Corruption and power in democracies

Abstract

We study the implications of Acton’s dictum that power corrupts when citizens vote (for three parties) and governments then form in a series of elections. In each election, parties have fixed tastes for graft, which affect negotiations to form a government if parliament hangs; but incumbency changes tastes across elections. Our model entails various plausible predictions about long-run patterns of government. Acton’s dictum results in possible government turnover, and in different predictions about possible government composition: for example, that the grand coalition may form.

Appendix (Proofs)

Lemma 2.2

(Equilibrium programs)

• a. If party $i$ is in power then it chooses a policy of $x_{i}$ and graft of $\alpha c_{i}^{2}/4\gamma ^{2}$; and sort $k$ citizens earn

  $$\begin{aligned} -\beta |x_{i}-x_{k}|-\frac{\alpha }{4\gamma }c_{i}^{2}. \end{aligned}$$

• b. If parties $i$ and $j$ share power then they agree to graft of $\alpha c_{i}^{2}/16\gamma ^{2}$ and $\alpha c_{j}^{2}/16\gamma ^{2}$ for parties $i$ and $j$ respectively, and to policy

  $$\begin{aligned} x=\left\{ \begin{array}{lrc} \frac{1}{2}(x_{i}+x_{j}) &{}\quad \mathrm{if} &{} c_{i}=c_{j} \\ x_{i} &{}\quad \mathrm{if} &{} c_{i}<c_{j} \\ x_{j} &{}\quad \mathrm{if} &{} c_{i}>c_{j} \end{array} \right. \end{aligned}$$

  Their agreement yields a surplus of

  $$\begin{aligned} 2K-\beta (1-\max \{c_{i},c_{j}\})|x_{j}-x_{i}|-\frac{\alpha }{8\gamma } (c_{i}^{2}+c_{j}^{2}) \end{aligned}$$

  and sort $k$ citizens earn

  $$\begin{aligned} -\beta |x-x_{k}|-\frac{\alpha }{16\gamma }(c_{i}^{2}+c_{j}^{2}) \end{aligned}$$

• c. If all three parties share power then they agree to graft of $\alpha c_{i}^{2}/36\gamma ^{2}$ for each party $i$ and to policy

  $$\begin{aligned} x=\left\{ \begin{array}{l@{\quad }l} x_{M} &{} \mathrm{if}\,parties\,M\,and\,R\,are\,pure\\ x_{i} &{} \mathrm{if}\,party\,i\,alone\,is\,pure\\ \frac{1}{3}\sum \nolimits _{l}x_{l} &{} \mathrm{otherwise} \end{array} \right. \end{aligned}$$

  Their agreement yields a surplus of

  $$\begin{aligned} 3K-\beta \sum _{l}(1-c_{l})|x_{l}-x|-\frac{\alpha }{6\gamma }\sum _{l}c_{l}^{2} \end{aligned}$$

  and sort $k$ citizens earn

  $$\begin{aligned} -\beta |x-x_{k}|-\frac{\alpha }{36\gamma }\sum _{l}c_{l}^{2} \end{aligned}$$

Proof

• a. We consider utility functions of the form

  $$\begin{aligned} v_{i}(x,y;c_{i})&= c_{i}(\alpha y_{i})^{1/2}-(1-c_{i})\beta |x-x_{i}|\\&-\gamma \sum _{l\in \{L,M,R\}}y_{l}-c_{i}^{2}\frac{\alpha }{4\gamma } +n_{i}+K-\varepsilon \left( x-x_{i}\right) ^{2} \end{aligned}$$

  for any $\varepsilon >0$. (We will take limits as $\varepsilon \rightarrow 0$.) When we have a single party $i$ in power this becomes

  $$\begin{aligned} v_{i}(x,y;1)&= (\alpha y_{i})^{1/2}-\gamma \sum _{l\in \{L,M,R\}}y_{l}-\frac{ \alpha }{4\gamma }+n_{i}+K-\varepsilon \left( x-x_{i}\right) ^{2} \\ v_{i}(x,y;0)&= -\beta |x-x_{i}|-\gamma \sum _{l\in \{L,M,R\}}y_{l}+n_{i}+K-\varepsilon \left( x-x_{i}\right) ^{2} \end{aligned}$$

  The bargaining process and the coalition formation process are irrelevant in this case, where party $i$ chooses $\left( x,y\right)$. Clearly, $x=x_{i}$, and $y_{j}=y_{k}=0$ for $j,k\ne i$ are optimal for $c_{i}\in \left\{ 0,1\right\}$. If $c_{i}=0$ then $y_{i}=0$ while if $c_{i}=1$ then $y_{i}= \frac{\alpha }{4\gamma ^{2}}$. Substituting into $\{u_{k}\}$ yields citizens’ utility.

• b. Now suppose that parties $i$ and $j$ form a coalition, and let $x_{i}<x_{j}$. Our assumption of transferable utility implies that any program which is agreed in equilibrium must maximize the surplus, as the proposer in that round could otherwise profitably deviate. We therefore determine the surplus maximizing policies first. The surplus from any agreement is

  $$\begin{aligned} \Sigma (x,y)&= \sum _{l=i,j}c_{l}(\alpha y_{l})^{1/2}-\sum _{l=i,j}(1-c_{l})\beta |x-x_{l}|-2\gamma \sum _{l=i,j}y_{l}\\&- \frac{\alpha }{4\gamma }\sum _{l=i,j}c_{l}^{2}+2K-\varepsilon \sum _{l=i,j}\left( x-x_{l}\right) ^{2}. \end{aligned}$$

  This is clearly strictly concave in $y_{l}$; so the surplus is maximized by setting $y_{l}=\frac{\alpha c_{i}^{2}}{16\gamma ^{2}}$: $l=i,j$. The maximized (wrt $y$) surplus therefore equals

  $$\begin{aligned} \Sigma (x)=-\sum _{l=i,j}(1-c_{l})\beta |x-x_{l}|-\varepsilon \sum _{l=i,j}\left( x-x_{l}\right) ^{2}-\frac{\alpha }{8\gamma } \sum _{l=i,j}c_{l}^{2}+2K. \end{aligned}$$

  Now consider the choice of $x$. The equilibrium policy must be in the interval $[x_{i},x_{j}]$. However, a more precise delineation requires consideration of various cases:

  • If both parties are corrupt then the policy minimizes $(x-x_{i})^{2}+(x-x_{j})^{2}$ for $x\in [x_{i},x_{j}]$, and is therefore equal to $\frac{1}{2}(x_{i}+x_{j})$.

  • If party $i$ alone is corrupt then the policy maximizes

    $$\begin{aligned} \beta (x_{j}-x)+\varepsilon \left[ (x-x_{i})^{2}+(x-x_{j})^{2}\right] , \end{aligned}$$

    and is therefore equal to $\min \{\frac{\beta }{4\varepsilon }+\frac{1}{2} (x_{i}+x_{j}),x_{j}\}$.

  • If party $j$ alone is corrupt then the policy maximizes

    $$\begin{aligned} \beta (x-x_{i})+\varepsilon \left[ (x-x_{i})^{2}+(x-x_{j})^{2}\right] , \end{aligned}$$

    and is therefore equal to $\max \{\frac{1}{2}(x_{i}+x_{j})-\frac{\beta }{4\varepsilon },x_{i}\}$.

  • If both parties are pure then the policy minimizes $(x-x_{i})^{2}+(x-x_{j})^{2}$ for $x\in [x_{i},x_{j}]$, and is therefore equal to $\frac{1}{2}(x_{i}+x_{j})$.

  Write the maximized surplus as $\Sigma$. Consider a subgame in which parties $i$ and $j$ have formed a coalition, and bargaining has reached round $S$. Our assumption that a caretaker government is equally bad for each party implies that the proposer would offer transfers which leave the other party with a utility of $0$. As each party is equally likely to propose, each party enters round $S$ with an expected utility of $\Sigma /2$. Accordingly, consider a subgame in which parties $i$ and $j$ have formed a coalition, and bargaining has reached round $S-1$ There is a subgame equilibrium in which no agreement is reached in this round; but the arguments above imply that any equilibrium agreement must maximize the surplus. Hence, irrespective of the equilibrium played, each party enters round $S$ with an expected utility of $\Sigma /2$, and the program eventually implemented in any equilibrium must maximize the surplus. The same argument applies to every earlier round. Taking limits as $\varepsilon \rightarrow 0$ and replacing the optimal policy specific for each of the above cases in $\Sigma (x)$ and in the utility of a sort $k$ citizen yields part b.

• c. Now suppose that the grand coalition forms. An argument in the last part again implies that any equilibrium agreement must maximize the surplus, which is

  $$\begin{aligned} \Sigma (x,y)&= \sum _{l}c_{l}(\alpha y_{l})^{1/2}-\sum _{l}(1-c_{l})\beta |x-x_{l}|-3\gamma \sum _{l\in \{L,M,R\}}y_{l}\\&-\frac{\alpha }{4\gamma } \sum _{l}c_{l}^{2}+3K-\varepsilon \sum _{l}\left( x-x_{l}\right) ^{2} \end{aligned}$$

  This is a strictly concave function of $y_{l}$; so the surplus is maximized when $y_{l}=\alpha c_{l}^{2}/36\gamma ^{2}$: all $l$. Substituting into the expression above yields surplus as the following function of $x$:

  $$\begin{aligned} \Sigma (x)=-\sum _{l}(1-c_{l})\beta |x-x_{l}|-\frac{\alpha }{6\gamma } \sum _{l}c_{l}^{2}+3K-\varepsilon \sum _{l}\left( x-x_{l}\right) ^{2}. \end{aligned}$$

  The surplus-maximizing value of $x$ must again be in $[x_{L},x_{R}]$.

  • If all parties are corrupt then the policy must maximize $-\varepsilon \sum _{l}\left( x-x_{l}\right) ^{2}$ and so $x=\frac{1}{3}\sum _{l}x_{l}$.

  • If all parties are pure then

    $$\begin{aligned} \frac{d\Sigma }{dx}=\left\{ \begin{array}{lcc} \beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in [x_{L},x_{M}) \\ -\beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in (x_{M},x_{R}] \end{array} \right. \end{aligned}$$

    so, for $\varepsilon$ small enough, the surplus function is strictly increasing [resp. decreasing] for $x\in [x_{L},x_{M})$ [resp. $x\in (x_{M},x_{R}]$]; and the surplus is maximized when $x=x_{M}$.

  • Now suppose that one party is corrupt. There are three cases to consider.

    • If $L$ is corrupt then

      $$\begin{aligned} \frac{d\Sigma }{dx}=\left\{ \begin{array}{lcc} 2\beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in [x_{L},x_{M}] \\ -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in [x_{M},x_{R}] \end{array} \right. \end{aligned}$$

      Hence, $\Sigma (x)$ is strictly increasing for $x<x_{M}$ when $\varepsilon$ is small enough, and $x_{M}>\frac{1}{3}\sum _{l}x_{l}$ implies that it is decreasing for $x>x_{M}$; so the surplus is maximized when $x=x_{M}$.

    • If $M$ is corrupt then the surplus equals a constant minus $\varepsilon \left[ \sum _{l}(x-x_{l})^{2}\right]$, which is maximized when $x=\frac{1}{3}\sum _{l}x_{l}$.

    • If $R$ is corrupt then

      $$\begin{aligned} \frac{d\Sigma }{dx}=\left\{ \begin{array}{lcc} -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in [x_{L},x_{M}] \\ -2\beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x\in [x_{M},x_{R}] \end{array} \right. \end{aligned}$$

      so the surplus is strictly decreasing for $x>x_{M}$ when $\varepsilon$ is small enough, and is maximized when $x=\frac{1}{3}\sum _{l}x_{l}$.

  • Finally, suppose that one party (say, $i$) is pure. Then

    $$\begin{aligned} \frac{d\Sigma }{dx}=\left\{ \begin{array}{lcc} \beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x < x_{i} \\ -\beta -2\varepsilon \sum _{l}(x-x_{l}) &{}\quad \text{ if } &{} x > x_{i} \end{array} \right. \end{aligned}$$

    so $x=x_{i}$ whenever $\varepsilon$ is small enough.

Taking limits as $\varepsilon \rightarrow 0$ and replacing the optimal policy specific for each of the above cases in $\Sigma (x)$ and in the utility of a sort $k$ citizen yields part c. $\square$

Proposition 3.1

If $d=0$ then, for generic parameters $\{\delta ,\xi ,\theta \}$, every subgame which starts at the beginning of some period $t$ has a unique equilibrium outcome. The government which forms in period $t$ depends on the state as follows:

• a. If $M$ is pure then it is in power.

• b. If $M$ and $R$ are corrupt then $M$ is in power if $\delta <4(\xi +\theta )$; and $L$ is otherwise in power.

• c. If $L$ and $M$ are corrupt then $M$ is in power if $\delta <4\xi$; and $R$ is otherwise in power.

• d. If $M$ is the only corrupt party then $M$ is in power if $\delta <4\xi$; $L$ and $R$ share power if $\theta <2\xi$ and $8(2\xi +\theta )<\delta$; and $R$ is otherwise in power.

• e. If all parties are corrupt then $M$ is in power if $\delta <4\xi$; and $M$ and $R$ otherwise share power.

Proof

There are four possible governments in any one-period game: each of the parties in power, and the coalition which would form in a hung parliament. We start with some useful preliminary results, which pin down the possible coalition governments.

Lemma 3.1

Suppose that $d=0$. If all parties secure some votes and no party receives a majority of votes then the government which forms in equilibrium consists of a pair of parties with maximal bilateral surplus.

Proof

Any equilibrium agreement must involve a program which maximizes the bargaining partners’ aggregate utility, else the proposer could profitably deviate to offering such a program. Our assumptions of equiprobable proposers and an equally bad caretaker government then imply that bargaining partners expect (at the end of the first phase) to split the surplus equally.

As parties are myopic, they evaluate each coalition according to its share of the surplus. Lemma 2.1a then implies the result if some two-party coalition yields a greater per-party surplus than the grand coalition. (We then say that the grand coalition is dominated by this coalition.) There are a number of cases to consider, all of which involve substituting for the per-party surplus in the proof of Lemma 2.2. If all parties are pure then the per-party surplus of $M$ and $R$ equals $K-\beta (x_{R}-x_{M})/2$, while that of the grand coalition is $K-\beta (x_{R}-x_{L})/3$; so the grand coalition is dominated. If all parties are corrupt then the per-party surplus of $M$ and $R$ equals $K-\alpha /8\gamma$, while that of the grand coalition is $K-\alpha /6\gamma$. If $i$ alone is pure then the per-party surplus of $i$ and another party equals $K-\alpha /16\gamma$, while that of the grand coalition is $K-\alpha /9\gamma$. Finally, suppose that $i$ alone is corrupt, so the grand coalition’s per-party surplus is $K-\alpha /18\gamma -\beta |x_{j}-x_{k}|/3$. The per-party surplus of a coalition of the two pure parties [resp. a pure and a corrupt party] is $K-\beta |x_{j}-x_{k}|/2$ [resp. $K-\alpha /16\gamma$]. It is easy to confirm that the grand coalition is then dominated by at least one of the coalitions.

$\square$

Lemma 3.2

Suppose that $d=0$, and that exactly two parties are corrupt. If all parties secure some votes then the two corrupt parties do not share power in any equilibrium.

Proof

Lemma 2.2b implies that an agreement between the two corrupt parties yields a lower surplus than an agreement between the pure and a corrupt party; so Lemma 2.1a implies that the corrupt parties would not share power in a hung parliament. $\square$

Lemma 3.3

Suppose that $d=0$. A corrupt and a pure party do not share power in any equilibrium.

Proof

Denote the pure party by $i$. Lemma 2.2b implies that the two parties would agree to policy $x_{i}$ and to some graft. All citizens must then prefer that $i$ be in power.    $\square$

We will now use the Lemmas above to characterize all equilibria:

Lemma 3.4

Suppose that $d=0$. Every game with generic parameters $\{\delta ,\xi ,\theta \}$ has a unique equilibrium outcome. The government which forms is top-ranked by moderates out of the three single-party governments and the coalition which would form were parliament to hang and all parties secured some votes.

Proof

The result is obvious if a majority of citizens are moderates; so suppose otherwise.

We first argue that moderates must top-rank the government which forms in any equilibrium.

If $M$ is pure then it is by top-ranked by moderates, and must be in power in every equilibrium: for any other government must agree to some $x\ne 0$ and/or some graft; so moderates and citizens whose ideal policy has the opposite sign to $x$ both prefer that $M$ be in power. Accordingly, we can focus on states in which $M$ is corrupt.

We start by precluding coalition governments which form because two parties share all votes equally (so all citizens are pivotal), but which would not form if all parties secured some votes. There are three cases to consider:

If $M$ and $R$ alone were corrupt and a leftist deviated to voting $L$ then Lemmas 3.1 and 3.2 imply that $L$ would share power with one of the corrupt parties. Lemma 2.2b then implies that this deviation would be profitable. An analogous argument precludes such an equilibrium when $L$ and $M$ alone are corrupt. Finally, suppose that $M$ alone is corrupt, and consider putative equilibria in which two parties each secure half of the votes and $\theta >2\xi$ or $\delta <8(2\xi +\theta )$. Lemma 3.3 implies that $M$ could not secure any votes in such an equilibrium; so the pure parties would have to share power, with some moderate(s) voting $L$. If such a moderate deviated to voting $R$ then $R$ would be in power; and $\theta >2\xi$ implies that this deviation is profitable. If $\delta <8(2\xi +\theta )$ and some citizen deviated to voting $M$ then $M$ would share power with one of the pure parties. If $L$ and $M$ [resp. $M$ and $R$] would share power then leftists [resp. rightists] could profitably deviate to voting $M$. Accordingly, we now focus on putative equilibria in which all parties secure some votes when a hung parliament is elected.

Suppose, first, that $k$ is in power, and that moderates prefer that $j\ne k$ be in power: $k\in \{L,M,R\}$. Citizens of sort other than $k$ who do not vote for $j$ must then have a profitable joint deviation to doing so. Hence, moderates either top-rank $k$ (and our claim is true) or they top-rank pure $L$ and $R$ sharing power.

Suppose, contrary to our claim, that $k$ is in power and that moderates prefer pure $L$ and $R$ to share power. If $k=R$ then leftists also prefer a hung parliament over $R$ in power; if $k=M$ then rightists also prefer a hung parliament over $M$ in power because Lemma 2.2b implies that $L$ and $R$ would agree to a negative policy; and if $k=L$ then rightists and moderates would prefer $L$ in power. Consequently, a majority of citizens must prefer a hung parliament; so citizens who are not sort $k$ and vote for $k$ have a profitable joint deviation.

Now suppose that $L$ and $R$ share power in an equilibrium, but moderates top-rank some party $k$ in power. Arguments analogous to those in the last paragraph imply that a majority of citizens then top-rank $k$ in power; so all such citizens who do not vote for $k$ have a profitable joint deviation to doing so.

In sum, moderates top-rank the government which forms in any equilibrium. If moderates top-rank a party in power then there is an equilibrium in which all citizens vote for that party; and if moderates top-rank pure parties $L$ and $R$ sharing power then there is an equilibrium in which each citizen of sort $i$ votes for party $i$.

The equilibrium is unique in generic games, where moderates have a strict preference ordering over the possible governments. $\square$

We now complete the proof of Proposition 3.1:

• a. This part follows immediately from Lemma 3.4 because moderates top-rank $M$ in power.

• b. If $M$ and $R$ are both corrupt then the only possible equilibrium governments are $L$ and $M$ in power. Moderates respectively earn $-\beta (\xi +\theta )$ and $-\alpha /4\gamma$ when $L$ and $M$ are in power, so this part follows from Lemma 3.4.

• c. The proof follows the same lines as that of part b.

• d. If $M$ is the only corrupt party then the only governments which can be formed in equilibrium are $M$ in power, $R$ in power and $L$ and $R$ sharing power. Now moderates prefer $M$ in power over $R$ in power if and only if $\delta <4\xi$. This condition implies that $M$ is in power because $L$ and $R$ only share power in a hung parliament if $\delta >8(2\xi +\theta )$. Conversely, Lemma 3.4 implies that $R$ is in power if $4\xi <\delta <8(2\xi +\theta )$. $L$ and $R$ can only share power if moderates prefer that $R$ share power with $L$ than that it be in power (viz. $\theta <2\xi$) and that they form a coalition in a hung parliament (viz. $\delta >8(2\xi +\theta )$), which implies that moderates prefer a coalition government over $M$ in power. The result then follows from Lemma 3.4.

• e. Moderates prefer to elect $M$ over any other party in power. All two-party coalitions yield the same surplus when $\varepsilon =0$, but a coalition of $M$ and $R$ maximizes the surplus whenever $\varepsilon >0$. The result then follows from Lemma 3.4. $\square$

Theorem 1

(Ergodic sets)

• a. The myopic ergodic set is an ergodic set for every $d\in [0,1)$;

• b. There is an ergodic set in which $M$ and $R$ always share power if and only if $4\xi <\delta <4(\xi +2\theta )$ and $d>1/3$;

• c. There is an ergodic set in which the grand coalition always forms if and only if $\delta >2\theta ,\,d>1/3$, and either party $M$ is the last to announce in a hung parliament or $\delta >4\xi$;

• d. Games with generic parameters $\{\delta ,\xi ,\theta \}$ have no other ergodic set.

Proof

• a. If a party is in power then it cannot profitably deviate from behaving myopically if its deviation does not change play in any subsequent period. Accordingly, we only need to focus on cases in which a hung parliament is elected on the putative equilibrium path. Proposition 3.1 implies that this occurs if and only if $M$ alone is corrupt, $\theta <2\xi$, and $8(2\xi +\theta )<\delta$. As parties in power expect to split their bilateral surplus equally, parties $L$ and $R$ respectively earn

  $$\begin{aligned} K-\beta \left[ \xi +\frac{1+2d}{2(1+d)}\theta \right] \quad \text{ and } K-\beta \left[ \xi +\frac{1}{2(1+d)}\theta \right] \end{aligned}$$

  on the putative equilibrium path. Consider a putative equilibrium in which $M$ announces $L$ and the other two parties announce each other. If $L$ or $R$ deviate to announcing themselves or $R$ deviates to announcing $M$ then a caretaker government forms in the first period and the deviating party earns the same return as in the putative equilibrium in every subsequent period. Consequently, such a deviation is unprofitable. Now suppose that $L$ deviates to announcing $M$. $L$ and $M$ then share power and split their bilateral surplus, with $L$ earning $K-\alpha /16\gamma$; as $\delta >4\xi ,\,R$ is in power next period, so $L$ earns $K-\beta (2\xi +\theta )$; and in every subsequent odd [resp. even] period, $L$’s returns are the same as it earns in even [resp. odd] periods in the putative equilibrium. Consequently, $L$’s payoff after so deviating is

  $$\begin{aligned} K-(1-d)\left[ \frac{\alpha }{16\gamma }+\beta d(2\xi +\theta )\right] -d^{2}\beta \left[ \xi +\frac{2+d}{2(1+d)}\theta \right] . \end{aligned}$$

  By construction, $L$ is better off splitting its bilateral surplus with $R$ than with $M$ in the first period; so its payoff after deviating ($K-\alpha /16\gamma$) is less than $K-\beta (\xi +\theta /2)$, its payoff in the putative equilibrium. Next period, $L$ earns $K-\beta (\xi +\theta )<K-\beta \left( 2\xi +\theta \right)$ on the equilibrium path. From then on, $M$ and a coalition of $L$ and $R$ alternate: as on the equilibrium path but in reverse order to the equilibrium path, yielding:

  $$\begin{aligned} d^{2}\left[ K-\beta \left[ \xi +\frac{2+d}{2(1+d)}\theta \right] \right] <d^{2} \left[ K-\beta \left[ \xi +\frac{1+2d}{2(1+d)}\theta \right] \right] \end{aligned}$$

  where the right hand is the payoff on the equilibrium path, proving this part. We now consider other ergodic sets. As noted above, the form that these can take is restricted by Markov perfection and by our supposition that citizens are myopic. Specifically, the program chosen by a given government is constant across all Markov perfect equilibria because subsequent payoffs only depend on subsequent states, which are unaffected by today’s program. A pure $M$ must therefore be in power in any ergodic set; and a pure party cannot share power with one or more corrupt parties in any ergodic set. Furthermore, an ergodic set can only be non-myopic if there is a state in which a different government forms than in the myopic ergodic set. As citizens are myopic, this is only possible if citizens elect a hung parliament in that state.

• b. The conditions on $\delta$ imply that moderates prefer corrupt $M$ and $R$ to share power over any party in power: so citizens cannot profitably deviate if they expect corrupt $M$ and $R$ to share power in a hung parliament. If $M$ and $R$ alone are corrupt then each party earns $K-\alpha /8\gamma$ on the putative equilibrium path, where they must announce each other. $M$ and $R$ will therefore share power if neither can profitably deviate to another announcement. Their incentive to so deviate depends on play after such a deviation. Suppose that the equilibrium specifies myopic play at every other state, and let $L$ announce $i\in \{M,R\}$ in this putative equilibrium. If $i$ deviated to announcing $L$ then it would earn $K-\alpha /16\gamma$ in the initial period. Proposition 3.2 and the bounds on $\delta$ imply that $M$ and $R$ would alternate in power without choosing any graft thereafter; so (corrupt) $i$ would earn $K-\alpha /4\gamma$ in every subsequent period. This deviation is unprofitable when $d>1/3$.

• c. $M$ must be corrupt in a state where the grand coalition forms, else it would be in power. Furthermore, there cannot be exactly one pure party in that state because the grand coalition’s program would be the pure party’s ideal policy and some graft; so citizens could profitably deviate to voting the pure party into power. Consequently, the grand coalition can either form when all parties are corrupt or when $M$ alone is corrupt. In the former case, Markov perfection implies that the grand coalition always forms. In the latter case, $M$ is in power before the grand coalition forms, and another government forms when all parties are corrupt. We consider these two possibilities in sequence: Suppose, first, that all parties are corrupt. The grand coalition would then choose a policy of $-\theta /3$, and graft of $\alpha /36\gamma ^{2}$ for each party, yielding moderates a payoff of $K-\beta \theta /3-\alpha /12\gamma$. Moderates would earn no more than $-\alpha /4\gamma$ by electing a single party ($M$) into power; so $\delta >2\theta$ implies that they prefer to elect a hung parliament in which the grand coalition forms. Each party earns $K-\alpha /6\gamma$ on the equilibrium path. Suppose that the equilibrium specifies myopic play at every other state. Any two parties which share power each earn $K-\alpha /8\gamma$ in that period, but $L$ and $R$ never subsequently take any graft. In particular, Proposition 3.1 implies that $M$ would always be in power if $\delta <4\xi$, yielding $L$ and $R$ a payoff of $K-\alpha /2\gamma$ in the subgame which follows a two-party government forming. If $\delta >4\xi$ then government composition depends on specific parameter values; but Proposition 3.1 implies that each government consists of pure parties. (Corrupt) $L$ and $R$ therefore earn $K-\alpha /4\gamma$ in the subgame which follows a two-party government forming. In sum, $L$ and $R$ earn no more than

  $$\begin{aligned} K-\frac{\alpha }{8\gamma }\left( 1-d\right) -\frac{\alpha }{4\gamma }d \end{aligned}$$

  if a two-party government forms in the first period; so $d>1/3$ implies that $L$ and $R$ both top rank the grand coalition. If $M$ announces last then Lemma 2.1b entails the result. If $\delta >4\xi$ then there is no graft after the first period, and $M$ also top ranks the grand coalition, regardless of the order of announcements. Lemma 2.1a then implies that the grand coalition always forms. The grand coalition can only share power in an ergodic set if they always share power: for the grand coalition can only form if $M$ is corrupt, and at least two parties are pure. If $L$ and $R$ are pure then the grand coalition would choose a policy of $-\theta /2$ and graft of $\alpha /36\gamma ^{2}$. Moderates would therefore earn $-\beta \theta /2-\alpha /36\gamma$, and would prefer the grand coalition to share power over $M$ in power when $\delta >9\theta /4$; whereas, some arguments above imply that moderates would only elect $M$ to power when all parties are corrupt if $\delta <2\theta$.

• d. We prove this part by considering the various alternatives:

  • Lemma 3.3 still holds because citizens are myopic: so the government chosen cannot consist of a pure and a corrupt party.

  • Pure $L$ and $R$ share power in the myopic ergodic set. Any other ergodic set in which two pure parties share power would include $M$; but this is impossible because a winning majority of citizens could profitably deviate to electing $M$ to power alone.

  • If only $L$ and $R$ were corrupt then moderates would vote for $M$ in power; and if only $L$ and $M$ were corrupt then moderates who prefer a coalition of $L$ and $M$ over $M$ in power would prefer $R$ in power over $L$ and $M$ sharing power. To see this note that, if $R$ alone is pure, then moderates earn

    $$\begin{aligned} -\beta \frac{\xi +\theta }{2}-\frac{\alpha }{8\gamma } \end{aligned}$$

    when $L$ and $M$ share power; whereas they earn $-\alpha /4\gamma$ and $-\beta \xi$ respectively when $M$ and $R$ are in power. Consequently, they prefer that $L$ and $M$ share power than that $M$ be in power whenever

    $$\begin{aligned} -\beta \frac{\xi +\theta }{2}-\frac{\alpha }{8\gamma }>-\frac{\alpha }{4\gamma }\Leftrightarrow \delta >4\left( \xi +\theta \right) \text{, } \end{aligned}$$

    which implies that they prefer $R$ in power over $L$ and $M$ sharing power.

$\square$

Proposition 3.4

(Improving cycles)

• a. The myopic ergodic set is suboptimal for moderates if and only if either $2\theta <4\xi <\delta <8(2\xi +\theta )$ or $2\theta <\delta <4\xi$. In both cases, $M$ alternates in power with a coalition of $L$ and $R$ in the only improving cycle.

• b. The myopic ergodic set improves on any ergodic set in which $M$ and $R$ always share power;

• c. A cycle in which $M$ alternates in power with a coalition of $L$ and $R$ improves on an ergodic set in which the grand coalition always forms.

Proof

• a. Proposition 3.1 implies that $M$ either alternates in power or is always in power. We consider these cases in sequence, starting with those parameters for which $M$ alternates in power. A government of pure $M$ chooses moderates’ ideal program; so any improving cycle must contain a period when $M$ is in power followed by a period when $M$ alone is corrupt. Whenever $M$ is not in power in the ergodic set: either $L$ and $R$ share power or $R$ alone is in power, depending on parameter values. If $L$ and $R$ share power in the ergodic set then Lemma 3.4 implies that moderates cannot be better off that period with any other government; so in this case, there is no improving cycle. On the other hand, there are parameter values for which $M$ alternates in power with $R$ in the ergodic set if $4\xi <\delta <8(2\xi +\theta )$; and moderates then prefer pure $L$ and $R$ to share power over $R$ alone in power if $\theta <2\xi$. This coalition could not occur in the ergodic set because it yields a smaller surplus than a coalition which includes $M$. Consequently, a cycle in which $M$ alternates in power with a coalition of $L$ and $R$ improves on the ergodic set for these parameter values. Now consider parameter values at which $M$ is always in power in the ergodic set. A cycle in which another corrupt party is in power can obviously not improve on this ergodic set. Furthermore, $\delta <4\xi$ implies that moderates prefer a corrupt $M$ in power over it sharing power with another corrupt party and over any other pure party in power. Now Lemma 3.3 implies that moderates prefer a pure party to be in power than to share power with a corrupt party. Consequently, this ergodic set is suboptimal for moderates if and only if a cycle in which $M$ alternates in power with a coalition of $L$ and $R$ is improving: that is, if $\delta >2\theta$. As before, this coalition could not occur in the ergodic set because it yields a smaller surplus than a coalition which includes $M$.

• b. Moderates earn $-\beta \xi /2-\alpha /8\gamma$ each period in an ergodic set where $M$ and $R$ always share power. This ergodic set only exists if $\delta >4\xi$, when (pure) $M$ and $R$ alternate in power in the myopic ergodic set, and moderates earn $0$ and $-\beta /2$ in alternate periods. As $\delta >4\xi$, moderates are better off in every period in the myopic ergodic set.

• c. Moderates earn $-\beta \theta /3-\alpha /12\gamma$ each period in an ergodic set where the grand coalition always forms. A cycle in which $M$ alternates with a coalition of $L$ and $R$ yields moderates a utility of $-\beta \theta /2$ in periods when $L$ and $R$ share power; and this exceeds $-\beta \theta /3-\alpha /12\gamma$ when $\delta >2\theta$: that is, whenever there is an ergodic set in which the grand coalition forms. $\square$