Veto players and equilibrium uniqueness in the Baron–Ferejohn model

See Snyder et al. (2005) and the references therein for further examples (especially footnote 6 on page 5). In addition, see Bowen (2014) and Celik et al. (2013) for its recent use in trade policy.

Eraslan (2002) proves uniqueness in SSPE payoffs.

Countries generally differ in procedures they follow in designing inter-party bargaining over a new government (see Diermeier and van Roozendaal 1998).

The selection of a formateur also differs from one country to another. In some countries it is the head of states (a monarch or an elected president) who appoints the formateur, in others it is an informateur (a senior, experienced, ‘elder statesman’).

The formation of coalition governments can be quite uncertain with respect to which party will be included in the coalition (see Müller and Strøm 2000; Laver and Schofield 1998).

A similar exercise for the open-rule version of the Baron–Ferejohn game is done by Primo (2007).

When the voting rule is unanimous, i.e., when \(q=n\), the Baron–Ferejohn game has a unique SSPE. Since this is obvious, we assume \(q<n\) for the rest of the analysis.

To eliminate unreasonable equilibria, weakly dominated strategies are ruled out.

Baron and Kalai (1993) argue that stationarity is an attractive restriction since it is the “simplest” equilibrium such that it requires the fewest computations by agents.

An example may be helpful. Consider a 5-player game with \(q=3\), and assume that player 1 is the proposer. There are 6 possible coalitions that player 1 may form: \(\psi _{1}^{1}=(1,1,0,0)\), \(\psi _{2}^{1}=(1,0,1,0) \), \(\psi _{3}^{1}=(1,0,0,1)\), \(\psi _{4}^{1}=(0,1,1,0)\), \(\psi _{5}^{1}=(0,1,0,1)\) and \(\psi _{6} ^{1}=(0,0,1,1)\) with corresponding probabilities \(g_{i}^{1}\) for \(i=1,\ldots ,6\) and \( {\sum \nolimits _{i=1}^{6}} g_{i}^{1}=1\). Hence, we have: \(p^{12}=g_{1}^{1}+g_{2}^{1}+g_{3}^{1}\), \(p^{13}=g_{1}^{1}+g_{4}^{1}+g_{5}^{1}\), \(p^{14}=g_{2}^{1}+g_{4}^{1}+g_{6}^{1}\) and \(p^{15}=g_{3}^{1}+g_{5}^{1}+g_{6}^{1}\).

One key uncertainty about coalition government formation can be the designation of a formateur. Diermeier and Merlo (2004) analyze formateur selection process for 11 parliamentary democracies over the period 1945–1997. They conclude that the data support the proportional selection, where formateurs are selected randomly proportional to the distribution of seat shares in the parliament as suggested by Baron–Ferejohn model.

This can also be seen from combining Eq. (5) with Lemma 3.

If \(r\geqslant q\), then only veto players are included in any winning coalition and the problem becomes trivial since there is no need to choose any coalition partners. Of course, there is no multiplicity of SSPE in this case. Hence, to make the problem interesting, we assume \(r<q\).

For example, after the recent general elections of June 2015 in Turkey, Nationalist Movement Party [Turkish: Milliyetci Hareket Partisi (MHP)] announced that it will not be involved in any coalition government that includes People’s Democratic Party [Turkish: Halkların Demokratik Partisi (HDP)]; see http://​www.​hurriyet.​com.​tr/​gundem/​29306673.​asp.

Other examples outside the coalition government context prevail as well. One such example is the amendment of Canadian Constitution, which is provided in Winter (1996). The British Parliament had the veto authority to overturn any proposal for the amendment of Canadian Constitution between the years 1867 and 1982. This veto power was changed in 1982 with another rule which required that the proposal for amendment must be supported at least two-thirds of the provinces in Canada and also that the supporting provinces must have 50 % of the population. At that time, Ontario and Quebec together had more than 50 % of the population. That means they together had a veto power without constituting a winning coalition. Another example is from finance, called “golden share”. Golden share grants minority shareholders veto rights on certain issues in shareholders’ meetings.

Note that in the standard Baron–Ferejohn game, continuation values are uniquely determined.

The discontinuity at \(r=q-1\) is due to the fact that when \(r=q-1\), non-veto players have no choice but form the winning coalition with veto players alone besides themselves.

Consider our previous example given in footnote 10, where \(n=5\), \(q=3\), and player 1 is the proposer. Recall that: \(p^{12}=g_{1}^{1}+g_{2}^{1}+g_{3} ^{1}\), \(p^{13}=g_{1}^{1}+g_{4}^{1}+g_{5}^{1}\), \(p^{14}=g_{2}^{1}+g_{4} ^{1}+g_{6}^{1}\) and \(p^{15}=g_{3}^{1}+g_{5}^{1}+g_{6}^{1}\). This implies that \( {\sum \nolimits _{j=2}^{5}} p^{1j}=2(g_{1}+g_{2}+g_{3}+g_{4}+g_{5}+g_{6})=2\).

As an example, consider the 3-player game we analyze in the main text. Since player-1 is assumed to be a veto player, it is true that \(p^{21}=p^{31}=1\) and \(p^{23}=p^{32}=0\). Therefore, the unknown randomization probabilities are \(p^{12}\) and \(p^{13}\). On the other hand, the linearly independent equations are: \(p^{12}+p^{13}=1\), implied by Eq. (31), and \(p^{12}=p^{13}\), implied by Eq. (33).