“The best alliances are those concluded between concealed intentions.”
Introduction
1The main focus of this article is to explore how electoral alliances are formed, using tools from cooperative game theory. The fundamental postulate of this study is that political coalitions have greater power than the sum of the powers of their members. Understanding why and how coalitions are formed in the political game is a subject which occupies an important place in the field of contemporary political science and several fundamental treatises have been devoted to these issues, including those of De Swaan (1973) and Riker (1962).
2It is not easy to model the strategic calculation leading to the formation of coalitions, and there are many ways to proceed.
3A noncooperative approach will privilege a description of the negotiation within the framework of a game under a normal (or extensive) form where the actions offered to the various players are precisely described. For example, if the issue on which the decision is based is that of the allocation of a set of ministry portfolios, the protocol will describe who is authorized to make proposals, at what moment they can do it and in which order (if they are more than one), who can block a proposal, when do the negotiations stop, etc. The work then consists of characterizing the equilibria of the game thus constructed.
4The cooperative approach will prefer a more summary description called the characteristic function of the game which consists of associating each potential coalition with the total payoffs it can achieve, irrespective of the actions taken by the players outside this coalition. The advantage of this more basic form is that the alliance forecasts are not too sensitive to the details of the negotiation process. In many cases, the extensive form is not, in fact, a constraint weighing over the players but rather the view of the modeler of the game unfolding. It is therefore important that the conclusions are robust with respect to artificial changes. The bargaining solutions examined by the cooperative game theory are numerous. We will focus here on two of them: the Shapley value and the nucleolus. These two solutions have received noncooperative foundations, i.e., they can be described as payoff vectors at equilibrium for certain negotiation protocols. There is thus no antagonism between the two approaches.
5This article has three objectives. Firstly, the objective is to introduce the concepts and tools from cooperative game theory to model the interactive, strategic decision context around the political groups between the two rounds of a tworound election. Our analysis is mainly motivated by the game of alliances resulting from the way of electing members of the regional councils in France. This is a tworound, party list system. The election is won in the first round if a list collects the absolute majority of the expressed votes. Otherwise, a secondround ballot is held. For a list to be able to be presented in the secondround ballot, it must have obtained at least 10% of the expressed votes during the first ballot. However, the composition of lists can be modified with respect to the first round by including candidates from the lists which obtained at least 5% of the expressed votes, and with the agreement of the candidate who is at the head of the list where these candidates were found during the first ballot. The candidates on the same list at the first ballot cannot be included on different lists on the second ballot. At the end of the election, the list having obtained the most votes in the region gets a quarter of the seats available. The other seats are distributed among all of the lists by proportional representation according to the rule of the highest average.
6At the end of the first ballot, the game of alliances is relatively sophisticated, for many reasons. Certain lists, having obtained between 5 and 10% of the votes during the first ballot, cannot continue to stand unless they make an alliance with one or more lists of which at least one exceeds 10% of the votes. Are they going to make alliances, and if yes, under which conditions? How are they going trade the electoral votes they are bringing into the basket? Some lists can continue to stand without alliances, but making an alliance can open the way for winning the bonus (to be received by the list with the most votes in the region), which corresponds to a quarter of the seats available. Making alliances increases the chances of winning this bonus, but might force the party to share it with its partners.
7Our first contribution aims to formulate this game of alliances as a transferable utility cooperative game (tu), i.e., the players do not a priori feel bound by special apportionment norms: everything is negotiable, and the seats serve, in this respect, as “compensation currency”. We should note that the game is a game between the lists present on the first ballot, and not between the individual persons forming these lists, since the lists cannot break up into several sublists with different strategies. The entire work consists of writing down the characteristic function of the game, i.e., to attribute to each coalition a number which reflects the number of seats this coalition can be sure of. The expression “to be sure of” is used because a coalition, once it forms, cannot ignore what the political groups will do outside the coalition. The larger this number, the stronger the coalition. If all coalitions would be politically feasible, a superadditive [1] game would be derived because unity makes strength. [2] When certain coalitions are politically inconceivable, the game is not necessarily superadditive, because the union of two feasible coalitions might itself be inconceivable. In any case, when superadditivity is at work, it finds its origin in two sources. First, in the existence of a critical threshold below which a list cannot continue to stand without an alliance (10% in the case of regional elections): a coalition including lists unable to continue to stand alone, by getting votes which would otherwise be lost, increases its relative number of votes. Second, in the existence of a bonus for the leading group (25% of the seats in the case of regional elections), because accepting additional partners inside the coalition can bring the thusformed coalition into the lead. We will describe these two effects in detail relying on a simple model describing the transfer of votes between the two ballots, a model which assumes that the votes obtained during the first ballot are “kept” by the parties who received them.
8The second contribution consists of characterizing the main solutions of this cooperative game in the case where the game of alliances includes at most three players. Our methodology easily extends to an arbitrary number of players, but the application to the regional elections of March 2010, which is found in the last part of this article, is applied in a context where only three players at most are active (the three leftwing parties: the Socialist Party, Europe Ecology, and the Left Front). We are introducing two very popular solutions – the nucleolus and the Shapley value – which describe two negotiation solutions within the framework of cooperative game theory. We are comparing these solutions to a solution derived from the mechanical application of a proportionality rule, according to which the members of a coalition share the seats obtained in proportion to their results after the first ballot. Even in the case where there are only three active players, the analysis of the global game is arduous, because the problem requires a careful and specific treatment of each of the strategic configurations which can occur. The major part of the results is found in the online appendix of the French version of the article. The theoretical analysis, however, enables us to propose a simple taxonomy of the possible configurations, reflecting the balance of power between coalitions.
9The third contribution is an application of this methodology to the regional elections held in France on the 14th and 21st of March 2010. Firstly, on the basis of the first ballot results, we proceed to building the characteristic function of a threeplayer game (the Socialist Party, Europe Ecology, and the Left Front) for three specific regions: Aquitaine, Auvergne, and Brittany. For each of these regions, we calculate the shares resulting from the proportionality rule, the nucleolus, and the Shapley value, and we compare these shares with the shares observed. We then use the taxonomy developed in the theoretical analysis to obtain a classification of the twentyone regions (Corsica excluded) of metropolitan France. These metropolitan regions provide a rich database; however, they do not cover all of the cases identified by the theory.
Relation to Literature
10The issue of forming coalitions between political players appears in various contexts other than the electoral environments. In many democracies, the multiparty system is the norm, and governments are almost always coalition governments. Many empirical and theoretical studies have been devoted to games describing the formation of a government after elections and the analysis of alliance strategies. In the literature relying on the spatial model, it is usual to describe a party or political group by a vector in a Euclidean space (the ideological space) and a number of seats won during an election. An alliance strategy between partners will consist of a shared program (a point in the ideological space) and a distribution of the ministerial portfolios (or other positions of responsibility). This work focuses on understanding which alliance(s) will form and which strategies will be implemented by the players. In the case where there is no ability to ally before elections, the voters have to anticipate the possible games of alliances (all of the possible continuations of the game after the elections) and vote accordingly. Of course, the opposite situation can be imagined, with certain players allying before the elections. Among the most relevant work on this subject, we can cite AustenSmith and Banks (1988, 1990), Brown and Franklin (1973), Brown and Frendreis (1980), Carroll, Cox, and Pachón (2004), Fréchette, Kagel, and Morelli (2004), Laver (1998), Laver and Schofield (1998), Laver and Shepsle (1996), and Warwick and Druckman (2001).
11The analysis of the formation and role of coalitions is also carried out in strategic contexts other than that of forming governments. For example, legislative coalitions form during certain elections and, here, the protocol describing the steps of forming the agenda and the vote legitimizes the use of noncooperative game theory (Baron and Ferejohn 1989; Eraslan 2002).
12In this article, as previously mentioned, we assume that the game has transferable utility, i.e., the players do not a priori feel bound by apportionment rules. We can nevertheless imagine a game of alliances of the same nature, taking place before the beginning of the vote. Everything depends upon the considered voting system and upon the allying capacities of the players. A remarkable example which was thoroughly studied (Lee, McKelvey, and Rosenthal 1979; Lee and Rosenthal 1976), is the “apparentement law,” an electoral law implemented in France in May 7, 1951, by the parties of the Third Force to reduce the influence of communists and Gaullists in the National Assembly. To this end, it has introduced the right of “apparentement” into the voting system. According to this system, each electoral list had to consist of as many candidates as seats available in the electoral district. The lists had the possibility of concluding deals before the elections: it was said they were becoming “apparented” (making allies). If the sum of votes obtained for apparented lists (or for a single list) was greater than or equal to 50% of the votes expressed, said lists or list would obtain all of the seats available in the district. Otherwise, the seats would be distributed proportionally among the various lists according to the method of the higher average. It should be noted that a coalition which would have concluded an apparentement agreement and would win an absolute majority would receive a very significant bonus. The cooperative game describing this law is a game without transferable utility because, once a preelectoral alliance is concluded, there is nothing left to negotiate. [3]
The Electoral Environment
Description of the Voting System and Notations
13We will now define more precisely the electoral environment. Let us consider an electoral district where the parties are in competition for nominating K representatives in an assembly whose domain is that of the electoral district. The election has (at most) two rounds. We will denote by N_{tot} the total number of registered voters.
14In the first round, after preelectoral alliances if any between various parties, P^{1} lists compete; each of them presents an ordered list of K names. For the voter, voting for candidates from more than one party is not allowed: there are thus P^{1} valid ballot papers in the first round. An electoral result in the first round is a vector of dimension P^{1} + 1 where N^{1}_{0} is the number of absentee voters and blank or invalid votes and N^{1}_{m} is the number of voters having voted for the list m (m = 1,…, P^{1}). Naturally: We will assume in the following that the lists are ordered in decreasing order of the results after the first ballot.
15If in the first round a list obtains an absolute majority of the votes expressed, the polls stop. A number ?K of seats, with 0 ? ? ? 1, is allocated on a proportional basis between all of the lists m with relative results above a threshold ?. The (1 – ?)K remaining seats are allocated as a bonus to the list arriving first.
16If no list obtains a majority, a second round is held (the following week in the case of French regional elections). The possible options open to the different political groups on the evening of the first round are characterized by two thresholds ? and ?, with 0 < ? < ? < 1. If the relative result of the political group m is below the first threshold ?, this group cannot participate in the second round. If its relative result is between the two thresholds, the group cannot run independently, but can merge with other lists, of which at least one must have risen above the higher of the two thresholds ?. Finally, if its relative result is above the threshold ?, the group can continue to run in the electoral competition, alone or by merging. [4]
17We will note by M^{2}, the number of political groups having reached the threshold ? and by R^{2} (R^{2} ? M^{2}), the number of those which are not forced to merge (i.e., having reached the threshold ?). [5] On the evening of the first round, the situation is summarized by the vector N^{1} and the integers M^{2} and R^{2} which derive from it. Following possible alliances between these M^{2} groups, P^{2} political coalitions are present in the second round, each presenting an ordered list of K candidates. To be included on a second round list, a candidate must be included on the firstround list of one of the groups forming the secondround coalition, and must have obtained the agreement of the head of his/her firstround list. The electoral result of this second round is then described by a vector , where N^{2}_{0} is the number of absentee voters and of blank or invalid votes on the second ballot, and N^{2}_{m} is the number of voters having voted for the list m.
18The K seats will then be distributed as follows: ?K (0 ? ? ? 1 seats are allocated on a proportional basis: the seats are attributed on a proportional basis among all lists m whose relative result is above ?. The number of remaining seats, (1 – ?)K, is allocated (in bulk) to the list arriving first, i.e., to list m^{*} such that:
20If the list m gets k seats, the first k candidates on this list are elected.
21In the voting system of the French regional elections of March 2010, ? = 75%, ? = 5%, ? = 10%.
22We should note that the voting system does not impose any constraint on the partners of a political coalition, or on the distribution of seats obtained by the coalition among its members (i.e., on the order of the candidates included on the merged list). The main question we are focusing on is the following: what agreements will be concluded between the two rounds of such an election?
Building the Characteristic Function
23To answer this question, we will introduce a methodology based on cooperative game theory. Potential coalitions must evaluate their total gain in terms of seats if they are to form.
Hypotheses on the Admissible Coalitions
24We will make the hypothesis that certain coalitions are judged as being impossible by all players in the game, notably because of the weight of ideological barriers. It is thus possible that the locations of certain groups in the political spectrum make impossible any form of compromise between the strategies/projects to be implemented between these groups, making their union unachievable within a coalition. One can also imagine that the longterm strategy of a party forbids alliances, even if they could prove to be beneficial on the shortterm, in terms of seats.
25We will assume that all of the lists in the first ballot share the same beliefs concerning the coalitions which may form. To simplify, we will assume that judging a coalition admissible or not is independent of what the other political groups do. We then denote by ? the set of admissible coalitions, i.e., those which are perceived by all of the players in the game as having a nonzero probability of happening. We will suppose a special structure for ?: we suppose that there is a partition of {1, 2,…, M^{2}} such that coalition S is admissible if and only if there is a j ? {1, 2,…, J}, such that S ? F_{j}. This hypothesis postulates that there is a certain number J of political families, that within each family j all alliances are possible, but that the alliances between lists from different families are impossible. We should note that ? contains singletons (a list can always decide not to form any alliance) and that, if a coalition S ? {1, 2,…, M^{2}} is in ?, then any coalition included in S is also found in ?. A particular case of set ? verifying this property is that where all alliances are perceived as admissible a priori: and ? is then the set of all subsets of {1, 2,…, M^{2}}.
Gains of a Coalition
26Let us consider S ? ? to be an admissible coalition. It would have to evaluate its total gain in terms of seats if it were to actually form, denoted V(S). The function V is called the characteristic function of the game.
27The first question raised for this coalition S is to know if it meets the conditions for actually presenting a list on the second ballot. If this is not the case, i.e., if none of its members crossed the threshold ? of the votes expressed during the first ballot, it does not win any seat after the second ballot. Formally, if S ? {1, 2,…, R^{2}} = Ø, V(S) = 0.
28If now the coalition S can present a list, the second question raised for it is to anticipate what the other political groups are likely to do. Once this forecast is formulated, it must anticipate the number of seats it would win within the various possible configurations. We will denote by ?(S) the set of partitions of {1, 2,…, M^{2}}\S that S estimates as possible. Formally, a partition of {1, 2,…, M^{2}}\S will be estimated as possible if each of its components belongs to ?. For each partition ? ? ?(S), the coalition S has now to anticipate the number of seats it would win if this partition is formed; we will denote it by V(S, ?).
The SecondRound Electoral Behavior
29To anticipate this number of seats, we now need to form hypotheses on how the voters will vote in the second round, were they to be faced with this configuration {S, ?}. In this article, we make the following hypothesis. The first round absentee voters will abstain in the second round. Voters who submitted blank or spoiled votes in the first round will vote the same in the second round. Voters who voted in the first round for a list that is not present in the second round, abstain. Those whose lists are still present, vote for the coalition including the list they voted for in the first round. These hypotheses describe a situation where the citizens whose favorite list is no longer in the competition lose interest in voting, and those whose list is present follow their leaders’ instructions and vote for the list, whatever the alliance concluded. Of course, we do not claim that the electoral mobility matrices describing the transfer of votes between the two rounds are always of this type: some voters can fail to follow the instructions of their favorite parties; some absentee voters might vote; some voters, whose favorites are absent on the second ballot, might decide to vote for their second or third choice. It is possible to build the characteristic function in the case of any matrix, but the notations become very cumbersome. [6]
30Furthermore, we assume here that the number of votes received by a coalition in the second round is a certain quantity. We thus eliminate the issues specific to uncertainty. To introduce them would force us to consider stochastic cooperative games. The approximation retained here appears to us as satisfactory at this stage of the work.
31Under these behavioral hypotheses, the number of votes received in the second round by a coalition does not depend on what the other lists do, but exclusively on its members. For any coalition S ? ?, the number of votes anticipated by S for the second round is:
Explicit Calculation of the Characteristic Function
33In such a scenario, the coalition S can estimate the number of seats it would obtain if the partition ? = {T_{1}…, T_{L}} ? ?(S) would form: if the coalition S arrives first, i.e., if N(S) > N(T_{l}) for any l = 1,…, L, the coalition S takes the bonus ((1 – ?)K seats) on top of the share of seats it wins via the proportional component of the voting system:
35otherwise, it obtains:
37The electoral gain of the coalition S thus depends on both the composition of the coalition itself and the alliances formed by the political groups outside the coalition. These coalitions are described by ?. To continue the reasoning, we will assume, as traditionally theoreticians of cooperative game theory do, that the members of the coalition S take as a reference value for V(S) the smallest possible value of V(S, ?) when all possible situations are considered for ?: [7]
39We will now give the explicit formula for V(S).
40If S ? {1, 2,…, R^{2}} = Ø, V(S) = 0;
41If now S ? {1, 2,…, R^{2}} ? Ø, one can verify that the worst which can happen to the coalition S is that all maximum coalitions are formed among the other lists. Formally, if S is an admissible coalition, it means that there is a j so that S ? F_{j}, and the worst partition possible for S is the partition {F_{1}…, F_{j – 1}, F_{j}\ S, F_{j + 1}…, F_{J}}. Indeed, this partition is that which, at the same time, minimizes S’s chances to arrive first and maximizes the number of votes expressed, thus minimizing the proportion of expressed votes going to S. The following explicit formula is then obtained for V(S):
Game Properties
Game Decomposition in a “Proportional” Component and a “Majority Bonus” Component
43The formula above clearly shows that the characteristic function is the sum of two functions, each being the characteristic function of a game.
44The proportional component of the game supposes as the characteristic function of the game:
46while the bonus component of the game equals, for any S ? ? (where prime signifies bonus in the following):
Superadditive Game
48We will now show that this game is superadditive, i.e., that for all coalitions S_{1}, S_{2} ? ? so that S_{1} ? S_{2} = Ø and S_{1} ? S_{2} ? ?, V(S_{1} ? S_{2}) ? V(S_{1}) + V(S_{2}).
49Let S_{1}, S_{2} be admissible so that S_{1} ? S_{2} = Ø and S_{1} ? S_{2} ? ?. Since S_{1} ? S_{2} ? ?, it implies that there is a j so that S_{1} ? F_{j} and S_{2} ? F_{j}.
Proportional Component of the Game
50Let us first examine the fraction of seats distributed according to the proportional component. This involves showing that:
52We note that if N(S_{1}) = N(S_{2}) = 0, then N(S_{1} ? S_{2}) = 0, and the inequality is satisfied (the two members are equal to 0).
53If N(S_{1}) > 0 and N(S_{2}) > 0, then the denominators of the two fractions of the righthand member are equal, and greater than or equal to the denominator of the lefthand member. Since, in addition, N(S_{1} ? S_{2}) = N(S_{1}) + N(S_{2}), the inequality is satisfied.
54The case where N(S_{1}) > 0 and N(S_{2}) = 0 remains to be examined. Two cases can be distinguished.
55Let us first consider the case where N(F_{j}\S_{1}) = 0, i.e., in F_{j}, the only lists able to continue alone belong to S_{1}. In this case, N(F_{j}\(S_{1} ? S_{2})) = 0 and, since N(S_{1} ? S_{2}) > N(S_{1}), the inequality is satisfied.
56Let us consider now the case where N(F_{j}\S_{1}) > 0:
58and the inequality is satisfied since N(S_{1} ? S_{2}) > N(S_{1}).
59We thus formally find the first source of (strict) superadditivity announced in the introduction: the existence of critical thresholds for being able to continue to stand alone in the second round.
Majority Bonus Component of the Game
60Let us now examine the fraction of seats corresponding to the bonus given to the list arriving first.
61It can be immediately verified that if the coalition S_{1} is able to guarantee the bonus, this is also the case for the coalition S_{1} ? S_{2}. Indeed, if the list S_{1} is able to be sure of the bonus, it means that N(S_{1}) > N(F_{j}\S_{1}) and N(S_{1}) > N(F_{l}) for any l = 1,…, J, l ? j. But then necessarily N(S_{1} ? S_{2}) > N(F_{j}\(S_{1} ? S_{2})) and N(S_{1} ? S_{2}) > N(F_{l}) for any l = 1,…, J, l ? j, which proves the result.
62Let us consider now the case where none of the coalitions gets the bonus alone: V_{prime}(S_{1}) = V_{prime}(S_{2}) = 0. It is immediately verified that V_{prime}(S_{1} ? S_{2}) ? V_{prime}(S_{1}) + V_{prime}(S_{2}), with a strict equality when the coalition S_{1} ? S_{2} arrives first, i.e., when N(S_{1} ? S_{2}) > N(F_{j}\(S_{1} ? S_{2})) and .
63We thus formally find the second source of (strict) superadditivity announced in the introduction: the existence of a majority bonus.
64Since on each of the parts of the game (the proportional part and the bonus part), the coalition S_{1} ? S_{2} guarantees for itself a payoff at least as high as the sum of the payoffs that the coalitions S_{1} and S_{2} are guaranteed separately, we have shown that the game is superadditive.
Bargaining Solutions
65In the previous section, we have built a cooperative game with transferable utility described by its characteristic function V, with players {1, 2,…, M^{2}}, where an alliance is possible if and only if it belongs to the set ? of admissible coalitions. At the end of the discussions/negotiations relating to the game of alliances, the political groups conclude one or more alliances, and within each of them they agree on the allocation of seats. This last decision is implemented through a sophisticated ordering [8] of the members of each list.
66Formally, an outcome thus consists of two parts: a division of the players into coalitions (alliances), (S^{q})_{1 ? q ? Q}, where S^{q} ? ? for any q, and an allocation x = (x_{1}, x_{2}, …, x_{M²}) so that:
68In this conventional definition in cooperative game theory, the prediction is twofold: which are the alliances formed and how do the members of these alliances share the seats when the power of coalition S is measured by V(S). We are going to skip here the first question and exclusively concentrate on the second, making the hypothesis that the maximum coalitions are formed.
69We have already indicated that this game is superadditive and thus, in principle, the players have nothing to lose by participating in the largest possible alliance. It is, however, unfair to claim that superadditivity alone leads to forming maximum coalitions. Indeed, if there is a threshold effect like in the case of a game with bonus only, the benefit from having additional players becomes zero once the threshold is crossed. Therefore, it should be expected that the bargaining solutions we are going to study will completely ignore certain players. In such a case, it might be more natural to consider that the alliance that will form is one with players receiving a positive playoff only. For example, consider the case where M^{2} = 3 groups are on an equal footing on the evening of the first ballot, and no alliance is taboo. It can be expected that two groups will ally to the detriment of the third, even if the game is superadditive. But which group will be left aside if they refuse to set a restrictive (i.e., binding) protocol such as the BaronFerejohn protocol? In this symmetrical context, it is also easy to imagine that, being worried that they could be the isolated group, the three players solemnly agree on the grand coalition and a corresponding allocation of the seats. It can also be imagined ad hoc (i.e., for reasons which are not described by the model itself) that external political “imperatives” force the formation of the grand coalition.
70Be that as it may, in the following we thus consider that maximum coalitions are formed, i.e., the partition F. [9] We have to explain how the allocation of seats will be determined. Contrary to a noncooperative game where the strategies of the players are clearly specified, the characteristic function V of a tu game simply summarizes the result produced by each coalition of players. According to the interpretation given by Moulin (1981):
Let us insist on the fact that the outcome of the game no longer depends on the strategic behavior of the players involved, but that, quite the opposite, the power of decision is put, indivisibly, in the hands of the community in charge of arbitration, without appeal, between the conflicts of opinion of its members. The essential point is the obligation to cooperate in making the decision: the power of the individual players and the coalitions of players only has an indirect impact. The players do not use this power as a threat, but the community takes it into account when determining if such or such apportionment is unfair to them.
The Proportionality Rule
72A first solution, whose link with the cooperative game theory presented above is not immediate, consists in retaining an apportionment where the share of each group m in the grand coalition is proportional to the value N^{1}_{m} representing its electoral score on the evening of the first round. This solution is not defined for any cooperative game, but applies directly to the cooperative game defined in the previous section. In a certain way, as we will see in the following, this rule represents the basis for discussions proposed by the leaders of the leftwing groups during the regional elections of 2010 to which the last part of this article is devoted.
73The definition of this very simple and natural solution does not at all require game theory terminology. It has nevertheless generated a series of articles in game theory, associated with the name of the sociologist Gamson (1960), who was the first to raise questions about forming coalitions in the context where the negotiators have their hands tied by the use of a rule and the resulting determination of how surpluses are allocated. The game thus built is, however, a game without transferable utility because the rules for allocating the surplus of a coalition once it is formed result from a rule which is exogenous to the model (as is, e.g., this proportionality rule). The cooperative game of Gamson is thus a game focused on the question: what coalition(s) will form? Gamson and his successors tried to answer this question and tested empirically or experimentally the proposed predictions. Sometimes, these predictions [10] are clearly defined (uniqueness), but they often generate indeterminacy. In the symmetrical case mentioned in the introduction to this section, e.g., the three structures where a player is left aside are retained. Here, the argument of the instability due to a possible treachery does not hold: the agreement with the partner does not put him in a better situation. On the other hand, any minimum deviation from symmetry breaks this multiplicity.
74Here, our reference to Gamson is therefore a little bit unfair, because we only mean by this the allocation of seats in the grand coalition resulting from the application of the proportionality rule. We adopt here a widespread political science terminology, as proved by, e.g., the work on the distribution of ministerial portfolios in the process of forming coalition governments (Brown and Frendreis 1980; Carroll, Cox, and Pachón 2004; Fréchette, Kagel, and Morelli 2004; Laver 1998; Laver and Schofield 1998; Laver and Shepsle 1996; Warwick and Druckman 2001). Although, in the case of regional elections, the proportionality rule has been cited many times by national leaders of the political groups, it is unclear if it acted as a rigid restriction in the negotiation processes held at the regional level.
The Nucleolus and the Shapley Value
75The nucleolus, like the Shapley value, assumes that the negotiators do not feel bound in advance by rules or practices that would limit their powers: the game is now a real transferable utility game. We want to test the resistance of a proposition x to the negotiating powers of the players as reflected by the values in the characteristic function V. [11]
76Let us start by some definitions. Consider an arbitrary cooperative game tu where with n ? 2 is a finite set of players and V is a function which associates a real number V(S) with each subset S of . [12] It is assumed that V(Ø) = 0.
The Nucleolus
77We will denote by the set of imputations, i.e., the set of allocations which are individually rational, where an allocation is called rational if it guarantees to each player what he can obtain alone if he does not conclude any alliance. An arrangement x ? X_{IR} is then a distribution of seats within the grand coalition based on the power of the players taken individually. For any distribution x ? X_{IR}, we calculate ?(x) the vector of dimension 2^{n} whose coordinates are the numbers
79for ordered from the largest to the smallest, i.e., so that ?^{i}(x) ? ?^{j}(x) for 1 ? i ? j ? 2^{n}.
80The number e(S, x) is called the excess of the coalition S: the higher this number, the more coalition S is entitled to be unhappy with the allocation reflected by x.
81The nucleolus of is the unique vector N_{u}(V) = x^{*} ? X_{IR}, so that ?(x^{*}) is a minimum, in the sense of the lexicographic order within the set {?(y)  y ? X_{IR}} [13].
Shapley Value
82The Shapley value Sh(V) (Shapley 1953) of the game V is the vector Sh(V) defined as follows:
An Illustration of the Relations between These Bargaining Solutions
84One of the questions one could ask is to what extent the nucleolus and the Shapley value are different or not, de facto, from the proportional allocation rule of Gamson, in the electoral game of alliances we are considering here.
85To understand the nature of these potential divergences, let us consider this very simple example. Let us use the notations introduced in the section devoted to the electoral environment, and let us consider the case where N_{tot} = 100, K is normalized to 1, [14] ? = {{1,2,3}, {4}}, ? = 0,05, ? = 0,10 and N^{1} ? (0, 30, 36 – x, 8, 26 + x) with 15 ? x ? 25. [15] The fourth political group is present in the second round, but, according to our hypothesis on ?, it cannot be part of any alliance, i.e., the alliance game is a threeplayer game. By choosing different values for the parameter x in the interval (15, 25), several possible configurations are covered. For example, when x = 25, N^{1} ? (0, 30, 11, 8, 51) and, in this case, the bonus goes to the group 4 even in the case where the maximum coalition {1, 2, 3} is formed. When x = 22, only the coalition {1, 2, 3} can guarantee the bonus for itself. Finally, when x = 19, the {1, 2} coalition can guarantee the bonus for itself (player 3 is not indispensable). We will successively consider these three cases.
86In the case where for the nucleolus, we obtain:
88On the other hand, for the Shapley value, we obtain:
90Although different, these two allocation solutions predict quite close shares for the first player. The total windfall here is . Expressed in proportions of this windfall accessible to the grand coalition {1, 2, 3}, the distribution vectors are equal to (63.649%; 24.873%; 11.475%) for the nucleolus and to (65.169%; 26.384%; 8.462%) for the Shapley value, while the Gamson key is equal to (61.224%; 22.449%; 16.327%). Here, the bargaining solutions differ from Gamson for the third player: they deviate by 5 percentage points (for the nucleolus) and 8 percentage points (for Shapley). In the bargaining solutions, the third player suffers from the fact that, if he does not conclude any alliance, he does not obtain any seat.
91For the nucleolus, we obtain:
93while for the Shapley value, we obtain:
95The total windfall of 0.64 is allocated here according to the proportions (50.486%; 31.736%; 17.77%) for Shapley and (51.302%; 32.552%; 16.145%) for the nucleolus. This can be compared with the Gamson key which equals (57.692%; 26.923%; 15.385%). In this case, the two bargaining solutions differ little from Gamson for the third player. The difference is significant, on the other hand, between the first two players in favor of the second who wins here on the order of 5 percentage points. How to explain these differences? Concerning the second player, he is indispensable in this configuration for obtaining the bonus. This enables him to claim a third of the bonus (at the same time for the nucleolus and for Shapley), which is a higher proportion than that represented by his share of votes in the first round. As for the third player, the two components of the game affect his negotiating power in a contradictory manner. As shown by the previous example, the proportional part of the game is unfavorable for him (with respect to the proportionality rule): indeed, without an alliance, he does not win any seat, which puts him in a position of weakness. On the other hand, the bonus part plays in his favor, insofar as, like the second player, he is necessary for winning the bonus. These two effects compensate each other, and we obtain for the bargaining solutions values which are close to the proportionality rule.
96For the nucleolus, we obtain:
98while for the Shapley value, we obtain:
100The total windfall of 0.6625 is allocated here according to the proportions (55.111%; 40.394%; 4.483%) for Shapley and (56.251%; 41.534%; 2.215%) for the nucleolus. This can again be compared to the Gamson key of (54.545%; 30.909%; 14.545%). In this case, the two bargaining solutions are also very close. The share of the big player is close to his Gamson share. On the other hand, the gap between bargaining and Gamson is significant for players 2 and 3. In particular, player 3 suffers from the fact that not only is he not able to stand alone, but moreover, in this configuration, the two other players do not need him to guarantee the bonus for themselves.
101These three cases are far from covering all configurations. Their purpose is to illustrate how to build the characteristic function, to show some computations for the two bargaining solutions, and to illustrate how they compare with Gamson. The calculations naturally reflect the qualitative implications of the evolution of the corresponding forces of the three players in the calculation of the bargaining solutions. The third player has a rather weak bargaining position because he has to count on at least one of the other two player to stay in the second round. From the point of view of the proportional part, this weakness costs him a lot in comparison with Gamson. On the other hand, when the bonus is at stake, he can do very well if his participation is essential (example 2). In this component of the game, it is more the structure of the winning coalitions that matters than the electoral results after the first round. In example 2, player 3 wins back a significant negotiating power that he loses again in example 3.
Introduction of a Typology of Electoral Configurations
102The illustrating examples proposed above emphasize well the importance of the two sources of superadditivity identified in the section devoted to the game properties. This superadditivity comes from the proportional part of the game (which are the lists that need alliances to be able to continue to stand in the second round?) and from the majority bonus component of the game (which are the coalitions that can guarantee the winning of the bonus?).
103The practical details of the bargaining solutions, [16] and the deviations of the bargaining solutions from Gamson’s allocation, will depend on the intersection of these two criteria: “Which are the lists who need an alliance to be able to continue to stand in the second ballot?” and “Which are the coalitions that can guarantee the winning of the bonus?”.
104In the next section devoted to the regional elections of March 2010, we will discuss in detail the configuration typology resulting from the intersection of these two criteria, and we will classify the metropolitan regions according to this typology.
Application to the Regional Elections of March 2010
105On the evening of the first round of the regional elections, on Sunday March 14, 2010, the different political groups discover their results, region by region. Among the political groups likely (ex ante) to continue to stand alone in at least one region (i.e., the groups m, so that in at least one region) are the lists formed around the Socialist Party (ps), the Union for the Presidential Majority (ump), the Movement for Democracy (MoDem), Europe Ecology (ee), the Left Front (Front de Gauche, fg), and the National Front (fn). Only the ps and the ump have crossed this threshold of 10% in all of the regions. [17] Among the political groups likely to be able to ally in certain regions (i.e., groups m so that in at least one region) are, in addition, the New Anticapitalist Party (npa).
106To what game of alliances will we devote our attention? For now, we leave aside Corsica and the three overseas regions which are more complex to study, because the game of alliances is not as simple as that retained for formalizing the electoral environment of the twentyone metropolitan regions. For these regions, if no alliance were “forbidden” (estimated as strategically or ideologically undesirable, politically incorrect, or demonized), the ump, the MoDem and the National Front would be players in their own right. In several regions, the MoDem crosses the threshold of 5% and, in one region, crosses that of 10%. The fn crosses the threshold of 5% in all of the regions and very often crosses the threshold of 10%. Assuming that an alliance of the MoDem with the ump or with the ps or with the ee, or that an alliance of the fn with the ump, are in the field of the possible alliances, we would get a fiveplayer game of alliances. But here, given the political context of these 2010 elections, we have retained the following hypotheses on the admissible alliances. We discard (according to the political context of these elections) the participation of the fn and of the MoDem in any alliance. The ump (or more exactly the presidential majority prealliance) thus becomes completely isolated. There are then only three potential players left, namely the three main leftwing groups: the ps, Europe Ecology, and the Left Front. In seventeen regions, these three parties presented separated lists on the first ballot. Among these seventeen regions, in twelve of them, the three parties cross the threshold of the 5% of the votes expressed during the first ballot, which makes the game of alliances between the two ballots a threeplayer game (but in one of these twelve regions, namely Languedoc, none of them crossed the 10% threshold, see Footnote 21). In five of them, only the ps and ee crossed this 5% threshold, which actually reduces the game to a twoplayer game. In the remaining four metropolitan regions (LowerNormandy, Burgundy, Champagne, and Lorraine – Corsica excluded), there was a preelectoral alliance (before the first ballot) between the ps and the fg, and Europe Ecology crosses the threshold of 5%; the game of alliances is there again a twoplayer game, this time between the joint list psfg and the ee list.
107Before proceeding to an examination of the permutations of the set of conceivable configurations (according to the intersection of criteria defined in the subsection “Introduction of a typology of the electoral configurations”) and to a more practical inventory of the relevant subset in the case of the regional data we have, we will study in more detail three regions and calculate for them, the shares allocated to each of the alliance partners. We report each time the shares observed, the shares resulting from the two bargaining models considered here (the nucleolus and the Shapley value), and finally the shares resulting from an a priori rule such as the proportionality principle (the Gamson rule). The interest of these calculations is to measure the deviations if any between the two bargaining solutions, and between those and the Gamson solution. [18]
108More specifically, we will consider in detail the cases of Aquitaine, Auvergne, and Brittany. The two regions of Aquitaine and Auvergne are chosen because they represent, among the regions where the Left Front is in the position of continuing to stand in the second round, two polar cases relative to the strength of the ps in the negotiations with its potential allies. In Aquitaine, the ps is extremely strong insofar as its two potential allies need it for continuing to stand in the second round, and as it does not need any alliance to win the majority bonus. In Auvergne, on the other hand, Europe Ecology and the Left Front can stand alone, and the ps needs an alliance with one of these two partners to win the majority bonus. The results obtained for the Shapley value and the nucleolus in these two regions are thus the extreme bounds of what is to be expected to be observed elsewhere in the other regions (cf. infra the typology of regions in “Systematical analysis of the metropolitan regions”). The case of Brittany is also interesting to consider, since it is one of the rare examples where the negotiations between the two rounds between the ps and Europe Ecology failed, and where the two lists stood separately in the second round.
Aquitaine
109Aquitaine is a region where the Socialist Party is in a position of strength during the negotiations between the two rounds, facing potential allies. Indeed, the Ecologists and the Left Front are incapable of standing in the second round without an alliance with the ps. Moreover, the ps has enough lead over the ump to be sure of winning the election even without any alliance.
Election Results
110In this region, K = 85 seats are available. The detailed results of the election are shown in Table A1 (page 61) of the working document of Le Breton and Van der Straeten (2012) (data obtained from the Internet site of the Ministry of the Interior).
First Round
111P^{1} = 11 lists run and N_{tot} = 2,280,634 voters were registered for the first round of March 14, 2010. The number of absentee voters was 1,150,523 (number of registered voters minus the number of voters who voted). The number of blank or invalid votes was 48,912. The sum of the absentee voters and blank or invalid votes was thus N^{1}_{0} = 1 199 435.
112Classifying the lists in the decreasing order of the results after the first round, the PS list obtained N^{1}_{1} = 406 871 votes, the ump list N^{1}_{2} = 238 367,…, down to the list led by X.P. Larralde who obtained N^{1}_{11} = 230 votes.
Second Round
113Here, R^{2} = 3 lists crossed the threshold of 10% and can continue to stand alone: the ps (37.6% of the votes expressed in the first round), ump (22.05%), and MoDem (10.4%) lists. Three lists have obtained a percentage of the votes expressed between 5 and 10% and are authorized to merge with one of the first three lists: the Europe Ecology (9.75%), fn (8.3%), and Left Front (5.95%) lists. The remaining five lists cannot participate in the second round. We thus have here M^{2} = 6 lists which can potentially participate in the second round.
114In this region, P^{2} = 3 coalitions are indeed running in the second round: the ump and MoDem lists stand as in the first round, and a Left Union list unites the Socialists, the Ecologists, and the Left Front. The fn does not merge with any list.
115The Left Union list wins N^{2}_{1} = 643 767 votes in the second round (which corresponds to 56.33% of the votes expressed in the second round), the ump list, N^{2}_{2} = 320 105 votes (28.01%), and the MoDem list, N^{2}_{3} = 178 852 votes (15.65%).
116The allocation of seats between these three coalitions is then as follows. The leftwing list wins the bonus for the leading list, which corresponds to a 1 – ? = 25% fraction of the seats. The remaining seats (in proportion of ? = 0.75) are allocated proportionally to the results in the second round. This gives in total for the leftwing list a theoretical percentage of seats equal to 0.25 + (0.75 * 0.5633) = 67.25%, for the ump list 0.75 * 0.2801 = 21.01%, and for the MoDem list 0.75 * 0.1565 = 11.74% of the seats. The number of available seats being K = 85, this gives a theoretical number of seats equal to 57.17 for the left, 17.86 for the ump, and 9.98 for the MoDem. The rule of rounding off to integers actually gives the following apportionment: 58 seats for the left, 17 for the ump, and 10 for the MoDem.
The Allocation of Seats within the LeftWing List
117Let us discuss now the distribution of the 58 seats within the leftwing coalition.
118To simplify the interpretation of the calculations, we will denote correspondingly by N_{PS}, N_{UMP}, N_{MoDem}, N_{EE,} and N_{FG} the number of votes after the first ballot of the ps, the ump, the MoDem, Europe Ecology, and the Left Front. [19] The sum of these votes is denoted by N = N_{PS} + N_{UMP} + N_{MoDem} + N_{EE} + N_{FG}.
Observed Apportionment
119How were the 48 seats won by the Left shared among the three allies? The following distribution took place: the ps got 45 seats, the Ecologists 10, and the Left Front 3. Expressed in percentages of seats that the coalition had to allocate, we obtain the numbers shown in the column “Observed” in Table 1.
Allocation of seats within the left union list in Aquitaine: (i) observed, and predicted by (ii) Gamson, (iii) Shapley, (iv) the nucleolus (in %)
Allocation of seats within the left union list in Aquitaine: (i) observed, and predicted by (ii) Gamson, (iii) Shapley, (iv) the nucleolus (in %)
120Our objective is now to compare this distribution to a Gamson allocation rule, to the Shapley value, and to the nucleolus.
Gamson’s Allocation Rule
121Gamson’s allocation rule states that the allocation of seats won by the coalition is proportional to the electoral weight of each member of this coalition. Admitting that the electoral weight of each party in the coalition is itself proportional to its result in the first round, one can see that, in this coalition, the ps has an electoral weight of the Ecologists of 18.3% and the Left Front of 11.2%. These numbers are shown in the column “Gamson” in Table 1.
The Characteristic Function
122The characteristic function of the game played within the left wing is the following (in percentages of seats available in the region):
124e.g., if the Left grand coalition would form, it would win, with our hypotheses on the behavior of voters, a fraction of the seats distributed according to the proportional rule, plus the seats allocated as a bonus. If the Ecologists do not conclude any alliance, they do not win any seat. If the Socialist Party does not conclude any alliance, the Socialists will get N_{PS} votes, the ump N_{UMP} votes, the MoDem N_{MoDem} votes, and the Ecologists and the Left Front will not able to continue to stand without an alliance with the ps. The ps thus wins of the votes expressed. It wins the fraction (1  ?) of the seats allocated as a bonus to the leading list, and of the fraction ? of remaining seats.
Shapley Value
125As an example, we are giving here the details of calculating the Shapley value. Knowing the characteristic function, we can now calculate the marginal contribution of each of the three parties ps, ee, and fg to the grand coalition, according to its order of formation. This contribution is expressed as a percentage of seats available in the region. The Shapley value is the average of these marginal contributions, and is calculated on the last line of the table.
126One can see in the table (in the column “PS Contribution”) that the Shapley value allocates to the ps all of the seats associated with the bonus.
127Let us look now at the share of seats allocated to the ps according to the proportional component of the game. One can see that, in all cases, the marginal contribution of the ps is larger than . This is the conjunction of two effects, which we will call “standing effect” and “dilution effect.” The standing effect corresponds to the fact that, by concluding an alliance with parties which did not cross the threshold of 10%, the ps enables them to continue to stand, and thus authorizes the taking up of the firstround electoral votes from these parties. It is visible on the lines corresponding to the eepsfg order of formation, where the ps enables ee to continue to stand, on the line fgpsee where the ps enables fg to continue to stand, and finally on the lines eefgps and fgeeps where it enables ee and fg at the same time to continue to stand. The second effect, called the “dilution effect” comes from the fact that if the ps does not conclude any alliance (or if all possible alliances are not concluded), it is in the position of excluding some groups from the second round. In the absence of these groups, for a given number of votes for the ps (N_{PS}), the ps votes from the first round represent a larger percentage of the votes in the second round. This effect is visible, e.g., in the pseefg or psfgee orders of formation where the parts of votes received by the ps equal , which is greater than . It also shows when the ps joins ee alone or the fg alone.
128Having calculated the Shapley values expressed in percentages of the seats in the region, we can now calculate how the seats obtained by the left would have been apportioned if the allocation had been proportional to the Shapley values. The ps would have won 95.7% of the seats, the Ecologists 2.7%, and the Left Front, 1.6%. These numbers are shown in the column “Shapley” in Table 1.
The Nucleolus
129Under the hypothesis retained here that the ps can be sure of the bonus without any alliance, the calculation of the nucleolus is a 3.A. case in the typology defined further (See Table 5). Using the formulas from Appendix II (page 41) of the working document by Le Breton and Van der Straeten (2012), we find for the nucleolus:
131i.e., here:
133Having calculated the nucleolus in percentages of the seats in the region, we can now calculate how the seats obtained by the left would have been apportioned if the allocation had been proportional to the values allocated to each player by the nucleolus. The ps would have won 96.2% of the seats, the Ecologists 2.5%, and the Left Front, 1.5%. These numbers are shown in the column “Nucleolus” in Table 1.
134The configuration in Aquitaine is an extreme case where the ps dominates its two potential partners on both counts: it does not need them for the bonus, and they need it to stand in the second round. Without any surprise, the negotiation logics, either Shapley or the nucleolus, lead to allocate to it the lion’s share. This is far from the results observed where the ps share was only 77.6%. This value is closer to Gamson’s model.
135As we will see in the last subsection devoted to a systematic analysis of the metropolitan regions, the configuration in UpperNormandy is qualitatively similar to that in Aquitaine.
Auvergne
136Auvergne is a region where, on the contrary, the ps is in a rather unfavorable position. Indeed, the Ecologists and the Left Front can continue to stand without making alliances. Furthermore, the ps needs at least one ally to gain the bonus of 25% of the seats assigned to the party arriving in the lead, and one cannot even exclude an alliance between the Ecologists and the Left Front arriving in the lead if the ps goes it alone.
The Results of the Election
First Round
137K = 47 seats are available. [20]
138P^{1} = 8 lists run and N_{tot} = 994,160 voters were registered for the first round.
139Classifying the lists in decreasing order of the results in the first round, the ump list obtained a number of votes N^{1}_{1} = 137 232, the ps list N^{1}_{2} = 133 925, the Left Front list N^{1}_{3} = 68 146, and the Ecologist list N^{1}_{4} = 51 106…
Second Round
140In Auvergne, R^{2} = 4 lists have crossed the threshold of 10% and are in position to stand alone: these are the ump, ps, the Left Front, and the Ecologist lists. The fn obtains 8.4% of the votes and cannot continue to stand without an alliance. The remaining three lists cannot participate in the second round. We thus have here M^{2} = 5 lists which can potentially participate in the second round.
141In this region, P^{2} = 2 coalitions are indeed running in the second round: the ump and MoDem lists stand as in the first round, and a Left Union list unites the socialists, the Ecologists, and the Left Front. The fn does not merge with any list.
142In the second round, the Left Union list arrives first with 59.68% of the votes expressed. It wins 33 seats, while the ump list wins 14.
The Allocation of Seats within the Left List
143Let us discuss now the distribution of seats within the leftwing coalition. We denote by N_{PS}, N_{UMP}, N_{MoDem}, N_{EE,} and N_{FG} the number of votes in the first round for the ps, ump, MoDem, ee, and fg. We denote by N = N_{PS} + N_{UMP} + N_{MoDem} + N_{EE} + N_{FG} the sum of these votes.
Observed Apportionment
144The following distribution took place: the ps won 17 seats, the Left Front, 9, and the Ecologists, 7. Expressed in percentages of seats that the coalition had to allocate, we obtain the numbers shown in the column “Observed” in Table 2 below.
Allocation of seats within the Left Union list in Auvergne: (i) observed, and predicted by (ii) Gamson, (iii) Shapley (two variants), (iv) the nucleolus (two variants), in %
Allocation of seats within the Left Union list in Auvergne: (i) observed, and predicted by (ii) Gamson, (iii) Shapley (two variants), (iv) the nucleolus (two variants), in %
Gamson’s Allocation Rule
145Gamson’s allocation rule states that the allocation of seats won by the coalition is proportional to the results in the first round. This rule involves a percentage of seats of for the ps, of 26.9% for the Left Front and of 20.2% for the Ecologists, shown in the column “Gamson” in Table 2.
The Characteristic Function
146What could the ecologists and the Left Front hope for if they together form a coalition excluding the ps? Seeing their results, the most likely scenario is that such an alliance could not win the region, and that the ump would win (we will call this hypothesis “variant 1”). The margin with the ump of this coalition is, however, less than 4 percentage points of the votes expressed in the first round, and it is not completely impossible that this coalition could win (we will call “variant 2” the scenario that a leftwing coalition excluding the ps could win the second round). We will successively study these two variants.
147In variant 1, the electoral gain of each of the possible coalitions within the left wing is the following (in percentages of seats available in the region):
149Let us note that the only difference between variants 1 and 2 resides in the electoral gain that the coalition formed by the Ecologists and the Left Front can be sure of. In variant 2, we have:
Shapley Value
151We can now calculate the marginal contribution of each of the three parties ps, ee, and fg to the grand coalition, according to its order of formation. This contribution is expressed in the percentage of seats available in the region. The Shapley value is the average of these marginal contributions, and it is calculated on the last line of each table:
152One can see, in these tables, that in variant 1, the ps wins twothirds of the bonus allocated to the leading list, and only onethird in variant 2.
153Having calculated the Shapley values expressed in percentages of the seats in the region, we can now calculate how the seats obtained by the left would have been apportioned if the allocation had been according to the Shapley value. These numbers are shown in the columns “Shapley (V1)” and “Shapley (V2)” of Table 2.
The Nucleolus
154In variant 1, we obtain for the nucleolus (See the working document by Le Breton and Van der Straeten (2012) for the details of the computation):
156i.e., here:
158Transformed into percentages of seats won by the left, these numbers are given in the fifth column of Table 2.
159In variant 2, we obtain for the nucleolus:
161which coincides with the Shapley value in this case.
162These numbers are shown in the columns “Nucleolus (V1)” and “Nucleolus (V2)” of Table 2.
163The comparison of the logic of the two variants is very instructive. In both cases, the two components of the game can be separated. For the first proportional component, the parties get their corresponding shares. For the bonus component, the calculation depends upon the variant retained. In the case of variant 1, the nucleolus gives the whole bonus to the ps. This results from the fact that the ps is in a privileged position with regard to the bonus. The two other groups are perfectly replaceable and thus “bear the brunt” of their competition with regard to the bonus. In the case of variant 2, the three groups play a symmetric role and each get a third of the bonus.
164One can also see that, in the case of variant 2, the nucleolus and the Shapley value coincide, while they do not coincide in the case of variant 1. In the case of variant 1, the Shapley value in fact allocates a sixth of the bonus to each of the two smallest groups.
165In the case of Auvergne, Gamson’s logic seems to have won over a logic of negotiation if variant 1, which seems the most plausible, is privileged.
Brittany
166The region of Brittany is an interesting case to study insofar as it is one of the rare regions where negotiations between the leftwing parties failed, and where there was no merging of lists before the second round. Understanding the balance of power just after the first round will help us grasp the reasons for this failure.
Election Results
First Round
167K = 83 seats are available. [21]
168P^{1} = 11 lists are in the running and N_{tot} = 2,332,945 voters were registered for the first round.
169Classifying the lists in decreasing order of firstballot results, the ps list led by JeanYves Le Drian won the number N^{1}_{1} = 408 551 votes, the ump list led by Bernadette Malgorn N^{1}_{2} = 260 731, and the Europe Ecology list led by Guy Hascoët N^{1}_{3} = 134 161…
Second Ballot
170In Brittany, R^{2} = 3 lists crossed the threshold of 10% and could have continued to stand alone: the ps (37.2%), ump (23.7%), and ee (12.2%) lists. The fn obtained 6.2% of the votes, the MoDem 5.4%, and these two lists could not continue to stand without an alliance. The other lists could not participate in the second round (in particular, the Left Front (3.5%) was not in the position of continuing to stand, even with an alliance). We thus have here M^{2} = 5 lists which can potentially participate in the second round.
171In this region, P^{2} = 3 coalitions are actually present in the second round: the ps, ump, and Europe Ecology lists stand as in the first round, the fn and the MoDem have not merged with any list.
172The results of the second ballot are such that the ps arrives first with 50.27% of the votes expressed. It wins 52 seats, while the ump list wins 20, and the Europe Ecology list wins 11.
The Theoretical Allocation of Seats within the Left List
173Let us now discuss the distribution of seats within the two leftwing lists present in the second round. Here again we denote by N_{PS}, N_{UMP}, and N_{EE} the number of votes in the first round for the ps, the ump, and the ee, correspondingly. We denote by N = N_{PS} + N_{UMP} + N_{EE} the sum of these votes.
Observed Apportionment
174In total, the two left lists (ps and ee) won 52 + 11 = 63 seats. The ps won 52 seats, meaning 82.5% of the seats held by a leftwing party, and Europe Ecology, 11, meaning 17.5% of the seats of the leftwing lists. These percentages are shown in the second column in the Table 3 below.
Allocation of seats for the leftwing lists in Brittany (i) observed, and predicted by (ii) Gamson, (iii) Shapley, (iv) the nucleolus, in %
Allocation of seats for the leftwing lists in Brittany (i) observed, and predicted by (ii) Gamson, (iii) Shapley, (iv) the nucleolus, in %
Gamson’s Allocation Rule
175Gamson’s allocation rule states that the allocation of seats won by the coalition is proportional to the electoral weight of each member of this coalition. Admitting that the electoral weight of each party in the coalition is itself proportional to its result in the first round, one can see that, in this coalition, the ps has an electoral weight of and the Ecologists of 24.7 %.
The Characteristic Function
176In the Breton case, the ps is14 percentage points ahead of the ump, and is thus in the position of winning alone the bonus to the leading list.
177The electoral gain of each of the possible coalitions within the left is as follows (in percentages of seats available in the region):
The Shapley Value and the Nucleolus
179In this case, the Shapley value and the nucleolus coincide and allocate to each player the payoff he can be sure of alone. We obtain:
181Transformed into percentages of the sum of the seats won by these two groups, we find the figures given in Table 3: the ps wins 83.5% of the left seats, which is very close to the apportionment observed.
Discussions between the LeftWing Lists of Two Ballots
182These theoretical considerations shed light on the discussions between the two rounds between Guy Hascouët, leader of Europe Ecology, and JeanYves Le Drian, leader of the ps in Brittany.
183The morning of March 15, Hascouët wrote to Le Drian:
We affirmed this morning our desire to form a union based on the results expressed by universal suffrage: the application of d’Hondt’s rule (12.21%/37.19%) would thus give us 14 seats. And yet, if in 2004, you applied d’Hondt’s rule, and if everywhere else in France this rule is applied, we are sorry to see that six hours after our first meeting, your proposal steps back from the initial proposal made this morning. While your socialist friends ask for our solidarity, you deny the result of the vote expressed by the Breton men and women. We thus reiterate our proposal: respect the universal suffrage by applying d’Hondt’s rule….
185J.Y. Le Drian answered in writing:
Following up on our various exchanges, I wanted to let you know about our second proposal which reflects the same desire: that of finding the ways and means of agreeing to an immediate merger. I offer ten eligible seats, which corresponds to more than 12% of the seats of the regional council (result obtained by your list) and three positions of meaningful responsibility, meaning 13.6% of the total executive power.
187There was no agreement. In an article entitled “À chacun son interprétation de la rupture” (To Everyone His Interpretation of the Rupture), the regional daily newspaper Le Télégramme wonders, in its March 17, 2010, issue, about the reasons for this failure and publishes some insightful statements: Who is wrong, who is right? Who is responsible for the rupture? Who most respects the Bretons’ vote? Who chose the right arithmetic formula? Guy Hascoët (GH) and JeanYves Le Drian (JYLD) obviously, pass the buck. They have conflicting grievances.
GH. – We have been humiliated by the president’s arrogance, who offered us no more than ten seats, while the proportionality rule entitled us to fourteen, even fifteen. This is thumbing your nose at democracy.
JYLD. – We came up against the intransigence of Europe Ecology. With 12% of the votes, they should have had nine seats. We offered ten, we went as far as eleven and even twelve… a little bit without my knowing. But they stuck to a request for fourteen. I am not irritated, but sorry.
GH. – Fourteen representatives for us, about fifty for them, this was the right distribution based upon the results obtained by our two lists in the first round. The rule retained between our two groups (Green and ps) provides an equitable allocation of all of the seats, including of those of the bonus to the first.
JYLD. – Nine seats for eeb, this is the result after the allocation of the bonus of 25% of the seats to the leading list, meaning ours. No, it is out of the question to share the bonus. In cycling, the stage winner shares the bonus with his teammates, not with the teammates of the second.
GH. – JeanYves Le Drian did not respect the distribution principles established by his party at the national level. This is the only region in France where it is happening.
JYLD. – I never took my orders from Paris. Brittany is the only region in France where a list was already uniting the ps, the pc, and the Ecologists in the first round.
JYLD. – During the negotiations, we spent about fifteen minutes on our project and they deemed it to be compatible as it was. The only issue were the seats, the seats, and again the seats!
189One can see that the second component of the game, the bonus to the first list, was the major stake in the argument between the ps list and the Europe Ecology list. What do the previous theoretical analyses show? Supposing that the fn and the MoDem were definitely pushed aside from the alliances, the game included only three players (ps, ump, and ee). We just saw above that from our behavioral hypotheses, the ps list had the majority and thus had no need to take into account the ump risk. As observed by Christian Guyonvarc’h, vice president of the udb (Union Démocratique Bretonne – Breton Democratic Union) of the exiting regional council: “No, we do not think that the right can win thanks to a triangular contest. There is no Bernadette Malgorn risk.” Moreover, given the ideological barrier forbidding an alliance between the ump and Europe Ecology, it is clear that the Shapley value and the nucleolus predict a total confiscation of the bonus by the ps list. Not – contrary to what J.Y. Le Drian says – because his list arrived first in the first round, but because the simple electoral game derived from the possible alliances on the eve of the second round is a simple dictatorial game.
Systematical Analysis of the Metropolitan Regions
190The analysis of the three regions of Aquitaine, Auvergne, and Brittany has emphasized the different balances of power between leftwing lists in the regions. To have a more synthetic view of what is happening in all French regions, we propose here to classify the regions according to these balances of power. The theoretical analysis of the section devoted to the bargaining solutions teaches us that these balances of power depend upon two crucial elements:
191• Can the lists of Europe Ecology and the Left Front continue to stand for the second round without making any alliance?
192• Which are the coalitions of the leftwing lists that can win the majority bonus?
A Typology of the Balances of Power between LeftWing Lists
193In this part, according to what was announced in the subsection devoted to the introduction of a typology of electoral configurations, we set out to take a closer look at these various configurations and classify the metropolitan regions according to this typology.
194Let us first consider the case where, on the evening of the first round, the three leftwing parties are separately present (i.e., did not make any preelectoral alliance) and are all in the position of continuing to stand (making alliances between the two ballots, if needed). We should note that in all regions we will study, the ps arrives first among the leftwing lists present in the first ballot (we have excluded Languedoc from the analysis – see footnote 21).
195From the point of view of the proportional component of the game, three scenarios are possible: the three lists can stand alone (case 1); only the ps and the strongest list after it can continue to stand without any alliance (case 2); only the ps is in a position to stand alone (case 3). [22]
196From the point of view of the balances of power generated by getting the bonus, five scenarios are possible: the ps does not need anybody (case A); the ps needs only one ally, and the two other groups cannot guarantee the bonus for themselves (these situations themselves split into two cases: the ps needs only the weaker list (case B), the ps needs the stronger list (case C)); the ps needs two allies and the two other groups cannot guarantee the bonus for themselves (case D); the two other groups can guarantee the bonus for themselves without the ps (case E). [23]
197Thus a total of 15 = 3 x 5 are, a priori, logically possible. In fact, one of these configurations cannot occur. It is that where ee and the fg cannot continue to stand without an alliance with the ps for the second round, but where the sum of their results is such that, if they could ally together (without the ps) for the second ballot, they would be in a position to win the bonus (Case 3.E).
198A situation where only two groups take part in the game of alliances should be added to these threeplayer configurations. This could happen because the three groups participated in the first round on separate lists, but, after the first round, one of them did not cross the 5% threshold required to continue to stand alone. Or this could happen because two parties have made preelectoral alliances and participated on a joint list in the firstround. The game of alliances is then a twoplayer game and not a threeplayer game. From the point of view of standing, two cases are possible: the two lists can stand alone (case 1?) or only the ps is in a position to stand alone (case 2?). From the point of view of the weight of the ps for guaranteeing the bonus, two cases are also possible: the ps does not need anybody (case A?); the ps needs its ally (case B?). [24] There are thus only four configurations of this type.
199Summing threeplayer and twoplayer configurations, there are therefore in total 14 = 4 = 18 theoretically possible configurations.
Classification of Metropolitan Regions
200With this typology in hand, we can now classify the French regions. For now, we leave aside overseas regions and Corsica, which are more complex to study, because the game of alliances is not as simple as that two or threeplayer game which has been retained for formalizing the electoral environment of the twentyone metropolitan regions. We will go back to these regions in the conclusion. We also leave aside Languedoc (see Footnote 21).
201As explained above, the typology results from the crossing of two criteria: possibilities of standing in the second round and leftwing coalitions able to guarantee the bonus for themselves. With regard to the first criterion (possibility of standing in the second round), it is very simple to determine which scenario fits the lists of a given region: it is enough to observe the firstround results of the different lists and compare them to the critical thresholds for standing (5% and 10%). With regard to the second criterion, we need to make hypotheses on how voters would behave in the second round. We will retain most of the hypotheses here that were made about the vote transfer in the subsection devoted to building the characteristic function, nevertheless nuancing them slightly. Under these hypotheses, it is very simple to determine if a leftwing coalition wins or not in the second round: it is enough to compare the sum of the firstround results to the firstround result of the main opponent, i.e., in practice, in all cases, the ump. [25] In this section, we will qualify these hypotheses and assume that the leftwing parties do not make the comparison between the firstround results so abruptly. More precisely, we will assume, as we did when we have looked at Auvergne, that they use a “margin of error” when they estimate their chances to win. In a somehow arbitrary manner, we will set this margin of error at 3 percent: a leftwing list estimates that it will win in the second round against the ump, if and only if the sum of its firstround results exceeds the firstround result of the ump by 3 percentage points. This is meant to capture the uncertainty [26] governing the exact behavior of the voters (e.g., imagine a situation where the National Front is not in a position to stand in the second round. Contrary to what has been assumed so far, it can be considered plausible that some of its firstround voters vote during the second round for the ump list) and/or incorporate a certain aversion to risk on the side of the parties.
202Under these new hypotheses, from the firstround results we can calculate the configuration in which the leftwing parties are found. Table 4 shows the firstround results of the three leftwing parties and the ump in each of the metropolitan regions (outside Corsica and Languedoc).
Firstround results of the three leftwing parties and the ump^{*}
Firstround results of the three leftwing parties and the ump^{*}
* In these regions, the PS and the Left Front have formed a preelectoral alliance, and participated in a joint list in the first round.203How are things in practice in the French regions? Tables 5a, 5b, and 5c classify the regions according to the configuration in which they are found.
Classification of configurations on the eve of the second round
Three leftwing parties (ps, ee, and fg) separately present in the first round, and all in a position of continuing to stand
Three leftwing parties (ps, ee, and fg) separately present in the first round, and all in a position of continuing to stand
Three leftwing parties (ps, ee, and fg) separately present in the first round, and the fg is not in a position to continue to stand
Three leftwing parties (ps, ee, and fg) separately present in the first round, and the fg is not in a position to continue to stand
Preelectoral alliance (before the first round) between the ps and the Left Front
Preelectoral alliance (before the first round) between the ps and the Left Front
Classification of configurations on the eve of the second round
204Let us first consider the eleven regions where the three groups are active on the evening of the first round. In two regions, the three leftwing parties can continue to stand for the second round without any alliance (Auvergne and NordPasdeCalais are in this case 1). In six regions, only the ps and Europe Ecology or the Left Front are able to stand without any alliance (case 2). Finally, in three regions, only the ps can continue to stand without any alliance (case 3). We have distinguished here five theoretically possible configurations for the possibility of winning the bonus. In practice, only two occur. Indeed, in all cases, the ps cannot be ignored; moreover, either it can win alone (case A), or an alliance with its smallest potential ally is enough to ensure victory (case B). By coupling these two modalities (standing in the second round and bonus awarded to the winner), we thus obtain 3 × 2 = 6 configurations over the fourteen theoretically possible; these are the six cells in Table 5a. One can see that these cells are not empty.
205In the situations with two players (Tables 5b and 5c), the four situations are covered. We are in the case 1?B? in one region (Pays de la Loire), in the case 1?A? in three regions (LowerNormandy, Brittany, PoitouCharente), in the case 2?B? in two regions (Champagne, FrancheComté) and in the case 2?A? in two regions (Burgundy, Lorraine). Alsace is the only region in France where the left grand coalition cannot win the bonus.
206What can be said about the bargaining solution predictions in all these regions?
207Detailed theoretical considerations (related in the working document by Le Breton and Van der Straeten [2012]) teach us that, in all situations met in practice, the Shapley value and the nucleolus are the sum of the solutions for each of the two components of the game (which is not true for the nucleolus in general). In those regions, we can thus analyze the game component by component.
208Let us look first at the bonus. In all the cases where the three players are active, the nucleolus predicts that the ps gets the whole bonus. Shapley allots it the whole bonus in type A situations, and twothirds in the other cases (type B). Thus in all regions, the negotiation solutions predict that the ps wins a considerable part (if not the entire bonus). What about the regions where only two players are active? The nucleolus and Shapley predictions coincide in this case: the ps gets the entire bonus in type A? situations, and half of it in the other cases (type B?: Pays de la Loire, FrancheComté, and Champagne).
209Let us look now at the proportional part of the game. In the case where all of the potential allies of the ps can continue to stand, the Shapley value and the nucleolus coincide with Gamson. In all of the other cases, the ps should win more than what the proportional rule predicts, due to the two effects of standing and dilution previously stated (cf. discussion of the Shapley value when showing the detailed example of the Aquitaine region).
210In the end, this analysis of the metropolitan regions teaches us that applying the proportionality rule to the allocation of seats won by the Left Wing is most of the time detrimental to the ps compared to what the bargaining solutions predict, such as the Shapley value and the nucleolus, sometimes in very significant proportions as shown by the extreme case of the Aquitaine region (where these solutions allocate more than 95% of the leftwing seats to the ps).
Conclusion
211In this article, we have developed a general method focusing on the strategic analysis of forming alliances on the eve of the second round of an election where competing lists are authorized (under certain conditions) to continue with or without an alliance with other lists. We focused our attention on two popular solutions of cooperative game theory: the nucleolus and the Shapley value. The methodology was applied to the regional elections of March 2010 in metropolitan France (with the exception of Corsica).
212We have shown that the position of strength of the Socialist Party with respect to its potential allies Europe Ecology and the Left Front during these elections means that bargaining solutions such as the Shapley value and the nucleolus predict for the ps a much larger part than that resulting from the application of a proportionality rule (according to which the allocation of seats within the leftwing parties is proportional to the results of the first round). The thorough quantitative analysis of Aquitaine shows that bargaining solutions can go as far as allocating the ps more than 95% of the leftwing seats.
213And yet the proportional rule has been preferred by the staff at the national level, as shown by the exchanges between the leaders of the ps and the ee in Brittany. In practice, it appears that this rule has indeed been quite broadly applied within the regions. Our study of Aquitaine and Auvergne show that the observed allocation is closer to Gamson than to the predictions of the bargaining solutions. More generally, in a recent working document, Dunz (2011) showed that simple regressions do not reject the hypothesis according to which the proportional rule was applied by the leftwing parties. [27] It thus appears that the bargaining solution was not dominant during the regional elections of March 2010. However, it would be more interesting to study the case when deviations from the proportionality rule are observed and if these deviations follow the predictions of the bargaining solutions or not. Brittany is an extreme example in this case, since the application of the bargaining logic in this region led to a failure to form a leftwing coalition (and the ps and ee lists standing separately in Brittany obtained a number of seats very close to the bargaining solutions). In the case of Aquitaine, it can be seen that the ps obtained more seats than predicted by Gamson, to the detriment of the Left Front; these deviations follow the direction predicted by the Shapley value and the nucleolus. Other analyses would be required to determine if this is also verified in the other metropolitan regions.
Case of Corsica and Overseas Regions
214In the cases of metropolitan France regions (with the exception of Corsica, and Languedoc), the alliance game was a two or threeplayer game (depending upon the regions).
215In the case of Corsica, the game was clearly more complicated, because the regionalist/proindependence political component is very significant. In the first round, the ump came in first with 21.34% of the vote, followed very closely by the pnc led by Gilles Simeoni. The ps came in only third with 15.50%. Four other lists came in behind them, able to stand in the second round without an alliance because the threshold ? is lowered from 10 to 7% in Corsica. These lists (fg, prg, csd, cl) had very respectable results. On the evening of the second round, there was thus (leaving aside the ump) a sixplayer game. In fact, four lists stood in the second round.
216The overseas regions also showed specificities which should be taken into account. From this point of view, Reunion Island is a textbook case. The Right stood divided in the first round. More precisely, four right lists were presented, out of which three (let us say d1, d2, andd3) were led by ump members. The fourth list (denoted by d4) had a separatist dimension. To the left, three important lists were presented: the pc (Communist Party), the ps, and ee. Three lists could continue to stand in the second round without an alliance (the pc, ps, and D1). The pc came in first with 30.2%, followed by d1 with 26.4%, and ps with 13.1%. Three other lists could continue to stand provided they made alliances: d2 with 6.7%, d3 with 5.4%, and d4 with 5.9%. The pc did not ally with the ps but with d3! The ps was left alone. d2 was left on the sidelines. Finally, d1 allied with d4. The rightwing list clearly won the second round with a result of 45.5% versus 35.5% for the main leftwing list. In addition to the subtlety of the alliance game, these figures also show that our model of vote transfer has to be reconsidered in the case of this election.
217Our methodology applies easily to electoral environments where the number of players participating in the alliance game is greater than or equal to four. This is important, because who can predict what the political landscape will be in France in the future, and the alliance games, if groups such as the MoDem and the National Front take an active part in this process? The difficulties of extending the scope are not conceptual, but computational. However, there are efficient algorithms for calculating the nucleolus, which could be used in real games. A complete characterization of the nucleolus, on the other hand, for all conceivable configurations as carried out here (in the working document version of this paper) seems to be out of reach.
Extensions and Future Work
218In our future work we would like to examine more carefully the process of vote transfer between the two rounds of the election. Here we made simple hypotheses, but they were certainly inadequate in many cases (as just mentioned with regard to Reunion Island). An interesting stochastic model could consist of drawing the list of the possible states of the system in the second round and writing down the electoral mobility matrices for all the configuration alliances allowed in the second round. With such a model, we could express the stochastic result of each list in the second round and thus calculate the probability that one list wins the winner’s bonus. The only thing to be done after that is to adjust the characteristic function by taking into account the mathematical expectations. If the attitude to risk should be taken into account, then the characteristic function for an ntu game should be formulated. Indeed, if in the bilateral case, the extension of the bargaining solution to the case of players having an aversion to risk does not raise major difficulties, in the multilateral case (at least three players as in the major part of our article), this extension would transform our game into a nontransferable utility cooperative game with its succession of difficulties.
219This first step would be required to build a real econometric test of the bargaining model proposed here. Writing an econometric version of the general bargaining model however, raises many questions.
220First of all, is this model testable? Can we determine from the observed data (in our case that would be mainly the allocation of seats) the behavioral implications of such a model? In such a case, it would then be possible to accept or reject the bargaining scenario as the explanation for the observed allocation of seats. In this article, this is the approach we have followed, matching however the bargaining solutions with specific hypotheses on the utility in case of agreement and in case of disagreement, and ignoring the stochastic component. We obtained specific formulas for the allocation of seats that we could compare with the observed allocations. At this stage, we see our work as a step towards an econometric model of electoral negotiations on the evening of the first round. [28] Our preliminary calculations are useful because they yield the order of magnitude of the two bargaining solutions. Comparison of the observed values leads to rejecting the bargaining model in the case where the negotiators have linear utility and where their values of reservation for all alliance configurations are based on the mobility matrices postulated in our article. The rejection is thus, strictly speaking, the rejection of this threehypothesis packet.
221It should then be asked if the model is identifiable, meaning if it is possible to identify from data the main parameters of the model. Again, this question is not treated in our article because it would require deriving a parametric and stochastic version of our model. We do not know of any econometric work on multilateral bargaining. When bargaining is bilateral, the situation is a little bit simpler: the cooperative bargaining solution is in fact Nash’s bargaining solution. This solution can be defined for arbitrary agreement and disagreement functions. The implications of Nash’s solution in the study of the data observed are very little studied. Actually, we have found only a very recent reference (Chiappori, Donni, and Komunjer 2012) who state testability and identifiability theorems. Among the hypotheses limiting the class of environments and bargaining situations considered, they assume that the utility in case of disagreement does not depend on the size of the cake to be shared in case of agreement. Clearly, this hypothesis is inappropriate in our context. Thus, there is still a long way to go to derive a thorough econometric version of our model.
Notes

[*]
Toulouse School of Economics (ut1gremaq) and Institut Universitaire de France. Correspondence: Toulouse School of Economics, cnrsgremaq, 21 allée de Brienne, 31000 Toulouse. Email: michel.lebreton@tsefr.eu.

[**]
Toulouse School of Economics (cnrsgremaq) and Institute for Advanced Study in Toulouse. Correspondence: Toulouse School of Economics, cnrsgremaq, 21 allée de Brienne, 31000 Toulouse. Email: karine.vanderstraeten@tsefr.eu.

[1]
A cooperative game is called superadditive if the payoff obtained by an alliance between two disjoint coalitions is at least as big as the sum of the payoffs of the two coalitions taken separately.

[2]
Given our assumptions about the transfer of votes between the two ballots, which will be described below.

[3]
This game is very close to a game analyzed by Gamson (1960), built on the basis of the proportionality rule.

[4]
Strictly speaking, a list having crossed the threshold ? can also withdraw from the competition and not run in the second round. This possibility will be neglected in the following and we will consider that all the lists above the threshold ? continue to stand for the second round (alone or merging).

[5]
To avoid trivial cases, we will assume that R^{2} ? 1: one list, at least, can continue to stand on the second ballot.

[6]
We consider more general mobility matrices in a work in progress with the political scientists Nonna Mayer and Nicolas Sauger.

[7]
In the presence of externalities between coalitions, the characteristic function becomes a more problematic notion (Rosenthal 1972). We discuss in the working document version of this article (Le Breton and Van der Straeten 2012) the more general concept of partition function.

[8]
Strictly speaking, the issue of the practical implementation of the bargaining solution is ignored here, because we discarded the difficulties relating to the uncertainty and to the integers. For an analysis of seat allocation functions, see Dunz (2011).

[9]
To paraphrase Aumann and Drèze (1974), we can say about our article that: “If the reader wishes, he may view the analysis here as part of a broader analysis, which would consider simultaneously the process of coalition formation and the bargaining of the payoff . . . Our analysis has been concerned with this last topic, and should thus be understood as a contribution to partial equilibrium analysis.” In fact, apart from some ad hoc articles on the topic, cooperative game theory has not yet developed a general and convincing approach to solve the former question. This is of course unfortunate, and we fully share the point of view expressed by Maschler (1992): “Consider a group of players who face a game. A basic question should be: What coalitions will form and how will their members share the proceeds? In my opinion, no satisfactory answer has so far been given to this important question. The current theory answers a more modest question: how would or should the players share the proceeds, given that a certain coalition structure has formed?”

[10]
See Le Breton, OrtuñoOrtin, and Weber (2008).

[11]
The reader will find in the working document of Le Breton and Van der Straeten (2012) a more detailed presentation of these two solutions and of their relations with other solutions such as, e.g., the kernel, the bargaining set, and the core. He will also find there an informal discussion on the objection/counterobjection logic at work in these solutions.

[12]
We will assume here that all coalitions are admissible.

[13]
More generally, we can consider any convex and compact subset X of R^{n} as a set of conceivable negotiation platforms.

[14]
The variable K does not play any role in this analysis; in addition, we neglect the integer problems.

[15]
We abandon here the convention according to which the lists are ordered in decreasing order of the electoral scores obtained in the first round.

[16]
To be found in the working document by Le Breton and Van der Straeten (2012).

[17]
Regarding the PS, this statement is not fully accurate. Indeed, in Languedoc, the official PS list only obtained 7.7% of the votes, far behind a dissident list led by Georges Frêche, which obtained 34.3% of the votes. In this region, EE got 9.1% of the votes and FG 8.6%. The dissident list led by Georges Frêche did not enter into any alliances with the PS, EE, or FG, and none of these lists could stand in the second round. We will exclude this region from the analysis when we classify all the French metropolitan regions.

[18]
This quantitative estimation is an essential and unavoidable step in building an econometric model aiming to explain how the apportionment is made when it is supposed to result from bargaining. Rejecting or accepting this hypothesis on the basis of the data observed by implementing a real statistical model would require more complex modeling though, and this is discussed in the conclusion.

[19]
For more readability, in this whole subsection, the parties’ lower index is replaced by their acronym. Moreover, we abandon the upper index 1 referring to the first round.

[20]
The detailed results of the election are shown in Table A2 (page 62) of the working document by Le Breton and Van der Straeten (2012).

[21]
The detailed results of the election are shown in Table A3 (page 63) of the working document of Le Breton and Van der Straeten (2012).

[22]
If the electoral law would allow two (or more) firstround lists who each received a minimum result of 5% to ally for the second round when the sum of their results is more than 10%, the game would be different. That could lead, e.g., to a PS at 15%, EE and FG at 9%, and UMP at 16%. In such a case, the strength of the EE and the FG would of course be greater.

[23]
We also have on top the case where even the grand alliance is not enough to win the bonus (cf. the case of Alsace).

[24]
Here too we have on top the case where even the grand alliance is not enough to win the bonus.

[25]
We should indeed recall that we made the hypothesis that neither the UMP nor any other party than the PS, EE, and FG can make any alliance.

[26]
In a future work, we intend to examine more rigorously the question of dealing with risk (margin of error, etc.). To do that, we will need to develop a stochastic model describing the voter flows between the two rounds (in all possible states of the system) and, if need be, switch from a transferable utility game to a nontransferable utility game.

[27]
This working document was discovered by the authors when they were preparing the revised version of this paper. Karl Dunz’s analysis is complementary to ours in that it carries out a more systematic quantitative treatment of the French regions. On the other hand, it does not include an explicit modeling of the negotiation games, nor the calculations of the bargaining solutions.

[28]
Even if our model is not econometric, it is perfectly suitable for an application preparing for a more thorough econometric work. We would like to draw attention to the fact that our approach is identical to that used in the work mentioned in the introduction on the allocation of ministerial portfolios in parliamentary democracies. The alliance game considered by these authors is a particular case of our model. It corresponds to the case where ? = 0, ? = 0, ? = 0, and where the vector with dimension P^{1} is interpreted as the vector of deputy seats in the assembly electing the government: N^{1}_{m} is the number of deputies affiliated to party m. In this particular model, to contrast Gamson’s model and a bargaining solution, it is required, as in our article, to list the mathematical implications of the bargaining solution. This is exactly what Laver and Schofield (1998) are doing for two bargaining solutions: the core and Schofield’s bargaining solution (1978). They accurately calculate the implications, as far as the allocation of portfolios is concerned, of the different solutions, and accept or reject a solution based on these calculations.