## Abstract

During the last decade unicameral proposals have been put forward in fourteen US states. In this paper we analyze the effects of the proposed constitutional reforms, in a setting where decision making is subject to ‘hard time constraints’, and lawmakers face the opposing interests of a lobby and the electorate. We show that bicameralism might lead to a decline in the lawmakers’ bargaining power vis-a-vis the lobby, thus compromising their accountability to voters. Hence, bicameralism is not a panacea against the abuse of power by elected legislators and the proposed unicameral reforms could be effective in reducing corruption among elected representatives.

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## Notes

- 1.
As reported by the Center for Public Integrity, over one billion dollars was spent in 2005 to lobby state politicians. Moreover, of the 2000 investigations on public corruption undertaken by the F.B.I. in 2006, most involve states and local officials (source: The New York Times May 11, 2006,

*F.B.I.’s Focus on Public Corruption Includes 2,000 Investigations*). - 2.
California unicameral legislature, October 4 2006, Attorney General File number 2600-034; State of New York Bill 9875, February 5 2010.

- 3.
As Madison (1788) had pointed out “...a senate, as a second branch of the legislative assembly, distinct from, and dividing the power with, a first,

*must be in all cases*a salutary check on the government. It doubles the security to the people, by requiring the concurrence of two distinct bodies in schemes of usurpation or perfidy, where the ambition or corruption of one would otherwise be sufficient”. - 4.
According to the New York unicameral bill proposal “A one house legislature will eliminate needless duplication and delay (...); it will speed up the budget process and facilitate the adoption of timely budgets” (source: State of New York, Bill Number A597, January 18 2005).

- 5.
Of course, bicameralism may also serve other purposes such as the representation of heterogenous interests that in modern democracies are associated with geographically distinct political jurisdictions such as for example, federal states. For a comprehensive view of bicameralism, see Tsebelis and Money (1997) and Voigt (2012). For an overview on the effects of federalism and bicameralism on corruption see instead Rose-Ackerman (2006).

- 6.
For an overview of bicameral arrangements in national states and a cross-country empirical analysis on bicameralism and corruption, see Testa (2010).

- 7.
For instance during the weeks preceding the approval of the 2005 New York state budget, it has been pointed out that “...winning on time passage from the legislature could be costly.... It might require Mr. Pataki to agree to hundreds of millions of dollars in extra spending” (

*The Calendar vs. the Purse for Albany’s Big 3*The New York Times, March 16 2005). For more details on late budget procedures in US federal states see Eckl (1998). - 8.
- 9.
The literature on bicameral legislative bargaining typically does not incorporate elections. One exception is Muthoo and Shepsle (2008) which lay out a model of optimal constitutional choice introducing elections in a reduced form, i.e. without explicitly modeling the voting strategy.

- 10.
The legacy motive represents a straightforward device to introduce non-pecuniary benefits enjoyed by politicians in power (Maskin and Tirole 2004). In a previous version of our paper (Facchini and Testa 2009), we had modeled the same idea by assuming that politicians derive instead a positive utility from implementing their own ideological agenda. While this allowed us to capture the role of political polarization, the main thrust of the analysis of bicameralism is not affected.

- 11.
We focus on a finite horizon game because it represents the most difficult scenario for electoral accountability, since in the last mandate politicians do not face elections. As in any finite horizon set up, the last period policy choice is trivial, and the second period only serves the purpose of modeling in a simple way the future

*electoral returns*from current policy choices. - 12.
The recent corruption charges against Jack Abramoff, one of the most influential lobbyists in Washington, has sparkled a wide debate on the large amounts of resources spent to gain influence on law making. As the Washington Post (June 22, 2005) points out “...companies are also hiring well-placed lobbyists to go on the offensive and find ways to profit from the many tax breaks, loosened regulations and other government goodies that increasingly are available.” In fact, professional lobbyists are usually hired for the exclusive purpose of constantly approaching legislators to promote the interests of their clients.

- 13.
- 14.
Note though that, as shown in the proof of Lemma 2, the alternative voting strategy \([\sigma '''(C^{H})=0,\sigma '''(C^{L})=1,\sigma '''({\mathbf {\varnothing }})=0]\) would deliver the same payoff for the voters.

- 15.
This rule would have allowed a member country, that had been outvoted on a proposal in Parliament, to ask for a new vote in the Council. This would have been equivalent to a system where the first body (Parliament) has proposal power and the second (Council) has final decision power.

- 16.
As pointed out in the literature (Persson et al. 1997) the advantage of a retrospective voting strategy conditioning on the last policy outcome is its simplicity. Moreover, in the context of our complex decision making process, it has the additional benefit of allowing the voter to hold multiple legislators accountable even if he does not punish or reward them differently when they undertake different actions.

- 17.
See Lemmas 6 and 8 in the Appendix. Note that our simple voting strategy \(\sigma ^{*}\) conditions on the final policy outcome, rather than on the behavior of each individual chamber. Alternatively, one could consider a more complex voting strategy, which would make the re-election of a given legislator dependent on the specific action he has undertaken, rather than only on the final policy outcome. Note that—as argued in Remark 1 in the Appendix—this more complex voting strategy does not allow the voter to obtain accountability if it cannot be already reached using our simple voting strategy. For this reason we focus the analysis on our simple voting strategy, which has the additional advantage of requiring less information on the entire policy formation process on the side of the voter.

- 18.
As for the equilibrium shares of profits, in the second period, if the legislator is the proposer, then \(\beta ^{2}_{g_1}=1\), if the lobby is the proposer then \(\beta ^{2}_{g_1}=\frac{C^H-\gamma (B+E)}{\pi }\). On the other hand, the second legislator, who cannot credibly us his veto in the second period, always gets \(\beta ^{2}_{g_2}=0\). Note that although the lobby’s outside option is worsened by the risk of binding time constraints, still the lobby always prefers agreement to disagreement. For this reason \(\gamma \) does not affect the share of profits paid to the lobby when \(g_1\) is the proposer. See also the proof of Lemma 3 in the Appendix.

- 19.
The asymmetry between the two chambers depends on the allocation of proposal powers, which in our setup lies with the first legislator. In an alternative setting, in which in the second period \(g_1\) retains proposal power only with probability

*p*, then the minimum shares of rent required by the two legislators to choose the high cost policy in \(t=1\) are \(\hat{\beta }_{g_1}^1=\overline{\beta }^1_{g_1}-\delta (1-p)\{q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }\}\) and \(\hat{\beta }_{g_{2}}^{1}=\overline{\beta }^1_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\) (see Lemma 6 in the Appendix). As we can immediately see, \(\hat{\beta }_{g_1}^1 \le \overline{\beta }^1_{g_1}\), whereas \(\hat{\beta }_{g_{2}}^{1} \ge \overline{\beta }^1_{g_2}\), i.e. if both chambers retain some proposal power in the second period, the difference in the minimum profit share they require to pass the high cost policy declines. We would like to thank a referee for suggesting this extension. - 20.
We focus on the cost of lobbying deriving from the electoral loss of multiple legislators because we are mainly interested in electoral incentives. However, it should be clear that having multiple chambers deciding sequentially rather than simultaneously can have a substantial impact on the lobby’s ability to bribe the legislator whenever lobbying is a costly, time consuming activity or the rents associated to an agreement decrease over time. Hence, our results on the positive effect of bicameralism on accountability hold

*a fortiori*if we introduce either a cost of lobbying or a profit that are time dependent. - 21.
Note that if there is uncertainty in the allocation of proposal power in the second period—see footnote 19—the second chamber can extract a larger rent, and thus the scenario in which bicameralism decreases accountability is less likely.

- 22.
Expressing the difficulty of buying legislators.

- 23.
This type of arrangement is very common. For instance, in the US only the House of Representatives can initiate budget legislation.

- 24.
Remember that in the second period the high cost policy will always be chosen. As for the profit shares, in \(t=2\), if amendment rights are restricted, the equilibrium profit shares are \(\beta _{g_{d}}^{2}=q+(1-q)\frac{C^H}{\pi }\) for all \(g_d\), with \(\sum _{d=1}^2 \beta _{g_{d}}^{2} \le 1\). If amendment rights are instead unrestricted, \(\beta _{g_{1}}^{2}=0\), whereas \(\beta _{g_{2}}^{2}=q+(1-q)\frac{C^H}{\pi }\). See proof in Appendix.

- 25.
See for instance Tsebelis and Money (1997).

- 26.
In most countries, this means that the text of a bill needs to be approved in the same form by both legislative bodies. Hence, in case of disagreement, the bill shuttles between the two chambers until an agreement is reached. However, in extreme cases of complete parliamentary deadlock, other mechanisms have been devised. For instance, in the US a conference committee can be called where delegates from each chamber meet to find a compromise. For more details see Tsebelis and Money (1997).

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## Author information

### Affiliations

### Corresponding author

## Additional information

We would like to thank assemblyman Richard L. Brodsky from the New York State Assembly for sharing with us the text of the Bill A597 (January 18 2005, last reintroduced as Bill 9875, February 5, 2010). We also wish to thank the participants of the Latin American Econometric Society meeting (Salvador de Bahia), the North American Econometric Society meeting (Chicago), the Midwest Political Science Association meeting (Chicago), the PSPE conference “Designing Democratic Institutions” (London School of Economics and Political Science) for useful comments. We especially thank an editor and an anonymous referee for suggestions that substantially improved the paper.

## Appendix

### Appendix

### Proof of Lemma 2

Note that in the second period the high cost policy is always chosen. Hence, we conclude that to show whether the voter’s expected payoff is maximized by \(\sigma ^{*}\), we only need to analyze the first period payoff for all \(\sigma \). Let us start by considering the following alternative strategy

Under \(\sigma ^{\prime }\), the high cost policy is preferred by any legislator receiving \(\beta _{g_{d}}\ge 0\), since he can receive lobby transfers and choose his most preferred policy in both periods. On the other hand, under the voting strategy \(\sigma ^{*}\) depending on the parameters of the model, the legislator will choose either \(C^{H}\) or \(C^{L}\). If \(C^{H}\) is chosen, then the expected payoff under the two alternative strategies is the same. On the other hand, if \(C^{L}\) is chosen then the voter prefers \(\sigma ^{*}\) to \(\sigma ^{\prime }\). Hence, we conclude that \(\sigma ^{\prime }\) is not an equilibrium strategy. Consider next the following alternative strategy

Under this voting strategy the incumbent is never reappointed. Therefore, since \(C^{H}\) generates a higher net profit to be shared, the legislator will always choose \(C^H\). Hence, \(\sigma ^{{\prime }{\prime }}\) is not an equilibrium voting strategy and more generally, by the same arguments, any strategy such that either \(\sigma (C^H)=1\) or \(\sigma (C^L)=0\), cannot be an equilibrium voting strategy. Finally, consider the strategy

Note that, as \( v_{g}^{1}(C^{L})>v_{g}^{1}(\varnothing )\), if the legislator does not receive transfers from the lobby, he always implements \(C^{L}\), i.e. \(C^{L}\) is the outside option. Since in the bargaining game the legislator chooses between \(C^{H}\) and the outside option \(C^{L}\), any voting strategy that punishes or rewards him for not choosing any policy does not affect his behavior and thus the policy outcome. As a result, the voter is indifferent between \(\sigma ^{*}\) and \(\sigma ^{{\prime }{\prime }{\prime }}\).

\(\square \)

### Proof of Lemma 3

In \(t=2\) the following holds. Under a closed rule arrangement, the second legislator can only approve or veto the policy chosen by the first. Furthermore, since \( v_{g_{2}}^{2}(C^H)>v_{g_{2}}^{2}(\varnothing )\forall \beta _{g_{2}}^{2}\), vetoing is not credible. As a consequence, if in \(t=2\) the lobby can induce the first legislator to choose \(C^{H}\), then she does not need to pay any positive transfer to convince the second to pass \(C^{H}\). Hence, \(\beta _{g_{2}}^{2}=0\). We can now determine the equilibrium transfers inducing the first legislator to choose \(C^{H}\). Given that the time constraint is binding with probability \(\gamma \), in case of disagreement, \(g_1\) outside option is \(\gamma v_{g_{1}}^{2}(\varnothing )+(1-\gamma )(B+E)\), and the lobby’s outside option is \(\gamma v_{l}^{2}(\varnothing )+(1-\gamma )B\). Remembering that \(v_{l}^{2}(\varnothing )=v_{g_{1}}^{2}(\varnothing )=0\), then the first legislator prefers agreement to disagreement if and only if \(\beta ^{2}_{g_1}\ge \frac{C^H-\gamma (B+E)}{\pi }\), whereas the lobby always prefers agreement. Hence, if the first legislator is the proposer then \(\beta ^{2}_{g_1}=1\), if the lobby is the proposer then \(\beta ^{2}_{g_1}=\frac{C^H-\gamma (B+E)}{\pi }\), and \(C^H\) is always chosen. Moving to \(t=1\), and remembering that in \(t=2\) the first legislator is the proposer with probability *q* and the lobby is the proposer with probability \(1-q\), the second period expected payoff for \(g_1\) is \(\left\{ (B+E-C^H)+q\pi + (1-q)[C^H-\gamma (B+E)] \right\} \). Hence, if in the first period the first legislator rejects the first lobby offer, he obtains the disagreement payoff \(\gamma v_{g_{1}}^{1}(\varnothing )+(1-\gamma )(B+E)+\delta \left\{ (B+E-C^H)+q\pi + (1-q)[C^H-\gamma (B+E)] \right\} \), whereas the agreement payoff is given by \(B+E-C^H+\beta ^1_{g_1}\pi +\delta (B-C^H)\). Hence, agreement is preferred to disagreement if and only if \(\beta ^{1}_{g_1}\ge \frac{C^H+\delta [E + q\pi +(1-q)C^H]}{\pi }-\gamma [1+\delta (1-q)]\frac{B+E}{\pi }\).

As for the second legislator, since \(\beta _{g_{2}}^{2}=0\), his disagreement payoff is \(v_{g_{2}}^{1}(\varnothing )+\delta \left[ (B+E-C^H) \right] \), whereas his payoff from agreement is \(B+E-C^H+\beta ^1_{g_2}\pi +\delta (B-C^H)\). Hence the second legislator can credibly threaten to veto the proposal passed by \(g_{1}\), unless he receives \(\overline{\beta } _{g_{2}}^{1}=\frac{C^{H}+\delta E-(B+E)}{\pi }\). On the other hand, if \(\frac{\delta E}{\pi }+\frac{C^{H}-(B+E)}{\pi }<0\), \(g_{2}\) cannot credibly veto any policy chosen by \(g_{1}\) and therefore \( \overline{\beta }_{g_{2}}^{1}=0\).

Finally, note that \(\overline{\beta }^1_{g_1} > \overline{\beta }^1_{g_2}\) if and only if \((1-\gamma )(B+E) + \delta \{q\pi + (1-q)[C^H-\gamma (B+E)]\} \ge 0\), which is always true because the second term is the expected lobby transfer, which is always weakly positive. \(\square \)

### Lemma 6

### Lemma 6

Under closed rule, the voting strategy \( \sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}({\mathbf {\varnothing }})=1]\) is the unique equilibrium voting strategy.

### Proof

Since the voting strategy depends only on the policy outcome, the optimality of the voting strategy relies on the same arguments as in the unicameral case. In particular, \(\sigma ^{*}\) is an equilibrium voting strategy since the voter is strictly better off by choosing \(\sigma ^{*}\) than under any alternative strategy \(\sigma ^{\prime }\) such that either \(\sigma ^{\prime }(C^H)=1\) or \(\sigma ^{\prime }(C^L)=0\). Moreover, under closed rule this is the unique equilibrium voting strategy because \(\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}(\varnothing )=1]\) is strictly preferred to \(\sigma ^{\prime }=[\sigma ^{\prime }(C^{H})=0,\sigma ^{\prime }(C^{L})=1,\sigma ^{\prime }(\varnothing )=0]\), since punishing or rewarding the legislators for not implementing any policy is not pay-off irrelevant. In fact, under \(\sigma ^{\prime }\), the second legislator cannot extract any rent since he cannot credibly veto any policy. This implies that \(C^H\) is more likely to be chosen because the feasibility constraint on lobby transfers is more easily satisfied when \(\beta _{g_2}=0\). Hence the voter strictly prefers \(\sigma ^{*}\) to \(\sigma ^{\prime }\). \(\square \)

### Remark 1

Consider an alternative voting strategy \(\sigma ^{\prime }_{g_d}\) whereby each legislator \(g_d\) is re-elected based on the policy he has passed and \(\sigma ^{\prime }_{g_d}=[\sigma ^{\prime }_{g_d}(C^H)=0, \sigma ^{\prime }_{g_d}(C^L)=1, \sigma ^{\prime }_{g_d}(\varnothing )=1]\). Note that if each legislator passes the same policy, \(\sigma ^{\prime }_{g_d}\) delivers the same outcome as \(\sigma ^{*}\). Consider now the case where the two legislators pass different policies. Since the second legislator can only veto the policy passed by the first, the only relevant scenario is the one in which the first legislator chooses \(C\in \{C^L,C^H\}\) and the second vetoes it so that no policy is passed. Since if the first legislator chooses \(C^L\), the second will always pass it, we only need to consider the case in which the first legislator chooses \(C^H\) and the second vetoes it. In this case under the voting strategy \(\sigma ^{*}\) both legislators are re-elected, whereas under the voting strategy \(\sigma ^{\prime }_{g_d}\), only the second legislator is re-elected. We can easily see that, given \(\sigma ^{*}\), if the first legislator chooses \(C^H\), which is subsequently vetoed, then his expected payoff is \(v_{g_1}(\varnothing )+\delta (B - C^H + q\pi + (1-q) C^H)\) which is strictly lower than the payoff from choosing \(C^L\), that is \(B+E+\delta [B+E-C^H+q\pi + (1-q) C^H]\). In other words, the first legislator will never find it optimal to choose \(C^H\) when the second legislator will veto it. The same is true under the alternative voting strategy \(\sigma ^{\prime }_{g_d}\) because the payoff for \(g_1\) from choosing \(C^H\) that is subsequently vetoed is \(v_{g_1}(\varnothing )+\delta (B-C^H)\), which is again strictly smaller than the payoff from choosing \(C^L\). Hence, punishing only the first legislator for choosing \(C^H\) when this outcome is subsequently vetoed does not make him more or less likely to choose \(C^H\) over \(C^L\).

### Lemma 7

### Lemma 7

Assume that the first chamber retains proposal power in the second period with probability *p*. Then in \(t=1\) the minimum shares of rent required by the two legislators to choose the high cost policy are \(\hat{\beta }_{g_1}^1=\overline{\beta }^1_{g_1}-\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\) and \(\hat{\beta }_{g_{2}}^{1}=\overline{\beta }^1_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\).

### Proof

If the first chamber loses proposal power in the second period, the share of rents required to choose the high cost policy in the first period is \(\widetilde{\beta }_{g_1}=\frac{C^H+\delta E-\gamma (B+E)}{\pi }\), whereas if she retains proposal power the share required is \(\widetilde{\beta }_{g_1}+\delta [q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\). Hence, when \(g_1\) retains proposal power with probability *p*, the share of rents required to choose the high cost policy is \(\hat{\beta }_{g_1}^1=\widetilde{\beta }_{g_1}+\delta p[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\) or equivalently \(\hat{\beta }_{g_1}^1=\overline{\beta }^1_{g_1}-\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\). By the same argument, the second chamber requires \(\widetilde{\beta }_{g_2}=\frac{C^H+\delta E- (B+E)}{\pi }\) when she does not gain proposal power and requires \(\widetilde{\beta }_{g_2}+\delta [q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\) if she gains proposal power in the second period. Hence \(\hat{\beta }_{g_{2}}^{1}=\widetilde{\beta }_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\), which can be rewritten as \(\hat{\beta }_{g_{2}}^{1}=\overline{\beta }^1_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]\). \(\square \)

### Proof of Proposition 2

If the minimum shares required under unicameralism and bicameralism to choose \( C^{H}\) are feasible (i.e. \(\overline{\beta }_{g}^{1}<1\) and \(\sum _{d=1}^{2} \overline{\beta }_{g_{d}}^{1}<1\)), then \(C^{H}\) is chosen. On the other hand, when the minimum shares are not feasible (i.e. \(\overline{\beta }_{g}^{1}>1\) and \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1\)), then \(C^{L}\) is chosen. Note that, when \(\gamma =0\), then \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}\). Therefore, when \(\overline{\beta }_{g}^{1}<\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<1\), both shares are feasible and \(C^H\) is chosen under both legislative arrangements, whereas if \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}>1\) none of the shares is feasible and \(C^L\) is chosen under both legislative arrangements. On the other hand, if \(\overline{\beta }_{g}^{1}<1<\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\), only the unicameral share is feasible implying that \(C^H\) is chosen under unicameralism and \(C^L\) under bicameralism. Consider now the scenario where the time constraint is binding with some probability (\(0<\gamma \le 1\)). Then two cases arise. If \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}\), then we obtain the same policy choice characterized when \(\gamma =0\). On the other hand, when \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<\overline{\beta }_{g}^{1}\) the following holds. If the minimum shares under the two legislative arrangements are feasible (\(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<\overline{\beta }_{g}^{1}<1\)) then \(C^H\) is chosen under both arrangements, if the same shares are not feasible (\(\overline{\beta }_{g}^{1}>\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1\), then \(C^L\) is chosen under both arrangements. Finally, if \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<1<\overline{\beta }_{g}^{1}\), then \(C^H\) is chosen under bicameralism whereas \(C^L\) is chosen under a unicameralism. \(\square \)

### Proof of Lemma 5

In \(t=2\), since \(v_{g_{d}}^{2}(C^{L})>v_{g_{d}}^{2}(\varnothing )\)
\(\forall d\), in the absence of lobby transfers the first legislator chooses \(C^{L}\) and the second legislator ratifies this choice. If amendment rights are restricted, once the policy \(C^{L}\) is chosen by the first legislator, it can be amended to \(C^{H}\) only if all legislators, including the first, approve the change. Remembering that in \(t=2\) each legislator and the lobby *l* make take-it-or-leave-it offers with probability *q* and \(1-q\), then \(\beta ^{2}_{g_d}=q+(1-q)\frac{C^H}{\pi }\), with \(\sum _{d=1}^2 \beta ^{2}_{g_d} \le 1\). Moving backward to the first period, for each legislator \(g_d\), the payoff from \( C^{H}\) is \(V_{g_{d}}(C^{H})=B+E+\beta _{g_{d}}^{1}\pi -C^{H}+\delta (B-C^{H})\) and the payoff from \(C^L\) is \(V_{g_{d}}(C^{L})=B+E+\delta (B+E+\beta _{g_d}^{2}\pi -C^{H})\). Therefore, if amendment rights are restricted, we find that each legislators prefers \(C^H\) to \(C^L\) if and only if \(\beta _{g_d}^{1}\ge \frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]\) with \(\sum _{d=1}^2 \beta ^{2}_{g_d} \le 1\).

On the other hand, in the case of unrestricted amendment rights, if in \(t=2\) the policy \(C^L\) is chosen by the first legislator, the lobby can still obtain \(C^H\) by paying \(\beta ^{2}_{g_2}=q+(1-q)\frac{C^H}{\pi }\) to the second legislator and \(\beta ^{2}_{g_1}=0\) to the first because \(v_{g_{1}}^{2}(C^{L})>v_{g_{1}}^{2}(\varnothing )\). In the first period, the second legislator obtains the expected payoff \(V_{g_{2}}(C^H)=B+E+\beta _{g_{d}}^{1}\pi -C^{H}+\delta (B-C^{H})\) by choosing \(C^H\), and the payoff \(V_{g_{2}}(C^L)=B+E+\delta (B+E+\beta _{g_2}^{2}\pi -C^{H})\) from choosing \(C^L\). Hence, the second legislator prefers \(C^H\) to \(C^L\) if and only if \(\beta _{g_2}^{1}\ge \frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]\). On the other hand, remembering that if \(C^L\) is passed by the first legislator, the lobby offers \(\beta _{g_2}^{1}=\frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]\) to the second legislator who amends \(C^L\) to \(C^H\), then by choosing \(C^L\), the first legislator obtains the expected payoff \(V_{g_{1}}(C^L)=B+E-C^H+\delta (B-C^H)\), whereas, by not passing any policy his expected payoff is \(V_{g_{1}}(\varnothing )=\delta (B+E-C^{H})\). Hence, if \(\delta E+C^H-(B+E)>0\) then \(V_{g_{1}}(\varnothing )>V_{g_{1}}(C^{L})\), which implies that the first legislator can credibly threaten not to pass any policy (i.e. \(V_{g_{1}}(\varnothing )>V_{g_{1}}(C^H)\)) unless he receives \(\beta _{g_1}^{1}\ge \frac{\delta E+C^H-(B+E)}{\pi }>0\). On the other hand, if \(\delta E+C^H-(B+E)\le 0\), the first legislator cannot credibly threaten not to choose any policy, and since \(V_{g_{1}}(C^{H})\ge V_{g_{1}}(C^{L})\) \(\forall \beta _{g_1}^{1}\), the lobby offers \(\beta _{g_1}^{1}=0\) and \(C^H\) is passed. \(\square \)

### Proof of Proposition 3

Suppose that Lemma 5 holds. If the sum of the minimum shares is feasible (\(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\le 1\)) then \(C^{H}\) is chosen, whereas if \(\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1\), \(C^{L}\) is chosen. Since the sum of the minimum shares under open rule is at least as high than under closed rule, then: (a) whenever the sum of the minimum shares is not feasible under closed rule, it will also not be feasible under open rule; (b) When the sum of the minimum share is feasible under closed rule, it may not be feasible under open rule. From (a) and (b) we conclude that whenever the low cost policy is chosen under closed rule it will also be chosen under open rule, while the reverse is not true.

Consider now the case where amendment rights are unrestricted, with \(0<\overline{\beta }^1_{g_d}<1\) for all \(g_d\) and \(\sum _{d=1}^{2} \overline{\beta }^1_{g_d} > 1\). In this case, the first legislator could choose \(C^{H}\) if he is offered the appropriate transfer. However, given that both legislators cannot be offered the transfer necessary to pass \( C^{H}\), the lobby will not find it optimal to carry out the transfer necessary to obtain \(C^{H}\) in the first legislative step, knowing that this proposal will be overridden by the subsequent legislator. As a consequence, the lobby offers \(\beta ^{1}_{g_1}=0\) to the first legislator. Since \(g_1\) anticipates that \( C^{L}\) will be overridden by the last legislator who can receive the appropriate rent share for choosing \(C^{H}\), then \( g_{1} \) rejects the offer and does not implement any policy. As a consequence, the second legislator legislators without proposal power will not be able to amend any proposal, and the mandate terminates with no policy implemented. \(\square \)

### Lemma 8

### Lemma 8

Under open rule, the voting strategy \( \sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}({\mathbf {\varnothing }})=1]\) is an equilibrium voting strategy. Moreover, under unrestricted amendment rights it is unique.

### Proof

Since the voting strategy depends only on the policy outcome, the optimality of the voting strategy relies on the same arguments as in the unicameral case. In particular, \(\sigma ^{*}\) is an equilibrium voting strategy since the voter is strictly better off by choosing \(\sigma ^{*}\) than under any alternative strategy \(\sigma ^{\prime }\) such that either \(\sigma ^{\prime }(C^H)=1\) or \(\sigma ^{\prime }(C^L)=0\). Moreover, under unrestricted amendment rights this is the unique equilibrium voting strategy because \(\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}(\varnothing )=1]\) is strictly preferred to \(\sigma ^{\prime }=[\sigma ^{\prime }(C^{H})=0,\sigma ^{\prime }(C^{L})=1,\sigma ^{\prime }(\varnothing )=0]\), since punishing or rewarding the legislators for not implementing any policy is not pay-off irrelevant. In fact, note that with unrestricted amendment rights, \(g_1\) might not find it optimal to pass any policy in order to prevent the final implementation of \(C^{H}\). As a result, \(\varnothing \) can be an outside option. Suppose now that the voter adopts the voting strategy \(\sigma ^{\prime }=[\sigma (C^{H})=0,\sigma (C^{L})=1,\sigma (\varnothing )=0]\). In this case, because of the punishment \(\sigma (\varnothing )=0\), not choosing any policy is strictly dominated by choosing either \(C^L\) or \(C^H\). Hence, either the first legislator chooses \(C^{L}\) and the last legislator amends it passing \(C^{H}\), or both legislators pass \(C^H\). Since for the voter \(v_{k}^{1}(C^{H})<v_{k}^{1}(\varnothing )\), then he strictly prefers \(\sigma ^{*}\) to \(\sigma ^{\prime }\). \(\square \)

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Facchini, G., Testa, C. Corruption and bicameral reforms.
*Soc Choice Welf* **47, **387–411 (2016). https://doi.org/10.1007/s00355-016-0969-9

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### Keywords

- Bargaining Power
- Vote Strategy
- Legislative Body
- Closed Rule
- Electoral Accountability