Information Markets, Elections and Threshold Contracts

Abstract

Our research on the use of information markets to foster welfare dates back to 2005. In 2006, we published a first paper on the use of information markets as a tool to assess the office-holders’ performance. We further expanded this theme and were able to show that information markets are useful when Political Contracts on a certain performance are difficult—if not impossible—to achieve.

Appendices

Appendix A: Proofs

Proof of Proposition 5.4

Suppose that voters use the robust election scheme RES. Both candidates decide simultaneously about their threshold contracts. We show that for $i=1,2$, the unique equilibrium of the politician’s contract choice is $C_i(p_i^1 = p_i^0 = \frac{1}{2})$.

Equilibrium Property: Given that candidate $g \in \{1,2\}$ offers $C_g(p_g^1 = p_g^0 = \frac{1}{2})$, politician $h \ne g$, $h \in \{1,2\}$ will not offer $p_h^k < \frac{1}{2}$ for any $k \in \{0,1\}$, since he would have no chance of winning the election. Furthermore, he has no incentive to offer $p_h^k > \frac{1}{2}$ for any $k \in \{0,1\}$, since this does not increase his chances of winning the election. Thus, given that candidate g offers $C_g(p_g^1 = p_g^0 = \frac{1}{2})$, a best response for candidate h is to offer $C_h(p_h^1 = p_h^0 = \frac{1}{2})$, independently of his type. Hence, offering $C_i(p_i^1 = p_i^0 = \frac{1}{2}) \ \forall i \in \{1,2\}$ is an equilibrium. In the next steps we show that it is unique.

Uniqueness: We know from Lemma 5.1 that $p_i^k \le \frac{1}{2} \ \forall k \in \{0,1\}, i = 1,2$, so we only have to examine whether there may exist other equilibria with threshold offers below $\frac{1}{2}$. Suppose that candidate g offers a contract with $p_g^k \le \frac{1}{2} \forall k \in \{0,1\}$ and $p_g^k < \frac{1}{2}$ for at least one $k \in \{0,1\}$. We distinguish three cases:

Case 1: First, consider a constellation with candidates g and h offering contracts $C_g(p_g^1< \frac{1}{2}, p_g^0 < \frac{1}{2})$ and $C_h(p_h^1< \frac{1}{2}, p_h^0 < \frac{1}{2})$. Then candidate h has an incentive to deviate by offering $C_h(p_h^1 < \frac{1}{2}, p_h^0 = \frac{1}{2})$. Indeed, under RES, his election chances are strictly higher when he offers $C_h(p_h^1 < \frac{1}{2}, p_h^0 = \frac{1}{2})$ and offering $p_h^0 = \frac{1}{2}$ does not reduce the reelection chances of h, whether he behaves congruently or dissonantly.²³

Case 2: Consider next a constellation with candidates g and h offering contracts $C_g(p_g^1 = \frac{1}{2}, p_g^0 < \frac{1}{2})$ and $C_h(p_h^1 = \frac{1}{2}, p_h^0 < \frac{1}{2})$. Then candidate h can profitably deviate by offering $C_h(p_h^1 = \frac{1}{2}, p_h^0 = \frac{1}{2})$ as this induces the same behavior (Lemma 5.2) and gives the same reelection chances, while increasing the election chances from $\frac{1}{2}$ to 1.

Case 3: We are left with the optimal response of politician $h \ne g$ if candidate g offers $C_g(p_g^1 < \frac{1}{2}, p_g^0 = \frac{1}{2})$. There are two possibilities for optimal responses: $C_h(p^1_h = p^0_h = \frac{1}{2})$ and $C_h(p^1_h < \frac{1}{2}, p^0_h = \frac{1}{2})$. We show now that both types of politicians will prefer to offer $C_h(p_h^1 = p_h^0 = \frac{1}{2})$ rather than $C_h(p^1_h < \frac{1}{2}, p^0_h = \frac{1}{2})$ in response to a contract $C_g(p_g^1 < \frac{1}{2}, p_g^0 = \frac{1}{2})$. Suppose that a candidate, say candidate 2, offers $C_2(p_2^1 < \frac{1}{2}, p_2^0 = \frac{1}{2})$.

Case 3a: We consider candidate 1 and assume first that he is of the congruent type. If a congruent politician offers a contract with $p_1^1 = p_1^0 = \frac{1}{2}$ and gets elected, then he will always behave congruently.²⁴ If a congruent politician offers a contract with $p_1^1 < \frac{1}{2}$ and $p_1^0 = \frac{1}{2}$ and gets elected,²⁵ then his behavior in state $s_1=0$ will depend on whether $R+G+ \mu [\delta R +\delta ^2 R]$ is larger or smaller than $(R + (1-\mu ) \delta R)$. Candidate 1 is better off by choosing $p_1^1 = p_1^0 = \frac{1}{2}$ if

$$\begin{aligned} \quad z [R+G+\delta R+\delta ^2 R]&+ (1-z) \{R+G+ \mu [\delta R +\delta ^2 R]\} \nonumber \\&\ge \\ \frac{1}{2} z[R+G+\delta R+\delta ^2 R]&+ \frac{1}{2} (1-z) \max \{R+G+\mu [\delta R +\delta ^2 R]; R + (1-\mu ) \delta R\}. \nonumber \end{aligned}$$

(5.10)

To analyze this inequality, we consider the two possible cases, starting with $R+G+\mu [\delta R +\delta ^2 R] \ge (R + (1-\mu ) \delta R)$. In this case, inequality (5.10) simplifies to $1 \ge \frac{1}{2}$ and thus holds. Next we look at $R+G+\mu [\delta R +\delta ^2 R] < (R + (1-\mu ) \delta R)$. Then inequality (5.10) can be simplified to

$$\begin{aligned} \frac{1}{2} z [R+G+&\delta R+\delta ^2 R] + (1-z) \{R+G+ \mu [\delta R +\delta ^2 R]\} \nonumber \\&\ge \frac{1}{2} (1-z) [R + (1-\mu ) \delta R]. \nonumber \end{aligned}$$

This condition is always fulfilled because $\frac{1}{2} z [R+ \delta R] > \frac{1}{2} (1-z) [R + (1-\mu ) \delta R]$ and the other terms on the left hand side of the condition are positive. Thus, a congruent politician 1 will offer a contract with $p_1^1 = p_1^0 = \frac{1}{2}$ in response to $C_2(p_2^1 < \frac{1}{2}, p_2^0 = \frac{1}{2})$.

Case 3b: Next, we analyze the behavior of politician 1 if he is dissonant and candidate 2 offers $C_2(p_2^1 < \frac{1}{2}, p_2^0 = \frac{1}{2})$. In contrast to our considerations for congruent politicians above, it is no longer clear this time whether politician 1 will behave congruently or dissonantly. Nevertheless, it still holds that he will offer a contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$. To substantiate this claim we distinguish four cases:

• (i) Suppose candidate 1 is elected and behaves in a dissonant manner regardless of the threshold contract he has offered.²⁶ Then we obtain

  $$\begin{aligned} EU^{1}\Big (p_1^1 = p_1^0 = \frac{1}{2} \Big ) = z(R+G) + (1-z)(R+G)= R+G \end{aligned}$$

  (5.11)

  and

  $$\begin{aligned} EU^{1}\Big (p_1^1<\frac{1}{2}, p_1^0 = \frac{1}{2} \Big )= & {} \frac{1}{2}\{z[R+G] + (1-z)[R+G + (1-\mu ) \delta R]\} \nonumber \\= & {} \frac{1}{2} [R + G + (1-z)(1-\mu ) \delta R] < R + \frac{G}{2}, \qquad \qquad \end{aligned}$$

  (5.12)

  where $EU^1$ denotes the expected utility of politician 1 depending on the contract he has offered. Hence, expected utility will be larger if he offers $p_1^1 = p_1^0 = \frac{1}{2}$.

• (ii) Suppose candidate 1 is elected and behaves in a congruent manner, regardless of the threshold contract he has offered.²⁷ For such circumstances we obtain

  $$\begin{aligned} EU^{1}\Big (p_1^1 = p_1^0 = \frac{1}{2} \Big )= & {} z[R+ \delta R + \delta ^2 R] + (1-z) [R+ \mu (\delta R + \delta ^2 R)] \nonumber \\= & {} R + [z+(1-z)\mu ](1+\delta ) \delta R \end{aligned}$$

  (5.13)

  and

  $$\begin{aligned} EU^{1}\Big (p_1^1 < \frac{1}{2}, p_1^0 = \frac{1}{2} \Big )= & {} \frac{1}{2} \{z[R+ \delta R + \delta ^2 R] + (1-z) [R+ \mu (\delta R + \delta ^2 R)]\} \nonumber \\= & {} \frac{1}{2} \{R + [z+(1-z)\mu ](1+\delta ) \delta R\}. \end{aligned}$$

  (5.14)

  As the expression (5.13) is larger than the expression in (5.14), candidate 1 is better off by offering $p_1^1 = p_1^0 = \frac{1}{2}$.

• (iii) Suppose candidate 1 is elected and behaves dissonantly with a contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$ and congruently with a contract $C_1(p^1_1 < \frac{1}{2}, p^0_1 = \frac{1}{2})$. According to Eqs. (5.12) and (5.14) acting congruently after having offered $p_1^1 < \frac{1}{2}$ is only optimal if $G < [z+(1-z)\mu ](1+\delta ) \delta R$. However, for $G {<} [z+(1-z)\mu ] (1+\delta ) \delta R$ the politician would act congruently after having offered $p_1^1 = \frac{1}{2}$ according to Eqs. (5.11) and (5.13). This is a contradiction and hence, case (iii) cannot occur.

• (iv) Suppose candidate 1 is elected and behaves congruently with the contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$ while behaving dissonantly with $C_1(p^1_1 < \frac{1}{2}, p^0_1 = \frac{1}{2})$. The utility of acting dissonantly with contract $C_1(p_1^1 < \frac{1}{2}, p_1^0 = \frac{1}{2})$ is smaller than the utility of acting dissonantly with contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$. As we have assumed that the candidate behaves congruently under $C_1(p_1^1 = p_1^0 = \frac{1}{2})$ and thus achieves higher or equal utility than by acting dissonantly, the utility of acting dissonantly with $C_1(p_1^1 < \frac{1}{2}, p_1^0 = \frac{1}{2})$ is smaller than the utility of behaving congruently with contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$. Hence, we can conclude that if politician 1 is of the dissonant type, he will always offer a contract $C_1(p_1^1 = p_1^0 = \frac{1}{2})$ given that candidate 2 offers a contract $C_2(p_2^1 < \frac{1}{2}, p_2^0 = \frac{1}{2})$.

To sum up, $C_i(p_i^1 = p_i^0 = \frac{1}{2}) \ \forall i \in \{1,2\}$ is the unique equilibrium under the election scheme RES.$\square$

Proof of Proposition 5.5

In Proposition 5.4 we have shown that both politicians will offer $C_i(p^1_i=p_i^0=\frac{1}{2})$ if they believe that voters will use RES. Now we show that RES is optimal for voters.

The proof of Proposition 5.3 in Appendix B shows that the equilibrium price on the information market will be larger than $\frac{1}{2}$ if the incumbent chooses the socially optimal action, while it will be smaller than $\frac{1}{2}$ if the incumbent chooses the socially undesirable action. So RES is optimal, as it induces the socially optimal action. Specifically, under RES a politician (say $i=2$) who offers a contract with a price smaller than $\frac{1}{2}$ will never generate a higher utility than a politician who offers thresholds $p_1^1$ and $p_1^0$ equal to $\frac{1}{2}$. Thus in this case electing politician 1 can never be worse than electing politician 2. Finally, we note that under RES a politician (say $i=2$) who offers a contract with a threshold strictly larger than $\frac{1}{2}$ will never generate a higher utility than a politician who offers thresholds $p_1^1$ and $p_1^0$ equal to $\frac{1}{2}$. In this case, electing politician 1 can never be worse than electing politician 2. This completes the proof. $\square$

Proof of Theorem 5.1

We start with dissonant politicians and look first at the case $s_1=1$ where the popular action is optimal from the voters’ point of view. The politician, however, would prefer the unpopular action. In the scenario with threshold contracts, the dissonant politician will undertake the socially optimal action if and only if

$$\begin{aligned} R + \delta R + \delta ^2 R&\ge R + G \nonumber \\ \Leftrightarrow \quad \delta R (1 + \delta )&\ge G. \end{aligned}$$

(5.15)

Comparison with the condition when threshold contracts are absent shows that Condition (5.15) is identical to Condition (5.4). The reason is that threshold contracts have no impact in state $s_1=1$.

We next consider the case $s_1=0$. In this state, voters prefer the unpopular action, while the dissonant politician prefers the popular action. The dissonant politician will only undertake the socially optimal action if

$$\begin{aligned} R + \mu (\delta R + \delta ^2 R)&\ge R + G \nonumber \\ \Leftrightarrow \quad \delta R \mu (1+\delta )&\ge G. \end{aligned}$$

(5.16)

Comparison with the condition in the scenario without threshold incentive contracts shows that Condition (5.5) is tighter than Condition (5.16), i.e. the set of parameter values fulfilling (5.16) is larger than the corresponding set for Condition (5.5). For instance, Eq. (5.16) is always fulfilled if R is sufficiently high, which is not true in general under Condition (5.5).

Next consider congruent politicians. In case $s_1=1$, a congruent politician will undertake the socially optimal action if

$$\begin{aligned} R + G + \delta R + \delta ^2 R \ge R. \end{aligned}$$

(5.17)

This condition is always fulfilled. In case $s_1=0$, a congruent politician will undertake the optimal action if

$$\begin{aligned} R + G + \mu (\delta R + \delta ^2 R) \ge R. \end{aligned}$$

(5.18)

Again, this condition is always fulfilled. Hence, in both states of the world, the politician will always pursue the policy optimal for the voters if he has offered a threshold contract with $p_i^1=p_i^0=\frac{1}{2}$. As showed in Eq. (5.7), this is not necessarily true for congruent politicians in the scenario without threshold contracts. $\square$

Proof of Proposition 5.6

Suppose that relative to R, G is sufficiently large to ensure that congruent politicians will always act congruently and dissonant politicians will always act dissonantly, irrespective of the threshold contracts they have offered. In Appendix B we show that, for G sufficiently large relative to R, the equilibrium price will be smaller than 1, even if politicians act in a socially optimal way. Thus, if both candidates offered contracts $p_i^1 = p_i^0 = 1$, neither of them would ever be able to fulfill their contract. This is an example of overpromising.

Suppose next that both candidates are of the congruent type. Then no candidate will deviate from the Nash equilibrium $p_i^1 = p_i^0 = 1$, as a deviating candidate would never be elected.

Now we show that the Nash equilibrium $p_i^1 = p_i^0=1$ is unique for certain parameters. Suppose both candidates offer threshold contracts with $p_1^1 = p_2^1~<~1$ and $p_1^0 = p_2^0 < 1$. Politicians face the trade-off between offering the largest thresholds that can be reached by acting congruently and deviating from this offer to higher values, thereby increasing election chances to 1. Deviation to higher threshold values is profitable if

$$\begin{aligned} \frac{1}{2}\{z [R+G+ \delta R + \delta ^2 R] + (1-z)[R+G+\mu (\delta R + \delta ^2 R)]\}< & {} (R+G) \nonumber \\ \Leftrightarrow R \{[z + \mu (1-z)] (\delta + \delta ^2) - 1\}< & {} G . \end{aligned}$$

(5.19)

We see that this condition will always be fulfilled if G is sufficiently large relative to R.²⁸ $\square$

Appendix B: Political Information Market

In this appendix we describe the functioning of the political information market in detail. First, we describe the assets and the investors. As investors receive information from two sources—the private signals and actions of politicians—we have to examine how both sources of information jointly determine the beliefs of the investors, step by step. Finally, we determine the equilibrium price in the market.

1.1 B.1 Assets

We assume that a political information market is organized during the first period after politicians have chosen their actions.

There are two assets, D and E, in which investors can trade. If the office-holder is reelected after the second period, the owners of asset D receive one monetary unit for a single unit of D. If the politician stands for reelection but is not reelected after the second period, the owners of asset E receive one monetary unit for a single unit of E. If the politician is not able to run for second reelection, e.g. if he was already deselected at the first reelection or if he does not want to stand for reelection, then all transactions that have occurred will be neutralized. This means that each investor will be paid back the money he has invested.²⁹

The information market works as follows: A bank or an issuer offers an equal amount of assets D and E. On the secondary market, traders can buy assets D or E.³⁰ Trading in the secondary market results in price p for one unit of asset D. As buying one unit of D and one unit of E pays one monetary unit with certainty, the price of asset E must be $1-p$, otherwise either traders or the issuer could make riskless profits. An equilibrium on the information market is a price $p^*$ such that traders demand an equal amount of assets D and E.³¹

It is useful to look more closely at the event tree associated with the assets. If, for example, an investor buys one unit of asset D at price p, then the event tree and the payoffs for the information market are given as in Fig. 5.3.

Fig. 5.3

Event tree and payoffs for the information market

In this chapter we specifically design information markets to allow for the design of reelection threshold contracts. If threshold contracts are offered, then the event tree and the payoffs for the information market have to be modified in the following way as depicted in Fig. 5.4.

Fig. 5.4

Event tree and payoffs for the information market including threshold contracts

Finally, note that with probability $\mu$ there is complete information in period 1. Then the price in the information market will be either 1 or 0, depending on whether the politician undertook the socially optimal action or not.

1.2 B.2 Investors

There are N potential investors.³² Investors are a subgroup of voters. We assume that there are many investors in the market. However, compared to the total number of voters investors constitute a minority and can not influence the voting outcome.

We assume that investors have log-utility with

$$\begin{aligned} U_j (Y_j +W_j) = \ln (Y_j + W_j), \end{aligned}$$

(5.20)

where $W_j$ is the investor’s wealth and $Y_j$ is gain or loss in the information market.³³ Each investor j obtains a signal $\sigma _j \in \{0; 1\}$ about the state of the world at the point in time when the politician in office discovers the state of the world.³⁴ The probability that investor j receives a correct signal, i.e. that $\sigma _j = s_1$, is given by $h_j \in (\frac{1}{2}, 1)$, where each investor j knows his personal signal quality $h_j$. Our assumption $h_j> \frac{1}{2}$ implies that the signals are not completely uninformative.³⁵ We assume that $h_j$ does not depend on the state that has occurred.³⁶

We first calculate the investors’ posterior probability estimations of the state after they have received their signals. We obtain:

$$\begin{aligned} Prob(s_1=1|\sigma _j=1) = \frac{z h_j}{z h_j + (1-z)(1-h_j)}, \end{aligned}$$

(5.21)

$$\begin{aligned} Prob(s_1=1|\sigma _j=0) = \frac{z (1-h_j)}{(1-z)h_j + (1-h_j)z}, \end{aligned}$$

(5.22)

$$\begin{aligned} Prob(s_1=0|\sigma _j=1) = \frac{(1-z)(1- h_j)}{z h_j + (1-z)(1-h_j)}, \end{aligned}$$

(5.23)

$$\begin{aligned} Prob(s_1=0|\sigma _j=0) = \frac{(1-z)h_j}{(1-z)h_j + (1-h_j)z}. \end{aligned}$$

(5.24)

1.3 B.3 Information from the Politician’s Choice

Investors may receive additional information about the state by observing the action of the incumbent. Recall that a politician of the congruent type will always behave congruently in equilibrium when threshold contracts are used. The behavior of a dissonant incumbent depends on the parameters R, G, $\delta$, and $\mu$, which are common knowledge among investors. Three cases can occur: First, the value of G may be sufficiently low relative to R. Then dissonant politicians will behave congruently. Second, the value of G is at an intermediate level, and dissonant politicians will behave congruently in the popular state $s_1=1$, while they will behave dissonantly in the unpopular state $s_1=0$. Third, the value of G may be rather high relative to the benefits from holding office. Then dissonant politicians will behave dissonantly in both states of the world. We summarize the three cases in Table 5.1, where $a_1^c$ denotes the action of a congruent politician and $a_1^d$ denotes the action of a dissonant politician.

Table 5.1 Actions of politicians

In the following we use $c \in \{1, 2, 3\}$ to denote the cases. In the next step we will calculate the conditional probabilities $Prob_c(s_1=1|a_1=1)$ and $Prob_c(s_1=0|a_1~=~0)$ for an individual investor without private signals updating his beliefs in the signalling game with politicians choosing their action. For example, we obtain $Prob_2(s_1=1|a_1=1)$ as $\frac{z}{z + \frac{1}{2}(1-z)} = \frac{2z}{z+1}$ for $c=2$. We summarize the conditional probabilities in Table 5.2.

Table 5.2 Conditional probabilities

1.4 B.4 Private Signals and Information from Politicians

Finally, we calculate the conditional probabilities $Prob_c(s_1|\sigma _j,a_1)$ for $c \in \{1,2,3\}$ when voters have received their private signals $\sigma _j$ and draw inferences from the signalling games among politicians.

Case $c=1$

Suppose $c=1$. Then investors will learn the state with certainty by observing the action of the incumbent and can disregard their signals $\sigma _j$. We obtain

$$\begin{aligned} Prob_1(s_1=1|\sigma _j=1, a_1 = 1) = Prob_1(s_1=1|\sigma _j=0, a_1 = 1) = 1, \end{aligned}$$

$$\begin{aligned} Prob_1(s_1=1|\sigma _j=1, a_1 = 0) = Prob_1(s_1=1|\sigma _j=0, a_1 = 0) = 0, \end{aligned}$$

$$\begin{aligned} Prob_1(s_1=0|\sigma _j=1, a_1 = 0) = Prob_1(s_1=0|\sigma _j=0, a_1 = 0) = 1, \end{aligned}$$

$$\begin{aligned} Prob_1(s_1=0|\sigma _j=1, a_1 = 1) = Prob_1(s_1=0|\sigma _j=0, a_1 = 1) = 0. \end{aligned}$$

Case $c=2$

In case 2, investors know with certainty that the true state of the world is $s_1=0$ when they observe $a_1=0$, i.e.

$$\begin{aligned} Prob_2(s_1=1|\sigma _j=1, a_1 = 0) = Prob_2(s_1=1|\sigma _j=0, a_1 = 0) = 0, \end{aligned}$$

$$\begin{aligned} Prob_2(s_1=0|\sigma _j=1, a_1 = 0) = Prob_2(s_1=0|\sigma _j=0, a_1 = 0) = 1. \end{aligned}$$

If investors observe $a_1=1$, then the signalling game reveals that the probability of $s_1=1$ after observing $a_1=1$ is equal to $\frac{2z}{z+1}$. Using the additional information from signal $\sigma _j$, investor j forms the following a posteriori belief³⁷:

$$\begin{aligned} Prob_2(s_1=1|\sigma _j=1, a_1 = 1) = \frac{\frac{2z}{z+1} h_j}{\frac{2z}{z+1} h_j+ \frac{1-z}{z+1} (1-h_j)} = \frac{2 z h_j}{2 z h_j + (1-z) (1-h_j)}. \end{aligned}$$

In a similar way we obtain

$$\begin{aligned} Prob_2(s_1=1|\sigma _j=0, a_1 = 1) = \frac{2 z (1 - h_j)}{2 z (1-h_j) + (1-z) h_j}, \end{aligned}$$

$$\begin{aligned} Prob_2(s_1=0|\sigma _j=1, a_1 = 1) = \frac{(1-z) (1 - h_j)}{2 z h_j + (1-z) (1-h_j)}, \end{aligned}$$

$$\begin{aligned} Prob_2(s_1=0|\sigma _j=0, a_1 = 1) = \frac{(1-z) h_j}{2 z (1-h_j) + (1-z) h_j}. \end{aligned}$$

Case $c=3$

In case 3, the investors do not gain any information from the politician’s action, as there is complete pooling. A congruent politicians behaves congruently, while all dissonant politicians behave dissonantly, and the probability for both types of politician equals $\frac{1}{2}$. Hence,

$$\begin{aligned} Prob_3(s_1=1|\sigma _j=1, a_1 = 1) = Prob_3(s_1=1|\sigma _j=1, a_1 = 0) = \frac{z h_j}{z h_j + (1-z) (1-h_j)}, \end{aligned}$$

$$\begin{aligned} Prob_3(s_1=1|\sigma _j=0, a_1 = 1) = Prob_3(s_1=1|\sigma _j=0, a_1 = 0) = \frac{z (1- h_j)}{(1-z) h_j + z (1-h_j)}, \end{aligned}$$

$$\begin{aligned} Prob_3(s_1=0|\sigma _j=1, a_1 = 1) = Prob_3(s_1=0|\sigma _j=1, a_1 = 0) = \frac{(1-z)(1- h_j)}{z h_j + (1-z) (1-h_j)}, \end{aligned}$$

$$\begin{aligned} Prob_3(s_1=0|\sigma _j=0, a_1 = 1) = Prob_3(s_1=0|\sigma _j=0, a_1 = 0) = \frac{(1-z) h_j}{(1-z) h_j + z (1-h_j)}. \end{aligned}$$

1.5 B.5 Price Formation Process

For ease of exposition, we assume that all investors are homogeneous concerning the quality of their signals $\sigma _j$, i.e. we assume that $h_j = h \ \forall j \in \{1,...N\}$.³⁸ Thus, investors only differ as to whether they receive signal $\sigma _j = 1$ or $\sigma _j =0$. When the number of investors is sufficiently large a fraction h of the investors will receive the correct signal, i.e. they receive $\sigma _j =1$ if $s_1=1$ or $\sigma _j =0$ if $s_1=0$, respectively.³⁹ A fraction $1-h$ will receive a misleading signal, i.e. they receive $\sigma _j =1$ if $s_1=0$ or $\sigma _j =0$ if $s_1=1$.

From Corollary 5.3 in Appendix C we know that the price in the information market will be a weighted average of the prices that would arise in the two subgroups of investors. This means that the price will be h times the price that would arise in a market where all investors receive a correct signal plus (1-h) times the price in a market where investors only receive incorrect signals. Again, we go through all three cases.

Case $c=1$

We start with case $c=1$. In this scenario, the action of the incumbent will perfectly reveal the state of the world. Thus, we obtain

$$\begin{aligned} p^{*1}_{1,1} = p^{*1}_{0,0} = 1 \end{aligned}$$

(5.25)

and

$$\begin{aligned} p^{*1}_{1,0} = p^{*1}_{0,1} = 0, \end{aligned}$$

(5.26)

where $p^{*c}_{a_1,s_1}$ denotes the equilibrium price in case c given action $a_1$ and state $s_1$. The equilibrium price will equal 1 if the incumbent chooses the socially optimal action, while the price will be 0 if the politician chooses the non-optimal action.

Case $c=2$

If $c=2$, we obtain

$$\begin{aligned} p^{*2}_{0,0} = 1 \end{aligned}$$

(5.27)

and

$$\begin{aligned} p^{*2}_{0,1} = 0, \end{aligned}$$

(5.28)

which reflects the fact that the equilibrium price will be equal to zero or one upon observing $a_1=0$, as this action reveals the true state of the world with certainty. If the incumbent undertakes $a_1=1$ in case $c=2$, then we obtain

$$\begin{aligned} p^{*2}_{1,1} = 1 - \frac{(1-z^2) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]} \end{aligned}$$

(5.29)

and

$$\begin{aligned} p^{*2}_{1,0} = \frac{2z(1+z) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]}. \end{aligned}$$

(5.30)

Case $c=3$

If $c=3$, we obtain

$$\begin{aligned} p^{*3}_{1,1} = 1 - \frac{(1-z) h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]}, \end{aligned}$$

(5.31)

$$\begin{aligned} p^{*3}_{1,0} = \frac{z h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]}, \end{aligned}$$

(5.32)

$$\begin{aligned} p^{*3}_{0,0} = 1 - \frac{z h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]}, \end{aligned}$$

(5.33)

$$\begin{aligned} p^{*3}_{0,1} = \frac{(1-z) h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]}. \end{aligned}$$

(5.34)

We observe that $p^{*3}_{1,1} = 1 - p^{*3}_{0,1}$ and $p^{*3}_{0,0} = 1 - p^{*3}_{1,0}$. The next proposition is the main result of Appendix B and the extended version of Proposition 5.5 in the text.

Proposition 5.3

(detailed version)

Suppose that $h > \hat{h}(z)$ with

$$\begin{aligned} \hat{h}(z) = {\displaystyle \frac{1 + \sqrt{\frac{3 z^2 + 2z -1}{-5 z^2 +10z -1}}}{2}} < 1. \end{aligned}$$

(5.35)

Then the equilibrium price in the information market fulfills the following conditions:

$$\begin{aligned} p^{*c}_{1,1} > \frac{1}{2} \ \forall c, \end{aligned}$$

$$\begin{aligned} p^{*c}_{0,0} > \frac{1}{2} \ \forall c, \end{aligned}$$

$$\begin{aligned} p^{*c}_{1,0} < \frac{1}{2} \ \forall c \end{aligned}$$

and

$$\begin{aligned} p^{*c}_{0,1} < \frac{1}{2} \ \forall c. \end{aligned}$$

Proposition 5.3 shows that for $h > \hat{h}(z)$, the equilibrium price will be larger than one-half in all circumstances, if the incumbent behaves congruently, while the equilibrium price will be smaller than one-half if the politician behaves dissonantly. Note that for $z \in (\frac{1}{2},1)$, $\hat{h}(z)$ is increasing in z and that $\hat{h}(z) \in (\frac{1}{2} + \sqrt{\frac{3}{44}},1)$. The intuition that $\hat{h}(z)$ must be larger than $\frac{1}{2}$ runs as follows: In the unpopular state $s_1=0$ in case $c=2$, where $Prob(s_1=0|a_1=1)$ is rather low, the signal must be sufficiently informative in order to detect dissonant behavior of a politician. A formal derivation and explanation for Condition (5.35) is given in the following proof of Proposition 5.3.

Proof of Proposition 5.3 (detailed version)

We will prove the statement in three steps:

Step 1: First, it is obvious that $p^{*1}_{1,1} > \frac{1}{2}$, $p^{*1}_{0,0} > \frac{1}{2}$, $p^{*1}_{1,0} < \frac{1}{2}$, $p^{*1}_{0,1} < \frac{1}{2}$, $p^{*2}_{0,0} > \frac{1}{2}$ and $p^{*2}_{0,1} < \frac{1}{2}$ for any values of $h \in (\frac{1}{2},1)$.

Step 2: The condition $p^{*2}_{1,1} > \frac{1}{2}$ is equivalent to

$$\begin{aligned} \frac{(1-z^2) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.36)

Some manipulations yield the condition

$$\begin{aligned} 2 z (1-z) + h(1-h) (11 z^2 - 6z -1) > 0. \end{aligned}$$

(5.37)

We note that $h(1-h) < \frac{1}{4} \ \forall h \in (\frac{1}{2},1)$ and that $2z(1-z) > - \frac{1}{4}(11 z^2 -6z -1) \ \forall z \in (\frac{1}{2}, 1)$. Thus, Condition (5.37) is always fulfilled.

Next we examine $p^{*2}_{1,0} < \frac{1}{2}$, which is equivalent to

$$\begin{aligned} \frac{2z(1+z) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.38)

Rearranging terms yields

$$\begin{aligned} h(1-h) < \frac{2z(1-z)}{-5 z^2 + 10z -1}. \end{aligned}$$

(5.39)

Solving for h leads to

$$\begin{aligned} h > \frac{1 + \sqrt{\frac{3 z^2 + 2z -1}{-5 z^2 +10z -1}}}{2}. \end{aligned}$$

(5.40)

Step 3: The next condition $p^{*3}_{1,1} > \frac{1}{2}$ is equivalent to

$$\begin{aligned} \frac{(1-z) h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.41)

This condition can be transformed to

$$\begin{aligned} z (1-z) + h(1-h) (4 z^2 - 2z -1) > 0. \end{aligned}$$

(5.42)

We note that $h(1-h) < \frac{1}{4} \ \forall h \in (\frac{1}{2},1)$ and that $z(1-z) > - \frac{1}{4}(4 z^2 -2z -1) \ \forall z \in (\frac{1}{2}, 1)$. Thus, Condition (5.42) is always fulfilled.

The condition $p^{*3}_{1,0} < \frac{1}{2}$ is equivalent to

$$\begin{aligned} \frac{z h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.43)

After some manipulations, we obtain

$$\begin{aligned} h(1-h) < \frac{z(1-z)}{- 4 z^2 + 6z -1}, \end{aligned}$$

(5.44)

which then yields

$$\begin{aligned} h > \frac{1 + \sqrt{\frac{2z -1}{-4 z^2 +6z -1}}}{2}. \end{aligned}$$

(5.45)

Condition (5.45) is a weaker condition than Condition (5.40) as the following inequality holds for all $z \in (\frac{1}{2},1)$:

$$\begin{aligned} \sqrt{\frac{2z -1}{-4 z^2 +6z -1}} < \sqrt{\frac{3 z^2 + 2z -1}{-5 z^2 +10z -1}}. \end{aligned}$$

(5.46)

Hence, if $h > \frac{1 + \sqrt{\frac{3 z^2 + 2z -1}{-5 z^2 +10z -1}}}{2}$, then Condition (5.45) will always be fulfilled.

Next we investigate $p^{*3}_{0,0} > \frac{1}{2}$, which leads to

$$\begin{aligned} \frac{z h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.47)

This condition is the same as (5.43) and thus always fulfilled for $h > \frac{1 + \sqrt{\frac{3 z^2 + 2z -1}{-5 z^2 +10z -1}}}{2}$.

Finally, we consider $p^{*3}_{0,1} < \frac{1}{2}$, which yields

$$\begin{aligned} \frac{(1-z) h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]} < \frac{1}{2}. \end{aligned}$$

(5.48)

This is identical to Condition (5.41), which always holds as shown above. $\square$

1.6 B.6 Sophisticated Election Scheme

In this subsection we prove Proposition 5.7 by proving the following detailed version of Proposition 5.7:

Proposition 5.7

(detailed version)

Suppose that $h > \hat{\hat{h}}$ with

$$\begin{aligned} \hat{\hat{h}} = {\displaystyle \frac{1 + \sqrt{\frac{z+1}{9z+1}}}{2} < \frac{1}{2} + \sqrt{\frac{3}{44}}} \ \approx 0.761. \end{aligned}$$

(5.49)

Then the equilibrium price in the information market fulfills the following conditions:

$$\begin{aligned} p^{*c}_{1,1} > z \ \forall c, \end{aligned}$$

$$\begin{aligned} p^{*c}_{0,0} > 1-z \ \forall c, \end{aligned}$$

$$\begin{aligned} p^{*c}_{1,0} < z \ \forall c \end{aligned}$$

and

$$\begin{aligned} p^{*c}_{0,1} < 1-z \ \forall c. \end{aligned}$$

Proof of Proposition 5.7

The proof follows the same line as the proof of Proposition 5.3.

Step 1: It is obvious that $p^{*1}_{1,1} > z$, $p^{*1}_{0,0} > 1-z$, $p^{*1}_{1,0} < z$, $p^{*1}_{0,1} < 1-z$, $p^{*2}_{0,0} > 1-z$ and $p^{*2}_{0,1} < 1-z$.

Step 2: We explore the condition $p^{*2}_{1,1} > z$, which is equivalent to

$$\begin{aligned} \frac{(1-z^2) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]} < 1-z. \end{aligned}$$

(5.50)

This can be rewritten into

$$\begin{aligned} 2 (1-z)^2 + h(1-h) (-9 z^2 +16z -7) > 0. \end{aligned}$$

(5.51)

Using $h(1-h) < \frac{1}{4} \ \forall h \in (\frac{1}{2},1)$ and $2(1-z)^2 > \frac{1}{4}(9 z^2 -16z +7) \ \forall z \in (\frac{1}{2}, 1)$ shows that Condition (5.51) is fulfilled for all $z \in \{\frac{1}{2},1\}$.

Next we examine $p^{*2}_{1,0} < z$, which yields

$$\begin{aligned} \frac{2(1+z) h (1-h)}{[2 z h + (1-z)(1-h)][2 z (1-h) + (1-z)h]} < 1. \end{aligned}$$

(5.52)

Rearranging terms leads to

$$\begin{aligned} h(1-h) < \frac{2 z(1-z)}{-9 z^2 + 8z +1} = \frac{2z}{9z+1}, \end{aligned}$$

(5.53)

which implies

$$\begin{aligned} h > \frac{1 + \sqrt{\frac{z+1}{9z+1}}}{2}. \end{aligned}$$

(5.54)

Step 3: Finally, conditions $p^{*3}_{1,1} > z$, $p^{*3}_{1,0} < z$, $p^{*3}_{0,0} > 1-z$ and $p^{*3}_{0,1} < 1-z$ are equivalent to

$$\begin{aligned} \frac{h (1-h)}{[z h + (1-z)(1-h)][z (1-h) + (1-z)h]} < 1, \end{aligned}$$

(5.55)

which in turn holds if and only if

$$\begin{aligned} h(1-h) < \frac{1}{4}. \end{aligned}$$

(5.56)

As $h \ge \frac{1}{2}$ Condition (5.56) is always fulfilled and the assertion is proven.$\square$

By comparing $\hat{h}$ and $\hat{\hat{h}}$ we obtain the following corollary:

Corollary 5.2

For all $z \$ with $\frac{1}{2}< z < 1$, $\hat{\hat{h}} < \hat{h}$.

Hence, for all values $z \in (\frac{1}{2},1)$ Condition (5.49) is easier to fulfill than Condition (5.35).⁴⁰ As a consequence, SES, which uses the results from Proposition 5.7, is applicable for signals with lower information content than RES. Note that Corollary 5.2 follows directly from comparing $\hat{h}$ and $\hat{\hat{h}}$. The claim $\hat{\hat{h}} < \hat{h}$ can be transformed to $2z^2 + z >1$, which proves the corollary.

Appendix C: General Price Formation Process

In this appendix we determine a general formula for an information market with heterogeneous agents. Suppose, without loss of generality, that politician 1 has been elected after offering a contract $C_1(p_1^1, p_1^0)$, that the politician undertakes $a_1=1$, and hence that $p_1^1$ applies.

For a price $p < p_1^1$, no investor will have a strict incentive to buy assets, as he will be paid back p. Thus, suppose $p \ge p_1^1$. An investor j with signal $\sigma _j$ has to weigh up the state of his information and the information the market price will reveal.⁴¹ One way of modeling the information aggregation process is as follows:

$$\begin{aligned} Prob_j (RE|p) = b_j \ Prob_j (RE) + (1-b_j) \ p, \end{aligned}$$

(5.57)

where $Prob_j(RE|p)$ is the probability assessment of investor j that the incumbent will be reelected, taking into account the information inferred from the market price. The term $Prob_j (RE)$ is given as the individual reelection probability estimation of an investor and depends on his signal $\sigma _j$, the signal quality $h_j$, the action $a_1$, and the case c. If, e.g., $c=3$, $a_1=1$, and $\sigma _j=1$, then $Prob_j (RE) = \frac{z h_j}{z h_j + (1-z)(1- h_j)}$, where we assume that z and $h_j$ are known to investor j. The weight $b_j$ (with $0 < b_j \le 1$) describes self-assessed confidence, i.e. the subjective confidence of an investor in his estimation $Prob_j (RE)$ relative to the market belief expressed by the price p.⁴² The information aggregation formula (5.57) is flexible. It captures the case $b_j=1$ when investors rely only on their own signal, which would occur if they can only submit a quantity (and not an entire demand/supply schedule depending on the price) to the market. For small values of $b_j$, investors rely mainly on the information aggregated by the market.⁴³

Given price p and signal $Prob_j (RE)$, an investor j aims to maximize his expected utility, which can be written as the following optimization problem:

$$\begin{aligned} \max \limits _{d_j} \ EU_j = Prob_j (RE|p) \ ln (W_j + d_j (1-p)) + (1-Prob_j(RE|p)) \ ln (W_j - d_j p), \end{aligned}$$

(5.58)

where $d_j$ denotes his demand. If $d_j$ is positive, investor j will want to buy $d_j$ units of asset D. If $d_j$ is negative, investor j will want to buy $d_j$ units of asset E. The solution of the investor’s optimization problem is

$$\begin{aligned} d_j^*= & {} W_j \ \frac{b_j \ Prob_j (RE) + (1-b_j) p - p}{p (1-p)} \nonumber \\ \Leftrightarrow \ \ d_j^*= & {} W_j \ \frac{b_j \ Prob_j (RE) - p \ b_j}{p (1-p)}. \end{aligned}$$

(5.59)

We thus obtain:

Proposition 5.8

There is a unique equilibrium in the information market given by

$$\begin{aligned} p^* = \sum \limits _{j=1}^{N} Prob_j (RE) \ \frac{W_j \ b_j}{\sum _{k=1}^{N} W_k \ b_k} \ . \end{aligned}$$

(5.60)

Proof of Proposition 5.8

Equilibrium in the information market requires that condition $\sum \limits _{j=1}^{N} d_j^* =0$ be fulfilled, which implies $\sum \limits _{j=1}^{N} W_j \ b_j \ Prob_j (RE) \ - p \sum \limits _{j=1}^{N} W_j \ b_j \ =0$. The assertion follows from this equation.$\square$

The market price is a wealth- and confidence-weighted average belief on the part of investors. We note that the market price is equal to the simple average belief of investors if traders are homogeneous with respect to wealth and confidence in their own belief. If confidence levels are homogeneous, the market price is a wealth-weighted average belief on the part of traders. We summarize both cases in the following corollary:

Corollary 5.3

• (i) Suppose $W_j = W \ \ \forall j$ and $b_j = b \ \ \forall j$. Then $p^* = \frac{1}{N} \sum \limits _{j=1}^{N} Prob_j (RE)$.

• (ii) Suppose $b_j = b \ \ \forall j$. Then $p^* = \sum \limits _{j=1}^{N} Prob_j (RE) \frac{W_j}{\sum _{k=1}^{N} W_k}$.

Appendix D: Welfare Gains

Here we provide an example of the welfare gains that can be achieved with the triple mechanism. Suppose that, at a time when this institution is introduced, it is only known that $\delta$ is equal to 1 and that $\mu$ is uniformly distributed in $[0, \frac{1}{2}]$. Since only the proportion of R and G is important for our analysis, we write $G=\alpha R$ with $0~ \le ~ \alpha ~ <~ \infty$. In the following, we calculate the values of $\mu$ that enable congruent behavior by the incumbent. We use eo to denote the case with elections only and tm to denote the scenario with the triple mechanism. From Condition (5.6) we conclude that, in the case of elections alone, a congruent politician will only behave congruently in state $s_1=1$ if

$$\begin{aligned} \alpha R + 3 R \ge R. \end{aligned}$$

This condition is equivalent to $\alpha \ge -2$. In the same way we obtain the other conditions summarized in Table 5.3.

Table 5.3 Conditions—behaviour of politicians

Note that congruent politicians will always behave congruently in the scenario with the triple mechanism, as conditions $\alpha \ge -2$ and $\mu \ge - \frac{\alpha }{2}$ are always fulfilled. Furthermore, if $\alpha \ge 1$ congruent politicians will always behave congruently in the scenario with elections only. Finally, it is apparent that a dissonant politician will never act congruently for $\alpha > 2$, which clearly derives from Corollary 5.1 and Theorem 5.1. In the next stage, we calculate expected utilities, starting with the triple mechanism scenario:

$$\begin{aligned} EU^{tm} = \frac{1}{2} + \frac{1}{2} z {\left\{ \begin{array}{ll} \int \limits _{0}^{\frac{1}{2}} 2 d\mu &{} \text { if } \alpha \le 2 \\ 0 &{} \text { if } \alpha> 2 \end{array}\right. } + \frac{1}{2} (1-z) {\left\{ \begin{array}{ll} \int \limits _{\frac{\alpha }{2}}^{\frac{1}{2}} 2 d\mu &{} \text { if } \alpha \le 1, \\ 0 &{} \text { if }\alpha > 1. \end{array}\right. } \end{aligned}$$

The reasoning for the above expression is as follows: A politician is of the congruent type with probability $\frac{1}{2}$. He always behaves congruently and thus generates a voter utility of 1. The probability that a politician is of the dissonant type and that state $s_1=1$ occurs is given by $\frac{1}{2} z$. In this case, the politician generates a utility of 1 for all feasible values of $\mu$, as long as $\alpha$ is not larger than 2. Finally, the probability that a politician is of the dissonant type and that state $s_1=0$ occurs is given by $\frac{1}{2} (1-z)$. In this case, the politician generates a utility of 1 for all values of $\mu$ with $\mu \ge \frac{\alpha }{2}$, as long as $\alpha$ is not larger than 1.⁴⁴ The calculation in the scenario with elections alone is similar and yields

$$\begin{aligned} EU^{eo}= & {} \frac{1}{2} z + \frac{1}{2} (1-z) {\left\{ \begin{array}{ll} \int \limits _{\frac{1-\alpha }{3}}^{\frac{1}{2}} 2 d\mu &{} \text { if }\alpha \le 1 \\ \int \limits _0^{\frac{1}{2}} 2 d\mu &{} \text { if }\alpha> 1 \end{array}\right. } \nonumber \\&+ \frac{1}{2} z {\left\{ \begin{array}{ll} \int \limits _0^{\frac{1}{2}} 2 d\mu &{} \text { if }\alpha \le 2 \\ 0 &{} \text { if }\alpha>2 \end{array}\right. } + \frac{1}{2} (1-z) {\left\{ \begin{array}{ll} \int \limits _{\frac{1+\alpha }{3}}^{\frac{1}{2}} 2 d\mu &{} \text { if }\alpha \le \frac{1}{2} \\ 0 &{} \text { if }\alpha > \frac{1}{2} . \end{array}\right. } \nonumber \end{aligned}$$

These expressions can be simplified to

$$\begin{aligned} EU^{tm}= & {} {\left\{ \begin{array}{ll} \frac{1}{2} + \frac{1}{2} [1-\alpha (1-z)] &{} \text { if }\alpha \le 1, \\ \frac{1}{2} + \frac{1}{2} z &{} \text { if }1 < \alpha \le 2, \\ \frac{1}{2} &{} \text { if }\alpha > 2 \end{array}\right. } \end{aligned}$$

(5.61)

and

$$\begin{aligned} EU^{eo}= & {} {\left\{ \begin{array}{ll} {\displaystyle z + \frac{1}{3} (1-z)} &{} \text { if }\alpha \le \frac{1}{2}, \\ {\displaystyle z + (1-z)\frac{(1+2\alpha )}{6}} &{} \text { if }\frac{1}{2}< \alpha \le 1, \\ \frac{1}{2} + \frac{1}{2} z &{} \text { if }1 < \alpha \le 2, \\ \frac{1}{2} &{} \text { if }\alpha > 2 . \end{array}\right. } \end{aligned}$$

(5.62)

We illustrate the relationships by calculating the utilities for four different values of $\alpha$. We choose one value of $\alpha$ that is smaller than 1, one value larger than 1, and $\alpha$ equal to 1. These values correspond to the cases where, for the politician, utility G is lower/higher than or equal to utility R. Furthermore, we add the special case $\alpha =0$, where the politician has no private benefits G. The expected utilities in these four cases are summarized in the following Table 5.4:

Table 5.4 Expected utilities

Note that in all cases we have $EU^{tm} \ge EU^{eo}$. Further, we see that $EU^{tm}$ is strictly larger than $EU^{eo}$ if $z < 1$ and $\alpha < 1$. The difference between $EU^{tm}$ and $EU^{eo}$ depends on z for $0<\alpha <1$. The last row in the table shows the relative welfare gains ($\Delta _{EU}$). $\Delta _{EU}$ is maximal for $\alpha = 0$. The example illustrates the following insights:

• (i) Threshold contracts have the highest effect in the case $\alpha =0$, i.e. if the politicians are only motivated by benefits R acquired from holding office and not from choosing his personally preferred action. Note that threshold contracts may reduce the reelection chances of the incumbent. Thus, threshold contracts will be more effective if politicians are mainly interested in getting reelected, which is expressed in a low value of $\alpha$.

• (ii) If $\alpha$ is at least equal to 1, i.e. if politicians are at least as motivated by G as by R, then there is no effect from threshold contracts. This is due to the fact that in state $s_1=0$ congruent politicians always behave congruently, while dissonant candidates always behave dissonantly. The conditions for congruent behavior in state $s_1=1$ are the same in the scenarios with or without threshold contracts. If $\alpha$ is at least equal to 2, then congruent politicians will always behave congruently, while dissonant candidates will always behave dissonantly. Thus, the expected utility is equal to $\frac{1}{2}$.

• (iii) Finally, for a given value of $\alpha$ we discover that $\Delta _{EU}$ is (weakly) decreasing in z. Thus, the higher the probability of the unpopular state $s_1=0$, the larger is the effect of threshold contracts.