in Multilateral Negotiations*

Rodney D. Ludema

The recent expansion and proliferation of preferential trade agreements (PTAs) in North America, Europe and elsewhere has led to a renewed interest in the topic of preferential trade and its implications for the world trading system. In a departure from the traditional concern about the welfare implications of preferential trade per se, this interest has largely focused on questions of a "systemic" nature: how does the proliferation of PTAs affect the evolution of a world trading system that was founded upon the principle of nondiscrimination, as embodied in the General Agreement on Tariffs and Trade (GATT)? Some view PTAs as antithetical to the GATT, inevitably leading to a world of warring trade blocs, while others view them as supplemental, being just one more path by which global free trade can be approached. While the recent success of the Uruguay Round of multilateral trade negotiations is arguably evidence for the latter view, no one really knows how PTAs, whether actual, anticipated or threatened, affect the distribution of gains from such negotiations. This paper shows that the potential for PTAs substantially alters the outcome of a multilateral negotiation, and depending on the type of PTAs in question, may effectively reduce the outcome to one of independent bilateral deals.

To keep track of links, we define the variable xij Î {0, 1}, such that xij = 0 if countries i and j are linked, and xij = 1 if they are not. The vector x = (x12, x13, x23) denotes the pattern of links across countries. For any two patterns, x and x', let xx' = (x12.x'12, x13.x'13, x23.x'23). Now define x12 = (0,1,1), x13 = (1,0,1), x23 = (1,1,0). In words, xij denotes the pattern in which only countries i and j are linked. Likewise xijxik denotes the pattern in which country i is linked with countries j and k , but j and k are not linked with each other (i is the hub and j and k are spokes). Note that such a pattern is only feasible if the links are of the FTA variety, because otherwise it would violate the common external tariff condition. To distinguish CU links, therefore, we denote them xcij. Thus, the set of possible link patterns are, Xf º {1, x12, x13, x23, x12x13, x12x23, x13x23, 0} for FTAs, and Xc º {1, xc12, xc13, xc23, 0} for CUs, where 0 = x12x13x23 = xc12xc13xc23 or world-wide free trade.

Let the flow of utility to country i from pattern x be denoted wi(x) Î Â. The joint utility of i and j is denoted wij(x) º wi(x) + wj(x), and world utility, w(x) º w1(x) + w2(x) + w3(x). The changes in utility (surpluses) generated by a new link are denoted: vi(x, x') º wi(xx') - wi(x'), vij(x,x') º wij(xx') - wij(x'), and v(x, x') º w(xx') - w(x'), for any two patterns x, x' Î X. When countries i and j link they generate for themselves a joint surplus of vij(xij, x) which we shall assume to be positive, but they also affect the third country k. The surplus of k generated by the ij link we refer to as an "externality," which can be either positive or negative. Further, the surplus generated by the first CU will generally differ from that of the first FTA, because the two links may have different external tariffs. To capture such considerations, define cij º wij(xcij) - wij(xij), and ek º wk(xcij) - wk(xij).

In addition to forming links, countries can make international transfers of utility. Let yij Î Â be the net transfer from j to i, let y be a vector of net transfers, and let z = (x, y), which we refer to as the "state." We assume that net transfers sum to zero across countries. The flow of utility to country i inclusive of transfers is defined as ui(z) º wi(x) + yij + yik.

The payoff of each country is its expected present-discounted utility over an infinite, discrete time horizon. Letting Ez denote the expectation over all sequences of states z = {z(t)}  , the payoff of country i is given by:

Ui = Ez (1 - d)S dtui(z(t)) (1)

where d Î (0, 1) is the discount factor.

Each period begins with an existing state z(t) and one country i Î N being selected at random to make a proposal to the other two. The selection probability is 1/3 for each country each period. A period contains two stages. In the first stage, the proposer i offers a pair of net transfers y'i = (y'ij, y'ik) to which j and k respond simultaneously by either accepting or rejecting. If both respondents accept the offer, then all three countries link and j and k make transfers y'ij, and y'ik, respectively, to i for eternity. At this point the game essentially ends, as the agreement is assumed to be binding and gains from linking are exhausted. If either respondent rejects the offer, the game moves to the second stage, in which proposer i makes another offer y"i = (y"ij, y"ik), and the respondents accept or reject as before. This time any respondent j accepting the offer links with the proposer and makes transfer y"ij, and any respondent rejecting the offer remains unlinked and makes no new transfers to country i this period. Note that by making offers which would surely be rejected (say, by requiring infinitely large transfers to the proposer), the proposer can effectively make a bilateral offer to a single country or no offer at all. At the end of the second stage, the game moves into the next period with a new state, and the process repeats itself.

Finally, two conventions are adopted to facilitate the exposition. First, we assume that whenever a proposer or respondent is indifferent between linking and not, they the break tie in favor of the link. Second, in order to solve the model and make the value comparisons discussed earlier, we will need to examine the equilibria arising from the subgame beginning in each state. Let Ui(z) be the expected value (1) in a stationary equilibrium, beginning in any state z prior to the selection of a proposer. It is useful to divide this value into two components as follows: Ui(z) = ui(z) + Vi(z), referring to ui(z) as the "immediate value" and Vi(z) the "bargaining value" of state z to country i. This convention facilitates the analysis, but more importantly, it helps to distinguish the immediate effect of preferential agreements, about which most of the existing literature is concerned, from the bargaining effect, which the contribution of this paper.

Which if these options the proposer will exercise depends on the expected world surplus generated by each one, less what is received by the other countries. Formally, if i is the proposer, j and k the respondents, z the current state, and we define zij = (xijx, y"ij, yik, yjk), then the proposer's expected payoff from option A is ui(z) + Ai(z), where Ai(z) is given by,

Ai(z) = (1 - d)v(xijxik,x) + dv(0,x) - Rj(z) - Rk(z). (2)

Rj(z) = vj(xik,x) + dVj(zik), and Rk(z) = vk(xij,x) + dVk(zij).

The first two terms in Ai consist of the world surplus generated by moving to xijxikx for one period plus that from global free trade thereafter (as next period's multilateral agreement is expected to be efficient). The term Rj(x) represents the surplus to j from rejecting, given that i and k will link, which consists of the externality from an ik link plus the bargaining value of state zik beginning next period. The term Rk(x) is k's surplus from rejecting.

Under option B (making an offer acceptable to j alone) the proposer receives ui(z) + Bij(z), where,

Bij(z) = (1 - d)v(xij,x) + dv(0,x) - dVj(z) - Rk(z). (3)

The difference between Bij and Ai is that the intermediate pattern is now xijx, and j's expected surplus from rejecting is the discounted bargaining value of the current state. This is because, by rejecting, country j can freeze the state at z until next period. Finally, under option C, since the state does not change, i would receive its immediate payoff plus the bargaining value of the current state when bargaining resumes next period, or ui(z) + dVi(z).

Vj(z | i) =   (4)

Vi(z | i) = v(0,x) - Vj(z | i) - Vk(z | i), (5)

Vi(z) = (1/3)[Vi(z | 1) + Vi(z | 2) + Vi(z | 3)]. (6)

Our primary focus will be on equilibria in which, although the agreement is reached in the multilateral stage, if the bilateral stage were reached, the proposer would choose to make as many acceptable bilateral offers as are possible given the initial pattern. This is a particularly interesting case because it implies that the potential bilateral deals completely determine to payoffs to the multilateral agreement. A sufficient condition for this behavior is, Ai(z) ≥ Bij(z) ≥ dVi(z), for all i and j, which is equivalent to,

(1 - d)min[v(xik,xijx), v(xik,x)] + dVj(z) - Rj(z) ≥ 0 (7)

for all i, j, and k. By making offer A instead of offer B (to j), the proposer enables the surplus v(xik,xijx) to be realized this period rather than next, resulting in an efficiency gain of (1 - d)v(xik,xijx) which is entirely captured by the proposer. Similarly, by exercising option B (to k) instead of C, the proposer enables the surplus v(xik,x) to be realized this period rather than next, for an efficiency gain of (1 - d)v(xik,x). The cost to the proposer of choosing A over B (to j) or of choosing B (to k) over C is Rj(z) - dVj(z), which is the difference in what it must give to country j to secure acceptance.

 i(z) = (1/3)[v(0,x) + (Ri(z) - Rj(z)) + (Ri(z) - Rk(z))] (8)

In (8) each country expects a third of the world surplus from liberalization plus some terms that can be interpreted as measures of relative bargaining power. Bargaining power depends upon how well a country does, relative to the other countries, when excluded from a bilateral agreement. Specifically, if i does better when excluded from an agreement between j and k than j does when excluded from an agreement with i and k (i.e., Ri(z) > Rj(z)), then i has greater bargaining power than j, other things equal.

If (7) is satisfied for Vj(z) =  j(z), then (8) is an equilibrium bargaining value. If (7) is satisfied for all Vj(z) satisfying (5), (6) and the condition Vj(z | i) Î {Rj(z), dVj(z)}, then (8) is the unique equilibrium bargaining value. In some cases, we can establish uniqueness using the following lemma:

Lemma 1: For all i, j, k Î N, if (1 - d)(1/3)(3 - 2d)min[v(xik,xijx), v(xik,x)] + d j(z) - Rj(z) ≥ 0, and Rj(z) is unique, then  i(z) is the unique equilibrium bargaining value for country i.

The first condition of lemma 1 is just (7), evaluated at Vj(z) = (1/3)[v(0,x) - Ri(z) - Rk(z) + 2dVj(z)], which is j's bargaining value when Vj(z | i) = Vj(z | k) = dVj(z). The second condition, that Rj(z) is unique, is obviously necessary for to  i(z) to be unique. However, it also means that lemma 1 is of limited usefulness in states where Rj(z) = dVj(z) for some j, because in such states establishing the uniqueness of Rj(z) is equivalent establishing the uniqueness of Vj(z) itself. However, as the analysis to follow will demonstrate, states in which Rj(z) ≠ dVj(z) for all j are by far the most complex, and there we will be well-served by lemma 1.

Suppose there are six goods, denoted Qij for i, j = 1, 2, 3 and i ≠ j. denote the (dutiable) imports of country i from country j, let Pij be the price of Qij in country j, and let Tij denote the specific tariff levied on Qij by country i. In addition, let each country be endowed with an ample quantity of a freely-traded numeraire good. The pattern of trade is illustrated in figure 1. We assume that the supply of exports from j to i is governed by the linear supply schedule, Qij = Pij, and that i's demand for imports from j is given by, Dij = aij - (Pij + Tij). Setting Dij = Qij gives the equilibrium volume of trade Qij = (1/2)(aij - Tij).

By assuming away all cross-price and income effects, we have made the model highly tractable but rather uninteresting for the study of preferential arrangements. The problem is that, without such effects, tariff changes on trade flows between any two countries have no effect on the third country, i.e., there are no externalities. If Qij were a substitute (complement) in consumption for Qik, then an increase in Qij brought on by a reduction in Tij would have a negative (positive) welfare effect on suppliers of Qik in k. Likewise, if Qij were a substitute (complement) in production for Qkj, then an increase in Qij would have a negative (positive) effect on consumers of Qkj in k. In order to include these important effects without sacrificing the model's simplicity, we introduce externalities in a transparent but ad hoc way: let the welfare of country k (≠ i, j) derived from Qij be given by (m/5)Q where m is a parameter which may be positive or negative. Thus, writing welfare as the sum of consumer surplus, producer surplus, externalities and tariff revenue gives:

wi =  +  + TijQ + TikQ  (9)

Following Kennan and Reizman (1990), we assume that payoffs of the governments correspond to (9) and, as an external tariff rule, that each government sets external tariffs so as to maximize its welfare, taking as given the tariffs of the other countries. Maximizing (9) with respect to Tik gives an optimal tariff of T*ik = (1/3)aik. This is the tariff county i applies to imports from any country k to which it is not linked, unless i is a member of a CU.

A customs union is assumed to set a common tariff to maximize the joint welfare of its members. Maximizing the joint welfare of the ij CU with respect to the common tariff Tck on imports from country k gives an optimal tariff of,

T*ck =  

where m = 4m/(15 - 2m). This highlights two important properties of a CU's tariff: it is both common and coordinated across its members. That the external tariff must be common is a constraint on the CU. The effect of this constraint is most evident when m = 0, in which case T*ck is the simple average of T*ik and T*jk. Coordination becomes relevant when m ≠ 0, in which case T*ck differs from average of T*ik and T*jk by the factor m. The intuition is that, by coordinating their external tariff, i and j are able to internalize the externalities associated with their trade with k. The greater the externality the lower will be the incentive for the CU to restrict trade with k, and this will tend to lower T*ck .

Using these tariffs in (9) gives the result that any time two countries i and j form an FTA, they jointly gain by an amount, vij º (1/36)(aij2 + aji2), while k experiences a change in welfare equal to rk º mvij. Three things are worth noting here. First, by construction, these surpluses are independent of the combination of FTAs in existence at the time the FTA is formed. Thus all of the terms, vij(xij, 1), vij(xij, xjk), and vij(xij, xikxjk) are equal to vij, and all of the terms vk(xij, 1), vk(xij, xjk), and vk(xij, xikxjk) are equal to rk . Second, the gain to an FTA is increasing in the terms aij and aji, which are measures of the size of each member country's market for its partner's good. This accords with the conventional wisdom that a country's most valuable bilateral liberalization is with its largest trading partner. Third, the externality is proportional to the trade gains of the two liberalizing countries.

When two countries i and j form a CU, they jointly gain by an amount vij + cij, and country k gains by rk + ek, where

cij =  -  2

ek =  m(m + 4) (aik2 + ajk2) +  2

The first term in each expression represents the immediate effect of external tariff coordination. This is always non-negative for the CU, and positive for the outside country if coordination results in the CU lowering its external tariff relative to the average FTA tariff (i.e., if m > 0). The coordination effect is proportional to aik2 + ajk2, which can be interpreted as a measure of the PTA's monopoly power in trade (monopsony power is perhaps more accurate). The second term in each expression represents the asymmetry effect. This measures the immediate welfare loss imposed on a CU, relative to an FTA, by the constraint that the external tariff be common. This loss increases with the difference in the sizes of the CU members' markets for the third country's good. As long as T*ck > 0 (m < 1), the third country gains from asymmetry, because in averaging its external tariffs, the CU lowers the tariff on the larger of its markets for third-country goods.

Consider the subgame beginning in a state with pattern xikxjk, where country k is the hub, bilaterally linked to spoke countries i and j. The only link yet to be negotiated is between i and j, which means that, as respondent, either spoke can freeze the pattern at xikxjk, regardless of the proposal. This implies that, Ri(xikxjk) = dVi(xikxjk) and Rj(xikxjk) = dVj(xikxjk). For country k, Rk(xikxjk) = rk. Substituting these values into (8) gives,

 i(xikxjk) =  j(xikxjk) = fvij (10a)

 k(xikxjk) = rk + (1 - d)fvij (10b)

where f = 1/(3-d).

Substituting (10) into (7) gives the condition (1 + df)vij ≥ 0, which implies that (10) are always equilibrium bargaining values. It is also straightforward to show that this equilibrium is unique if m ≤ (3 - 2d)/2d. This condition is always satisfied low enough d, and for d near 1 it is satisfied if m ≤ 1/2. The reason uniqueness requires an upper bound on m is that when m is postive, spokes i and j would like to grab a share of the positive externality which their link confers on hub k . If they choose to link in the bilateral stage, they forfeit this externality. The opportunity cost of this forfiture is the discounted share of the externality they would expect receive by postponing their link to the next multilateral stage. In the equilibrium represented by (10), this opportunity cost is zero, and thus the spokes would always link in the bilateral stage instead of waiting for a discounted share of vij. An alternative equilibrium would be one in which the spokes always forgo a bilateral-stage link, in which case the equilibrium bargaining value for each country would be (1/3)(vij + rk). The opportunity cost in this equilibrium would then be (2d/3)rk. This must be compared to the spokes' joint benefit of linking in the bilateral stage of the current period instead of waiting for the next multilateral round, which is vij - (2d/3)vij. The result is that this alternative equilibrium can only exist if (2d/3)rk > vij - (2d/3)vij, or m > (3 - 2d)/2d. In what follows, we assume m ≤ 1/2, so that equilibrium on any hub-and-spoke subgame is unique.

In (10), the spoke countries expect a share f of their joint surplus, while the hub gets the externality as well as a share of the spokes' surplus that vanishes as d approaches unity. To appreciate the term f, it is helpful to think of the hub-and-spoke subgame as a peculiar sort of lottery. Suppose each country has a 1/3 chance of winning a dollar, but either spoke can veto the outcome and force a new drawing next period. In order to collect the dollar, therefore, the winner would have to pay the spoke(s) their discounted value of the lottery to prevent a veto. Defining value of the lottery to a spoke to be f, it must that f = (1/3)(1 - df) + (2/3)df, simplifying to f = 1/(3-d). The value to the hub is (1/3)(1 - 2df), which simplifies to (1 - d)f.

Next let us consider the subgame beginning with the initial pattern xjk, where countries j and k have the only FTA in place but continue to negotiate with country i for the formation of the final two links. Again consider the bilateral stage first. If country i is the proposer and chooses option A, then the pattern becomes 0. If j were to reject this offer, then k would link with i, the pattern would become xikxjk, and bargaining would continue for the final link in the next period. From (10), the value to j of rejecting A is Rj(xjk) = rj + dfvij, and likewise Rk(xjk) = rk + dfvik. When not proposing, country i may obstruct any link, so that Ri(xjk) = dVi(xjk). As before, substituting these values into (8) gives,

 i(xjk) = f(vij + vik) (11a)

 j(xjk) = rj + fvij + (1 - d)fvik (11b)

 k(xjk) = rk + fvik + (1 - d)fvij (11c)

Lemma 2: Equations (11) are the unique equilibrium bargaining values for x = xjk.

The country left out of the original FTA can expect a share f of each of the remaining surpluses. An FTA member receives f of the surplus from its link with the third country, plus a share (1-d)f of that between its original FTA partner and the third country. For d near zero, both f and (1-d)f are close to 1/3, while as d approaches unity, f approaches 1/2 and (1-d)f approaches zero. Intuitively, the lower is d, the greater is the power of the proposer, as the responding countries are willing to give up more today to avoid waiting a period to get what they expect in the future. As d nears zero, the proposer gets everything, and since each country is the proposer with probability 1/3, ex ante each country expects a third of the total surplus. As d approaches unity, the proposer's advantage diminishes and each FTA member gets half of the surplus created by its link with the third country; since an FTA member cannot impede the link between its original FTA partner and the third country, it gets none of the surplus from this link.

Consider negotiations between a CU, consisting of countries j and k, and the rest of the world, country i. Because the only alternative reachable trade state in Xc is 0, any country can prevent the state from changing, by rejecting an offer. Thus, Vi(z|j) = dVi(z), for all i and j, just as in the standard Rubinstein bargaining model. Using this in (5) and (6) and assuming v(0,xcjk) > 0, the solution is,

Vh(xcjk) = (1/3)v(0,xcjk) = (1/3)[vij + vik + rj + rk - (cjk + ei)] (12)

for all h Î N.

The expression (12) can be readily compared with (11), the case of the pre-existing FTA. Let the difference in joint payoff between a CU and an FTA consisting of countries j and k be Djk º Ujk(xcjk) - Ujk(xjk). Note that this measure includes the difference in immediate values. Thus,

Djk = [cjk + Vjk(xcjk)] -  jk(xjk)

The first term in (13) represents the extent to which, for a given surplus, a CU is a more effective bargaining coalition than an FTA. By precluding further bilateral deals between individual CU members and the rest of the world, a CU collectively extracts two thirds of the available world surplus, while an FTA collectively extracts (2 - d)f, which ranges from two thirds (as d ® 0) to one half (as d ® 1). We refer to this advantage as the "committment value" of a CU, and note that it depends positively on the volume of trade between the PTA and the rest of the world. There is, however, an important caveat: while a CU may have an advantage in bargaining over a given surplus, it does not necessarily bargain over the same surplus as does an FTA. This is reflected by the two other terms in (13).

The second term of (13) reflects that the common external tariff of a CU has the effect of removing cjk and ei from the available surplus, and hence from the bargaining table, prior to the multilateral negotiations. The CU gains by cjk immediately, but had cjk remained as part of the surplus, the CU would have received two thirds of it through bargaining anyway. Thus, the net gain is (1/3)cjk. The common external tariff also gives ei immediately to the outside country, only two thirds of which the CU would have been able to extract had ei remained as part of the surplus. Hence, ei produces a net loss to the CU of (2/3)ei. Of course, if ei is negative, as when m is negative and the PTA members are relatively symmetric, then the immediate effect of the common external tariff translates into an unambiguously positive bargaining effect for the CU. In other words, the long-run advantage of a CU is greater the greater is the short-run harm the common external tariff does to the rest of the world.

The intuition behind the third term in (13) is similar to that of the second but less direct. While the externalities rj and rk are, strictly speaking, part of the surplus regardless of the form of PTA, they are not shared equally among the countries when the PTA is an FTA. By rejecting a multilateral offer, an FTA member j knows that the other member k will link with i, and hence j's receipt of rj is inevitable. Thus it is as if rj were removed from the surplus and given to j prior to negotiations. Viewed in this way, the CU can be thought of as adding rj + rk to the negotiable surplus, from which it receives (2/3)(rj + rk). The net loss to a CU is therefore (1/3)(rj + rk), which can be positive (m > 0) or negative (m < 0).

We can take this comparison further by using the definitions of vij, vik, cjk, ei, rj and rk:

Djk =  

Several results from (14) are evident by inspection. The value of a CU relative to an FTA is higher the higher is the discount factor and the lower is the asymmetry between PTA members. The effect on Djk of an increase in the total surplus vij + vik depends on the sign of df - m. If df > m, which will occur if m is small or d is close to one, then the commitment value of the CU either dominates or works in the same direction as the externalities, and therefore any increase in total surplus benefits the CU.

The effect on Djk of the PTA's monopoly power aji2 + aki2 depends on the sign of the externality. If m > 0, the coordination effect of the common external tariff ultimately harms the CU relative to an FTA, because it induces the CU to lower its external tariff and benefits the outside country so that 2ei > cjk. If m < 0, the CU is relatively better off, because coordination results in an increase in the external tariff that harms the outside country. Finally, the sign of the derivative of Djk with respect to m itself is ambiguous.

An important feature of this model is that the two CU members continue to negotiate as two players (though the common external tariff restricts their choices). If instead we had treated the CU as being a single negotiator, it would have done worse than (13). This is because if the CU and third country each propose with probability 1/2, the expected payoff is an even split between the CU and third country. The third country gets, Vjk(xcjk) = (1/2)v(0,xcjk) and thus it clearly pays for a customs union not to act as a single country.

In this section we take up the more systemic question of the effect of allowing CUs and FTAs on the outcomes of multilateral trade negotiations. The "outside option principle" of Binmore, Shaked and Sutton [1989) states that an outside option will be irrelevant to the outcome of a bargaining game, unless it is better for at least one player than the outcome of a game with no outside options. While they established this for a single outside option of exogenous value, our concern is with multiple options of endogenous value, and therefore whether PTAs are relevant in this sense remains an open question. As before we shall consider FTAs first and then compare them with CUs, this time also with the benchmark,  i(1) = (1/3)v(0,1), which is the payoff each country would receive if PTAs were ruled out entirely.

Let the initial pattern be x = 1 and the set of available states be Xf. Each country is in a position similar to that of country i (the country with no initial links) in section B.1. Each proposer in the bilateral stage of a period can offer two acceptable bilateral deals, one such deal or none. This gives quite a number of option permutations to consider. Fortunately, lemma 2 guarantees that Ri(1) = ri + df(vij + vik) for all i, and thus lemma 1 can be used. Using Ri(1) in (8) gives,

 i(1) = ri + f(vij + vik)+ (1 - d)fvjk (15)

for all i. The sufficient condition for uniqueness in lemma 1 reduces to (1/2)[(1 + d2f2)/df] ≥ m, which is guaranteed by m ≤ 5/4.

Expression (15) establishes the relevance of FTAs as outside options for the multilateral negotiations. Note that as d approaches unity, expression (15) becomes  i(1) = (1/2)(vij + vik). Thus each bilateral surplus is spilt evenly between each pair of countries. This is same payoff that would result from three independent bilateral negotiations.

Ujk(xjk) = rj + rk + vjk + (2 - d)f(vij + vik), (16)

Ujk(1) = rj + rk + 2fvjk+ (2 - d)f(vij + vik). (17)

The difference between (16) and (17) is (1-d)fvjk, which is negligible for high d. Thus countries receive almost the same equilibrium payoff whether they are members of an FTA or not. The only reason countries will sign bilateral FTAs is to salvage the short-term (one-period) gains from those links if the multilateral talks fail, and this determines the distribution of gains in the multilateral agreement.

Now consider the case in which there are no pre-existing agreements and the set of possible links is Xc. The game is the same as in the FTA case, except that in the bilateral stage, if i and j both accept bilateral offers from k, then they must link not only with k (and pay y"k) but also with each other. If only j accepts a bilateral offer from k, then the rejecting country i receives, Rci(1) = ri + ei + (d/3)v(0,xcjk), which is the gain from having j and k link plus the discounted expected value of the agreement to be reached with the jk CU next period. The relationship between this and the Ri(1) in the FTA case is:

Rci(1) = Ri(1) + (1 - d)ei - dDjk (18)

 ci(1) =  i(1) + (1/3)[(1 - d)(2ei - ej - ek) - d(2Djk - Dij - Dik)] (19)

To guarantee the uniqueness of  ci(z), lemma 1 must be modified as follows:

Lemma 3: For all i, j, k Î N, if (1 - d)(1/3)(3 - 2d)min[v(0,xcij), v(xcik,1)] + d cj(z) - Rcj(z) ≥ 0, then  ci(z) is the unique equilibrium bargaining value for country i.

For an arbitrary discount factor, determining whether or not the condition for lemma 3 is satisfied is not an easy, or particularly worthwhile, task. However, as d approaches one, it becomes straightforward:

Corollary: For sufficiently high d, Vci(z) =  ci(z), if and only if, D12 + D13 + D23 ≥ 0.

That is, all countries will offer CUs to each other in the bilateral stage, if and only if CUs are better than FTAs on average. The reason is that for high d the efficiency differences between alternative proposals become negligible, so proposer i will choose option A if and only if  cj(z) ≥ Rcj(z) and  ck(z) ≥ Rck(z). From (18) and (19),  cj(z) - Rcj(z) approaches (1/3)SijÎNDij as d approaches one.

 ci(1) = (2/3) i(1) + (1/3) i (1) + (1/9){(cij - cjk) + (cik - cjk) - 2[(ej - ei) + (ek - ei)]} (20)

The first two terms in on the right-hand side of (20) make up a weighted average of  i(1) and  i(1). Thus, looking only at these terms, the CU regime is ex ante more egalitarian than the FTA regime, tending to draw the countries towards the mean and closer to regime without PTAs. It also implies that countries responsible for a disproportionate amount of the world surplus would ex ante prefer an FTA regime to a CU regime.

What happens when SijÎNDij is negative? As d approaches one, the only equilibria that survive are those in which at least one country would not offer a CU to at least one other country in the bilateral stage of negotiations. Three different types of such equilibria are possible. The first type is where one country, say 1, exercises option A, while 2 and 3 either offer a CU exclusively to 1 or offer no bilateral deal at all. The limit values in this case are, V1(1) = Rc1(1) + SijÎNDij, V2(1) = Rc2(1), and V3(1) = Rc3(1). The second type of equilibrium is where two countries, say 1 and 2, make exclusive bilateral offers to each other, while 3 makes no offer at all. The limit values in this case are, V1(1) = Rc1(1) + (1/2)SijÎNDij, V2(1) = Rc2(1) + (1/2)SijÎNDij, and V3(1) = Rc3(1). The third possible equilibrium is that in which no country links in the bilateral stage at all. In this equilibrium, the bargaining values are just  i(1) for all i. These values are worked out in the appendix. It is not possible to rule out any of these equilibria without further assumptions.

Perhaps the most natural next step in the analysis is to put FTAs and CUs together and consider a game in which the set of available patterns is XfÈXc. Only two things can said about this game that follow directly from the analysis above: as d approaches one, if a CU is better for each pair than an FTA (Dij > 0 for all i and j), then the unique equilibrium bargaining values are  ci(1); if an FTA is better for each pair than and a CU (Dij < 0 for all i and j), then the unique equilibrium bargaining values are i(1). The more interesting, though exceedingly more complex, cases are those in which some pairs prefer FTAs and others prefer CUs. Few general conclusions can be stated about such cases; however, it is possible to construct examples that convey some meaning.

Suppose m = 0 and d is near one, so that the two factors determining the relative advantage of a CU are the committment value and asymmetry. Let the demand parameters be as follows: a12 = a13 = a, and a21 = a23 = a31 = a32 = ab, where b Î [0, 1]. Country 1 is can be thought of as the "big" country with demand for imports from 2 and 3 parameterized by a. Countries 2 and 3 are symmetric in their demands from imports from 1 as well as from each other. These are parameterized by ab, where b measures the size 2 and 3 relative to 1. Choosing a so that v(0, 1) = 1, the surpluses become,

v12 = v13 =  , v23 =  , (21)

c12 = c13 =  , c23 = 0, (22)

e3 = e2 =  , e1 = 0 (23)

D12 = D13 =  -  , D23 =  , (24)

It is clear from (24) that the PTA consisting of 2 and 3 is always better off as a CU, while the other two pairs may be better-off as FTAs, if b is low enough. Let  denote critical value of b, such that for b <  , D12 < 0. It can be shown that  = [(13 - 4(13 - 3df)1/2]/(13 - 12df), which for d = 1 is equal to .487.

Making use of the symmetry between 2 and 3, we make another dimensional simplification by restricting attention to symmetric equilibria, which we take to mean V2(1) = V3(1). This allows us to define q º V2(1)/V1(1), as the bargaining value of 2 (or 3) as a share of the bargaining value of 1. With v(0, 1) = 1, this implies V1(1) = 1/(1 + 2q).

The game begins with x = 1. In the subgame of the bilateral-stage, each proposer now has essentially five options: Ac) propose CUs to both respondents; A) propose FTAs to both respondents; Bc) propose a CU to one respondent; B) propose an FTA to one respondent; and C) make no acceptable offer. Let the value of the best such option for proposer i be yi = max{Aci, Ai, Bcij, Bij, Bcik, Bik, dVi}. It is possible to compute yi for any pair (q, b). Doing so enables us to partition the space of (q, b) into six sets:

MI = {q, b | y1 = A1, y2 = Bc23}, MII = {q, b | y1 = A1, y2 = Ac2}

MIII = {q, b | y1 = Ac1, y2 = Ac2}, MIV = {q, b | y1 = Ac1, y2 = Bc21}

MV = {q, b | y1 = A1, y2 = B21} and MVI = {q, b | y1 = dV1, y2 = Bc23}

These sets are illustrated in Figure 2, as regions circumscribed by gray lines. Each of these regions has an intuitive explanation. Except for region VI, country 1's proposal strategy is determined by the sign of D12. To the left of  , country 1 proposes FTAs and to the right, CUs. In region VI, the value of q, and hence V2, is so low that 1 finds it optimal to give the respondents dV2 instead of R2. Likewise, for low q, countries 2 and 3 prefer to give each other dV2 instead of R2, which they can do by each offering an exclusive bilateral CU to the other. For high q as in regions IV and V, countries 2 and 3 prefer to give country 1 dV1 instead of R1, which they can do by each offering 1 an exclusive bilateral deal. Whether that deal should be a CU or an FTA depends on the sign of D12, which is what distinguishes IV from V. Regions I, II and III feature less extreme values of q, and so the primary determinant of y2 is b. Whenever 2 and 3 propose a bilateral deal to each other, they propose a CU. The main issue, then, is whether or not to also propose a CU to 1. For b not too small, a CU with 1 is not very costly, and so 1 is proposed a CU in regions II and III. For b very small it is costly and thus 2 and 3 would rather leave 1 out, and this determines region I.

The next step in finding equilibria is to observe that for each pair (y1, y2) we can use equations (4), (5) and (6) to work out the corresponding bargaining values, as functions of b. This allows us to derive six functions ql(b), l = I, II, .. VI. These are illustrated in figure 2 as thin solid lines. For the final step, we determine equilibrium bargaining values by matching up the functions with the sets. More accurately, ql(b) is an equilibrium payoff ratio, if and only if (ql(b), b) Î Ml. The points satisfying this condition are represented in figure 2 by thick solid lines. The equilibria lie in regions I, II, and III.

For b >  , CUs are better than FTAs for each pair, and thus Vi(1) =  ci(1). For b <  , country 1 proposes FTAs to both 2 and 3 in the bilateral stage, while countries 2 and 3 either propose CUs to both respondents or propose an exclusive 23 CU. By virtue of the symmetry of the example, both of the proposal strategies of 2 and 3 yield the same payoffs for each country. Those payoffs are:

V1(1) = (1/3)[1 + 2(Rc1(1) - R2(1))] (25a)

V2(1) = (1/3)[1 + R2(1) - Rc1(1)] (25b)

We are now in a position to compare this example with the results of the two previous sections, in which we restricted X to being either Xc or Xf but not both. One technical issue that must be dealt with before making this comparison is the determination of equilibria in the CU regime when SijÎNDij < 0 (this occurs when b is less than about .373). It turns out that the only equilibrium one can rule out is that of the third type mentioned above, the one in which no country links in the bilateral stage. The first type of equilibrium exists for .232 ≤ b ≤ .373 and produces a payoff ratio equal to qIV(b) from figure 2. The second type of equilibrium exists for all b ≤ .373 and the corresponding payoff ratio is qVI(b) from figure 2. The proof of this is found in the appendix.

Piecing together the equilibria across the domain of b for the four different regimes produces figure 3. The payoff ratio in the FTA regime is  2(1)/ 1(1), from (19), and is depicted by the gray line, labeled Xf. The CU regime consists of qIV(b) and qVI(b) for low b and followed by qIII(b) and is represented by thin solid lines, labeled Xc. The open regime is represented by thick solid lines, labeled XfÈXc. Finally, the payoff ratio in a regime of no PTAs (the "closed" regime) is simply  2(1)/ 1(1) = 1. It is clear from figure 3 how the different countries would rank these regimes. The big country always prefers the FTA regime, while the small symmetric countries prefer the CU regime for small b and the closed regime for high b.

An interesting feature of this example is that there is no general presumption about which country would prefer the open regime to the closed one. For b >  , V1(1) = (1/3)[1 - 2(Rc1(1) - Rc2(1))] in the open regime and therefore the big country gains from the open regime whenever Rc1(1) > Rc2(1). For b <  , (25) implies that the big country gains from the open regime whenever Rc1(1) > R2(1). The lower is b is larger is the share of the world surplus due to trade with country 1, and thus the less the big country is harmed by being left out of the 23 CU (i.e., the higher is Rc1(1)). On the other hand, as b falls, it also reduces the common external tariff of any CU of which country 1 is a member, and this works to the advantage of 2 and 3 (i.e., Rc2(1) rises). As b falls, Rc2(1) rises faster than Rc1(1), working to the advantage of 2 and 3, until b falls below  , at which point country 1 prefers FTAs and the relevant rejection value for 2 and 3 becomes R2(1). For b <  , smaller b always benefits the big country.

While noncooperative bargaining models have the advantage over cooperative solution concepts of explicitly representing the bargaining environment and the behavior of its agents, noncooperative models can be unwieldy. Hence, it is sometimes desirable to use a cooperative solution concept instead, provided the two approaches can be shown to be mutually consistent (see Sutton, 1986, for discussion). The problem is that most N-player cooperative solution concepts are ill-suited for direct application to the present context, because they rely on two components of the underlying game which do not fit our model: coalition structures (partitions of the players into disjoint coalitions) and characteristic functions (which assign to each coalition a value independent of the actions of its compliment). Because FTA links are not transitive, the set Xf cannot be adequately represented by a set of partitions (see Aumann and Myerson, 1988, Myerson, 1991). Moreover, characteristic functions do not allow for externalities.

(P1) (1 - d)(1/3)(3 - 2d)min[v(xik,xijx), v(xik,x)] + d j(z) - Rj(z) ≥ 0 for all i, j, k Î N, i ≠ j ≠ k,

The only combination satisfying P1 is that in which each player chooses A. Since this combination produces bargaining value  i(z) for all i, it follows that  i(z) is unique if Rj(z) is unique for all j.

1a) Countries 2 and 3 also choose A. A sufficient condition for this to be an equilibrium is (7) evaluated at Vj(z) =  j(z) , which is consistent with P1, because (1/3)(3 - 2d) < 1.

1b) Countries 2 and 3 make exclusive bilateral offers to each other. For this to be an equilibrium it must be that B23 > A2, and B32 > A3. Summing these conditions gives,

(P2) (1 - d)[v(x13,x23x) + v(x12,x23x)] + dV2(z) + dV3(z) - R2(z) - R3(z) < 0

Applying B23 > A2, and B32 > A3 to equations (4), (5) and (6) gives V1(z) =  1(z), which implies

V2(z) + V3(z) =  2(z) +  3(z). Substituting  2(z) +  3(z) into P2 reveals that P2 voliates P1.

For every other combination of strategies by countries 2 and 3 in case 1, it must be that, V1(z) Î { 1(z), f[v(0,x) + R1(z) - R2(z) - R3(z)], [1/(3-2d)][v(0,x) - R2(z) - R3(z)]} and either,

(P3) (1 - d)v(x23,x1jx)+ dV1(z) - R1(z) < 0, for j = 2, 3, or

(P4) (1 - d)v(x23,x) + dV1(z) - R1(z) < 0.

Evidently, P3 and P4 violate P1, when V1(z) =  1(z). When V1(z) = f[v(0,x) + R1(z) - R2(z) - R3(z)], the term dV1(z) - R1(z) in P3 and P4 becomes 3f[d 1(z) - R1(z)], and when V1(z) =[1/(3-2d)][v(0,x) - R2(z) - R3(z)], the term dV1(z) - R1(z) becomes [3/(3-2d)][d 1(z) - R1(z)]. Thus, P3 and P4 violate P1 for all possible V1(z) in this case.

2a) Countries 2 and 3 make exclusive bilateral offers to 1. This case requires, among other things, that B12 > A1, and B21 > A2. Solving equations (4), (5) and (6) for this case gives,

V1(z) = [1/(3-2d][v(0,x) - R3(z) - dV2(z)]. Thus, the condition B21 > A2, becomes

(1 - d)v(x23,x12x) + d[1/(3 - 2d)][v(0,x) - R3(z) - dV2(z)] - R1(z) < 0, or

(3 - 2d)(1 - d)v(x23,x12x) + d[v(0,x) - R2(z) - R3(z)] - (3 - 2d)R1(z) - d[dV2(z) - R2(z)] < 0, or

(P5) (3 - 2d)(1/3)(1 - d)v(x23,x12x) +  1(z) - R1(z) - (d/3)[dV2(z) - R2(z)] < 0

The sum of the first three terms in P5 is positive by P1. So for P5 to hold it must be that dV2(z) - R2(z) > 0, which contradicts the condition B12 > A1.

2b) One player, say country 2, makes an exclusive bilateral-stage offer to 1, and country 3 makes no acceptable offer. Like case 2a this requires B12 > A1, and B21 > A2. Summing these conditions gives,

(P6) (1 - d)[v(x23,x12x) + v(x13,x12x)] + dV1(z) + dV2(z) - R1(z) - R2(z) < 0

Solving equations (4), (5) and (6) for this case gives, V1(z) = V2(z) = f[v(0,x) - R3(z)]. This enables us to re-write P6 as,

(3 - d)(1/3)(1 - d)[v(x23,x12x) + v(x13,x12x)] + d 1(z) - R1(z) + d 2(z) - R2(z) < 0

2c) Country 2 makes an exclusive bilateral offer to 3, and country 3 makes an exclusive bilateral offer to 1. This requires that B12 > A1, B23 > A2 and B31 > A3. Summing these conditions together gives,

(P7) (1 - d)[v(x13,x12x) + v(x12,x23x) + v(x23,x13x)] + dSiÎNVi(z) - SiÎNRi(z) < 0

As all equilibria are efficient, SiÎNVi(z) = SiÎN i(z). This implies P7 violates P1.

For every other combination of strategies by countries 2 and 3 in case 2, it must be that Bij ≥ dVi and Bji < dVj for at least one pair of countries i and j. From (3) it follows that Bij - dVi = Bji - dVj. Thus, Bij ≥ dVi and Bji < dVj cannot simultaneously hold.

Case 3: No country makes an acceptable offer. This requires that dVi > Bij, for all i, j in N, which implies,

(P8) (1 - d)[v(x12,x) + v(x13,x) + v(x23,x)] + dSiÎNVi(z) - SiÎNRi(z) < 0

It follows from (7), that if the following equations are satisfied for all Vj(z) satisfying (5), (6) and the condition Vj(z | i) Î {Rj(z), dVj(z)}, then (11) is the unique equilibrium bargaining value:

(P9) (1 - d)(vik+ rj)+ dVj(z) - rj - dfvij ≥ 0

(P10) (1 - d)(vij+ rk) + dVk(z) - rk - dfvik ≥ 0

P9 holds iff Ai ≥ Bij, Bik ≥ dVi, and Bki ≥ dVk. P10 holds iff Ai ≥ Bik, Bij ≥ dVi, and Bji ≥ dVj. Thus there are only three possible equilibrium strategy combinations to consider.

Case 1: Both P9 and P10 hold. In this case, i chooses Ai, k chooses Bki, and j chooses Bji. The bargaining values in this equilibrium are found in equations (11). Subsituting (11) into P9 and P10 gives,

(P11) (1 - d)(vik+ rj)+ d[rj + fvij + (1 - d)fvik] - rj - dfvij ≥ 0

(P12) (1 - d)(vij+ rk) + d[rk + fvik + (1 - d)fvij] - rk - dfvik ≥ 0

These reduce to (1 - d)(1 + dfvik) ≥ 0 and (1 - d)(1 + dfvij) ≥ 0, respectively.

Case 2: P9 holds, P10 does not. In this case, i chooses Bik, k chooses Bki, and j chooses dVj, which implies: Vj(z | i) = Vj(z | k) = Rj(z), and Vk(z | i) = Vk(z | j) = dVk(z). Using this information in equation (6) and solving gives: Vi(z) = Vk(z) = f[v(0,xjk) - Rj],Vj(z) = f[(1-d)v(0,xjk) + 2Rj]. Now if P10 fails, it must be that,

(P13) (1 - d)(vij + rk) + df[(vik + vij + rj + rk) - (rj + dfvij)] - rk - dfvik < 0

Case 3: P1, P2 do not hold. In this case, i, j and k choose dVi, dVj, dVk, repectively, which implies: Vj(z | i) = Vj(z | k) = dVj(z), and Vk(z | i) = Vk(z | j) = dVk(z). Using this information in equation (6) and solving gives: Vi(z) = Vj(z) =Vk(z) = (1/3)v(0,x). For this to be an equilbrium requires:

(P14) (1 - d)(vik + rj + vij + rk)+ (2d/3)(vik + vij + rj + rk) - rj - rk - dfvij - dfvik < 0

Simply redo the proof of lemma 1, replacing v(xik,xijx) with v(0,xcij) and v(xik,x) with v(xcik,1).

(P15) (1 - d)(1/3)(3 - 2d)min[v(0,xcij), v(xcik,1)] + d i(1) - Ri(1) +

+ (d/3)[(1 - d)(2ei - ej - ek) - d(2Djk - Dij - Dik)] - [(1 - d)ei - dDjk] ≥ 0

From (8) and (15), we know that  i(1) - Ri(1) ® 0 as d ® 1, thus P15 ® (1/3)[Dij + Dik - 2Djk] + Djk ≥ 0, or Dij + Dik + Djk ≥ 0. QED

As d ® 1, Vi(1| j) ® min[Vi(1), Rci(1)] = Rci(1) + pi, where pi = min[Vi(1) -Rci(1), 0] ≤ 0. Thus,

Vi(1) = (1/3)[v(0,1) - Rcj(1) - Rck(1) + Rci(1)] + (1/3)(2pi - pj - pk), and

(P16) Vi(1) - Rci(1) = (1/3)SijÎNDij + (1/3)(2pi - pj - pk)

Case 0: pi = pj = pk = 0. This corresponds to the case in which each country makes acceptable bilateral-stage offers to both other countries. This case was ruled out by the corollary above.

Case 1: p1 < 0, p2 = p3 = 0. In this case, V1(1| j) = V1(1) for j = 2,3 and Vj(1| k) = Rcj(1) for all k, which means 2 and 3 must offer a CU exclusively to 1 or offer no bilateral deal at all, while 1 makes acceptable bilateral-stage offers to both 2 and 3. For this equilibrium to exist, it is necessary that, V2(1) - Rc2(1) = (1/3)SijÎNDij - (1/3)p1 ≥ 0 or SijÎNDij ≥ p1, and that p1 = (1/3)SijÎNDij + (2/3)p1 or p1 = SijÎNDij. From this it follows that: V1(1) = Rc1 + SijÎNDij, and Vj(1) = Rcj, for j = 2, 3.

Case 2: p1 < 0, p2 < 0, p3 = 0. In this case, V1(1| j) = V1(1), V2(1| j) = V2(1) and V3(1| j) = Rc3(1) for all j, which means that 1 and 2 make exclusive bilateral offers to each other, while 3 makes no offer at all. For this equilibrium to exist, it is necessary that, V3(1) - Rc3(1) = (1/3)SijÎNDij - (1/3)(p1 + p2) ≥ 0 or SijÎNDij ≥ p1 + p2, that p2 = (1/3)SijÎNDij + (2/3)p2 - (1/3)p1, and that p1 = (1/3)SijÎNDij + (2/3)p1 - (1/3)p2. Thus, p1 = p2 = (1/2)SijÎNDij. From this it follows that: V1(1) = Rc1 + (1/2)SijÎNDij, V2(1) = Rc2 + (1/2)SijÎNDij V3(1) = Rc3.

Case 3: p1 < 0, p2 < 0, p3 < 0. In this case, V1(1| j) = V1(1), V2(1| j) = V2(1) and V3(1| j) = V2(1) for all j, which means no country links in the bilateral stage. For this equilibrium to exist, it is necessary that

pi = (1/3)SijÎNDij + (1/3)(2pi - pj - pk) for all i, j, k, or pi = (1/3)SijÎNDij for all i. This implies Vi(1) =  i(1) for all i.

The above discussion establishes the properties of those equilibria that are not necessarily ruled out as d ® 1. It does not imply that they exist for any d < 1. Next, we establish the existence of equilibria in the symmetric example of section III.C.3.

For an arbitrary discount factor, the payoffs in case 1 are: V1(1) = [1/(3-2d)](1 - 2Rc2(1)) and V2(1) = [1/(3-2d)](1 - d + Rc2(1)). For this to be an equilibrium it must be that A1 ≥ max[B12, dV1], and that either B21 or dV2 > max[A2,B23]. Begin by showing A1 - B12 ≥ 0. From (2) and (3) this condition is:

A1 - B12 = (1 - d)v(0,xc12) + dV2 - Rc2 ≥ 0

= (1 - d)v(0,xc12) + d[1/(3-2d)](1 - d + Rc2) - Rc2 ≥ 0

= (3 - 2d)v(0,xc12) + d - 3Rc2 ≥ 0

Substituting Rc2 = e2 + (d/3)v(0,xc12) and taking the limit as d ® 1, gives the condition: 1 - 3e2 ≥ 0. Using the definition of e2 from (23) the condition becomes, b ≥ .232. It is straightforward to establish the conditions under which A1 - dV1 ≥ 0. Using A1 = 1 - 2Rc2, A1 - dV1 = [1 - d/(3-2d)](1 - 2Rc2). This is positive whenever 1 - 2Rc2 > 0, which is true for b > .194. Finally, we show that B21 and dV2 > max[A2,B23]. Note that as d ® 1, both B21 and dV2 ® Rc2, while A2 = 1 - Rc1 - Rc2 and B23 ® 1 - Rc1 - Rc2. So the limiting condition is Rc2 > 1 - Rc1 - Rc2, which is equivalent to SijÎNDij < 0. Thus, case 1 equilibria exist for all b such that b ≥ .232 and SijÎNDij < 0.

For an arbitrary discount factor, the payoffs in case 2 are: V1(1) =f(1 - d + 2Rc1(1)) and V2(1) = f(1 - Rc1(1)). For this to be an equilibrium it must be that dV1 > max[A1, B12], and B23 > max[A2,B21,dV2]. As d ® 1, dV1 ® Rc1, while A1 = 1 - 2Rc2 and B12 ® 1 - 2Rc2. Thus, the limiting condition for dV1 > max[A1, B12] is Rc1 > 1 - 2Rc2, which is equivalent to SijÎNDij < 0. Next we show B23 > max[A2,B21,dV2]. We have already shown that dV1 > B12, which implies that dV2 > B21, so that we only need to show, B23 > max[A2, dV2]. As d ® 1, B23 ® Rc2, while A2 = 1 - Rc1 - Rc2. Thus, in the limit B23 > A2 is equilvlent to Rc2 > 1 - Rc1 - Rc2, or SijÎNDij < 0. Next, consider the condition B23 > dV2. Written out this is:

(1 - d)v23 + d - dV2 - Rc1 > dV2

(1 - d)v23 + d > 2df(1 - Rc1) + Rc1

(1 - d)v23 + (1 - d)df > (1 - d)3fRc1

(3 - d)v23 + d > 3(d/3)(1 - v23)

(3 - d)v23 > - dv23

This is true for any d. Thus, case 2 equilibria exist for all b such that SijÎNDij < 0.

Finally, for an arbitrary discount factor, the payoffs in case 3 are: Vi(1) = 1/3 for all i. For this to be an equilibrium it must be, among other things, that dV1 > Bc23. This condition is, dV1 > (1 - d)v23 + d - dV2 - Rc1. Using Vi = 1/3 and Rc1 = d/3(1 - v23) gives,

(d/3)(1 - v23) > (1 - d)v23 + (d/3)

- (d/3)v23 > (1 - d)v23

This is false for for any d. Thus, case 3 equilibria do not exist.