a prisoners' dilemma game if ea h player has a stri tly dominant a tion andthe stri tly dominant equilibrium is not Pareto-e� ient in the weak sense.15The last statement is a dire t onsequen e of the fa t that for every 2 × 2-bimatrix prisoners' dilemma gamethe Nash equilibrium payo� for ea h player equals his minimax payo�.16A more dire t proof of Theorem 4 would be wel ome.8

Theorem 3 If, in ase M = 2, for ea h of the isolated stage games ea h point of the Paretoboundary of its feasible set is individually rational and at least one of these Pareto boundaries isa singleton, then PB(Hα) = PB(Hag) and therefore there is expansion nowhere. ⋄For the proof of this theorem see Appendix A.4. The on lusion of expansion nowhere in Theorem 3even holds for general M without the singleton assumption:Theorem 4 If for ea h of the isolated stage games ea h point of the Pareto boundary of its feasibleset is individually rational, then there is expansion nowhere. ⋄Also for the proof of this theorem see Appendix A.4.Figure 2 illustrates Theorem 4 and Figure 4 shows that there are situations of expansionnowhere that are not overed by Theorem 4. Note that in Figure 2 there even is no enri hment ofthe aggregated negotiation set (and that for player 2 the �rst isolated stage game 'is half the se ondone'). An important issue for further resear h is whether for the ases spe i�ed in Theorem 4 therealways is no enri hment of the aggregated negotiation set.Figures 3 and 5 show ases where there is partial expansion. Note that in Figure 1 thereis expansion everywhere. Another interesting question for further resear h is whether expansioneverywhere always holds in Theorem 2. An even more basi question is whether or not an expansionpoint always exists in Theorem 2.Finally we note that even in ase ea h isolated stage game is symmetri , there may be partialexpansion as Figure 5 shows.6 Con lusionIn this paper we have presented some general results on more ooperation and Pareto improvementswhi h an be a hieved by linking of repeated games. We have de�ned `more ooperation' by thenotion of enri hment of the aggregated negotiation set and `Pareto improvement' by the notion ofexpansion point of the Pareto boundary of the aggregated negotiation set. Using these notions wehave formalized for tensor games the theme `linking may sustain more ooperation' and `linkingmay bring Pareto improvements'.We have shown that in the ase linking brings Pareto improvements, it also sustains more ooperation but that the reverse does not hold in general. We have identi�ed a lass of isolatedstage games for whi h linking does not sustain more ooperation and a lass for whi h it does.In order to identify this last lass we formalized the basi idea that an ex hange of on essionsmay enhan e ooperation if the issues have ompensating asymmetries of similar magnitude. Forthis lass all isolated stage games are asymmetri and permutations of ea h other and all have theproperty that ea h full- ooperative payo� ve tor is not individually rational. Con erning Paretoimprovements, we derived (in the appendix) a hara terization of expansion points in terms ofpositive normal ones and used this in order to identify a lass where linking does not bringPareto improvements. We showed that also in the ase all isolated stage game are symmetri (butnot identi al), more ooperation and even partial expansion is possible.The �gures that we used for illustrating our results lead to interesting questions for furtherresear h:A. How far an one deviate in Theorem 2 from the situation of (exa t) permuted games? Thiswould model the notion of `similar magnitude' in the expression `an ex hange of on essionsin issues that have ompensating asymmetries of similar magnitude'.B. Derive (interesting) su� ient onditions (like the onje ture in C) for the existen e of expansionpoints.C. If the isolated stage games have ompensating asymmetries of exa tly the same magnitude andone of them has a Nash equilibrium and a defe t, is there then always expansion everywhere?More basi ally, we onje ture that there then always is at least one expansion point.9

D. If for ea h of the isolated stage games ea h point of the Pareto boundary of its feasible set isindividually rational, is there then no enri hment of the aggregated negotiation set?Finally, we observe that although this paper is about game theory, the problems we deal with arein fa t geometri problems related to Minkowski sums and interse tions of onvex sets. Therefore,basi resear h on linking should (also) relate to these topi s.A Appendi esBefore turning to the proofs in Appendix A.4 we present some de�nitions and useful results. Forthose for whi h it is di� ult to tra e them in the literature we also give a proof.A.1 Permuted gamesGiven a Cartesian produ t of sets A1 × . . .×AN , we de�ne for a permutation κ of {1, . . . , N} themapping Tκ : A1 × · · · × AN → Aκ(1) × · · · × Aκ(N) by Tκ(a1, . . . , aN ) := (aκ(1), . . . , aκ(N)).Let Γ be a game in strategi form and π a permutation of N . We de�ne the game in strategi form π(Γ) ( alled a permuted game of Γ) as the game in strategi form where the a tion set Zi ofplayer i is Xπ(i) and his payo� fun tion hi is fπ(i) ◦ Tπ−1. So,hi(z1, . . . , zN) = fπ(i)(zπ−1(1), . . . , zπ−1(N)).Finally, a game in strategi form Γ where ea h player has the same a tion set X is alledsymmetri if for ea h permutation π of N one has Γ = π(Γ).A.2 Normal onesLet A be a non-empty subset of RN and x ∈ A, i.e. x is an element of the topologi al losure of

A. ThenNA(x) := {d ∈ RN | (y − x) · d ≤ 0 for all y ∈ A}.

NA(x) is a onvex one and is alled the normal one of A in x. Moreover, we de�ne for x ∈ Athe positive normal one of A in x asN+

A (x) := {d ∈ NA(x) | d > 0}.Note that 0 ∈ NA(x), but that N+A (x) may be empty.Let kA (1 ≤ k ≤ M) be subsets of RN . It is straightforward to prove hat for ka ∈ kA (1 ≤ k ≤

M), with a :=∑N

k=1 ka, one hasNP

M

k=1 kA(a) = ∩Mk=1NkA(ka). (8)A.3 Pareto boundariesDe�ne the fun tion C : RN → R by C(x) :=

∑N

l=1 xl. For a subset A of RN we de�ne A as the setof maximizers of the restri ted fun tion C ↾ A, i.e. of the fun tion C : A → R. Moreover, de�nes(A) ∈ R ∪ {−∞, +∞} as the supremum of the fun tion C ↾ A. Closedness (boundedness) of Aimplies losedness (boundedness) of A and if A is a non-empty ompa t subset of RN , then A isnon-empty and ompa t as well.It is also straightforward to prove the following properties for all subsets A, B of RN : o(A) = o(A); (9)

s( o(A)) = s(A); (10)s(A + B) = s(A) + s(B). (11)10

For a subset A of RN its (strong) Pareto boundary PB(A) is de�ned as the set of elements a ofA for whi h there does not exist c ∈ A with c > a whereas its weak Pareto boundary PBw(A) isde�ned as the set of elements a of A for whi h there does not exist c ∈ A with c ≫ a. Of ourse,PB(A) ⊆ PBw(A). For ∂A, the topologi al boundary of A, we have

A ⊆ PB(A) ⊆ PBw(A) ⊆ ∂A.So PB(A) 6= ∅ if A is ompa t and non-empty.Let Ak (1 ≤ k ≤ M) be subsets of RN . It is easy to show that for ak ∈ Ak (1 ≤ k ≤ M), witha :=

∑N

k=1 ak, one hasa ∈ PB(

M∑

k=1

Ak) ⇒ ak ∈ PB(Ak) for all k.Thus in parti ularPB(

M∑

k=1

Ak) ⊆M∑

k=1

PB(Ak). (12)Lemma 1 Let A be a ompa t subset A of RN . For ea h a ∈ A there exists b ∈ PB(A) withb ≥ a. ⋄Proof.� Z := {z ∈ RN | z ≥ x} is losed. This implies that Z∩A is ompa t. Be ause x ∈ Z ∩A,Z ∩ A 6= ∅ and therefore also PB(Z ∩ A) 6= ∅. Take y ∈ PB(Z ∩ A). Then y ∈ Z, so y ≥ x. Alsoy ∈ PB(A), be ause otherwise there would exist b ∈ A with b > y. Then we had b > y ≥ x, sob ∈ Z ∩ A and b > y, whi h is a ontradi tion with y ∈ PB(Z ∩ A). Q.E.D.Lemma 1 now will be used to derive further properties.Lemma 2 For two non-empty subsets A and B of RN with A ⊆ B and a ∈ A one has:

B ompa t and PB(B) ⊆ A ⇒ N+B (a) = N+

A (a). ⋄Proof.� Be ause A ⊆ B one has N+B (a) ⊆ N+

A (a). By ontradi tion we prove that N+B (a) ⊇

N+A (a). So suppose γ ∈ N+

A (a) \N+B (a). Now (w− a) ·γ ≤ 0 for all w ∈ A, but not for all z ∈ B.This implies that there is a w ∈ B \ A su h that γ · (w − a) > 0. Be ause B is ompa t, there is,by Lemma 1, b ∈ PB(B) su h that b ≥ w. Be ause γ > 0, also γ · (b − a) > 0. So b 6∈ A. But

b ∈ PB(B) ⊆ A, whi h is a ontradi tion. Q.E.D.In general the in lusion in (12) is not an equality. Here is a spe ial ase where equality holds:Lemma 3 [A, B ⊆ RN , B ompa t and #PB(B) = 1] ⇒ PB(A + B) = PB(A) + PB(B). ⋄Proof.� Only ⊇` remains to be proved. This we do by ontradi tion. So suppose x ∈ PB(A) +PB(B), but x 6∈ PB(A + B). Write PB(B) = {b}. Let a ∈ PB(A) su h that x = a + b.Be ause B is ompa t, there is for ea h y ∈ B an element of PB(B), i.e. b, su h that y ≤ b. Sob − y ≥ 0 (y ∈ B. Be ause x ∈ A + B and x 6∈ PB(A + B), there is d ∈ A + B with d > x. Leta′ ∈ A and b′ ∈ B su h that d = a′ + b′, Then a′ > a + (b− b′) ≥ a, so a′ > a. But a ∈ PB(A),a ontradi tion. Q.E.D.Lemma 4 Let B, C ⊆ RN su h that for no c ∈ C there exists d ∈ Cc with d > c. ThenPB(B ∩ C) = PB(B) ∩ C. ⋄Proof.� �⊆�: by ontradi tion. So suppose a ∈ PB(B ∩ C) and a 6∈ PB(B) ∩ C. Be ausea ∈ B ∩ C ⊆ C, it follows that a 6∈ PB(B). Now there is b ∈ B with b > a. Be ausea ∈ PB(B ∩ C), it follows that b 6∈ B ∩ C. Thus b ∈ Cc, a ∈ C and b > a, whi h is a ontradi tion.�⊇�. Suppose d ∈ PB(B) ∩ C. One has d ∈ B ∩ C. If we would have a ∈ B ∩ C su h thata > c, then, noting that a ∈ B and d ∈ B, we would have a ontradi tion. Q.E.D.11

Lemma 5 Let A be a non-empty onvex subset of RN . Then a ∈ PBw(A) ⇒ N+A (a) 6= ∅. ⋄Proof.� De�ne B := {x ∈ RN | x ≥ a}. One has B◦ = {x ∈ RN | x ≫ a} and thus B◦ ∩ A = ∅.

B◦ and A are onvex, non-empty and disjoint. Using a separation theorem, there exists an a�nehyperplane that A and B◦ separates. Therefore there exists γ ∈ Rn\{0} su h that γ ·z ≤ γ ·b (z ∈A, b ∈ B◦). Even now

γ · z ≤ γ · b (z ∈ A,b ∈ B). (13)With b = a it follows that γ · z ≤ γ · a (z ∈ A). Now we prove by ontradi tion that γ > 0. So(remembering that γ 6= 0) suppose γi < 0 for some i. For b ∈ B de�ned by bj := aj (j 6= i) andbi := x aar x ≥ ai, we have

γ · b =

n∑

j=1,j 6=i

γjaj + γix.For x large enough this number is less than γ · a, whi h is a ontradi tion with (13). Q.E.D.A.4 Remaining proofsProof of (2). Be ause the game is regular, co(U) is losed, and bounded. So it is ompa t.17Be ause it is also non-empty, co(U) also is non-empty and therefore, by (9), also U 6= ∅. Be auseof the general identityU = f(Y ), (14)also Y 6= ∅. Q.E.D.Proof of (1). `⊆': by ontradi tion. So suppose u ∈ PB(H) and u 6∈ PB( o(U))∩RN

+ . Be auseu ∈ RN

+ , it follows that u 6∈ PB( o(U)), Noting that u ∈ o(U), there exists w ∈ o(U) withw > u. Therefore w ∈ RN

+ and thus w ∈ H , whi h is a ontradi tion with w ∈ PB(H).`⊇': suppose u ∈ PB( o(U)) ∩ RN+ . Then u ∈ H and there does not exist w ∈ o(U) with

w > u. Thus there also dos not exist w ∈ H with w > u. Q.E.D.Proof of (7). `⊆': suppose u ∈ EXP. Then u ∈ PB(Hag) and there exists w ∈ Hα su h thatw ≫ u. By (6), u ∈ Hα. Therefore w 6∈ PBw(Hα).`⊇': suppose u ∈ PB(Hag) \ PBw(Hα). By (6), u ∈ Hα. Be ause u 6∈ PBw(Hα), there is anw ∈ Hα with w ≫ u. Thus u ∈ EXP. Q.E.D.Proof of Theorem 1. 1. We he k that equality in (5) holds. For r :=

∑k kr one has (withsums on k ∈ M)

∑(RN

+ ∩ co(kU)) =∑

(RN+ ∩ kr co(1U)) =

∑(krRN

+ ∩ kr co(1U)) =∑

kr(RN+ ∩ co(1U)) =

r(RN+ ∩ co(1U)) = rRN

+ ∩ rco(1U)) = RN+ ∩ rco(1U) = RN

+ ∩∑

(kr co(1U)) = RN+ ∩

∑co(kU).We observe that the fourth equality holds be ause RN

+ ∩ co(1U) is onvex and the seventh holdsbe ause co(1U) is onvex.2. Using kU ⊆ RN+ and ∑

k co(kU) ⊆ RN+ we obtain ∑

k(RN+ ∩ co(kU)) =

∑k co(kU) =

co(∑

k kU) = RN+ ∩ co(

∑k kU) = RN

+ ∩∑

k co(kU).3. Be ause of (6). Q.E.D.Proof of Theorem 2. First a lemma:Lemma 6 Suppose the following two onditions hold:17Note that U need not be ompa t. 12

A. There exists an l su h that no element of the onvex hull of the full- ooperative payo� ve torsof lΓ is individually rational,B. The trade-o� dire t sum game (⊕Γ)α has an individually rational full- ooperative payo� ve tor.Let b be su h a payo� ve tor.Then b ∈ Hα \ Hag and thus there is an enri hment of the aggregated negotiation set. ⋄Proof.� Condition A omes down to co(lU) ∩ RN+ = ∅ and Condition B to b ∈ Uα ∩ RN

+ . Using(11) and the s-notation of Appendix A.3, we obtains(Uα) =

∑

k

s(kU).Of ourse, b ∈ Hα.Next we prove by ontradi tion that b 6∈∑

k kH . Suppose that b =∑

k kh with the kh ∈ kH .Using (10) we have for ea h k ∈ M

∑

j

khj ≤ s(co(kU)) = s(kU). (15)Be ause lh ∈ RN+ it follows that lh 6∈ o(lU) and so lh ∈ o(lU) \ o(lU). By virtue of (9) we have

co(lU) = co(lU) and so lh ∈ o(lU) \ o(lU). Therefore, in (15) we have a stri t inequality fork = l. Be ause b ∈ Uα, one has ∑

j bj = s(Uα). It follows that s(Uα) =∑

k s(kU) >∑

k

∑j khj =∑

j

∑k khj =

∑j bj = s(Uα), whi h is a ontradi tion. Q.E.D.Now we will prove Theorem 2. We start by observing that if a regular game in strategi form has a j-defe t, then no element of the onvex hull of the full- ooperative payo� ve tors isindividually rational. Indeed, let Ij be the set of individually rational payo� ve tors for player j.Having a j-defe t means that U ∩Ij = ∅. Note that this is equivalent to co(U)∩Ij = ∅.18 Finally,using (14) it follows that co(f(Y )) ∩ RN

+ = ∅.Be ause of the above observation and 1Γ = Γ, ondition A of Lemma 6 holds for l = 1.The proof is omplete if we show that (⊕Γ)α has a full- ooperative multi-a tion Y and a Nashequilibrium N su h that Y is a Pareto improvement of N. Indeed, denoting the payo� fun tionsof (⊕Γ)α with g1, . . . , gN , g(N) is individually rational and therefore g(Y) too. Let n be a Nashequilibrium of 1Γ. By virtue of (2), 1Γ has a full- ooperative multi-a tion y. Be ause kΓ = πk(Γ),Tπk

(n) is a Nash equilibrium of kΓ and Tπk(y) is a full- ooperative multi-a tion of kΓ. Let

N := Ψ(

Tπ1(n)...

TπN(n)

), Y := Ψ(

Tπ1(y)...

TπN(y)

).By (3) and (4) we have that N is a Nash equilibrium of (⊕Γ)α and Y is a full- ooperative multi-a tion of (⊕Γ)α. Be ause 1Γ has a j-defe t, n is not full- ooperative; (4) implies that N is notfull- ooperative either. The payo�s in N are

gi(N) =

N∑

k=1

(fπk(i) ◦ Tπ−1

k

)(Tπk(n) =

N∑

k=1

fπk(i)(n) =

N∑

l=1

f l(n).So ea h player has the same payo�, say a, in N. In the same way one shows that ea h player hasthe same payo�, say b, in Y. The total payo� in N is Na and that in Y is Nb. Be ause N is notfull- ooperative it follows that Na < Nb, i.e. a < b whi h implies that Y is a unanimous Paretoimprovement of N. Q.E.D.18Here we use that for two subsets A and B of RN with Bc onvex: A ∩ B = ∅ ⇔ o(A) ∩ B = ∅.13

Proof of Theorem 3. We may assume that #PB(co(2U)) = 1. Next note that by (1)PB(co(kU)) = PB(kH) (k = 1, 2).So also #PB(2H) = 1. And be ause, using (1 and (12), PB(co(Uα)) = PB(co(1U) + co(2U)) ⊆

PB(co(1U)) + PB(co(2U)) ⊆ RN+ , also

PB(co(Uα)) = PB(Hα).Now we obtain, noting that feasible sets and negotiation sets are ompa t, using Lemma 3,PB(Hα) = PB(co(Uα)) = PB(co(1U) + co(2U)) =

PB(co(1U)) + PB(co(2U)) = PB(1H) + PB(2H) = PB(1H + 2H) = PB(Hag). Q.E.D.Proof of Theorem 4. First a lemma:Lemma 7 Suppose a ∈ PB(Hag). Thena ∈ EXP ⇔ N+

co(Uα)(a) = ∅. ⋄Proof.� ⇒. Let c ∈ PB(Hα) su h that c ≫ a. For all γ > 0 one has γ · (c − a) > 0. Be ausec ∈ co(Uα), it follows that γ 6∈ N+

co(Uα)(a).⇐. By Lemma 5 one has a 6∈ PBw(co(Uα)). Let c ∈ co(Uα) with c ≫ a. Sin e a ∈ RN

+ , alsoc ∈ RN

+ . This implies c ∈ Hα. Thus a ∈ EXP. Q.E.D.Now we prove Theorem 4. A ording to Lemma 7 the proof is omplete if we an prove thatN+

co(Uα)(a) 6= ∅ for all a ∈ PB(Hag).So suppose a ∈ PB(Hag) = PB(∑

k kH). By Lemma 5 one has N+P

k kH(a) 6= ∅. Be ause

a ∈∑

k kH , there exists ka ∈ kH(k ∈ M) su h that a =∑

k ka. With (8) one obtains∩kN+

kH(a) 6= ∅.By assumption PB(co(kU)) ⊆ RN+ for all k. Therefore PB(co(kU)) ⊆ RN

+ ∩ co(kU) = kH . So we an apply Lemma 2 with A = kH and B = co(kU) and getN+

co(kU)(ka) = N+kH(ka) (k ∈ M)and therefore

∩kN+co(kU)(a) 6= ∅.Applying again (8) one obtains N+

co(Uα)(a) 6= ∅. Q.E.D.Referen esJ. Benoît and V. Krishna. The folk theorems for repeated games: A synthesis. Te hni al report,New York University, Penn State University, 1996.B. Bernheim and M. Whinston. Multimarket onta t and ollusive behavior. Rand Journal ofE onomi s, 21(1):1�26, 1990.M. Botteon and C. Carraro. Strategies for environmental negotiations: Issue linkage with hetero-geneous ountries. In N. Hanley and H. Folmer, editors, Game Theory and the Environment.Edward Elgar, Cheltenham, 1998. 14

C. Carraro and D. Sinis al o. International environmental agreements: In entives and politi ale onomy. European E onomi Review, 42(3�5):561�572, 1999.M. Finus. Game Theory and International Environmental Cooperation. Edward Elgar, Cheltenham(UK), 2001. ISBN 1 84064 408 7.H. Folmer and P. von Mou he. Inter onne ted games and international environmental problems,II. Annals of Operations Resear h, 54:97�117, 1994.H. Folmer and P. von Mou he. Transboundary pollution and international ooperation. In T. Ti-etenberg and H. Folmer, editors, The International Yearbook of Environmental and Resour eE onomi s 2000/2001, pages 231�266. Edward Elgar, Cheltenham, 2000. ISBN 1 84064 337 4.H. Folmer, P. von Mou he, and S. Ragland. Inter onne ted games and international environmentalproblems. Environmental and Resour e E onomi s, 3:313�335, 1993.R. Just and S. Netanyahu. The importan e of stru ture in linking games. Agri ultural E onomi s,24:87�100, 2000.S. Ragland. International Environmental Externalities and Inter onne ted Games. PhD thesis,University of Colorado, 1995.G. Spagnolo. On interdependent supergames: Multimarket onta t, on avity and ollusion. Jour-nal of E onomi Theory 89/1, 89(1):127�139, 1999.

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