Power and welfare in bargaining for coalition structure formation

Abstract

We investigate a noncooperative bargaining game for partitioning n agents into non-overlapping coalitions. The game has n time periods during which the players are called according to an exogenous agenda to propose offers. With probability $\delta$, the game ends during any time period $t<n$. If it does, the first t players on the agenda get a chance to propose but the others do not. Thus, $\delta$ is a measure of the degree of democracy within the game (ranging from democracy for $\delta =0$, through increasing levels of authoritarianism as $\delta$ approaches 1, to dictatorship for $\delta =1$). We determine the subgame perfect equilibrium (SPE) and study how a player’s position on the agenda affects his bargaining power. We analyze the relation between the distribution of power of individual players, the level of democracy, and the welfare efficiency of the game. We find that purely democratic games are welfare inefficient and that introducing a degree of authoritarianism into the game makes the distribution of power more equitable and also maximizes welfare. These results remain invariant under two types of player preferences: one where each player’s preference is a total order on the space of possible coalition structures and the other where each player either likes or dislikes a coalition structure. Finally, we show that the SPE partition may or may not be core stable.

Appendix: The Monte Carlo sampling procedure

The basic idea of Monte Carlo sampling is to consider only a subset of all games $\mathcal {G}' \subset \mathcal {G}$ and use $\mathcal {P}_{\rho _i}(\mathcal {G}')$ and $\mathbb {E}_{\rho _i}(\mathcal {G}')$ as estimators of $\mathcal {P}_{\rho _i}(\mathcal {G})$ and $\mathbb {E}_{\rho _i}(\mathcal {G})$. Our sampling procedure can be described as follows. For a given N, $\rho$, $\delta$, we first chose the number of Monte Carlo iterations, that is, $|\mathcal {G}'|$. Next, for each iteration:

• (i) We sample a game G from $\mathcal {G}$ (to be included in $\mathcal {G}'$) by sampling a random combination of preference orderings of all the players in N (see, for instance, Table 2). To this end use Knuth’s shuffle algorithm [14].

• (ii) For the combination of preferences generated in (i), we identify the welfare maximizing coalition structure(s), $\pi _{SW}(G)$;

• (iii) We solve the game with backward induction and compute the equilibrium coalition structure, $\pi ^{*}_1({G})$;

• (iv) For each player $i \in N$ we record his expected rank for the equilibrium coalition structure, $er_{\rho _i}(\pi ^*_1(G))$. We also compute the efficiency of the game G, $er_{\rho _i}(\pi ^*_1(G))$.

Finally, having generated G, we compute both $\mathcal {P}_{\rho _i}(\mathcal {G}')$ and $\mathbb {E}_{\rho _i}(\mathcal {G}')$.