Gradualism and uncertainty in international union formation: The European Community’s first enlargement

Abstract

This paper introduces a new theoretical framework of international unions qua coalitions of countries adopting a common policy and common supranational institutions. I make use of a three-country spatial bargaining game of coalition formation, in order to examine the endogenous strategic considerations in the creation and enlargement of international unions. Why would we observe a gradualist approach in the formation of the grand coalition even if the latter is assumed to be weakly efficient? I propose asymmetric information about the benefits of integration as a mechanism that can generate gradual union formation in equilibrium. As it turns out, it may well be in the ‘core’ countries’ interest to delay the accession of a third, ‘peripheral’ country in order to (1) stack the institutional make-up of the initial union in their favor and (2) signal their high resolve to wait out the expansion of their bilateral subunion. A related case from the European experience provides an interesting illustration.

Appendix

Proposition 1

For any δ ∈ [0,1] and y _c  ≥ 0, c ∈ 2 ^N \∅ subject to assumptions 1, 2, and 3, there can be no gradual coalition-formation path in any subgame-perfect Nash equilibrium of the baseline game with complete information.

Proof

To characterize the subgame-perfect Nash equilibria of this bargaining game, we first need to define the players’ proposal and acceptance strategies, such that they satisfy sequential rationality in every subgame. Following a common policy proposal $x_t^\iota$ by proposer ι ∈ {A,B,C} at time t = 0, 1, then players j,k ≠ ι need to simultaneously decide whether to accept or reject. Depending on the proposal and the underlining parameters of the model, accepting may be a (weakly) dominant strategy or just a best response. In a subgame-perfect Nash equilibrium, the acceptance strategies are characterized by conditional and unconditional acceptance sets defined for ι ≠ j ≠ k as follows:

$${\text{Conditional acceptance sets}}:\quad \quad CA_t^j \left( {\pi _t ,y_{ij} ,y_{jk} } \right) = \left\{ {x_t^\iota :V_t^j \left( {\alpha _t^j \left( {x_t^\iota } \right) = 1} \right) \geqslant V_t^j \left( {\alpha _t^j \left( {x_t^\iota } \right) = 0} \right)\left| {\alpha _t^k \left( {x_t^\iota } \right)} \right. = 1} \right\}$$

$${\text{Unconditional acceptance sets}}:\quad UA_t^j \left( {\pi _t ,y_{ij} ,y_{jk} } \right) = \left\{ {x_t^\iota :V_t^j \left( {\alpha _t^j \left( {x_t^\iota } \right) = 1} \right) \geqslant V_t^j \left( {\alpha _t^j \left( {x_t^\iota } \right) = 0} \right),\forall \alpha _t^k \left( {x_t^\iota } \right)} \right\}$$

In light of our assumption of weak superadditivity, it quite obvious that $UA_t^j \subseteq CA_t^j$, ∀t, π _t , j, k ≠ ι, since the marginal contribution of any additional member to the orthogonal worth of a coalition can only be positive. Moreover, given the unidimensionality of the model and the linearity of the utility functions, these acceptance sets are convex and compact.

The simultaneity of the acceptance game gives rise to a multiplicity of equilibria in subgames with common policy proposals $x_t^\iota \in \left( {CA_t^j \cap CA_t^k } \right)\backslash \left( {UA_t^j \cup UA_t^k } \right)$, ι ≠ j, k, i.e., such that pure acceptance is only a best response but not a dominant strategy for either non-proposing player j or k. This implies that an action profile of (no, no) would also constitute a pure Nash equilibrium in such a proposal subgame, albeit Pareto (weakly) dominated by the (yes, yes) equilibrium profile. The indeterminacy of such coordination voting games generates a continuum of subgame-perfect Nash equilibria, whereby the proposer’s optimal proposal is conditioned by the responders’ mutually reinforced expectations of equilibrium play in any proposal subgame with two distinct pure strategy Nash equilibria. This gives rise to the possibility of credible rejection threats by the responders for such proposed common policies that make them both better off if and only if the grand coalition ABC forms.

Now turning to optimal proposal strategies, subgame-perfection and the underlying coalition value parameters of the game, i.e., y _c  ≥ 0, ∀c ∈ 2^N\∅, would imply that in equilibrium no autarchic player would make a common policy proposal that would never be acceptable to either of his/her coalition partners. There always exists a potential non-singleton coalition which any country i prefers to the autarchic state. However, the content of that proposal can determine the size of the proposer’s optimally preferred coalition at time t and for given coalition structure π _t .

Gradual coalition-formation in this model occurs whenever a player joins an existing coalition in the second period of bargaining or any new coalition forms after a first-period bargaining impasse. To show that gradualism may not arise in any subgame-perfect Nash equilibrium of the complete-information workhorse model, I examine each possible coalition-formation-path and then use proof by contradiction:

• 1. Let π _{t =0} = {A|B|C}, i.e., no coalition has formed after the first round of bargaining for some δ ∈ [0,1] and some first-period proposer ι _{t =0} ∈ {A,B,C}. Then in any subgame-perfect equilibrium, for given conjectures about play off the equilibrium path, proposer ι _t=1 will propose some common policy to players j, k ≠ ι such that

  $$x_1^{\iota * } = \mathop {\arg \max }\limits_{x_t^\iota } \left\{ {y_{j:\alpha _1^{j * } \left( {x_1^\iota } \right) = 1} - \left| {x_1^\iota - m_\iota } \right|\left| {x_1^\iota \in \left( {CA_1^{j * } \cap CA_1^{k * } } \right)} \right. \cup \left( {UA_1^{j * } \cup UA_1^{k * } } \right)} \right\},j \ne k \ne \iota .$$

  Depending on the coalition values of the underlying cooperative structure, the proposer will choose the best possible two-country or three-country coalition in accordance with sequential rationality and subgame perfection. Note that the non-proposing players j and k may only threaten to coordinate on a suboptimal (no, no) equilibrium for $x_1^\iota \in \left( {CA_1^{j * } \cap CA_1^{k * } } \right)\backslash \left( {UA_1^{j * } \cup UA_1^{k * } } \right)$, since any policy proposal within a player’s unconditional acceptance set will surely be accepted in equilibrium (dominant strategy). Using backwards induction, one may infer the players’ optimal acceptance sets in the first period in anticipation of equilibrium play in the second period. If second-period acceptance sets are non-empty, then so are first-period sets. Hence, because of intertemporal discounting and random recognition, it would never be optimal for the first-period proposer to make a proposal unacceptable to either coalition partner, since that would imply foregoing the immediate benefits of coalition formation as well as one’s proposal prerogative. First-period acceptance sets are always non-empty even for very low values of bilateral coalition formation (y _AB , y _BC , y _AC ), since we know by assumption that there always exists a non-degenerate set of policies such that the grand coalition ABC is always Pareto superior to the autarchic state; therefore, autarchy may not persist as a first-period bargaining outcome within any subgame-perfect Nash equilibrium.

• 2. Now let π _{t =0} = {AB(x _AB )|C}, i.e., partial union AB has formed after round one at some common policy within the Pareto set [m _A , m _B ] of coalition AB. That can only be the outcome of a first-period proposal by A, such that $x_0^{A * } = \mathop {\arg \max }\limits_{x_0^A } \left\{ {V_0^{A * } \left( {x_0^A } \right)\left| {x_0^A \in UA_0^B \backslash CA_0^C } \right.} \right\}$, since B’s subgame-perfect first-period proposal of m _B would have been accepted by both A and C. We need to show that ABC(x _ABC ) cannot be the outcome of the second round of bargaining in a subgame-perfect NE for any x _AB , x _ABC , and δ. Assume by contradiction that π _{t =1} = {ABC(x _ABC )}. Subgame perfection would imply that x _ABC  ≥ m _C  − y _ABC (C’s participation constraint) and x _ABC  ≤ y _ABC  − y _AB  + x _AB (A’s participation constraint). For the grand coalition unanimity acceptance set to be non-empty, we need that x _AB  ≥ m _C  + y _AB  − 2y _ABC (*). This effectively rules out any x _AB  < m _C  − 2y _ABC as possible equilibrium first-period proposals. We proceed to prove the contradiction by showing that profitable deviations exist for any other possible x _AB :

  • a. Let x _AB  ∈ [m _C  − 2y _ABC , m _A ): this is a Pareto-dominated set of proposal for both A and B, since both would be unambiguously better off with a proposed policy of x _AB = m _A ; it is not only closer to their ideal positions, but it also enhances their bargaining leverage vis-à-vis C by shrinking the ABC Pareto set.

  • b. Let x _AB  = m _A : from (*) this implies that y _AB  ≤ 2y _ABC  − (m _C  − m _A ). For low types y _AB  ∈ [0, y _ABC  − (m _B  − m _A )), the contradiction follows by showing that $V^{A}_{0} {\left( {{\left\langle {ABC{\left( {x^{ * }_{{ABC}} } \right)},ABC{\left( {x^{ * }_{{ABC}} } \right)}} \right\rangle }} \right)} \geqslant EV^{A}_{0} {\left( {{\left\langle {AB{\left( {m_{A} } \right)}\left| C \right.,ABC{\left( {x^{{j * }}_{1} } \right)}} \right\rangle }} \right)}$, ∀δ, y _ABC , where the expectation is taken over the identity of the proposer j at t = 1 (hence $x_1^{j * }$ is ex ante unknown in equilibrium) and $x_{ABC}^ * = \mathop {\arg \max }\limits_{x_0^A } \left\{ {V_0^A \left( {x_0^A } \right)\left| {\alpha _0^{j * } } \right.\left( {x_0^A } \right) = 1,\forall j \ne A,x_0^A \in X} \right\}$. A simple algebraic calculation shows that this holds for any y _AB within the above interval. Hence, A would have an incentive to deviate to a better proposal given the subgame-perfect acceptance strategies of B and C. Similarly, for intermediate types y _AB  ∈ [y _ABC  − (m _B  − m _A ),2y _ABC  − (m _C  − m _A )], there always exists a globally acceptable, Pareto efficient grand coalition first-period proposal $x_{ABC}^ *$ that makes A weakly better off compared to the gradual coalition-formation subgame. Going through all the possible subgames and subcases for y _AB  ∈ [y _ABC  − (m _B  − m _A ),2y _ABC  − (m _C  − m _A )] and δ ∈ [0,1], it turns out that for $x_{ABC}^ * = m_A + y_{ABC} - y_{AB} - \varepsilon$, ε ≥ 0, there always exists an ε ≥ 0 such that the following conditions are satisfied:

    • i. $V_0^A \left( {x_0^A = x_{ABC}^ * } \right) \geqslant EV_0^A \left( {x_0^A = m_A } \right)$ (A’s optimization problem)

    • ii. $x_{ABC}^ * \in UA_0^{B * } \cap CA_0^{C * }$ (B and C’s mutual acceptance forms a Nash equilibrium)

    • iii. $x_{ABC}^ * = m_A + y_{ABC} - y_{AB} - \varepsilon \geqslant m_C - y_{ABC}$ (C’s participation constraint)

• c. Finally, for any x _AB  > m _A it would be enough to show that, given that only B will accept, i.e., $x_{AB} \in UA_0^{B * } \backslash CA_0^{C * }$, A would profit from deviating to a proposal $x_0^{A * } = \mathop {\arg \max }\limits_{x_0^A } \left\{ {V_0^A \left( {x_0^A } \right)\left| {x_0^A \in UA_0^{B * } } \right.\backslash CA_0^{C * } } \right\}$, i.e., the one closest possible to m _A , such that only country B would accept. Indeed, this is a profitable deviation, since it would bring both immediate policy gains and enhanced second-period bargaining leverage (because of a restricted grand coalition Pareto set). For some $x_0^A \geqslant x_{ABC}^ * \geqslant m_C - y_{ABC}$, subgame perfection and sequential rationality also imply that C should also have accepted A’s first-period proposal.

• 3. Now let π _{t =0} = {A|BC(x _BC )}, i.e., partial union BC has formed after round one (following a proposal by country C). Using a similar reasoning as above, the assumption that π _{t =1} = {ABC(x _ABC )} leads to a contradiction, since that coalition-formation path cannot be part of a subgame-perfect Nash equilibrium.

• 4. Finally the case of π _{t =0} = {AC(x _AC )|B} may never arise in equilibrium, since either A’s or C’s unconditional acceptance of each other’s policy proposals implies that the median country B is always better off accepting too. QED

Proposition 2

In the two-period bargaining game with asymmetric information there exists a perfect Bayesian Nash equilibrium, whereby the representative of either of the extreme countries i  = A or C will propose $x_0^{i * } = m_i$ at time t = 0 if and only if $y_{iB} \in \left( {\tilde y_{iB} ,y_{ABC} } \right]$ for some $\tilde y_{AB} \in \left[ {y_{ABC} - \left( {m_B - m_A } \right),2y_{ABC} - \left( {m_C - m_A } \right)} \right]$ or $\tilde y_{BC} \in \left[ {y_{ABC} - \left( {m_C - m_B } \right),2y_{ABC} - \left( {m_C - m_A } \right)} \right]$ , in which case coalition AB (or BC respectively) will form right away and may later expand to the grand coalition ABC at time t = 1 with strictly positive probability. Otherwise, for $y_{iB} \in \left[ {0,\tilde y_{iB} } \right]$ , i = A or C will propose $x_0^{i * } = \hat x^j \left( \delta \right)$ such that j ≠ i, j = A, C is just indifferent between accepting and rejecting at time t = 0, in which case the grand coalition ABC will form immediately. If median country B gets to propose first, then all its types will pool on an equilibrium proposal $x_0^{B * } = m_B$ , which will lead to the immediate formation of the grand coalition.

Proof

Having characterized how the proposed equilibrium plays out in the subgames, where A is the first-period proposer, I will now proceed to derive the equilibrium threshold type $\tilde y_{iB}$ for either extreme country i = A, C as the first-period proposer. Lemma 1 below completes the characterization by deriving the threshold point $\tilde x_0^i \left( \delta \right)$, i = A, C and the corresponding compromising first-period proposal $\hat x^j \left( \delta \right)$, j = A, C, j ≠ i.

The unique threshold type $\tilde y_{AB}$ has to be such that the representative of country A would be ex ante indifferent between the immediate $\left\langle {ABC\left( {\hat x^C } \right),ABC\left( {\hat x^C } \right)} \right\rangle$ and the gradual 〈AB(m _A )|C, ABC(x _ABC )〉 equilibrium coalition formation paths. Hence, to ensure the incentive-compatible truthful revelation of types at t = 0 within the context of the above semi-separating equilibrium, the cutoff type $\tilde y_{AB} \in \left[ {y_{ABC} - \left( {m_B - m_A } \right),2y_{ABC} - \left( {m_C - m_A } \right)} \right]$ for country A will have to satisfy the following incentive constraint:

$$\begin{aligned} V_0^A \left( {x_0^A = m_A \left| {y_{AB} } \right. = \tilde y_{AB} } \right) = V_0^A \left( {x_0^A = \hat x^C \left| {y_{AB} } \right. = \tilde y_{AB} } \right) \Leftrightarrow \\ \tilde y_{AB} + \frac{\delta }{3}\left[ {2y_{ABC} - \left( {m_C - m_A } \right) + \tilde y_{AB} + m_A - \frac{{m_A + m_C - \tilde y_{AB} }}{2} + y_{ABC} } \right] = \left( {1 + \delta } \right)\left[ {m_A - \hat x^C + y_{ABC} } \right] \Leftrightarrow \\ \tilde y_{AB} = m_A + \frac{\delta }{{2 + \delta }}m_C - \frac{{2\left( {1 + \delta } \right)}}{{2 + \delta }}\hat x^C + \frac{2}{{2 + \delta }}y_{ABC} \\ \end{aligned} $$

(2)

In order to derive the above expression for $\tilde y_{AB}$, I make use of the optimal proposal and acceptance strategies in the second-period subgames analyzed in the model and also of the fact that the lowest of high types $\tilde y_{AB}$ will always accept C’s second-period proposal $x_1^{C * } = \frac{{m_A + m_C - \tilde y_{AB} }}{2}$ with certainty. Also note that each second-period subgame equilibrium utility is discounted by a factor ${\raise0.7ex\hbox{$\delta $} \!\mathord{\left/ {\vphantom {\delta 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}$ reflecting temporal discounting and equal recognition probabilities. Given the spatial location of $\hat x^C \left( \delta \right) \in \left[ {m_C - y_{ABC} ,m_B } \right]$ derived in Eq. 5 below, it is fairly straightforward to confirm that the cutoff type $\tilde y_{AB}$ indeed lies within the interval [y _ABC  − (m _B  − m _A ),2y _ABC  − (m _C  − m _A )]. Using a similar approach for C, where proposal $\hat x^A \left( \delta \right)$ is computed in Eq. 6 below, one can derive the following cutoff proposer type:

$$\widetilde{y}_{{BC}} = \frac{2}{{2 + \delta }}y_{{ABC}} + \frac{{2{\left( {1 + \delta } \right)}}}{{2 + \delta }}\widehat{x}^{A} - \frac{\delta }{{2 + \delta }}m_{A} - m_{C} .$$

(3)

Finally, for all subgames starting with country C as the first-period proposer, extreme country A’s equilibrium beliefs are as follows:

$$\sigma ^{{A*}} {\left( {y_{{BC}} \left| {x^{C}_{0} } \right.} \right)} \to \left\{ {\begin{array}{*{20}l} {{y_{{BC}} \sim U{\left[ {0,\widetilde{y}_{{BC}} } \right]},} \hfill} & {{{\text{for}}\;x^{C}_{0} \in {\left( { - \infty ,\widehat{x}^{A} } \right)} \cup \left( {\widehat{x}^{A} ,m_{A} + y_{{ABC}} } \right]} \hfill} \\ {{y_{{BC}} \sim U\left( {\widetilde{y}_{{BC}} ,y_{{ABC}} } \right]} \hfill} & {{{\text{for}}\;x^{{`C}}_{0} \in {\left( {m_{A} + y_{{ABC}} ,m_{C} } \right)} \cup {\left( {m_{C} , + \infty } \right)}} \hfill} \\ \end{array} } \right.$$

Lemma 1

There exists for country C a threshold point on the real line, $\tilde x_0^C \left( \delta \right) \in \left[ {m_C - y_{ABC} ,m_B } \right]$ , such that for $x^{A}_{0} < \widetilde{x}^{C} {\left( \delta \right)}$ and $\pi _0 = \left\{ {AB\left( {x_0^A } \right)\left| C \right.} \right\}$ a positive support of low types $y_{AB} \leqslant \tilde y_{AB}$ will reject C’s optimal second-period proposal $x_1^{C*}$ given its equilibrium beliefs, while for $x_0^A \geqslant \tilde x_0^C \left( \delta \right)$ all low types $y_{{AB}} < \widetilde{y}_{{AB}}$ will accept $x_1^{C * }$ . A similar threshold point $\tilde x_0^A \left( \delta \right)$ exists for country A.

Proof

C’s optimal second-period proposal given its updated equilibrium beliefs following a first-period proposal $x_0^A \in \left[ {m_C - y_{ABC} ,m_B } \right]$ that falls within the grand union Pareto set of common policies becomes

$$x_1^{C * } = \mathop {\arg \max }\limits_{x_1^C \in \left[ {m_C - y_{ABC} ,m_C } \right]} \left\{ {\Pr \left( {\alpha _1^{A * } \left( {x_1^C } \right) = 1\left| {y_{AB} \sim U\left[ {0,\tilde y_{AB} } \right]} \right.} \right) \times \left( {x_1^C - m_C + y_{ABC} } \right)} \right\} = \frac{{m_C + x_0^A }}{2}.$$

(4)

Hence, $\tilde x^C \left( \delta \right)$ has to be such that the threshold y _AB type is just indifferent between accepting and rejecting $x_1^{C * }$, i.e.,

$$u_{t = 1}^A \left( {{\text{Acc}}\;x_1^{C * } = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\left( {m_C + \tilde x^C } \right)\left| {y_{AB} } \right. = \tilde y_{AB} } \right) = u_{t = 1}^A \left( {{\text{Rej}}\;x_1^{C * } = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\left( {m_C + \tilde x^C } \right)\left| {y_{AB} } \right. = \tilde y_{AB} } \right).$$

So for $x^{A}_{0} < \widetilde{x}^{C} {\left( \delta \right)}$ and $\pi _0 = \left\{ {AB\left( {x_0^A } \right)\left| C \right.} \right\}$, there is a positive support of low country A types $y_{AB} \leqslant \tilde y_{AB}$ that reject $x_1^{C * } = \frac{{m_C + x_0^A }}{2}$, which implies that C’s first-period rejection continuation value $V_{t = 0}^C \left( {{\text{Rej}}\;{\text{x}}_0^{\text{A}} \left| {B\;{\text{acc}}} \right.} \right)$ becomes quadratic. For $x_0^A \geqslant \tilde x_0^C \left( \delta \right)$ and $\pi _0 = \left\{ {AB\left( {x_0^A } \right)\left| C \right.} \right\}$, all low types $y_{AB} \leqslant \tilde y_{AB}$ will accept $x_1^{C * } = \frac{{m_C + x_0^A }}{2}$, since C’s optimal second-period proposal would make the highest possible low type $\tilde y_{AB}$ just indifferent. This implies that C’s rejection continuation value becomes linear, in which case it is quite straightforward to derive the value of $\hat x^C \left( \delta \right)$ in the following manner:

$$\hat x^C \left( \delta \right) = \frac{{6 + 2\delta }}{{6 + 5\delta }}\left( {m_C - y_{ABC} } \right) + \frac{{2\delta }}{{6 + 5\delta }}m_B + \frac{\delta }{{6 + 5\delta }}m_C .$$

(5)

Using a similar approach to derive $\hat x^A \left( \delta \right)$ for country C as the first-period proposer, it turns out that:

$$\hat x^A \left( \delta \right) = \frac{{6 + 2\delta }}{{6 + 5\delta }}\left( {m_A + y_{ABC} } \right) + \frac{{2\delta }}{{6 + 5\delta }}m_B + \frac{\delta }{{6 + 5\delta }}m_A \leqslant \tilde x^A \left( \delta \right).$$

(6)

Proposition 3

For a non-degenerate support of high y _AB types $\left( {\tilde y_{AB} ,\tilde y_{AB} + \varepsilon } \right]$ , ε > 0, there exists an immediate grand coalition formation policy $x_{ABC}^ * \in \left[ {m_C - y_{ABC} ,m_B } \right]$ such that $\left\langle {ABC\left( {x_{ABC}^ * } \right),ABC\left( {x_{ABC}^ * } \right)} \right\rangle \underline \succ _i \left\langle {AB\left( {m_A } \right)\left| C \right.,\left. {ABC\left( {x_1^{j * } } \right)} \right)} \right\rangle$ for all i, j = A, B, C and $\left\langle {ABC\left( {x_{ABC}^ * } \right),ABC\left( {x_{ABC}^ * } \right)} \right\rangle \succ _i \left\langle {AB\left( {m_A } \right)\left| C \right.,\left. {ABC\left( {x_1^{j * } } \right)} \right)} \right\rangle$ for at least one i, where $\underline \succ$ and $\succ$ denote the weak and strict preference relations respectively and $x_1^{j * }$ denotes the optimal second-period proposal for any representative j to get recognized.

Proof

Let $y_{AB} \in \left( {\tilde y_{AB} ,\tilde y_{AB} + \varepsilon } \right]$, ε > 0. We first need to find the set of grand union policy proposals that make A weakly better off in an immediate coalition formation path rather than a gradual one, i.e., find x _ABC ≥ m _A such that

$$V_0^A \left( {\left\langle {ABC\left( {x_{ABC} } \right),ABC\left( {x_{ABC} } \right)} \right\rangle } \right) \geqslant EV_0^A \left( {\left\langle {AB\left( {m_A } \right)\left| C \right.,ABC\left( {x_1^{j * } } \right)} \right\rangle } \right),j = A,B,C.$$

So we need the following inequality to hold:

$$\begin{aligned} \left( {1 + \delta } \right)\left( {m_A - x_{ABC} + y_{ABC} } \right) \geqslant y_{AB} + \frac{\delta }{3}\left[ {2y_{ABC} - \left( {m_C - m_A } \right) + y_{AB} + m_A - \frac{{m_A + m_C - \tilde y_{AB} }}{2} + y_{ABC} } \right] \\ \Leftrightarrow m_A \leqslant x_{ABC} \leqslant \frac{1}{{1 + \delta }}\left( {\frac{{2 + \delta }}{2}m_A + \frac{\delta }{2}m_C + y_{ABC} - \frac{{3 + \delta }}{3}y_{AB} - \frac{\delta }{6}\tilde y_{AB} } \right) = \overline x _{ABC} . \\ \end{aligned} $$

Given that $\tilde y_{AB} \leqslant 2y_{ABC} - \left( {m_C - m_A } \right)$, it turns out that $\overline{x} _{{ABC}} \geqslant m_{C} - y_{{ABC}}$, which implies that there may be such an immediate grand coalition proposal that could make C weakly better off. Since B, the moderate country, will trivially have a strict preference to participate in an immediate grand union with a common policy much closer to its own ideal point, all we need to show is that C is weakly better off under such an immediate agreement compared to the gradual equilibrium, whereby C believes A and B to be of a high type, i.e.,

$$\begin{aligned} V_0^C \left( {\left\langle {ABC\left( {\bar x_{ABC} } \right),ABC\left( {\bar x_{ABC} } \right.} \right\rangle } \right) \geqslant EV_0^C \left( {\left\langle {AB\left( {m_A } \right)\left| C \right.,ABC\left( {x_1^{j * } } \right)} \right\rangle } \right) \\ \Leftrightarrow \left( {1 + \delta } \right)\left( {\bar x_{ABC} - m_C + y_{ABC} } \right) \geqslant \frac{\delta }{3}\left[ {\left( {m_A + y_{ABC} - y_{AB} } \right) - m_C + y_{ABC} + \frac{{m_A + m_C - \tilde y_{AB} }}{2} - m_C + y_{ABC} } \right] \\ \Leftrightarrow y_{AB} \leqslant 2y_{ABC} - \left( {m_C - m_A } \right). \\ \end{aligned} $$

Since we know from before that $\tilde y_{AB} \leqslant 2y_{ABC} - \left( {m_C - m_A } \right)$, then the latter expression has to be true for some $y_{AB} \in \left( {\tilde y_{AB} ,\tilde y_{AB} + \varepsilon } \right]$, ε > 0 and will hold as a strict inequality for any interior cutoff type $\widetilde{y}_{{AB}} < 2y_{{ABC}} - {\left( {m_{C} - m_{A} } \right)}$. We have thus shown that an immediate grand coalition formation path under $\overline{x} _{{ABC}}$ will be a Pareto superior solution, hence the interim inefficiency of the gradualist equilibrium. QED