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Empirical Economics

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01-09-2014

Estimating noncooperative and cooperative models of bargaining: an empirical comparison

Authors: Masanori Mitsutsune, Takanori Adachi

Published in: Empirical Economics | Issue 2/2014

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Abstract

This paper examines the issue of model selection in studies of strategic situations. In particular, we compare estimation results from a noncooperative formulation of government formulation à la (Baron and Ferejohn in Am Poli Sci Rev 87:34–47, 1989) with those from two alternative cooperative formulations (Nash in Econometrica 18:155–162, 1950; Shapley and Shubik in Am Poli Sci Rev 48:787–792, 1954). Although the estimates of the ministerial ranking are similar, statistical testing suggests that the noncooperative formulation is best fitted to the observed data among the alternative models. This result implies that modeling the noncooperative structure in bargaining situations is crucially important at the risk of possibly misspecifying the details of the game.

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The Nash solution concept is axiomatically constructed, and thus is free of negotiation procedure. See, e.g., Okada (2010) for a noncooperative foundation for the Nash solution and Okada (2007) for its application.

When a symmetric two-person Nash solution is analyzed, the maxmin values or the Nash equilibrium is customarily set as a breakdown point. However, such good ways do not exist to model the breakdown situation in the analysis of asymmetric $n$-person Nash solution.

See Hu (2006) for an extension of the Shapley–Shubik power index.

Note that a voter here is a party. Considering the reality, it would be more natural that a member of parliament, not a party, has a vote. We assume a hypothetical situation where each party as a whole decides on its vote, binding its members’ decisions.

How the distribution of party size affects the determination of $ v_{i}^{SS}(W)$ is understood in the following example. Suppose that there are four parties with $W=(40,25,20,15)$. In the grand coalition formation starting party $1$, followed by parties $2$, $3$, and $4$ (this is one permutation), party $2$ is pivotal and the winning coalition consists of parties $1$ and $2$, if the simple majority rule is employed. Party $i$’s Shapley–Shubik power index is the average of it being pivotal of all permutations.

It is obvious that $v_{i}^{SS}(W)$ also depends on the majority quota rule (for example, the rule can require that more than two thirds, rather than half, of all votes be necessary for a winning coalition). We assume the simple majority within the ruling party of the data period (a faction of the party is a player of the bargaining game; see 3.4 below). We later consider the robustness of our empirical results by checking the case of the simple majoriy within all parliamentary members of the lower house, which is given a decisive power by law (see 4.4.1 below).

Hart and Mas-Colell (1996), Laruelle and Valenciano (2008), Miyakawa (2008), Britz et al. (2010) and Predtetchinski (2011), among others, study the asymptotic coincidence of a noncorporative equilibrium to a cooperative solution. In particular, stationary subgame perfect equilibria in a $n$-player game where a proposer drops from the game with a small probability if the agreement is not reached converge to asymmetric Nash solution with each player’s recognition probability being his or her power index.

See, e.g., Okada (2011) and Kawamori (2013) for recent extensions of Baron and Ferejohn (1989) model. Ray (2007) is an excellent survey of the noncooperative theory for coalition formation.

Eraslan (2002, p. 15) shows that proposer $i$’s stationary equilibrium payoff, $v_{i}^{*}$, is obtained by

$$\begin{aligned} \left\{ \begin{array}{l} \underset{(z_{j})_{j\ne i}}{\!\!\!\!\!\max }1-\sum _{j\ne i}z_{j}v_{j}^{*} \\ \text {subject to }\sum _{j\ne i}=q-1 \\ z_{j}\in \{0,1\}\text {,} \end{array}\right. \end{aligned}$$

where $q=n/2$ (when $n$ is even) or $q=(n+1)/2$ ($n$ is odd), and $ v_{j}^{*}$ is nonproposer $j$’s stationary equilibrium payoff ($ v_{j}^{*}=\delta p_{j}$ if $j$ is such that $z_{j}=1$, and $v_{j}^{*}=0$ if $j$ is such that $z_{j}=0$). Clearly, there are some players who gain nothing if the unanimous agreement is not implemented.

Admittedly, the specification of bargaining power is a “reduced form” because we do not model how it is determined in bargaining models. However, our main results seem robust to alternative specifications. We are grateful to one of the referees for making us realize this issue.

See also McDonald (1984, p. 653, eq.(14)). We believe that there is a typo in equation (14); $p+q$ in the denominator should be $p+q-1$.

Note that we use the information on ex-post payoffs (i.e., after a proposer is selected). If we use functional forms of ex-ante payoffs, the scale effect parameter $\alpha $ would be identified from the probability of becoming a proposer given the party size distribution. However, the identification of $\delta $ would not be guaranteed.

During the period of the study, although the Prime Minister and the Cabinet Secretary were often occupied by the same party, there were still four exceptions (out of 44 government formations).

The page’s URL is: http://hpcgi2.nifty.com/TOMOMI/index-e.cgi (last accessed: November 2013).

Because the LDP attained a majority in the Lower House during the period of this study, this assumption is natural. See Appendix A of Adachi and Watanabe (2008) for details.

On the other hand, Ono (2012) uses detailed data on LDP leaders’ factions to empirically argue that they took tougher stance on ministerial allocation when their political positions were more stable.

As in Adachi and Watanabe (2008), the standard errors are calculated using the bootstrap method. The number of iteration is 500.

Notice that the values of log likelihood in Table 2 are positive. In Adachi and Watanabe (2008), it is incorrectly reported as a negative value. The reported estimates in Adachi and Watanabe (2008) are modified by Adachi and Watanabe (2010): the latter reestimates the model with the corrected data.

See, e.g., Greene (2008, pp. 498–500) and Wooldridge (2010, p. 481).

The chi-square values at different significance levels are given by $2.7055$ ($10\%), 3.8415 (5\,\%$), and $6.6349$ ($1\%$).

See, e.g., Greene (2008, pp. 140–142) and Wooldridge (2010, pp. 505–509).

In calculating these numbers with the help of Tomoki Matsui’s website (see Footnote 14), we assume the total number of LDP politicians is $280$: the mean number of LDP politicians in the Lower House in each term is $280.78$ (with the total number being $18$): the Prime Minister’s faction has $84$ politicians, while each of the other factions have $28$ politicians.

We owe this idea to Yusaku Horiuchi.

The likelihood statistics are calculated as $21.8236$ (Adachi and Watanabe), $21.8305$ (Nash), and $19.3363$ (Shapley and Shubik), respectively. These values are all greater than $11.34$ (at the $1\%$ significance level with the degree of freedom being three).

Japan adopts a parliamentary regime with a bicameral legislature. If the House of Councilors (the upper house) and the House of Representatives (the lower house) disagree on the designation for 10 days, the latter’s decision is enforced.

For example, Ramseyer and Rosenbluth (1993:211) cite an interview with one faction leader of the LDP faction who answered that factions were not distinguishable based on policy areas.

See, e.g., Merlo and Wilson (1995, 1998), Merlo (1997), and Hanazono and Watanabe (2012) for bargaining models with equilibrium delays.

Adachi T, Watanabe Y (2008) Ministerial weights and government formation: estimation using a bargaining model. J Law Econ Org 24:95–119

Adachi T, Watanabe Y (2010) Corrigendum to “ Ministerial weights and government formation: estimation using a bargaining model”. http://www.soec.nagoya-u.ac.jp/adachi.t/aw_jleo_corrigendum.pdf

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Title: Estimating noncooperative and cooperative models of bargaining: an empirical comparison
Authors: Masanori Mitsutsune
Takanori Adachi
Publication date: 01-09-2014
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 2/2014
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-013-0755-9

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