Estimating noncooperative and cooperative models of bargaining: an empirical comparison

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.

Appendices

Appendix A

The Nash solution is characterized as follows. Notice first that

$$\begin{aligned}&\underset{v}{\max }\quad \prod _{i=1}^{n}(v_{i}-c_{i})^{b_{i}} \\&\Leftrightarrow \underset{v}{\max } \sum _{i=1}^{n}b_{i}\log (v_{i}-c_{i}). \end{aligned}$$

Using the normalization $\sum _{i=1}^{n}v_{i}=1$, we can rewrite the expression above as

$$\begin{aligned} \underset{v}{\max }\sum _{i=1}^{n-1}b_{i}\log (v_{i}-c_{i})+b_{n}\log \left( 1-\sum _{i=1}^{n-1}v_{i}-c_{n}\right) . \end{aligned}$$

By solving this maximization problem, we have

$$\begin{aligned} \frac{\partial }{\partial v_{i}}\left( \sum _{i=1}^{n-1}b_{i}\log (v_{i}-c_{i})+b_{n}\log (1-\sum _{i=1}^{n-1}-c_{n})\right) =0, \end{aligned}$$

which leads to the desired result:

$$\begin{aligned} \frac{v_{i}-c_{i}}{b_{i}}-\frac{v_{n}-c_{n}}{b_{n}}=0 \end{aligned}$$

for each $i=1,..,n-1$.

Appendix B

In this appendix, we briefly describe institutional and noninstitutional characteristics of government formation of Japan during the period of 1958 to 1993 (see Appendix A of Adachi and Watanabe (2008) for more details). We below focus on the features that are incorporated into our modeling of cooperative and noncooperative bargaining games.

The forming process of a new government (i.e., a cabinet) with a new Prime Minister begins when s/he is designated by the legislature.²⁵ Then, s/he selects the cabinet members. Finally, the Emperor appoints him or her as the Prime Minister. The following characteristics are considered in modeling our bargaining games.

• The process of choosing a Prime Minister and the cabinet formation was an internal process within the Liberal Democratic Party (LDP) . This is because, during the period between 1955 and 1993, the LDP maintained a majority in the House of Representatives, and the Prime Minister-designates were always LDP presidents.

• LDP factions played the main role in government formation. No LDP faction left the LDP, and LDP factions of significant size obtained cabinet posts in most of the cabinets. Thus, both in cooperative and noncooperative bargaining models, we assume that players are factions of the LDP. In estimation, we use the variation in the faction size within the LDP.

• LDP factions had few differences with respect to their preferences on policy issues. ²⁶ We thus assume that a minister’s weight, $\beta _{j}$, is common for all factions.

• Delays were not important in the cabinet formation.All of the cabinets were formed within 3 days following the selection of a Prime Minister-designate. Thus, we do not consider bargaining models that generate equilibrium delays.²⁷