# AN AVERSION TO RISK POWER INDEX IN A SIMPLE GAME

Anselme Njocke

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Anselme Njocke. AN AVERSION TO RISK POWER INDEX IN A SIMPLE GAME. Finance India, 2014, XXVIII (3). hal-05361216

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# « AN AVERSION TO RISK POWER INDEX IN A SIMPLE GAME »

Anselme NJOCKE<fnref n="1" />

## SUMMARY

In this work, we propose an aversion to risk power index in a simple game based on the precautionary index (or the capacity of influence) of a player in a negotiation, as well as the order in which coalitions are formed. Let us note that some classical power indexes encountered in literature such as the Shapley-Shubik power index (1954), the Johnston index (1978) and the Deegan-Packel power index (1978) are generalized, as it were, by taking risk aversion into account. When submitting the aversion to risk power index to the test of a number of axioms, it appears that the violation of these axioms is strongly linked to the vector of precautionary indexes of the players. We show that in a society exclusively consisting of absolute, timorous players, the aversion to risk power index leads to an equal sharing out of power among its various members, regardless of their marginal contributions. We also show, that when the decision rule is unanimity and when there are only two classes of players, i.e. absolute, timorous players on the one hand, and absolute, courageous players in the other, the aversion to risk power index assigns a zero to each, absolute timorous player, and total power (divided equally) to absolute, courageous players. Eventually we show that a dictator game is akin to a game where the dictator would be the only absolute, courageous player, while all the other players would be absolute, timorous players, whatever the rule of decision requiring a quota strictly greater than the highest individual number of votes, in a simple (or weighted) game.

**KEY WORDS** : Absolute courageous player, absolute timorous player, aversion to risk power index, capacity of influence, marginal claim of a player, marginal contribution, precautionary index.

**JEL Classification Number – D71**

<footnote n="1">¹ Université de Pau et des Pays de L'Adour, Faculté de Droit, d'Economie et de Gestion, Centre d'analyse théorique et de traitement des données (CATT), Avenue du Doyen Poplawski, B.P. 1633 - 64016, Pau Cedex, France, e-mail : anselme.njocke@univ-pau.fr</footnote>
# 1. INTRODUCTION

Major of power indexes<fnref n="2" /> encountered in literature such as the Shapley-Shubik power index (1954), the Johnston power index (1978) and the Packel-Deegan power index (1978) are built from the principle of "to each according to his contribution" weighted by a weight based on a certain fair allocation approach.

Generally, the amount that a critical (or decisive) player receives when sharing his marginal contribution between the members of the (minimal) winning coalition which he just joined, is identical to the share of any other critical player, even if the initial number of votes held by each is not identical. It is the same for non critical players. Such sharing of these marginal contributions between the members of the (minimal) winning coalition or not, can be subject to refusal or objection of some members (critical or not) of the grand coalition. In addition, the protesters may not see the merits of ownership of all the different marginal contributions of any player only by the critical players, to the extent that for a player to be critical, he must generally rely on the previous members of the coalition that he has just joined. It means that being critical results from the contributions of the members who preceded him. This challenge can also be based on the rationale for choosing a qualified majority, which plays a role in the fact that a player can be critical.

The existence of pressure groups (lobbies) reflects somehow the reluctance of some members to accept that the marginal contribution of a critical player should come back to him completely, or that the whole marginal contribution should come back to some critical members of the grand coalition. These pressure groups are somehow institutionalized in the United States of America, a lobby may even be awarded a meeting place in the premises of

<footnote n="2">² There are many other power indices in the literature and are generally extensions of the Shapley-Shubik index (such as the Banzhaf power index (1965), the normalized Banzhaf index, also called the Banzhaf-Coleman power index (1971), the Dubey-Shapley power index (1979), the Packel-Holler power index (1983) to name only a few) and they could have been examined in this analysis. But for simplicity we have chosen not to retain all these indices.</footnote>
Congress to prepare to challenge the use of votes of some members (players) of Congress in order to safeguard their interests.

Such a practice exists in Europe, as the European Parliament and the German Bundestag where lobbying is somehow regulated as access to both meetings is subject to registration which is made public. However this practice is not generally institutionalized in many states of the European Union.

The existence of such groups thus reveals, if need be, that the allocation of power to the players is not solely based on their marginal contributions. Also in a given simple game, these allocations of power can mainly rely, on marginal claims on the marginal contributions of these players. The end result of the negotiation of power sharing will depend on each player's ability to influence the sharing of the marginal contribution of any one member of the Assembly of players. So we assume that each individual (or each player) has a bargaining power index characterizing his cautious approach to the pressures or challenges from other players. Each player should temper (or not) his claims against the protests of the other players. This bargaining power index will be called, in this analysis, his precautionary index.

Outside the Shapley-Shubik index that reflects the order in which a winning coalition is formed, the other indices do not take into account the interest of such an order in the formation of winning or minimal winning coalitions. In this analysis, the order of formation of winning or minimal winning coalitions (in calculating the power of a player) is essential to understand the dispute about or claim on a portion of the marginal contribution of a player by the players already in the coalition.

A player who contests the least, or who does not claim a small portion of his marginal contribution is considered a timid player, whereas a brave player will claim a significant portion of his marginal contribution.
The purpose of this work is to question the equal sharing of the marginal contribution between the members of a (minimal) and winning coalition taking into consideration the precautionary indexes of the players; it is tantamount to questioning the assumption of a form of solidarity among the members of the (minimal) winning coalition. In other words, it aims at relaxing the strong or restrictive assumption of uniform character of the valor of the players whether it is in a (minimal) winning coalition or not: everyone does not evenly claim what they should receive in the negotiation Similarly, there exists a variety of capacities of influence (precautionary indexes) among the members of a coalition, the two extreme cases are a maximum capacity of influence (null precautionary index) and a minimal ability to influence (precautionary index equal to unity).

Subsequently, we characterize a player with a capacity of maximum influence as a valiant absolute player, and the fearful player as a player with a minimum capacity of influence.

The study will show that the aversion to risk power index generalizes a number of power indices (conventional or classic) in a simple game, such as the Shapley-Shubik index, the Johnston index and the Deegan-Packel index.

Generally, in the proposed theories of power in a voting game, the question was to know, within eligible coalitions (i.e. coalitions likely to form), how to assign the weight which a decisive player (or critical player) should receive in a qualifying coalition. Various answers have been put forward through a series of power indices found in literature. It turned out that in the face of this multiplicity of indices, another question came up, namely that of the best predictor (in the sense that the real power of a player is better understood). Very often, these various indices differ, both in the ordinal level and cardinal level, in the evaluation of the power of players. The question of what is the best index leads to the choice of criteria to be used to decide between these different indices.
In this work, the approach of assigning authority to the players is based, firstly, on endogenous factors which are the precautionary indexes of the players and, secondly, on the order of formation of the coalitions.

In a society consisting exclusively of timorous absolute members, the aversion to risk power index leads to equal sharing of power between the various members, whatever their marginal contributions. We show that such equal sharing of collective power can be achieved between the different members of society, if they are all absolute, valiant players and if the decision rule is unanimity.

We show that when the decision rule is unanimity, and that there are only two classes of players, timid absolute players on the one hand, and valiant absolute players on the other, the aversion to risk power index assigns no power to each absolute timid player, and offers an equal distribution of the total power to each absolute valiant player. Whereas when the decision rule is unanimity, the Shapley-Shubik index (which implicitly takes into account that the zero vector of precautionary indexes of players, i.e. when all the players are brave absolute player) offers an equal distribution of the total power among all the players. Finally, we show that a dictatorial game could be likened to a game where the dictator is the only absolute valiant player, while all the other players would be timid, absolute players, for any simple game where the decision rule requires a higher quota strictly greater than the highest individual number of votes.

This paper is organized as follows: in the second section, we present some major indices of power (Shapley-Shubik, Johnston and Deegan-Packel) which are generalized by the aversion to risk power index in a simple game. In the third section, we present the marginal claim of a player in integrating the precautionary index of that player. In the fourth section, we present the model for sharing the marginal contribution of a player. In the fifth section, we present the aversion to risk power index in a simple game. In the same fifth section, we
analyze some properties that can be verified by the aversion to risk power index. It appears
also that the aversion to risk power index in a simple game can violate the axioms of
invariance (symmetry), additivity, and the dummy when it does not coincide with the
Shapley-Shubik index, the Johnston index and the Deegan-Packel index; the aversion to risk
power index can verify the transfer and block axioms when it does not coincide with the
indices of Johnston and Deegan-Packel.

2. SOME MAJOR POWER INDICES IN A SIMPLE GAME

**2.1. Definition of a simple game (or weighted voting game)**

A simple game is defined as a pair (N, Γ), where Γ is a nonempty set of nonempty
subsets of N called winning coalitions (or a set of majorities on N). The non-empty set N of
agents (or voters) is called society. A winning coalition of (N, Γ) has the power to exclude
any outcome of the collective choice, regardless of what other coalitions do.

It is generally considered a simple game is a weighted majority game (or a quota game) if we have:

a) A given natural number *M* (called a quota) less than or equal to the cardinality of the set of players *N* (i.e. *M* ≤ |*N*|).

b) The given number of votes (or voices) *m*<sub>*i*</sub> available to each player *i* ∈ *N* knowing each player in turn vote.

c) A coalition *S* is said to be a winning coalition if the sum of the votes available to its members is greater than or equal to the quota (i.e. $\sum_{i \in S} m_i \ge M$).

And a coalition *S* is said to be a losing coalition if the sum of the voices available to
its members is strictly less than the quota (i.e. $\sum_{i \in S} m_i < M$).
In this simple game $(N, \Gamma)$, we can associate the characteristic function, noted $v$ and defined as follows:

$$ \forall S \subset N \Rightarrow \begin{cases} v(S) = 1 & \text{if } \sum_{i \in S} m_i \ge M \\ v(S) = 0 & \text{otherwise} \end{cases} $$

Any coalition $S$ such that $v(S) = 1$ is called a winning coalition, and any coalition $S$ such that $v(S) = 0$ is called a losing coalition. The game $v$ is often called a weighted majority game (or a quota game), and often noted $v: [M; m_1, m_2, \dots, m_i, \dots, m_n]$.

A player $i$ is said to be critical, decisive or pivot in the coalition $S$ if $v(S) - v(S - \{i\}) = 1$. In other words if the player $i \in S$, then $S$ is the winning coalition, and the coalition $(S - \{i\})$ is a losing coalition.

## 2.2. A general formulation of power indices in a simple game

Most major indices of power found in literature, of which we make a brief presentation below, appear as a weighted average, noted $\phi_i(v)$, of the marginal contributions of a player. This general formulation can be given by the expression:

$$ \phi_i(v) = \sum_{\substack{S \in \Gamma \\ i \in S}} g(i, S) [v(S) - v(S - \{i\})] $$

The difference $[v(S) - v(S - \{i\})]$ represents the marginal contribution of the player $i$; and this difference, when it is equal to unity, can also be interpreted as the fact that a coalition $S$ is decisive because it contains the decisive player $i$ (or pivot player). The function $g(i, S)$ represents the weight assigned to the marginal contribution of the player $i$. It appears that this weight depends on two arguments: the first argument is the player $i$ himself, on the one hand, and the second argument, on the other hand, being the winning coalition $S$ which includes the
player *i*, this function *g* may depend only on one argument (and often only depends on the
winning coalition *s*).

**2.3. Some major indices of power in a single game**

**2.3.1. The Shapley-Shubik power index (1954)**

The first application of the Shapley index dates back to 1954 when Shapley himself
and Shubik tried to analyze the distribution of power in the Security Council of the United
Nations. Shapley and Shubik considered a special class of voting games called simple games,
defined above. They believed that once a majority was reached, the player who had voted last
received a unit of power (i.e. its marginal contribution) as the decision by a qualified majority
was adopted by him. They assumed that the order in which individuals take part in the vote
was determined randomly and equiprobable. If denotes s the cardinality of the set s, then the
Shapley-Shubik index of a player i, noted φ_i^{ss}(v), is defined by:

$$
\varphi_i^{ss}(v) = \sum_{\substack{S \subseteq N \\ i \in S}} \frac{(s-1)!(n-s)![v(S) - v(S - \{i\})]}{n!}
$$

The number $\varphi_i^{ss}(v)$ represents the fraction of power of player $i$ in the simple game $(N, \Gamma)$. The index $\varphi_i^{ss}(v)$ is obtained by dividing the number of permutations of the players for which the player $i$ is pivot by the total number of possible permutations: it is the probability of drawing at random a permutation in which the player $i$ is the pivot.

Note that, in Shapley-Shubik, the weight $g(i, S)$ assigned to the marginal contribution of the player $i$ is equal to unity, either $g(i, S) = 1$.
### 2.2.2. The Johnston power index (1978)

Johnston takes into account the fact that in a winning coalition, a player *i* may not be the only critical player. We note that when Banzhaf<fnref n="3" /> and Coleman determine the power of the player *i*, they do not pose the question of the existence of one (or more) critical player(s) rather than the player *i* belonging to the coalition *s*. If the player *i* is the only critical player in *s*, we can consider that his power is considerable, unlike the situation where there was critical players other than the player *i* in *s*. Johnston's idea is that the power of a player in a coalition should depend on the number of critical players in that coalition, and that his power is inversely proportional to the total number of critical players in the said coalition. In other words, the power of a winning coalition *s* (which is equal to unity) is divided equally between the critical players in this winning coalition. The procedure for calculating the Johnston index in a simple game is as follows:

- We begin by enumerating the set $C_i(v)$ of all winning coalitions where the player *i* is critical.

$$ \text{number } \beta_i^{(v)} \text{ total is such that: } \hat{\beta}_i(v) = \frac{\left| \sum_{\substack{S \subseteq N \\ i \in S}} [v(S) - v(S - \{i\})] \right|}{\left| \sum_{\substack{S \subseteq N \\ i \in S}} [v(S) - v(S - \{i\})] \right|} $$

The power of a player so depends on the number of times he is a decisive player or a critical player.

The Banzhaf power index-Coleman (1971): the Banzhaf-Coleman index is obtained by normalizing the Banzhaf

$$ \text{index, namely: } \bar{\beta}_i(v) = \frac{1}{\sum_{\substack{j \in N \\ S \subseteq N}} \sum_{\substack{i \in S \\ j \in S}} [v(S) - v(S - \{j\})]} \cdot \frac{\left| \sum_{\substack{S \subseteq N \\ i \in S}} [v(S) - v(S - \{i\})] \right|}{\left| \sum_{\substack{S \subseteq N \\ i \in S}} [v(S) - v(S - \{i\})] \right|} $$

This index is defined as the number of times the player is a critical player divided by the number of all winning coalitions.

<footnote n="3">³ The Banzhaf power index (1965): Banzhaf assumes that each player vote freely and independently. Let $s \in \Gamma$ and let $i \in s$. We say that *i* is a critical player in *s* if $(s - \{i\}) \notin \Gamma$. In other words, if *i* resigned from *s*, while the remaining coalition without the player *i* is no longer a winning coalition. Banzhaf total power of a player *i* ∈ *s* is the total number $\beta_i$ of winning coalitions where the player *i* is a critical player (and not the number of permutations of the players for which the player *i* is pivot player, given that the players vote as a block) and the</footnote>
- Then we count the number $c(S)$ of players in each winning coalition of the set $C_i(v)$ of all winning coalitions.

- We determine the Johnston's score or the total power of a player $i$ using the formula:

$$J_i(v) = \sum_{S \in C_i(v)} \frac{1}{c(S)} [v(S) - v(S - \{i\})]$$

- We then determined by normalizing the Johnston index $\bar{J}_i(v)$:

$$\bar{J}_i(v) = \frac{\sum_{S \in C_i(v)} \frac{1}{c(S)} [v(S) - v(S - \{i\})]}{\sum_{j \in N} J_j(v) = \sum_{j \in N} \sum_{S \in C_j(v)} \frac{1}{c(S)} [v(S) - v(S - \{j\})]}$$

Note also that, in Johnston, the weight $g(i, s)$ assigned to the marginal contribution of the player $i$ is equal to the inverse of the number of critical players in the winning coalition $s$ where the player $i$ is critical, either $g(i, s) = \frac{1}{c(S)}$.

### 2.2.3. The Packel-Deegan power index (1978)

-Deegan and Packel base their power index on three assumptions:

- We must take into account that the minimal coalitions. A winning coalition $s \subset N$ is called minimal coalition if $v(T) = 0$ for any coalition $T \subset s$. In other words, a minimal winning coalition contains only critical players.

- The probability of forming a minimal coalition winning is the same for all minimal winning coalitions.

- The power of a player in a minimal winning coalition is exactly the same as any other player in this minimal winning coalition.
The procedure for calculating the Deegan-Packel index in a simple game (N, Γ) is as follows:

- We start by determining the set $M(v)$ of minimal winning coalitions

- We determine then the set $M_i(v)$ of all minimal winning coalitions $s$ to which the player $i$ belongs.

- Then we count the number $s = |s|$ of players in each minimal winning coalition of the set $M_i(v)$.

- We determine the score of Deegan-Packel or and the total power of a player $i$ using the formula: $$DP_i(v) = \sum_{S \in M_i(v)} \frac{1}{|v(S) - v(S - \{i\})|}$$

This means that each player belonging to the minimal winning coalition receives a share of power inversely proportional to the size of this $s$ coalition which is either $s$.

- We then determined by normalizing the Deegan-Packel index:

$$\overline{DP}_i(v) = \frac{DP_i(v)}{\sum_{j \in N} DP_j(v)} = \frac{\sum_{S \in M_i(v)} \frac{1}{|v(S) - v(S - \{i\})|}}{\sum_{j \in N} \sum_{S \in M_j(v)} \frac{1}{|v(S) - v(S - \{j\})|}} = \frac{1}{|M(v)|} \sum_{S \in M_i(v)} \frac{1}{|S|}$$

where $|M(v)|$ denotes the cardinality of the set $M(v)$ of all minimal winning coalitions.

Finally, we note that in Deegan and Packel the weight $g(i, s)$ assigned to the marginal contribution of a player $i$ is equal to the inverse of the number of decisive players in the minimal winning coalition $s$ where the player $i$ is critical, either $g(i, s) = \frac{1}{s}$.
### 2.2.4. Note in conclusion of this presentation to major indices

This brief overview of Shapley-Shubik index, Johnston index and Deegan-Packel index, it is clear that for each, the marginal contribution of a player was weighted by a weight which is justified by considerations specific to each author. The problem of sharing this marginal contribution is placed in normative term, that is to say in a sense that we would like to know what should or seem to be a fair sharing of power between the members of an assembly in a collective decision-making. In addition, it is as if Johnston power index and Deegan-Packel power index appear as power indices of solidarity within the meaning of the veil of ignorance in Rawls, but solidarity, however, restricted to the members of the winning coalition in Johnston and members of the minimal winning coalition in Deegan-Packel. What we propose here is to consider the sharing of such a marginal contribution in terms not normative but positive term, that is to say in a way that would explain what a player at the end of a negotiation on its marginal contribution from its own factors and factors that are unique to other players, i.e. from the vector of precautionary indexes characterizing players.

## 3. PRECAUTIONARY INDEX AND THE MARGINAL CLAIM OF A PLAYER WITH RISK CONSIDERATION

### 3.1. Definition 1: The precautionary index of a player or his capacity to influence the final share

The precautionary index of a player is defined as the capacity or ability to influence the final share in the negotiation for his marginal contribution and marginal contributions of other players. This precautionary index for a player $i$, noted $\alpha_i$, is such that: $\alpha_i \in [0,1]$.

### 3.2. Definition 2: A timid absolute player

We say that a player $i$ is a timid absolute player if his precautionary index is such that $\alpha_i = 1$ (that is a player unable to claim any portion of its marginal contribution).
### 3.3. Definition 3: A valiant absolute player (or a courageous absolute player)

We say that a player $i$ is a valiant absolute player if his precautionary index is such that $\alpha_i = 0$ (that is a player who can claim all of his marginal contribution).

### 3.4. Definition 4: Marginal claim in the sense of Shapley-Shubik

Let's suppose that the order of entries to form the grand coalition $N$ is $\omega$. And let $\omega_{k+1}$ be the player who joins the grand coalition $N$ in the rank $k+1$. This player $\omega_{k+1}$ generates a marginal contribution $e_{\omega, k+1} = v(S_{\omega, k+1}) - v(S_{\omega, k})$ where $S_{\omega, p}$ represents the whole of $p$ first players to enter following $\omega$. This marginal contribution matches the marginal claim of this player in the sense of Shapley-Shubik.

### 3.5. Definition 5: Marginal claim taking into account the precautionary index of a player

The level of demand (or marginal claim) of a player $\omega_{k+1}$ who joins the grand coalition $N$ in the rank $k+1$, when he announces the starting point for negotiation, will not be the level of demand $e_{\omega, k+1} = v(S_{\omega, k+1}) - v(S_{\omega, k})$ in the sense of Shapley-Shubik, but a level of requirement that integrates its precautionary index. Player $\omega_{k+1}$ will have a level requirement rather defined by the expression $e_{\omega, k+1}^* = (1 - \alpha_{k+1})(v(S_{\omega, k+1}) - v(S_{\omega, k}))$. This last expression represents the marginal claim of player $\omega_{k+1}$ when he factors in the risk, that is to say when he integrates its precautionary index.

## 4. MODEL OF SHARING THE MARGINAL CONTRIBUTION OF A PLAYER

The sharing model of the “aversion to risk power index”, in a simple game is based on the vector of precautionary indexes of players in the grand coalition $N$, and that we note

$$ \alpha = (\alpha_i)_{i \in N} = (\alpha_1, \alpha_2, \dots, \alpha_i, \dots, \alpha_n). $$

The assumptions underlying this model of sharing are:
Hypothesis 1: All the players gather in $N$ by successive entries and randomly.

Hypothesis 2: It is assumed that the various possible orders of entry to form the grand coalition $N = \{1,2,..., n-1, n\}$ are equally likely.

Hypothesis 3: Let $\Omega$ be the set of permutations of $N$. Let us Consider a permutation $\omega \in \Omega$ and define the gain $G_{\alpha}(i, \omega, v)$ of a player $i$ belonging to the grand coalition $N$ when the order of the entries to form grand coalition is $\omega$ and that the vector of precautionary indexes is $\alpha = (\alpha_i)_{i \in N}$.

Hypothesis 3.a: Player $\omega_1$, who joins the grand coalition $N$ first, generates a marginal contribution $v(\omega_1)$ which he gets in full.

Hypothesis 3.b. Player $\omega_{k+1}$ who enters the grand coalition $N$ in the rank $k+1$ generates a marginal contribution $e_{\omega,k+1} = v(S_{\omega,k+1}) - v(S_{\omega,k})$ where $S_{\omega,p}$ represents the set of the first $p$ players to integrate the grand coalition following $\omega$. This marginal contribution $e_{\omega,k+1} = v(S_{\omega,k+1}) - v(S_{\omega,k})$ is split between all the players who joined in as far as rank $k+1$ as follows:

Hypothesis 3.b.1.: The last player to join in $\omega_{k+1}$ receives a share $e_{\omega,k+1}^* = (1-\alpha_{k+1})(v(S_{\omega,k+1}) - v(S_{\omega,k}))$ of his marginal contribution $e_{\omega,k+1}$.

Hypothesis 3.b.2.: Any player $l$ of $S_{\omega,k}$, other than the first-in $\omega_1$, receives as his share

$$e_{\omega,l}^* = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k}))$$

Hypothesis 3.b.3.: The first player $\omega_1$, who joins the grand coalition $N$, receives as his share

$$e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) \text{, that is to say, all of the remaining amount.}$$
Therefore, $G_\alpha(i, \omega, v)$ is the total sum of the shares obtained by the player $i$ in this sharing process.

# 5. THE AVERSION TO RISK POWER INDEX IN A SIMPLE GAME

## 5.1. The aversion to risk power index in a simple game

The aversion to risk power index in a simple game is vector: $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ with $\psi_i(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} G_\alpha(i, \omega, v)$.

The share $\psi_i(\alpha, v)$ is the fraction of power that a player $i$ receives, based on precautionary indexes of players: that is, somehow, the probability of drawing at random a permutation in which the player $i$ is pivot to the consideration of precautionary indexes of players.

## 5.2. Notations

The ratings below are used in the properties of the following paragraph:

$n = |N|$: Cardinal of the grand coalition $N$.

$n!$: Number of permutations of the grand coalition $N$.

$2^N$: The set of all nonempty subsets of $N$.

$\Gamma(v)$: The set of winning coalitions.

$\Gamma_i(v)$: The set of all winning coalitions containing the player $i$.

$|\Gamma_i(v)|$: Number of winning coalitions containing the player $i$.

$c_i(v)$: The set of winning coalitions where player $i$ is critical.

$c(S)$: Number of critical players in the coalition $S \in C_i(v)$.
$M(v)$ : The set of minimal winning coalitions

$M_i(v)$ : The set of minimal winning coalitions containing the player $i$.

$s = |S|$ : Number of players in a minimal winning coalition of the set $M_i(v)$.

$|M_i(v)|$ : Number of minimal winning coalitions containing the player $i$.

### 5.3. Equivalence between the aversion to risk power index and some major indices of power

The general formulation of indices of power in a simple game proposed in paragraph 2.2 is such that: $\psi_i(v) = \sum_{S \in \Gamma} g(i, S)[v(S) - v(S - \{i\})]$. This formulation allows us to construct the table of equivalence between the aversion to risk power index and some major indices of power retained in this work.

<table><caption>Table 1: Equivalence between the aversion to risk power index and some major indices of power</caption><thead><tr><th>Some major indices of power</th><th>Eligible coalitions</th><th>g(i, S) : part of marginal contribution of player i in [v(S) - v(S - {i})]</th><th>α<sub>i</sub> = (1 - g(i, S)) : precautionary index of player i</th></tr></thead><tbody><tr><td>Shapley-Shubik</td><td>Winning coalitions</td><td>g(i, S) = 1 ∀i ∈ S ∈ Γ(v) and<br>g(i, S) = 0 ∀i ∈ S ∉ Γ(v)</td><td>α<sub>i</sub> = 0 ∀i ∈ S ∈ Γ(v) and<br>α<sub>i</sub> = 1 ∀i ∈ S ∉ Γ(v)</td></tr><tr><td>Johnston</td><td>Winning coalitions</td><td>g(i, S) = <sup>1</sup>&frasl;<sub>c(S)</sub> ∀i ∈ S ∈ C<sub>i</sub>(v) ≠ Ø<br>and g(i, S) = 0 if C<sub>i</sub>(v) = Ø</td><td>α<sub>i</sub> = <sup>1 - <sup>1</sup>&frasl;<sub>c(S)</sub></sup>&frasl;<sub>1</sub> ∀i ∈ S ∈ C<sub>i</sub>(v) ≠ Ø<br>and α<sub>i</sub> = 1 if C<sub>i</sub>(v) = Ø</td></tr><tr><td>Deegan-Packel</td><td>Minimal winning coalitions</td><td>g(i, S) = <sup>1</sup>&frasl;<sub>s</sub> ∀i ∈ S ∈ M<sub>i</sub>(v) ≠ Ø<br>And g(i, S) = 0 if M<sub>i</sub>(v) = Ø</td><td>α<sub>i</sub> = <sup>1 - <sup>1</sup>&frasl;<sub>s</sub></sup>&frasl;<sub>1</sub> ∀i ∈ S ∈ M<sub>i</sub>(v) ≠ Ø<br>and α<sub>i</sub> = 1 if M<sub>i</sub>(v) = Ø</td></tr></tbody></table>
Table 1, above, shows the idea: in Shapley-Shubik, for a given player $i \in S$, there is no discussion possible on his share $g(i, S)$ from which it returns when he belongs to the winning coalition. He is a valiant absolute player ($\alpha_i = 0$).

By cons (on the other side), in Johnston, and in Packel and Deegan, the debate on sharing the marginal contribution of the player $i$, $[v(S) - v(S - \{i\})]$, is reduced exclusively on critical player of $s$.

In addition, we find that this solidarity in sharing among critical players of $s$, results in equal sharing of the marginal contribution of player $i$: which means that all critical players in $s$ have the same precautionary index, i.e. one character for valor uniform ($\alpha_i = \left(1 - \frac{1}{c(S)}\right) \forall i \in S \in C_i(v) \neq \emptyset$ in Johnston and $\alpha_i = \left(1 - \frac{1}{s}\right) \forall i \in S \in M_i(v) \neq \emptyset$ in Deegan and Packel).

Considering that players belonging or not to winning coalition or to minimal winning coalition does not have the same precautionary index in claiming their marginal contribution, it will be possible to deform (or rather to generalize) classical indices of power such as Shapley-Shubik index, Johnston index and Deegan-Packel index.

We propose a first example (section 5.3.1) where the aversion to risk power index coincides with Shapley-Shubik index, Johnston index and Deegan-Packel index.

Then, in a second example (section 5.3.2), we offer Shapley-Shubik power index distorted, Johnston power index distorted and Deegan-Packel power index deformed, when the vector of precautionary indexes of players no longer coincides with each vector of precautionary indexes of those classical indices of power.
### 5.3.1. Examples of game where the aversion to risk power index coincides with some major indices of power

Consider the previous vote game: $v: [9; m_1 = 6, m_2 = 3, m_3 = 3, m_4 = 3]$

Table 2 below shows the equivalence between the aversion to risk power index and some classical indices of power such as the Shapley-Shubik index, Johnston index and Deegan-Packel index.

<table><caption>Table 2: Equivalence between the aversion to risk power index and some classical power indices in the game $v: [9; m_1 = 6, m_2 = 3, m_3 = 3, m_4 = 3]$</caption><thead><tr><th>Vector of precautionary indexes of players<br>α = (α<sub>1</sub>, α<sub>2</sub>, α<sub>3</sub>, α<sub>4</sub>)</th><th>Share of first player :<br>ψ<sub>1</sub>(α, v)</th><th>Share of second player :<br>ψ<sub>2</sub>(α, v)</th><th>Share of third player :<br>ψ<sub>3</sub>(α, v)</th><th>Share of fourth player :<br>ψ<sub>4</sub>(α, v)</th><th>Some classical indices of power</th></tr></thead><tbody><tr><td>α<sub>i</sub> = 0 ∀ i ∈ S ∈ Γ(v) and<br>α<sub>i</sub> = 1 ∀ i ∈ S ∉ Γ(v)</td><td><sup>3</sup>⁄<sub>6</sub></td><td><sup>1</sup>⁄<sub>6</sub></td><td><sup>1</sup>⁄<sub>6</sub></td><td><sup>1</sup>⁄<sub>6</sub></td><td>Shapley-Shubik index</td></tr><tr><td>α<sub>i</sub> = (1 - <sup>1</sup>⁄<sub>c(S)</sub>) ∀ i ∈ S ∈ C<sub>i</sub>(v) ≠ ∅<br>and α<sub>i</sub> = 1 if C<sub>i</sub>(v) = ∅</td><td><sup>27</sup>⁄<sub>42</sub></td><td><sup>5</sup>⁄<sub>42</sub></td><td><sup>5</sup>⁄<sub>42</sub></td><td><sup>5</sup>⁄<sub>42</sub></td><td>Johnston index</td></tr><tr><td>α<sub>i</sub> = (1 - <sup>1</sup>⁄<sub>s</sub>) ∀ i ∈ S ∈ M<sub>i</sub>(v) ≠ ∅<br>and α<sub>i</sub> = 1 if M<sub>i</sub>(v) = ∅</td><td><sup>9</sup>⁄<sub>24</sub></td><td><sup>5</sup>⁄<sub>24</sub></td><td><sup>5</sup>⁄<sub>24</sub></td><td><sup>5</sup>⁄<sub>24</sub></td><td>Deegan-Packel index</td></tr></tbody></table>

### 5.3.2. Shapley-Shubik, Johnston and Deegan-Packel power indices distorted

In paragraph 5.3 above, the players involved in the sharing of the marginal contribution of a player, whether in Shapley-Shubik, in Johnston or in Deegan-Packel had the same precautionary index.
In Shapley-Shubik, the precautionary index any player belonging to a winning coalition is $\alpha_i = (1 - g(i, S)) = (1 - 1) = 0$

In Johnson the precautionary index any player belonging to a winning coalition is

$$ \alpha_i = (1 - g(i, S)) = \left(1 - \frac{1}{c(S)}\right). $$

In Deegan-Packel the precautionary index any player belonging to a minimal winning coalition is $\alpha_i = (1 - g(i, S)) = \left(1 - \frac{1}{s}\right)$.

We consider that the hypothesis of uniformity classic valor players in a winning coalition or minimal winning coalition is too strong: in fact, there is a priori no reason for members of the same coalition $s$ have each the same level of demand (the same precautionary index) power $v(S) = 1$ generated by this coalition winning or this minimal winning coalition.

We will assume that in Shapley-Shubik, the player $i$ often critical (or pivot) is a timid absolute player ($\alpha_i = 1$), while the other players, often less critical than him are valiant absolute players ($\forall j (j \neq i), \alpha_j = 0$).

Similarly, in Johnston, we will consider the player $i$ the most critical, is a timid absolute player ($\alpha_i = 1$), while the other players ($\forall j (j \neq i) \in S \in C_i(v)$), often less critical than him, share the power, $v(S) = 1$, equally $\left\{ \alpha_j = \left(1 - \frac{1}{c(S) - |\{i\}|}\right) \right\}$, excluding player $i$.

And, in Deegan-Packel, we will consider the player $i$ the most critical, is a timid absolute player ($\alpha_i = 1$), while the other players ($\forall j (j \neq i) \in S \in M_i(v)$), often less critical than him, share the power, $v(S) = 1$, equally $\left\{ \alpha_j = \left(1 - \frac{1}{|S| - |\{i\}|}\right) \right\}$, excluding player $i$.
Consider the previous vote game: $v: [9; m_1 = 6, m_2 = 3, m_3 = 3, m_4 = 3]$. Player 1 is the one that is often critical in this game, by associating a precautionary index $\alpha_1 = 1$, we have Shapley-Shubik power index distorted, Johnston power index distorted and Deegan-Packel power index distorted, as follows:

<table><caption>Table 3: The classical power indexes distorted: non identical precautionary indexes among members of a (minimal) winning coalition in the game $v: [9; m_1 = 6, m_2 = 3, m_3 = 3, m_4 = 3]$</caption><thead><tr><th>Vector of precautionary<br>indexes of players<br>α = (α<sub>1</sub>, α<sub>2</sub>, α<sub>3</sub>, α<sub>4</sub>)</th><th>Share of<br>first<br>player:<br>ψ<sub>1</sub>(α, v)</th><th>Share of<br>second<br>player:<br>ψ<sub>2</sub>(α, v)</th><th>Share of<br>third<br>player:<br>ψ<sub>3</sub>(α, v)</th><th>Share of<br>fourth<br>player:<br>ψ<sub>4</sub>(α, v)</th><th>Classical<br>power<br>indexes<br>distorted</th></tr></thead><tbody><tr><td>α<sub>1</sub> = 1, with 1 ∈ S ∈ Γ(v),<br>and α<sub>2</sub> = α<sub>3</sub> = α<sub>4</sub> = 0<br>avec 2,3,4 ∈ S ∈ Γ(v)</td><td>0</td><td><sup>1</sup>&frasl;<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>3</sub></td><td>Shapley-<br>Shubik<br>power index<br>distorted</td></tr><tr><td>α<sub>1</sub> = 1, with C<sub>1</sub>(v) ≠ ∅,<br>and<br>α<sub>j</sub> = (1 - <sup>1</sup>&frasl;<sub>(c(S) - |{i}|)</sub>)</td><td><sup>3</sup>&frasl;<sub>42</sub></td><td><sup>13</sup>&frasl;<sub>42</sub></td><td><sup>13</sup>&frasl;<sub>42</sub></td><td><sup>13</sup>&frasl;<sub>42</sub></td><td>Johnston<br>power index<br>distorted</td></tr><tr><td>α<sub>1</sub> = 1, with M<sub>1</sub>(v) ≠ ∅,<br>and<br>α<sub>j</sub> = (1 - <sup>1</sup>&frasl;<sub>(|S| - |{i}|)</sub>)<br>∀j(j≠i) ∈ S ∈ M<sub>i</sub>(v)</td><td>0</td><td><sup>1</sup>&frasl;<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>3</sub></td><td>Deegan-<br>Packel<br>power index<br>distorted</td></tr></tbody></table>

It appears in the table 3 that, when player 1 is a timid absolute player, even though it is part of a winning coalition or minimal winning coalition (and yet it is the player most often critical), Shapley-Shubik power index distorted, Johnston power index distorted, and Deegan-Packel power index distorted, make him drop his power. The Johnston power index distorted assigns a non-zero amount of power while the latter relative ranking players 2, 3 and 4. The
Shapley-Shubik power index distorted and the Deegan-Packel power index distorted the last
rank by assigning a share of power anywhere!

**5.4. Axioms (or postulates) can be verified by power index**

Consider that $\phi(v) = (\phi_i(v))_{i \in N}$ be the power index associated to the simple game $(N, \Gamma)$. The axioms below can be considered as the minimum requirement should verify that an index of power so that it can be considered as a good index.

**5.4.1. The normalization axiom**

The power index $\phi(v) = (\phi_i(v))_{i \in N}$ is normalized if $\sum_{i=1}^{n} \phi_i(v) = v(N) = 1$.

Normalization means that the total power of the vote is completely distributed among
the members of the community $N$.

**5.4.2. The dummy axiom**

The dummy axiom states that if a player does not belong to any minimal winning
coalition, then the relative power of the dummy player in the game $v$ is zero.

Formally, we have: $\forall i \in N / i \notin M(v) \Rightarrow \phi_i(v) = 0$ where $M(v)$ represents the set of minimal winning coalitions.

**5.4.3. The symmetry axiom**

The symmetry axiom states that: $\forall i, j \in N$, if $m_i = m_j$ so $\phi_i(v) = \phi_j(v)$: this means that the same power must be attributed to two players $i$ and $j$ therefore that each has the same number of voices or votes in a meeting.
### 5.4.4. The monotonicity axiom

The monotonicity axiom states that, if the number of votes held by a player *i* is more important than the number of votes held by a player *j* in an assembly, then the power of the player *i* must be greater than the power of the player *j*.

Formally, we have: $\exists i, j \in N / m_i < m_j$ and $\phi_i(v) < \phi_j(v)$

### 5.4.5. The additivity axiom

Let $(N, v)$ and $(N, c)$ be two simple games. We call amount of these two games, the game noted $u = (v + c)$ and defined by: $\forall S \subset N, (v + c)(S) = v(S) + c(S)$

The index $\phi(u)$ is additive if: $\forall i \in N, \phi_i(u) = \phi_i(v + c) = \phi_i(v) + \phi_i(c)$

### 5.4.6. The transfer axiom

The transfer axiom states that when moving from one game $v$ to game $w$ as the number of votes held by a player (the donor) decreases during this passage in favor of at least one other player (the recipient), then its power (the donor) must decrease in the new game $w$.

Formally we have:

If, on the one hand, $v: [M, m_1, m_2, ..., m_i, ..., m_n] \to [M, m'_1, m'_2, ..., m'_i, ..., m'_n]$ and $w: [M, m'_1, m'_2, ..., m'_i, ..., m'_n] \to [M, m_1, m_2, ..., m_i, ..., m_n]$ such that $\sum_{j \in N} m_j = \sum_{j \in N} m'_j$, and if, on the other hand, $\exists i, j \in N / m'_i < m_i \text{ et } m'_j > m_j$ then $\phi_i(w) < \phi_i(v)$.

### 5.4.7. The block axiom

The block axiom suggests that, if in a game, individuals *i* and *j* decide to merge their forces or their voice by voting every time, keeps, then the power generated by such a merger must be greater than or equal to the power of the player *i* before the formation of the block.
The axiom of block so states that if when going from one game $v: [M, m_1, m_2, ..., m_i, ..., m_n]$ to a game $w: [M, m'_1, m'_2, ..., m'_{j-1}, m'_{j+1}, ..., m'_n]$ and $(N - \{j\}, \Gamma)$ with a distribution such as voice

$\forall k \neq i, j \ m_k = m'_k$ and $m'_i = m_i + m_j$ then $\varphi((i \oplus j)(w)) \geq \varphi_i(v)$.

This means that the player $(i \oplus j)$ must see his increased power in the new game

$w: [M, m'_1, m'_2, ..., m'_{j-1}, m'_{j+1}, ..., m'_n]$ from the power of the player $i$ in the former game

$v: [M, m_1, m_2, ..., m_i, ..., m_n]$.

**5.4.8. The dictator axiom**

In a simple game $(N, \Gamma)$, if $\exists i_o \in N, \forall S \in 2^N, i_o \in S \Leftrightarrow S \in \Gamma$. The player $i_o$ is called the dictator.

This axiom means that any coalition containing the dictator wins and any coalition not
containing it is not winning. In other words the player $i_o \in N$ is, by itself, a winning coalition.
It is said that the simple game $(N, \Gamma)$ is dictatorial

**5.5. A characterization of the aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ in a**
**simple game**

We see that the breach or non-infringement of a number of axioms by the aversion to
risk the power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is closely related to the vector of precautionary
indexes associated with it.

**THEOREM 1:** The aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ satisfies normalization axiom

**Proof:**

Let us show that $\sum_{i=1}^{n} \psi_i(\alpha, v) = v(N) = 1$ this is to say that $\psi(\alpha, v)$ is normalized.
Consider a permutation $\omega \in \Omega$; we know that $G_\alpha(i, \omega, v)$ is the power of a player $i$
belonging to the grand coalition $N$ when the order of entries to form $N$ is $\omega$.

By summing the individual shares of power obtained by all the players of the grand
coalition in a permutation $\omega \in \Omega$, the sharing model of the marginal contribution of a player,
according to hypothesis 3, requires that: $$ \sum_{i=1}^{n} G_{\alpha}(i, \omega, v) = v(N) = 1. $$

We know that $\psi_i(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} G_\alpha(i, \omega, v)$.

Summing over $i$, we have: $$ \sum_{i=1}^{n} \psi_i(\alpha, v) = \sum_{i=1}^{n} \left\{ \frac{1}{n!} \sum_{\omega \in \Omega} G_\alpha(i, \omega, v) \right\} = \frac{1}{n!} \sum_{\omega \in \Omega} \left\{ \sum_{i=1}^{n} G_\alpha(i, \omega, v) \right\} $$

Since $\sum_{i=1}^{n} G_{\alpha}(i, \omega, v) = v(N) = 1$, it follows that:

$$ \sum_{i}^{n} \psi_{i}(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} \left\{ \sum_{i=1}^{n} G_{\alpha}(i, \omega, v) \right\} = \frac{1}{n!} \sum_{\omega \in \Omega} v(N) = \frac{1}{n!} (n!v(N)) = v(N) = 1 $$

Therefore, the risk aversion value $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is normalized.

**THEOREM 2:** In a weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$, the aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ leads to equal sharing of collective power between the members of the grand coalition $N$ if all its members are fearful absolute players.

Formally we have: $\forall v \in \Gamma$, $\psi_i(\alpha, v) = \frac{1}{|N|} = \frac{1}{n}$ if $\alpha_i = 1, \forall i \in N$.

**Proof:**

Consider any player $i \in N$ such that $\alpha_i = 1$. Let's examine the share $\psi_i(\alpha, v)$ he receives in
a given permutation $\omega$ in the formation of the grand coalition $N$.
i) Suppose that the player i is the player $\omega_1$, he receives a share $v(S_{\omega,1}) = v(\{\omega_1\})$, the assumption 3.a.

ii) Suppose that the player i is the player $\omega_k$. By the assumption 3.b.1, he receives a share $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 0$.

iii) Assume that the player i is the player $\omega_l$ belonging to $S_{\omega,k}$, he receives, by the assumption 3.b.2, a share. $e_{\omega,l}^* = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = 0$.

iv) Suppose that the player i is the player $\omega_1$, he receives, based on the assumption 3.b.3, a share:

$$e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = v(S_{\omega,k+1}) - v(S_{\omega,k}) \cdot \text{since } \alpha_t = 1, \forall t \in \{2, \dots, k+1\}$$

Note that the part of the player i (absolute timid player) is zero in a given permutation, except when he entries the first.

And so $G_\alpha(i, \omega, v) = ((v(S_{\omega,k+1}) - v(S_{\omega,k})) + v(S_{\omega,1}))$.

Considering all the permutations $\omega \in \Omega$, there are $(n-1)!$ ways to permute the other members of the grand coalition between when the player i entries the first.

Therefore we have $\forall i \in N$:

$$\psi_i(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} G_\alpha(i, \omega, v) = \frac{1}{n!} \sum_{\omega \in \Omega} (n-1)! (v(S_{\omega,k+1}) - v(S_{\omega,k}) + v(S_{\omega,1})) = \frac{v(S_{\omega,n})}{n} = \frac{v(N)}{n} = \frac{1}{n}$$
**THEOREM**<sup>4 3</sup>: In a weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$, when, on the one hand, the grand coalition $N$ is a partition of two classes of players $N_1$ and $N_2$ where $N_1$ is the set valiant absolute players and $N_2$ the set of timid absolute players; and when, secondly, the decision rule is unanimity, then the aversion to risk the power index assigns a zero to each timid absolute player and total power, divided equally, players brave.

Formally we have:

$$ 
\left. 
\begin{array}{l}
N = N_1 \cup N_2, \quad N_1 \neq \emptyset, N_2 \neq \emptyset, \text{ et } N_1 \cap N_2 = \emptyset \\
\forall j \in N_1, \quad \alpha_j = 0 \\
\forall h \in N_2, \quad \alpha_h = 1 \\
M = |N|
\end{array}
\right\} \Rightarrow 
\left\{ 
\begin{array}{l}
\psi_h(\alpha, v) = 0, \forall h \in N_2 \\
\psi_j(\alpha, v) = \frac{1}{|N_1|}, \forall j \in N_1
\end{array}
\right.
$$

**Proof :**

Suppose that the set $N$ is such that:

<table><caption>Table 4: Distribution of powers of different members of the EEC (1958 version) with a qualified majority of 17 votes (unanimously).</caption><caption>Game z: [17:4,4,4,2,2,1] : Unanimity as the decision rule</caption><thead><tr><th>Power indice</th><th>Power of player 1 France</th><th>Power of player 2 Germany</th><th>Power of player 3 Italy</th><th>Power of player 4 Belgium</th><th>Power of player 5 Netherlands</th><th>Power of player 6 Luxembourg</th></tr></thead><tbody><tr><td>Shapley-Shubik power index<br>φ<sup>SS</sup>(α, v)<br>α<sub>i</sub> = 0<br>∀i ∈ S ∈ Γ(v)</td><td>φ<sup>SS</sup><sub>1</sub>(α, v) = 0,1666</td><td>φ<sup>SS</sup><sub>2</sub>(α, v) = 0,1666</td><td>φ<sup>SS</sup><sub>3</sub>(α, v) = 0,1666</td><td>φ<sup>SS</sup><sub>4</sub>(α, v) = 0,1666</td><td>φ<sup>SS</sup><sub>5</sub>(α, v) = 0,1666</td><td>φ<sup>SS</sup><sub>6</sub>(α, v) = 0,1666</td></tr><tr><td>Aversion to risk power index<br>ψ(α, v)<br>α<sub>i</sub> = 1, i ∈ {1,2,3}<br>and<br>α<sub>i</sub> = 0, i ∈ {4,5,6}</td><td>ψ<sub>1</sub>(α, v) = 0</td><td>ψ<sub>2</sub>(α, v) = 0</td><td>ψ<sub>3</sub>(α, v) = 0</td><td>ψ<sub>4</sub>(α, v) = 0,3333</td><td>ψ<sub>5</sub>(α, v) = 0,3333</td><td>ψ<sub>6</sub>(α, v) = 0,3333</td></tr></tbody></table>

Under the Table 4, it appears that the aversion to risk power index attributed to France, Germany and Italy (when they are the timid absolute players) a null power when the decision rule is unanimously! And the aversion to risk power index attributed to Belgium, the Netherlands and Luxembourg (when they are the valiant absolute players) all the power they share equally. While the Shapley-Shubik power index (each player being in a valiant absolute claim of its marginal contribution) shares the total power equally between all members of the EEC (1958 version) when the decision rule is unanimously (that is to say when a qualified majority is 17 votes).

<footnote n="4"><sup>4</sup> The numerical example below shows that Theorem 3.</footnote>
$N = N_1 \cup N_2, N_1 \neq \emptyset, N_2 \neq \emptyset \text{ et } N_1 \cap N_2 = \emptyset.$

Suppose that $\alpha_h = 1, \forall h \in N_2$, and that, $\alpha_j = 0, \forall j \in N_1$.

As the decision rule is unanimity, a player $i \in N$, by the model of sharing power of the aversion to risk power index, receives:

- A share $v(S_{\omega,1}) = v(\{\omega_1\}) = 0$, if he is the player $\omega_1$ by the assumption 3.a. insofar as it cannot be alone unanimously.

- Suppose that the player $i$ is the player $\omega_k$. By the assumption 3.b.1, he receives a share $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 1$, since unanimity is reached when the last to enter.

- Suppose that the player $i$ is the player $\omega_l$ belonging to $S_{\omega,k}$, he receives by the assumption 3.b.2, a share $e_{\omega,l}^* = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = 0$, since unanimity is not reached yet when entering the grand coalition $N$.

- Suppose that the player $i$ is the player $\omega_1$, he receives, based on the assumption 3.b.3 a share $e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = v(S_{\omega,k+1}) - v(S_{\omega,k}) = 0$ since unanimity is not reached yet when one considers the set $\{t = 2, \dots, t = k+1\}$.

In summary, each player $i \in N$ receives $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 1$ as a share when he comes last.

We have $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 1$, since $\alpha_j = 0, \forall j \in N_1$; in other words, each player $j \in N_1$ receives the same marginal contribution when he comes last.
But as $\alpha_h = 1, \forall h \in N_2$, we get: $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 0$. This implies that

$$ \psi_h(\alpha, \nu) = 0, \forall h \in N_2, \text{ and also that } \sum_{h=1}^{|N_2|} \psi_i(\alpha, \nu) = 0. $$

We know that the aversion to risk power index, $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$, is normalized, which is formally written: $\sum_{i=1}^n \psi_i(\alpha, \nu) = 1$.

But since, on the one hand, $\sum_{i=1}^n \psi_i(\alpha, \nu) = \sum_{h=1}^{|N_2|} \psi_h(\alpha, \nu) + \sum_{j=1}^{|N_1|} \psi_j(\alpha, \nu)$ and, on the other hand, $\sum_{h=1}^{|N_2|} \psi_i(\alpha, \nu) = 0$ then we get $\sum_{j=1}^{|N_1|} \psi_j(\alpha, \nu) = |N_1|\psi_j(\alpha, \nu) = 1 \Leftrightarrow \psi_j(\alpha, \nu) = \frac{1}{|N_1|}$.

**COROLLARY 1:** In a weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$, when, on the one hand, the grand coalition $N$ is a partition of two classes of players $N_1$ and $N_2$, such as $N_1 = \{j\}$ constituted only one player $j$ called valiant absolute player, and $N_2$ consists of the set of all other players all timid absolute players, and when, on the other hand, the decision rule is unanimity, then the aversion to risk power index $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$ assigns a zero to each timid absolute player and total power the only valiant absolute player $j$.

Formally we have:

$$ \begin{array}{l} N = N_1 \cup N_2, \ N_1 \neq \emptyset, N_2 \neq \emptyset, \text{ et } N_1 \cap N_2 = \emptyset \\ N_1 = \{j\} \text{ et } N_2 = N - \{j\} \\ j \in N_1, \alpha_j = 0 \\ \forall h \in N_2, \alpha_h = 1 \\ M = |N| \end{array} \quad \Rightarrow \quad \left\{ \begin{array}{l} \psi_h(\alpha, \nu) = 0, \forall h \in N_2 \\ \psi_j(\alpha, \nu) = \frac{1}{|N_1|} = 1 \end{array} \right. $$

It is indeed of Theorem 2 for the class $N_1$ of players brave reduced to $N_1 = \{j\}$, that is to say for the only player $j$.
**COROLLARY 2:** In a weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$, when, on the one hand, the grand coalition $N$ is such that all players are valiant absolute players and, secondly, the decision rule is unanimity, then the aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ gives all the power, divided equally, the players of the grand coalition $N$.

Formally we have: $\forall j \in N, \alpha_j = 0$} 
$M = |N|$} $\Rightarrow \psi_j(\alpha, v) = \frac{1}{|N|}, \forall j \in N$

It is indeed of theorem 3 for the class $N_1$ of players brave such that $N_1 = N: \alpha_j = 0, \forall j \in N$.

**THEOREM 4:** For any weighted game $v: [M > \max(m_k)_{k=1,...,n}; m_1, m_2, ..., m_i, ..., m_n]$, if a player $j \in N$ is the only valiant absolute player in a society $N$, and therefore any other player $h \in (N - \{j\})$ is a timid absolute player, then the player focuses his hands the entire collective power. This player $j$ is called the universal dictator.

Formally we have:

$N = N_1 \cup N_2, N_1 \neq \emptyset, N_2 \neq \emptyset, \text{ et } N_1 \cap N_2 = \emptyset$} 
$N_1 = \{j\} \text{ et } N_2 = N - \{j\}$} 
$j \in N_1, \alpha_j = 0$} 
$\forall h \in N_2, \alpha_h = 1$} 
$M > \max(m_k)_{k=1,...,n}$} 
$\Rightarrow \left\{ \begin{array}{l} \psi_h(\alpha, v) = 0, \forall h \in N_2 \\ \psi_j(\alpha, v) = 1 \end{array} \right.$

**Proof:**

Consider the set $N$ is partitioned as follows: $N = N_1 \cup N_2$ and $N_1 \cap N_2 = \emptyset$, such that $N_1 = \{j\}$ constituted only one valiant absolute player $j$.

Suppose, first, that $\alpha_h = 1, \forall h \in N_2$ and the other that $\alpha_j = 0$.

As the decision rule based on a quota $M > \max(m_k)_{k=1,...,n}$, then a player $i \in N$, by the sharing model of the aversion to risk power index, receives:
- A share $v(S_{\omega,1}) = v(\{\omega_1\}) = 0$, if he is the player $\omega_1$, with the assumption 3.a. insofar as he cannot by himself achieve this quota.

- Suppose that the player $i$ is the player $\omega_k$. By the assumption 3.b.1, he receives a share $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1}))$, since the quota is reached when the last to enter.

- Suppose that the player $i$ is the player $\omega_l$ belonging to $S_{\omega,k}$, he receives by the assumption 3.b.2, a share $e_{\omega,l}^* = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k}))$, since the quota is reached when entering the grand coalition $N$.

- Suppose that the player $i$ is the player $\omega_1$, he receives, based on the assumption 3.b.3 a share $e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = v(S_{\omega,k+1}) - v(S_{\omega,k})$ since the quota is reached when one considers the set $\{t = 2, \dots, t = k+1\}$.

We have $e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1}))$, since $\alpha_j = 0, \forall j \in N_1$: in other words, player $j \in N_1$ receives all his marginal contribution when he enters in last.

By summarizing, we have:

$$ 
\left\{ 
\begin{aligned}
e_{\omega,k}^* &= (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 0 && \text{because } \alpha_h = 1 \quad \forall h \in N_2 \\
e_{\omega,l}^* &= (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = 0 && \text{because } \alpha_h = 1 \quad \forall h \in N_2 \\
e_{\omega,1}^* &= \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = v(S_{\omega,k+1}) - v(S_{\omega,k}) = 0 && \text{because } \alpha_j = 0, \quad j \in N_1
\end{aligned}
\right. 
$$

Therefore, $\psi_h(\alpha, v) = 0, \forall h \in N_2$, and thus that $\sum_{h=1}^{|N_2|} \psi_i(\alpha, v) = 0$ with $|N_2| = n-1$.
We know that $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is normalized: $\sum_{i=1}^{n} \psi_i(\alpha, v) = 1$.

Since $\sum_{i=1}^{n} \psi_i(\alpha, v) = \sum_{h=1}^{n-1} \psi_h(\alpha, v) + \psi_j(\alpha, v)$, and that $\sum_{h=1}^{|N_2|} \psi_i(\alpha, v) = 0$, then we have
$\psi_j(\alpha, v) = 1$.

**COROLLARY 3:** In a weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$ when the decision rule is unanimity, Shapley-Shubik power index, equivalent (Theorem 4) to the aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$, when all players are valiant absolute players, distributes equally the collective power (Corollary 2), between players from the grand coalition $N$.

Corollary 3 is a consequence of Theorem 4 and Corollary 2.

**NOTE:** Corollary 3 and Theorem 2 show that the aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ leads to the same allocation of powers to the players of the grand coalition $N$. Each player $i \in N$ gets the same amount of power, $\psi_i(\alpha, v) = \frac{1}{|N|}$, or when all players of the grand coalition $N$ are brave absolute players ($\alpha_i = 0 \ \forall i \in N$) and the decision rule is unanimity ($M = |N|$) (Corollary 2), or when all the players of the grand coalition $N$ are timid absolute players ($\alpha_i = 1 \ \forall i \in N$) in any weighted game $v: [M; m_1, m_2, ..., m_i, ..., m_n]$ (theorem 2).

**THEOREM 5:** The aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ satisfies axioms of normalization, additivity, symmetry (or neutrality), dummy, block and transfer if all the players are brave absolute players that is to say, if $\alpha_i = 0 \ \forall i \in N$.
**Proof:**

The number $g(i, S)$ is the weight assigned to the marginal contribution of the player $i$ after the general formulation of power indices presented in section 2.2. Now we have associated with each player $i$ a precautionary index $\alpha_i = (1 - g(i, S))$. In Shapley-Shubik, this weight is equal to unity, either $g(i, S) = 1$. It follows that each player is characterized by a zero precautionary index, so the aversion to risk power index coincides with Shapley-Shubik index. However, we know that the Shapley-Shubik index satisfies$^5$ axioms of normalization, additivity, symmetry (or neutrality), dummy, transfer and block.

**THEOREM 6:** The aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ satisfies axioms of normalization, additivity, symmetry (or neutrality) and dummy, and does not verify the axioms of block and transfer if, on the one hand, any player belonging to a winning coalition and critical in this winning coalition ($\forall i \in S \in C_i(v) \neq \emptyset$), is characterized by a precaution index $\alpha_i = \left(1 - \frac{1}{c(S)}\right)$; and if on the other hand, any player not belonging to any winning coalition ($\forall i \in S \in C_i(v) = \emptyset$), is characterized by a precaution index $\alpha_i = 1$.

**Proof:**

The precaution index associated with each player $i$ is such that $\alpha_i = (1 - g(i, S))$. In Johnston, we note, first, that the weight $g(i, S)$ assigned to the marginal contribution of the player $i$ is equal to the inverse of the number of decisive players in a winning coalition $S$ where the player $i$ is critical, i.e. $g(i, S) = \frac{1}{c(S)}$, $\forall i \in S \in C_i(v)$.

<footnote>$^5$ For the demonstration, see G. Owen, Game Theory (1995), Third Edition, Academic Press, (pp. 262-265).</footnote>
We note, on the other hand, the weight $g(i, s)$ assigned to the player's marginal contribution is zero ($g(i, s) = 0$), when the player $i$ is not critical in any winning coalition ($\forall i \in S$, $C_i(v) = \emptyset$).

Thus, the aversion to risk power coincides with Johnston index. One knows that Johnston index check the properties$^6$ of normalization, neutrality (symmetry), additivity and dummy, and it does not verify transfer axiom and block axiom.

**THEOREM 7:** The aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ satisfies axioms of normalization, additivity, symmetry (or neutrality) and dummy, and does not verify the block axiom and transfer axiom if, on the one hand, any critical player in this minimal winning coalition ($\forall i \in S \in M_i(v) \neq \emptyset$), is characterized by a precautionary index $\alpha_i = \left(1 - \frac{1}{c(S)}\right)$, and if on the other hand, any player not belong to any minimal winning coalition ($\forall i \in S \in M_i(v) = \emptyset$), is characterized by a precautionary index such that $\alpha_i = 1$.

**Proof:**

The precaution index associated with each player $i$ is such that $\alpha_i = (1 - g(i, S))$. In Deegan-Packel, we note on one hand that the weight $g(i, S)$ assigned to the marginal contribution of the player $i$ is the inverse of the number of decisive players in the minimal winning coalition where the player $i$ is critical, so that $g(i, S) = \frac{1}{|S|}$ ($\forall i \in S \in M_i(v)$). We note, on the other hand, the weight $g(i, S)$ assigned to the player's marginal contribution is zero ($g(i, S) = 0$), when the player $i$ is critical in no minimal winning coalition $S$ ($\forall i \in S, M_i(v) = \emptyset$).

<footnote>$^6$ For the demonstration, see G. Owen, Game Theory (1995), Third Edition, Academic Press, (pp. 262-265).</footnote>
Thus, the aversion to risk power index coincides with Deegan-Packel index. One knows that Deegan-Packel index verifies the properties<fnref n="7" /> of normalization, neutrality (symmetry), additivity and dummy, and does not verify transfer axiom and block axioms.

**THEOREM 8:** The aversion to risk power index $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ satisfies axioms of normalization, additivity can not verify the axioms of individual rationality, symmetry (or neutrality), dummy, transfer and block, if there is at least one player which is not valiant absolute player, that is to say $\exists i \in N$, such as $\alpha_i \neq 0$.

**Proof:**

i) We have already showed that $\sum_{i=1}^{n} \psi_i(\alpha, v) = v(N) = 1$, this is to say that $\psi(\alpha, v)$ is normalized. (theorem 1)

ii) Let us show that $\psi(\alpha, u) = (\psi_i(\alpha, u))_{i \in N}$ is additive

Let $(N, v)$ and $(N, c)$ be two simple games. We call sum (or amount) of these two games, the game noted $u = (v + c)$ and defined by: $\forall S \subset N$, $(v + c)(S) = v(S) + c(S)$. The value $\psi(\alpha, u)$ is additive if: $\forall i \in N \quad \psi_i(\alpha, u) = \psi_i(\alpha, (v + c)) = \psi_i(\alpha, v) + \psi_i(\alpha, c)$.

Let us examine the part of a player $i$ following the sharing model $\psi(\alpha, u)$ in the game $u$.

iia) Let us consider that player $i$ is player $\omega_1$ and that there is not another new entrant in the formation of the grand coalition $N$ in the game $u$. Following assumption 3.a, he receives as his share the whole marginal contribution he generates, i.e. $u(\{\omega_1\}) = (c+v)(\{\omega_1\}) = c(\{\omega_1\}) + v(\{\omega_1\})$ by the definition of the game $u$.

iib) Let player $i$ be player $\omega_{k+1}$, he generates a marginal contribution $e_{\omega, k+1} = u(S_{\omega, k+1}) - u(S_{\omega, k})$. Based on assumption 3.b.1 he receives as his share:

<footnote n="7">⁷ For the demonstration, see G. Owen, Game Theory (1995), Third Edition, Academic Press, (pp. 262-265).</footnote>
$$
\begin{align*}
e_{\omega, k+1}^* &= (1 - \alpha_{k+1})(u(S_{\omega,k+1}) - u(S_{\omega,k})) = (1 - \alpha_{k+1})((c+v)(S_{\omega,k+1}) - (c+v)(S_{\omega,k})) \\
&= (1 - \alpha_{k+1})(c(S_{\omega,k+1}) + v(S_{\omega,k+1}) - c(S_{\omega,k}) - v(S_{\omega,k})) \\
&= (1 - \alpha_{k+1})(c(S_{\omega,k+1}) - c(S_{\omega,k})) + (1 - \alpha_{k+1})(v(S_{\omega,k+1}) - v(S_{\omega,k}))
\end{align*}
$$

iic) If the player *i* is neither the player *ω*₁ nor the player *ω*ₖ, and that he is a player *l*
belonging to *S*<sub>*ω*,k*</sub>, following the assumption 3.b.2 he receives as his share:

$$
\begin{align*}
e_{\omega,l}^* &= (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (u(S_{\omega,k+1}) - u(S_{\omega,k})) = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t ((c+v)(S_{\omega,k+1}) - (c+v)(S_{\omega,k})) \\
&= (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t ((c(S_{\omega,k+1}) + v(S_{\omega,k+1})) - (c(S_{\omega,k}) + v(S_{\omega,k}))) \\
&= (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t ((c(S_{\omega,k+1}) - c(S_{\omega,k})) + (v(S_{\omega,k+1}) - v(S_{\omega,k}))) \\
&= (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (c(S_{\omega,k+1}) - c(S_{\omega,k})) + (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k}))
\end{align*}
$$

iid) If player *i* is player *ω*₁ and there is after him, the entry of a player *ω*ₖ, following
assumption 3.b.3 he receives as his share the remaining amount:

$$
\begin{align*}
e_{\omega,1}^* &= \prod_{t=2}^{k+1} \alpha_t (u(S_{\omega,k+1}) - u(S_{\omega,k})) = \prod_{t=2}^{k+1} \alpha_t ((c+v)(S_{\omega,k+1}) - (c+v)(S_{\omega,k})) \\
&= \prod_{t=2}^{k+1} \alpha_t ((c(S_{\omega,k+1}) + v(S_{\omega,k+1})) - (c(S_{\omega,k}) + v(S_{\omega,k}))) \\
&= \prod_{t=2}^{k+1} \alpha_t ((c(S_{\omega,k+1}) - c(S_{\omega,k})) + (v(S_{\omega,k+1}) - v(S_{\omega,k}))) \\
&= \prod_{t=2}^{k+1} \alpha_t (c(S_{\omega,k+1}) - c(S_{\omega,k})) + (1-\alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k}))
\end{align*}
$$

We deduce, from iia), iib), iic) and iid), that:

$$
\forall i \in N, \psi_i(\alpha, u) = \psi_i(\alpha, (v+c)) = \psi_i(\alpha, v) + \psi_i(\alpha, c).
$$

The aversion to risk power index $\psi(\alpha, u) = (\psi_i(\alpha, u))_{i \in N}$ is therefore additive.

iii) Let us prove that $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is not always individually rational.
Saying that $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is not always individually rational amounts to showing

that: $\exists i \in N$ such that $\psi_i(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} G_\alpha(i, \omega, v) \le v(\{i\})$

Let us suppose, on the one hand, player $i$ is player $\omega_k = i$, and let us consider, on the other hand, that he is not dummy and also that $v(\{\omega_k\}) = v(\{i\}) \ge 0$.

### iiia) Based on assumption 3.b.1, player $\omega_k$ receives a portion

$$e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) \text{ of his marginal contribution } e_{\omega,k}.$$

Assuming that player $\omega_k$ is a timid absolute $(\alpha_k = 1)$, he receives

$$e_{\omega,k}^* = (1 - \alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1})) = 0.$$

### iiib) According to hypothesis 3.a, if player $i$ is player $\omega_k = \omega_1$, he gets as his share

$$v(\{\omega_k\}) = v(\{\omega_1\}) = v(\{i\})$$

### iiic) If he is not the first to join in, that is to say he is a player $\omega_l$ belonging to $S_{\omega,k}$ and that he is a timid absolute $(\alpha_l = 1)$, then $\forall \alpha_t(t=l+1,..., k+1) \in [0,1]$, according to hypothesis 3.b.2, he

$$\text{receives a share } e_{\omega,l}^* = (1 - \alpha_l) \prod_{t=l+1}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = 0.$$

### iiid) If player $i$ is player $\omega_1$ and there is a player $\omega_k$ (with $k > 1$), based on assumption 3.b.3

$$\text{he receives as his share, the remaining amount } e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) = 0 \text{ . if } \alpha_t = 0 \\ \forall t \in \{2,..., k+1\}.$$

Therefore, in the permutation $\omega$, player $i$ only receives $G_\alpha(i, \omega, v) = v(\{i\})$. Considering all permutations $\omega \in \Omega$, as there are $(n-1)!$ ways to permute the other players after the entry of $i = \omega_1$, that player $i$ receives as his share, following the aversion to risk power index
$$ \psi_i(\alpha, v) = \frac{1}{n!} \sum_{\omega \in \Omega} (n-1)! v(\{i\}) = \frac{v(\{i\})}{n} \le v(\{i\}). $$ We deduce that $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$ is not always individually rational.

iv) Let us show that the aversion to risk power index, $\psi(\alpha, v) = (\psi_i(\alpha, v))_{i \in N}$, it is not always neutral (symmetric)

Let $\omega_k$ be the player who entry at rank $k$ in the permutation $\omega$. Let $s_{\omega,k-1}$ be the set of the $(k-1)$ first players to have entered before player $\omega_k$.

Let $\omega_l$ be the player whose rank is $l$ in the permutation $\omega$. Let $s_{\omega,l-1}$ be the set of the $(l-1)$ first players to have entered before player $\omega_l$.

Let us consider that: $e_{\omega,k} = v(S_{\omega,k}) - v(S_{\omega,k-1})$ and $e_{\omega,l} = v(S_{\omega,l}) - v(S_{\omega,l-1})$. Can we say that the value $\varphi(\alpha, v) = (\varphi_i(\alpha, v))_{i \in N}$ is always neutral? In other words, do we have the inference: $\forall (S = S_{\omega,k-1} = S_{\omega,l-1})_{k,l \in S} \subset N, (e_{\omega,k} = e_{\omega,l}) \Rightarrow (\psi_k(\alpha, v) = \psi_l(\alpha, v))$?

Based on assumption 3.b.1, player $\omega_k$ receives a portion $e_{\omega,k}^* = (1-\alpha_k)(v(S_{\omega,k}) - v(S_{\omega,k-1}))$ of his marginal contribution $e_{\omega,k}$; similarly, player $\omega_l$ receives a portion $e_{\omega,l}^* = (1-\alpha_l)(v(S_{\omega,l}) - v(S_{\omega,l-1}))$ of his marginal contribution $e_{\omega,l}$.

To get $\varphi_k(\alpha, v) = \varphi_l(\alpha, v)$, that would require $e_{\omega,k}^* = e_{\omega,l}^*$. Or say that: $e_{\omega,k}^* = e_{\omega,l}^* \Rightarrow ((1-\alpha_k) = (1-\alpha_l)) \Rightarrow \alpha_k = \alpha_l$.

But, as generally, the players' precautionary indexes are not necessarily identical, and in particular, this could be the case for players $k$ and $l$: so that $\alpha_k \ne \alpha_l$. It is therefore $e_{\omega,k}^* \ne e_{\omega,l}^*$, thus $\psi_k(\alpha, v) \ne \psi_l(\alpha, v)$.
So $\forall (S = S_{\omega,k-1} = S_{\omega,l-1})_{k,l \in S} \subset N$, the equality $e_{\omega,k} = e_{\omega,l}$ does not always implies the equality $\psi_k(\alpha, \nu) = \psi_l(\alpha, \nu)$. We can thus infer that the value $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$ is not always neutral.

v) Let us show that $\varphi(\alpha, \nu) = (\varphi_i(\alpha, \nu))_{i \in N}$ it is not always dummy.

Player i is said dummy if: $v(S \cup \{i\}) = v(S) \forall S \subset N \Rightarrow v(\{i\}) = 0$ for $S = \emptyset$.

Does such a dummy player always have as a nil part following the risk aversion value
$\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$? In other words, do we have $\psi_{i-dummy}(\alpha, \nu) = 0$?

Consider a permutation $\omega$ such that the dummy player is player $\omega_1$.

Let us consider player $\omega_2$ who entered after $\omega_1$, generates a marginal contribution $e_{\omega,2} = (v(S_{\omega,2}) - v(S_{\omega,1})) \neq 0$. If player $\omega_2$ is not a valiant absolute player, that is to say if $\alpha_2 \neq 0$, then according to assumption 3.b.1, he receives only $e_{\omega,2}^* = (1-\alpha_2)(v(S_{\omega,2}) - v(S_{\omega,1}))$ and leaves the part $\alpha_2(v(S_{\omega,2}) - v(S_{\omega,1}))$ to dummy player $\omega_1$. The dummy player can then end up with a non-zero part according to the aversion to risk power index.

In addition, generally, under the assumption 3.b.3, the player $\omega_1$ receives as his part

$$e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})).$$

If we have, on the one hand, $v(S_{\omega,k+1}) - v(S_{\omega,k}) \neq 0$ and if, on the other hand, we have

$$\alpha_t \neq 0 \quad \forall t \in \{2, \dots, k+1\}, \text{ then } e_{\omega,1}^* = \prod_{t=2}^{k+1} \alpha_t (v(S_{\omega,k+1}) - v(S_{\omega,k})) \neq 0.$$

We deduce from the foregoing that $\psi_{i-dummy}(\alpha, \nu) \neq 0$. Consequently, the risk aversion value $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$ is not always dummy.
vi) Let us show that $\varphi(\alpha, \nu) = (\varphi_i(\alpha, \nu))_{i \in N}$ does not always verify transfer axiom.

By Theorem 2, the power conferred by the aversion to risk power index, $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$, to a member of the set $N_2$ of all timid absolute players $(\forall h \in N_2, \alpha_h = 1)$ is zero $(\psi_h(\alpha, \nu) = 0, \forall h \in N_2)$ whatever its marginal contribution.

Similarly, the power conferred by the aversion to risk power index, $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$, to

a member of the set $N_1$ of all valiant absolute players is equal to the inverse of cardinal $N_1$

$(\psi_j(\alpha, \nu) = \frac{1}{|N_1|}, \forall j \in N_1)$ whatever its marginal contribution.

And any transfer of votes between members of $N_1$ or between members of $N_2$ does not change the problem moment that their brave or timid attitude remains the same.

This is also true for the transfer of voice from some members of $N_1$ to some members of $N_2$, and vice versa.

We can deduce that the aversion to risk power index, $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$, does not verify the transfer axiom.

vii) Let us show that $\varphi(\alpha, \nu) = (\varphi_i(\alpha, \nu))_{i \in N}$ it is not always verifies the bloc axiom.

By Theorem 2, in the old game $v: [M, m_1, m_2, ..., m_i, ..., m_n]$, it was $\psi_j(\alpha, \nu) = \frac{1}{|N_1|}$, $\forall j \in N_1$ and $\psi_h(\alpha, \nu) = 0, \forall h \in N_2$.

Consider a player $i$ of the set $N_1$ of all valiant absolute players $(\alpha_i = 0)$ and a player $j$ of the set $N_2$ of all timid absolute players $(\alpha_j = 1)$ who decide to form the block $(i \oplus j)$.
This merger resulted in a new game $w$, where the society is now
$N' = N_1' \cup N_2' \cup \{i \oplus j\}$ with $N_1' = N_1 - \{i\}$ and $N_2' = N_2 - \{j\}$. The cardinality of the set $N_1'$
being equal to $|N_1'| = |N_1| - 1$, that of the set $N_2'$ being equal to $|N_2'| = |N_2| - 1$, and of the set $N'$
being equal to $|N'| = |N| - 1$.

Assume that during the merger $(i \oplus j)$, the player $j \in N$ communicates any fear to the player $i \in N$ so that the player $(i \oplus j)$ becomes a timid absolute block or as so $\alpha_{(i \oplus j)} = 1$.

Thus the index vector precautionary index of players becomes

$\alpha' = (\alpha_h)_{h \in N'} = (\alpha_1, \alpha_2, \dots, \alpha_{i-1}, \alpha_{(i \oplus j)}, \alpha_{j+1}, \dots, \alpha_n)$

Prior to the formation of the block $(i \oplus j)$, the player $i$ would receive a share of power $\psi_i(\alpha, v) = \frac{1}{|N_1|}$, and the player $j$ received a share of power $\psi_j(\alpha, v) = 0$, as mentioned above$^{}$ by Theorem 2.

In the new game $w$, by Theorem 2, the aversion to risk power index,
$\psi(\alpha', w) = (\psi_i(\alpha', w))_{i \in N'}$ assigns each member $l \in N_1'$, power $\psi_l(\alpha', w) = \frac{1}{(|N_1'| - 1)}$, and each
timid absolute member $k \in N_2' \cup \{i \oplus j\}$, power $\psi_k(\alpha', w) = 0$. So it happens that
$\psi((i \oplus j)(\alpha', w)) = 0$.

After the formation of the block $(i \oplus j)$, it appears that we have the following
inequality: $(\psi_{(i \oplus j)}(\alpha', w) = 0) < (\psi_i(\alpha, v) = \frac{1}{|N_1|})$. This is a violation of the axiom of block
which states that we should have $\psi_{(i \oplus j)}(\alpha', w) > \psi_i(\alpha, v)$.
The aversion to risk power index, $\psi(\alpha, \nu) = (\psi_i(\alpha, \nu))_{i \in N}$, does not always check the axiom of block.

It would have been different if, during melting $(i \oplus j)$, the player $i \in N_1$ had communicated all his courage to the player $j \in N_2$ so that the block $(i \oplus j)$ would become a valiant absolute player as $\alpha_{(i \oplus j)} = 0$. One can checks, by following the same reasoning, aversion to risk power index, in this case, verifies the axiom of block.

# 6. Conclusion

In this paper, we proposed an aversion to risk power index based on a precautionary index of a player and the order of formation of coalitions (an order only taken into consideration by the Shapley-Shubik index).

We have shown that in a simple game, the aversion to risk power index generalized some major indices such as the Shapley-Shubik index, the Johnston index and the Deegan-Packel index. With the aversion to risk power index, the "dummy" player can be the sole beneficiary of the total power in any coalition to which he belongs, even though he is not a pivot in any given permutation of the grand coalition. Such granting of power to the "dummy" player is impossible in the case of traditional power indices: they give him a zero share of power.

When a player is a timid absolute player (absolute coward player), regardless of his initial contribution in terms of the number of votes, the aversion to risk power index assigns a zero share of power in any game where the weighted majority requires at least two qualified votes. Valour characterized by a precautionary index equal to zero (the only case considered in the Shapley-Shubik traditional power index) appears as a key element in a power sharing in
any simple game. The timorous attitude of a player may be extremely detrimental when claiming his marginal contribution.

We have also shown that in every simple game (or weighted game), when a society was exclusively made up of timid, absolute players, then the risk aversion power index shared collective power equally between the various members, whatever their marginal contributions. We have shown that such equal sharing of collective power was also achieved between the different members of society, when they were all brave players and that the rule of decision was unanimous.

We also showed that when the decision rule is unanimity, and there are only two classes of players, absolute timid players on the one hand, and absolute valiant players on the other hand, the aversion to risk power index assigns zero to each absolute timid player and total power (distributed in an egalitarian way) to valiant players. It also appeared that the dictatorial game could be likened to a dictator game where the dictator was the only absolute valiant player (concentrating in his hands all the power group), while all the other players would be absolute timid players, regardless of the decision rule requiring a quota strictly greater than the highest individual number of votes in a simple (or weighted) game.

We have only included in this work, three conventional indices of power (Shapley-Shubik, Johnston and Deegan-Packel) when it came to emphasize the equivalence between the latter and the aversion to the risk power index. Such equivalence (generalization) can also be highlighted with many other indexes of power found in literature (which are mostly extensions of the Shapley-Shubik power index).

It appeared that the violation or non-violation of a number of axioms or postulates (minimum properties a reasonable index should verify to be considered as a good indicator) by the aversion to risk power index was strongly linked to the vector of precautionary indexes associated with it.
The aversion to risk power index appears, somehow, as a reflection of a form of social
organization (or a form of constitution defining the power of each member in the process of
collective decision) based on the greater or lesser cowardice (or more or less valor) evinced
by the members of a community. The distribution of collective power, which is itself, a basis
(or bases) of choice mechanism sharing the fruits or the wealth of a society.

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