Relation to Literature

10The issue of forming coalitions between political players appears in various contexts other than the electoral environments. In many democracies, the multi-party 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 Austen-Smith 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 non-cooperative 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 pre-electoral alliance is concluded, there is nothing left to negotiate. [3]

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 pre-electoral alliances if any between various parties, P¹ 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 P¹ valid ballot papers in the first round. An electoral result in the first round is a vector of dimension P¹ + 1 where N¹₀ is the number of absentee voters and blank or invalid votes and N¹_m is the number of voters having voted for the list m (m = 1,…, P¹). 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 M², the number of political groups having reached the threshold ? and by R² (R² ? M²), 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 N¹ and the integers M² and R² which derive from it. Following possible alliances between these M² groups, P² 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 first-round list of one of the groups forming the second-round coalition, and must have obtained the agreement of the head of his/her first-round list. The electoral result of this second round is then described by a vector , where N²₀ is the number of absentee voters and of blank or invalid votes on the second ballot, and N²_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:

19

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.

The Second-Round 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:

32

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 ? = {T₁…, 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:

34

35otherwise, it obtains:

36

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]

38

39We will now give the explicit formula for V(S).

40If S ? {1, 2,…, R²} = Ø, V(S) = 0;

41If now S ? {1, 2,…, R²} ? 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 {F₁…, 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):

42

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:

45

46while the bonus component of the game equals, for any S ? ? (where prime signifies bonus in the following):

47

Superadditive Game

48We will now show that this game is superadditive, i.e., that for all coalitions S₁, S₂ ? ? so that S₁ ? S₂ = Ø and S₁ ? S₂ ? ?, V(S₁ ? S₂) ? V(S₁) + V(S₂).

49Let S₁, S₂ be admissible so that S₁ ? S₂ = Ø and S₁ ? S₂ ? ?. Since S₁ ? S₂ ? ?, it implies that there is a j so that S₁ ? F_j and S₂ ? 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:

51

52We note that if N(S₁) = N(S₂) = 0, then N(S₁ ? S₂) = 0, and the inequality is satisfied (the two members are equal to 0).

53If N(S₁) > 0 and N(S₂) > 0, then the denominators of the two fractions of the right-hand member are equal, and greater than or equal to the denominator of the left-hand member. Since, in addition, N(S₁ ? S₂) = N(S₁) + N(S₂), the inequality is satisfied.

54The case where N(S₁) > 0 and N(S₂) = 0 remains to be examined. Two cases can be distinguished.

55Let us first consider the case where N(F_j\S₁) = 0, i.e., in F_j, the only lists able to continue alone belong to S₁. In this case, N(F_j\(S₁ ? S₂)) = 0 and, since N(S₁ ? S₂) > N(S₁), the inequality is satisfied.

56Let us consider now the case where N(F_j\S₁) > 0:

57

58and the inequality is satisfied since N(S₁ ? S₂) > N(S₁).

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 S₁ is able to guarantee the bonus, this is also the case for the coalition S₁ ? S₂. Indeed, if the list S₁ is able to be sure of the bonus, it means that N(S₁) > N(F_j\S₁) and N(S₁) > N(F_l) for any l = 1,…, J, l ? j. But then necessarily N(S₁ ? S₂) > N(F_j\(S₁ ? S₂)) and N(S₁ ? S₂) > 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(S₁) = V_prime(S₂) = 0. It is immediately verified that V_prime(S₁ ? S₂) ? V_prime(S₁) + V_prime(S₂), with a strict equality when the coalition S₁ ? S₂ arrives first, i.e., when N(S₁ ? S₂) > N(F_j\(S₁ ? S₂)) 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 S₁ ? S₂ guarantees for itself a payoff at least as high as the sum of the payoffs that the coalitions S₁ and S₂ are guaranteed separately, we have shown that the game is superadditive.

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 N¹_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 left-wing 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 2ⁿ whose coordinates are the numbers

78

79for ordered from the largest to the smallest, i.e., so that ?ⁱ(x) ? ?^j(x) for 1 ? i ? j ? 2ⁿ.

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:

83

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 N¹ ? (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 three-player game. By choosing different values for the parameter x in the interval (15, 25), several possible configurations are covered. For example, when x = 25, N¹ ? (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.

Example 1

x = 25. In this case, N¹ ? (0, 30, 11, 8, 51), and we obtain:

86In the case where for the nucleolus, we obtain:

87

88On the other hand, for the Shapley value, we obtain:

89

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.

Example 2

x = 22. Let us consider now the second case. The vote distribution is then N¹ ? (0, 30, 14, 8, 48) and:

91For the nucleolus, we obtain:

92

93while for the Shapley value, we obtain:

94

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.

Example 3

x = 19. Let us consider now the third case. The vote distribution is then N¹ ? (0, 30, 17, 8, 45) and:

96For the nucleolus, we obtain:

97

98while for the Shapley value, we obtain:

99

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.

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.

The Allocation of Seats within the Left-Wing List

117Let us discuss now the distribution of the 58 seats within the left-wing 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.

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 %)

Party Observed Gamson Shapley Nucleolus PS 77.6 70.6 95.7 96.2 EE 17.2 18.3 2.7 2.5 FG 5.2 11.2 1.6 1.5

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):

123

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 first-round electoral votes from these parties. It is visible on the lines corresponding to the ee-ps-fg order of formation, where the ps enables ee to continue to stand, on the line fg-ps-ee where the ps enables fg to continue to stand, and finally on the lines ee-fg-ps and fg-ee-ps 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 ps-ee-fg or ps-fg-ee 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:

130

131i.e., here:

132

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 Upper-Normandy 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 Allocation of Seats within the Left List

143Let us discuss now the distribution of seats within the left-wing 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.

Table 2

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 %

Observed Gamson Shapley (V1) Shapley (V2) Nucleolus (VI) Nucleolus (V2) PS 51.5 52.9 57.6 46.3 68.9 46.3 FG 27.3 26.9 23.4 29.1 17.8 29.1 EE 21.0 20.2 19.0 24.6 13.3 24.6

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 left-wing 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):

148

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:

150

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 two-thirds of the bonus allocated to the leading list, and only one-third 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):

155

156i.e., here:

157

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:

160

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 left-wing 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]

168P¹ = 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 first-ballot results, the ps list led by Jean-Yves Le Drian won the number N¹₁ = 408 551 votes, the ump list led by Bernadette Malgorn N¹₂ = 260 731, and the Europe Ecology list led by Guy Hascoët N¹₃ = 134 161…

Second Ballot

170In Brittany, R² = 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 M² = 5 lists which can potentially participate in the second round.

171In this region, P² = 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 left-wing 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 left-wing party, and Europe Ecology, 11, meaning 17.5% of the seats of the left-wing lists. These percentages are shown in the second column in the Table 3 below.

Table 3

Allocation of seats for the left-wing lists in Brittany (i) observed, and predicted by (ii) Gamson, (iii) Shapley, (iv) the nucleolus, in %

Observed Gamson Shapley Nucleolus PS 82.5 75.3 83.5 83.5 EE 17.5 24.7 16.5 16.5

Allocation of seats for the left-wing 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):

178

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:

180

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.

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 left-wing 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 left-wing lists that can win the majority bonus?

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 three-player game which has been retained for formalizing the electoral environment of the twenty-one 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 left-wing 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 first-round 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 left-wing coalition wins or not in the second round: it is enough to compare the sum of the first-round results to the first-round 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 left-wing parties do not make the comparison between the first-round 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 left-wing list estimates that it will win in the second round against the ump, if and only if the sum of its first-round results exceeds the first-round 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 first-round 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 first-round results we can calculate the configuration in which the left-wing parties are found. Table 4 shows the first-round results of the three left-wing parties and the ump in each of the metropolitan regions (outside Corsica and Languedoc).

Table 4

First-round results of the three left-wing parties and the ump^*

PS Ecologists Left Front UMP Alsace 19.0 15.6 1.9 34.9 Aquitaine 37.6 9.75 5.6 22.0 Auvergne 28.0 10.7 14.3 28.7 Lower-Normandy* 32.6 12 . 27.7 Burgundy* 36.3 9.8 . 28.8 Brittany 37.2 12.2 3.5 23.7 Centre 28.2 11.7 7.5 29.0 Champagne* 31.0 8.5 . 31.8 Franche-Comté 29.9 9.4 4.0 32.1 Upper-Normandy 34.9 9.1 8.4 25.0 Île-de-France 25.3 16.6 6.6 27.8 Limousin 38.1 9.7 13.1 24.2 Lorraine* 34.4 9.2 . 23.8 Midi-Pyrénées 40.9 13.5 6.9 21.7 Nord-Pas-de-Calais 29.2 10.3 10.8 19.0 Pays-de-la-Loire 34.4 13.6 4.99 32.8 Picardy 26.6 9.98 5.35 25.9 Poitou-Charente 39.0 11.9 4.7 29.4 Provence-Alpes-Côte-d’Azur 25.8 10.9 6.1 26.6 Rhône-Alpes 25.4 17.8 6.3 26.4

First-round results of the three left-wing parties and the ump^*

* In these regions, the PS and the Left Front have formed a pre-electoral 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.

Tables 5 a.b.c

Classification of configurations on the eve of the second round

5a

Three left-wing parties (ps, ee, and fg) separately present in the first round, and all in a position of continuing to stand

Case 1 Case 2 Case 3 the three parties can continue to stand without any alliance only parties 1 and 2 can continue to stand without an alliance only party 1 can continue to stand without an alliance Case A Nord-Pas-de-Calais Limousin Aquitaine 1 Midi-Pyrénées Upper-Normandy wins without any alliance Provence-Alpes-Côte-d’Azur Case B Auvergne Centre Picardy 1 needs an alliance, Île-de-France alliance with 3 is sufficient Rhône-Alpes

Three left-wing parties (ps, ee, and fg) separately present in the first round, and all in a position of continuing to stand

5b

Three left-wing parties (ps, ee, and fg) separately present in the first round, and the fg is not in a position to continue to stand

Case 1? Case 2? the two parties can only party 1 can continue to stand without any alliance continue to stand without any alliance Case A? Brittany 1 wins without any alliance Poitou-Charentes Case B? Pays-de-la-Loire Franche-Comté 1 needs an alliance the left wing Alsace cannot win

Three left-wing parties (ps, ee, and fg) separately present in the first round, and the fg is not in a position to continue to stand

5c

Pre-electoral alliance (before the first round) between the ps and the Left Front

Case 1? Case 2? the two parties can only party 1 can continue to stand without any alliance continue to stand without any alliance Case A? Lower-Normandy Burgundy 1 wins without any alliance Lorraine Case B? Champagne 1 needs an alliance

Pre-electoral 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 left-wing parties can continue to stand for the second round without any alliance (Auvergne and Nord-Pas-de-Calais 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 (Lower-Normandy, Brittany, Poitou-Charente), in the case 2?B? in two regions (Champagne, Franche-Comté) 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 two-thirds 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, Franche-Comté, 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 left-wing seats to the ps).

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 three-player game (depending upon the regions).

215In the case of Corsica, the game was clearly more complicated, because the regionalist/pro-independence 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 six-player 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 right-wing list clearly won the second round with a result of 45.5% versus 35.5% for the main left-wing 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 non-transferable 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 three-hypothesis 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.