The farsighted core in a political game with asymmetric information

Abstract

We consider a three-player-political game that modifies a weighted majority game by having farsighted players. We follow Acemoglu et al. (Rev Econ Stud 75:987–1009, 2008) in assuming that the power of each player is exogenously given but allow a winning coalition to choose any wealth distribution, as in Jordan (2006). Under complete information, the farsighted core is enlarged by the heterogeneity of player’s powers. In addition, we introduce asymmetric information about powers among players and define a corresponding concept of the farsighted core under asymmetric information which is a variant of the credible core in Dutta and Vohra (Math Soc Sci 50:148–165, 2005).

Appendix

1.1 Proof of Proposition 1

• Consider the game at state (N, (w, w, w)).

Let us show that allocations $(x_1, x_2, x_3) = (\frac{1}{2}, \frac{1}{2}, 0), (\frac{1}{2}, 0, \frac{1}{2}), (0, \frac{1}{2}, \frac{1}{2})$ are in the farsighted core. These are allocations in which two of the three players obtain $\frac{1}{2}$. We denote them by $(x_i, x_j, x_k ) = (\frac{1}{2}, \frac{1}{2}, 0)$. First, consider an objection by coalition $\{i, j \}$. By (1), we have that

$$\begin{aligned}&y_i ( \{ i, j \} | (N, (w,w,w)) ) = \frac{1}{2}, \\&y_j ( \{ i, j \} | (N, (w,w,w) )) = \frac{1}{2}. \end{aligned}$$

Because

$$\begin{aligned}&x_i = \frac{1}{2} = y_i ( \{ i, j \} | (N, (w, w,w) ) ), \\&x_j = \frac{1}{2} = y_j ( \{ i, j \} | (N, (w, w,w)) ), \end{aligned}$$

coalition $\{ i , j \}$ does not have an objection to $(x_i, x_j, x_k )$. Next, consider an objection by coalition $\{ i, k \}$. Because

$$\begin{aligned}&y_i ( \{ i, k \} | ( N, (w,w,w)) ) = \frac{1}{2}, \\&y_k ( \{i, k \} | (N, (w,w,w) ) ) = \frac{1}{2}, \end{aligned}$$

player i does not join coalition $\{i, k \}$ because $y_i ( \{ i , k \} | ( N , (w,w,w) ) ) = \frac{1}{2} = x_i$, while player k has an incentive to join coalition $\{ i, k \}$. Therefore, coalition $\{i, k \}$ cannot form under $(x_i, x_j, x_k)$. Finally, consider an objection by coalition $\{ j, k \}$. Applying the same argument as in $\{ i, k \}$, there is no coalition formation at $(x_{i}, x_{j}, x_{k})$. Thus, allocation $(x_i, x_j, x_k ) = (\frac{1}{2}, \frac{1}{2}, 0)$ is in the farsighted core.

Second, let us show that any allocation except $(x_i, x_j, x_k ) = (\frac{1}{2}, \frac{1}{2}, 0)$ does not belong to the farsighted core. If $x= (x_1, x_2, x_3 ) \ne (\frac{1}{2}, \frac{1}{2}, 0), (\frac{1}{2}, 0, \frac{1}{2}), (0, \frac{1}{2}, \frac{1}{2})$, then there exists $\{ i, j \} \subset N$ such that $x_i < \frac{1}{2}$ and $x_j < \frac{1}{2}$. Because

$$\begin{aligned}&y_i ( \{ i, j \} | (N, (w,w,w) ) ) = \frac{1}{2}> x_i, \\&y_j ( \{ i, j \} | (N, (w,w,w) ) )= \frac{1}{2} > x_j, \end{aligned}$$

coalition $\{ i, j \}$ has an farsighted objection to x at (N, (w, w, w) ) . This implies that $x \in X {\setminus } \{ (\frac{1}{2}, \frac{1}{2}, 0), (\frac{1}{2}, 0, \frac{1}{2}), (0, \frac{1}{2}, \frac{1}{2}) \}$ is not in the farsighted core.

We can apply the same argument to state (N, (s, s, s) ) as state (N, (w, w, w) ) because $y_i ( \{i, j \} | (N , (s, s, s )) ) = y_i ( \{i, j \} | (N, (w, w, w)) )$ and $y_j ( \{ i, j \} | (N, (s, s, s)) ) = y_j ( \{ i, j \} | (N, (w, w,w )) )$. Therefore, only $(x_i, x_j , x_k) = (\frac{1}{2}, \frac{1}{2}, 0)$ is in the farsighted core in the political game with complete information at (N, (s, s, s)).

• Consider the political game at state (N, (s, s, w) ). Two of the three players, players 1 and 2, are the strong type.

• (a) Let us show that any allocation $x=(x_{1},x_{2},x_{3})$ such that $x_1 + x_2 + x_3 = 1$ and $x_1 \ge \frac{1}{2}$ is in the farsighted core of a political game with complete information. Since $x_1 \ge \frac{1}{2}$, $x_2 + x_3 \le \frac{1}{2}$. Moreover, $x_2 \le \frac{1}{2}$ and $x_3 \le \frac{1}{2}$. Because

  $$\begin{aligned} y_1 ( \{1, 2 \} | ( N , (s, s, w) ) ) = \frac{1}{2} \le x_1, \end{aligned}$$

  player 1 never joins coalition $\{ 1, 2 \}$ which has an objection to x. Therefore, coalition $\{ 1, 2 \}$ does not form because players 1 and 2 must agree unanimously to form coalition $\{ 1, 2 \}$. Next, consider formations of coalition $\{ 1, 3 \}$ and $\{ 2, 3 \}$. Because player 3 is the weak type, we have that

  $$\begin{aligned} y_3 ( \{1, 3 \} | ( N, (s, s, w )) ) = y_3 ( \{2, 3 \} | ( N , (s, s, w) ) ) =0 \le x_3. \end{aligned}$$

  Thus, coalitions $\{ 1, 3 \}$ and $\{2, 3 \}$ do not have an objection to x. There is no coalition formation at x. This implies that $x \in X$ such that $x_1 \ge \frac{1}{2}$ is in the farsighted core.

• (b) We can prove that an allocation $x \in X$ satisfying $x_2 \ge \frac{1}{2}$ is in the farsighted core in a same way because players 1 and 2 are symmetric.

• (c) Finally, consider the rest of the allocations; that is, $\hat{X} = X {\setminus } \{ x \in X | (x_1 \ge \frac{1}{2}) \text { or } (x_2 \ge \frac{1}{2}) \}$. Let $\hat{x} = ( \hat{x}_1, \hat{x}_2, \hat{x}_3 ) \in \hat{X}$. These allocations satisfy that $\hat{x}_1 < \frac{1}{2}$ and $\hat{x}_2 < \frac{1}{2}$. Because players 1 and 2 are strong, both players obtain the expected payoff of $\frac{1}{2}$ through forming coalition $\{ 1, 2 \}$. Then, we have that

  $$\begin{aligned}&y_1 ( \{ 1, 2 \} | ( N , (s, s, w) ) ) = \frac{1}{2}> \hat{x}_1, \\&y_2 ( \{ 1, 2 \} | ( N, ( s, s, w) ) ) = \frac{1}{2} > \hat{x}_2. \end{aligned}$$

  Thus, coalition $\{ 1, 2 \}$ has an objection to $\hat{x}$.

• Consider the political game at state (N, (w, w, s)). Two players are weak and one player is strong.

• (a) Let us show that any allocations $x \in X$ such that $x_1 \ge \frac{1}{2}$ is in the farsighted core. Because player 3 is strong, the expected payoff for player 1 when coalition $\{ 1, 2 \}$ forms is

  $$\begin{aligned} y_1 ( \{ 1, 3 \} | ( N, (w, w, s) )) = 0 \le x_1. \end{aligned}$$

  Therefore, player 1 does not join coalition $\{1, 3 \}$. If coalition $\{ 1, 2 \}$ forms, the expected payoff for player 1 is

  $$\begin{aligned} y_1 ( \{ 1, 2 \} | ( N, (w, w, s) )) = \frac{1}{2}\le x_1. \end{aligned}$$

  Therefore, player 1 also does not join coalition $\{ 1, 2 \}$. Finally, let us check that coalition $\{ 2, 3 \}$ does not form. Note that $x_1 \ge \frac{1}{2}$ implies $x_2 \le \frac{1}{2}$. If coalition $\{ 2, 3 \}$ forms, then player 2 obtains

  $$\begin{aligned} y_2 ( \{ 2, 3 \} | ( N, (w, w, s) ) = 0 \le x_2 \end{aligned}$$

  because player 3 is strong. Therefore, player 2 does not join coalition $\{ 2, 3 \}$. There is no coalition formation at $x \in X$ such that $x_1 \ge \frac{1}{2}$. Thus, these allocations are in the farsighted core.

• (b) Next, consider the set of allocations $x \in X$ such that $x_2 \ge \frac{1}{2}$. Because players 1 and 2 are symmetric, we can apply the same argument of (a) to the set of allocations $x \in X$ such that $x_2 \ge \frac{1}{2}$. Therefore, these allocations are in the farsighted core.

• (c) Finally, consider the set of allocations $x \in X$ such that $0 \le x_1 < \frac{1}{2}$ and $0 \le x_2 < \frac{1}{2}$. Because

  $$\begin{aligned}&y_1 ( \{ 1, 3 \} | (N, (w, w, s) ) ) = 0 \le x_1, \\&y_2 ( \{ 2, 3 \} | ( N, (w, w, s) ) ) = 0 \le x_2, \end{aligned}$$

  players 1 and 2 never form a coalition with player 3. However, if players 1 and 2 form coalition $\{ 1, 2 \}$, then

  $$\begin{aligned}&y_1 ( \{ 1, 2 \} | (N, (w, w, s) ) )= \frac{1}{2}> x_1, \\&y_2 ( \{ 1, 2 \} | (N, (w, w, s) )) = \frac{1}{2} > x_2. \end{aligned}$$

  Coalition $\{ 1, 2 \}$ has an objection to x. Thus, these allocations are not in the farsighted core.

1.2 Proof of Proposition 2

(i) Let us show that three allocations $x = ( \frac{1}{2}, \frac{1}{2}, 0)$, $(\frac{1}{2}, 0, \frac{1}{2})$, and $(0, \frac{1}{2}, \frac{1}{2})$ are in the farsighted credible core. Here, we will prove it for $(\frac{1}{2}, \frac{1}{2}, 0)$. Note that $x_1 = \frac{1}{2} \in \left[ \frac{1}{2} - \frac{p}{2}, 1 - \frac{p}{2} \right)$ and $x_2 = \frac{1}{2} \in \left[ \frac{1}{2} -\frac{p}{2}, 1 - \frac{p}{2} \right)$. By Lemma 1, information that players i and j are the strong type is transmitted among the members of coalitions if they agree to join it. First, consider an objection by coalition $\{ 1, 2 \}$. In this case, both players have commonly known that they are the strong type. The event E for coalition $\{1, 2 \}$ is given by $E = E_1 \times E_2 \times \Theta _3 = \{ s \} \times \{ s \} \times \Theta _3$. Let us check that the objection by coalition $\{ 1, 2 \}$ is self-selective over E. Because $R_1 (E_1) = \{ w \}$, $R_2 ( E_2 ) = \{ w \}$ and both player 1 and 2 believe with probability 1 that its partner is strong, the expected payoffs of player 1 and 2 are zero and satisfy

$$\begin{aligned}&Y_1 ( \{ 1, 2 \} | N, w, \{ s\} \times \Theta _3 ) = 0< x_1 = \frac{1}{2}, \\&Y_2 ( \{ 1, 2 \} | N, w, \{ s \} \times \Theta _3 ) = 0 < x_2 = \frac{1}{2}. \end{aligned}$$

Thus, the objection is self-selective for coalition $\{ 1, 2 \}$ over E.

Because $\theta _1 = s$ and $E_{-1} = \{(s , s) , (s, w) \}$, the conditional probability of player 1’s belief about $\theta _{-1}$ is given by $q ( s, s | s, E_1 ) =p$ and $q ( s, w | s, E_{-1} ) = 1-p$. Furthermore, taking an account that $y_1 ( \{ 1, 2 \} | (N, (s, s, s) ) = \frac{1}{2}$ and $y_1 ( \{ 1, 2 \} | ( N , (s, s, w) ) ) = \frac{1}{2}$, we have that

$$\begin{aligned} Y_1 ( \{ 1, 2 \} | N, s, E_{-1} ) = \frac{1}{2}. \end{aligned}$$

Therefore, player 1 does not join coalition $\{1, 2 \}$ because $Y_1 ( \{ 1, 2 \} | N , s, E_{-1} ) = \frac{1}{2} = x_1$. Player 2 also does not join coalition $\{1, 2 \}$ for the same reason.

Next, consider an objection by coalition $\{ 1, 3 \}$. Player 3 believes that player 1 is the strong type with probability one. If player 3 is the weak type, he does not join coalition $\{ 1, 3 \}$. Then, players 1 and 3 have a common belief that player 3 is also the strong type if he joins coalition $\{ 1, 3 \}$. Under their beliefs, players 1 and 3 expect the payoff of $\frac{1}{2}$ through an objection by coalition $\{1, 3 \}$. Thus, $Y_1 ( \{1, 3 \} | N, s, \Theta _2 \times \{s \} ) = \frac{1}{2}$. Because $x_1 = \frac{1}{2}$, player 1 would not join the coalition. An objection by coalition $\{1, 3 \}$ does not occur. We can apply the same argument to coalition $\{ 2, 3 \}$ as coalition $\{1, 3 \}$. Because there is no objection, $( x_1 , x_2, x_3 ) = (\frac{1}{2}, \frac{1}{2}, 0)$ is in the farsighted credible core.

(ii) We show that any allocation $x \in X$ except $\{ (\frac{1}{2}, \frac{1}{2}, 0), (\frac{1}{2}, 0, \frac{1}{2}), (0, \frac{1}{2}, \frac{1}{2} )\}$ is not in the farsighted credible core. For any x, there are two players i and j such that $0 \le x_i < \frac{1}{2}$ and $0 \le x_j < \frac{1}{2}$.

• (a) If $0 \le x_i < \frac{1}{2} - \frac{p}{2}$ and $0 \le x_j < \frac{1}{2} - \frac{p}{2}$, there is no information transmission among the members of coalition $\{i, j \}$. Suppose that $E = \Theta _i \times \Theta _j \times \Theta _k$. Because

  $$\begin{aligned} Y_i ( \{ i , j \} | N, s, E_{-i} )&= 1- \frac{p}{2}> x_i, \\ Y_j ( \{ i, j \} | N, s, E_{-j} )&= 1- \frac{p}{2} > x_j, \end{aligned}$$

  coalition $\{ i, j \}$ has an objection to x.

• (b) If $\frac{1}{2} - \frac{p}{2} \le x_i < \frac{1}{2}$ and $\frac{1}{2} - \frac{p}{2} \le x_j < \frac{1}{2}$, information that players i and j are the strong type is commonly known by the members of coalition $\{ i, j \}$. Define $E = \{ s \} \times \{ s \} \times \Theta _k$. Because $R_i (E_i) = \{ w \}$, $R_j ( E_j ) = \{ w \}$, and

  $$\begin{aligned} Y_i ( \{ i , j \} | N, w, E_{-i} )&= 0< x_i, \\ Y_j ( \{ i, j \} | N, w, E_{-j} )&= 0 < x_j, \end{aligned}$$

  an objection to x is self-selective for coalition $\{ i, j \}$ over E. Moreover, we have that

  $$\begin{aligned} Y_i ( \{ i, j \} | N, s , E_{-i} )&= \frac{1}{2}> x_i, \\ Y_j ( \{ i, j \} | N, s, E_{-j} )&= \frac{1}{2} > x_j. \end{aligned}$$

  Therefore, coalition $\{ i, j \}$ has a farsighted objection to x over E. Thus, coalition $\{i, j \}$ has a farsighted credible objection to x.

• (c) If $\frac{1}{2} - \frac{p}{2} \le x_i < \frac{1}{2}$ and $0 \le x_j < \frac{1}{2} - \frac{p}{2}$, information that player i is the strong type is transmitted. Therefore, $E = E_i \times E_j \times \Theta _k = \{s \} \times \Theta _j \times \Theta _k$. Because

  $$\begin{aligned} Y_i ( \{ i, j \} | N , w, \Theta _j \times \Theta _k )&= \frac{1}{2} - \frac{p}{2} \le x_i, \end{aligned}$$

  an objection to x is self-selective for coalition $\{ i, j \}$ over $E = \{s \} \times \Theta _j \times \Theta _k$. Furthermore, because

  $$\begin{aligned} Y_i ( \{ i, j \} | N , s, \Theta _j \times \Theta _k )&=1 - \frac{p}{2}> x_i, \\ Y_j ( \{ i, j \} | N, s , \{ s \} \times \Theta _k )&= \frac{1}{2} > x_j, \end{aligned}$$

  coalition $\{ i, j \}$ has a farsighted objection to x over E. Thus, coalition $\{ i, j \}$ has a farsighted credible objection to x.

1.3 Proof of Proposition 3

(i) We first show that the set of allocations represented by $F_1$ is in the farsighted credible core. Because $x_1 \ge \frac{1}{2} - \frac{p}{2}$ and $x_2 \ge \frac{1}{2} - \frac{p}{2}$ in $F_1$, it is commonly known that players 1 and 2 are the strong type if they join a coalition. Therefore, player 3 of the weak type does not join coalitions $\{1, 3\}$ and $\{ 2, 3 \}$ because his expected payoff by forming a coalition is zero, which is less than $x_3$. When coalition $\{1 , 2 \}$ forms, players 1 and 2 have the expected payoff of $\frac{1}{2}$ because they believe that their partner is the strong type with probability one. Because player 1 has his wealth of $x_1 \ge \frac{1}{2}$, he does not agree to form coalition $\{1, 2 \}$. Therefore, there is no objection to $x \in F_1$. Thus, $F_1$ is a part of the farsighted credible core at (N, (s, s, w)).

Next let us show that any allocation in $F_2$ is in the farsighted credible core. By Lemma 1, player 3 of the weak type does not join any coalition because he has already obtained $x_3 \ge \frac{1}{2} - \frac{p}{2}$. Only an objection by coalition $\{ 1, 2 \}$ is possible. However, player 1 does not agree to join the coalition because $x_1 \ge \frac{1}{2}$. We conclude that any allocation in $F_2$ is in the farsighted credible core.

Furthermore, we can apply the same argument to allocations in $F_3$ and $F_4$ as in $F_1$ and $F_2$ if interchanging player 1 for player 2. As a result, $F_1$, $F_2$, $F_3$ and $F_4$ belong to the farsighted credible core at (N, (s, s, w) ).

(ii) Next, we prove that

$$\begin{aligned} D_1&= \left\{ x \in X {\big |} ~0 \le x_1< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_2< \frac{1}{2} - \frac{p}{2} \right\} , \\ D_2&= \left\{ x \in X {\big |} ~0 \le x_1< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_3< \frac{1}{2} - \frac{p}{2} \right\} , \\ D_3&= \left\{ x \in X {\big |} ~0 \le x_2< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_3 < \frac{1}{2} - \frac{p}{2} \right\} \end{aligned}$$

are not in the farsighted credible core.

Consider any allocation x in $D_1$ and an objection by coalition $\{ 1, 2 \}$. By Lemma 1, no information about their types is transmitted between players 1 and 2. Therefore, $E = \Theta _1 \times \Theta _2 \times \Theta _3$. Because

$$\begin{aligned} Y_1 ( \{ 1, 2 \} | N, s, \Theta _2 \times \Theta _3 )&= p \times \frac{1}{2} + (1-p) \times 1 = 1- \frac{p}{2}> x_1, \\ Y_2 ( \{ 1, 2 \} | N, s, \Theta _1 \times \Theta _3 )&= 1 - \frac{p}{2} > x_2, \end{aligned}$$

coalition $\{ 1, 2 \}$ can form.

Consider any allocation x in $D_2$ and an objection by coalition $\{ 1, 3 \}$. By Lemma 1, no information about their types is transmitted between players 1 and 3. Therefore, $E = \Theta _1 \times \Theta _2 \times \Theta _3$. Because

$$\begin{aligned} Y_1 ( \{ 1, 3 \} | N, s, \Theta _2 \times \Theta _3 )&= p \times \frac{1}{2} + (1-p) \times 1 = 1- \frac{p}{2}> x_1, \\ Y_3 ( \{ 1, 2 \} | N, w, \Theta _1 \times \Theta _2 )&= p \times 0 + (1- p ) \times \frac{1}{2}= \frac{1}{2} - \frac{p}{2} > x_2, \end{aligned}$$

coalition $\{ 1, 3 \}$ can form.

For any allocation in $D_3$, applying the same argument as in $D_2$ by interchanging player 1 to player 2, we can see that coalition $\{ 2, 3 \}$ forms.

Because any allocation in $D_1$, $D_2$ and $D_3$ has an objection, $D_1 \cup D_2 \cup D_3$ is not the farsighted credible core.

(iii) Repeating a part of (ii) in the proof of Proposition 2 and considering an objection by coalition $\{ 1, 2\}$, we can prove that $\{ x \in X | x_1< \frac{1}{2} \text { and } x_2 < \frac{1}{2} \}$ is not comprised of the farsighted credible core because players 1 and 2 are the strong type.

1.4 Proof of Proposition 4

We can prove that the sets of $D_1 = \{ x \in X | 0 \le x_1< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_2 < \frac{1}{2} - \frac{p}{2} \}$, $D_2 = \{ x \in X | 0 \le x_1< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_3 < \frac{1}{2} - \frac{p}{2} \}$ and $D_3 = \{ x \in X | 0 \le x_2< \frac{1}{2} - \frac{p}{2} \text { and } 0 \le x_3 < \frac{1}{2} - \frac{p}{2} \}$ are not in the farsighted credible core by applying the almost same argument as (ii) in the proof of Proposition 3.

For any allocation except $D_1$, $D_2$ and $D_3$, there are at least two players i and j whose wealth allocations $x_i$ and $x_j$ are greater than or equal to $\frac{1}{2} - \frac{p}{2}$. A player of the weak type does not join any coalition if his wealth is greater than or equal to $\frac{1}{2} -\frac{p}{2}$. Therefore, any coalition cannot form. Thus, these allocations are in the farsighted credible core.