Three essays on game theory and bargaining

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

Dynamic matching and bargaining: the role of deadlines

We consider a dynamic model where traders in each period are matched randomly into pairs who then bargain about the division of a fixed surplus. When agreement is reached the traders leave the market. Traders who do not come to an agreement return next period in which they will be matched again, as long as their deadline has not expired yet. New traders enter exogenously in each period. We assume that traders within a pair know each other's deadline. We define and characterize the stationary equilibrium configurations. Traders with longer deadlines fare better than traders with short deadlines. It is shown that the heterogeneity of deadlines may cause delay. It is then shown that a centralized mechanism that controls the matching protocol, but does not interfere with the bargaining, eliminates all delay. Even though this efficient centralized mechanism is not as good for traders with long deadlines, it is shown that in a model where all traders can choose which mechanism to

Search and bargaining in large markets with homogeneous traders

We study pair-wise decentralized trade in dynamic markets with homogeneous, non-atomic, buyers and sellers that wish to exchange one unit. Pairs of traders are randomly matched and bargaining a price under rules that offer the freedom to quit the match at any time. Market equilbria, prices and trades over time, are characterized. The asymptotic behavior of prices and trades as frictions (search costs and impatience) vanish, and the conditions for (non) convergence to walrasian prices are explored. As a side product of independent interest, we present a self-contained theory of non-cooperative bargaining with two-sided, time-varying, outside options.

Mediation incomplete information bargaining with filtered communication

We analyze a continuous-time bilateral double auction in the presence of two-sided incomplete information and a smallest money unit. A distinguishing feature of our model is that intermediate concessions are not observable by the adversary: they are only communicated to a passive auctioneer. An alternative interpretation is that of mediated bargaining. We show that an equilibrium using only the extreme agreements always exists and display the necessary and sufficient condition for the existence of (perfect Bayesian) equilibra which yield intermediate agreements. For the symmetric case with uniform type distribution we numerically calculate the equilibria. We find that the equilibrium which does not use compromise agreements is the least efficient, however, the rest of the equilibria yield the lower social welfare the higher number of compromise agreements are used.

Bargaining and Conflict with Incomplete Information

This paper studies bargaining and conflict under incomplete information, provides an overview and a critical account of the literature on the topic and contributes with original research. We first revise models of mechanism design and sequential bargaining that take confrontation as final. Conflict and inefficiencies are to be expected in these models whenever parties have optimistic prospects on the outcome of the all-out conflict. After examining the causes and reasons for this optimism, we move to the analysis of the recent literature that considers the existence of limited confrontations that allow bargaining to resume. In the presence of private information, these limited conflicts convey information and thus become a bargaining instrument. The paper closes with a discussion on the related empirical literature, the challenges that it faces and some potential avenues for further research.

Optimism and commitment: An elementary theory of bargaining and war

We propose an elementary theory of wars fought by fully rational contenders. Two parties play a Markov game that combines stages of bargaining with stages where one side has the ability to impose surrender on the other. Under uncertainty and incomplete information, in the unique equilibrium of the game, long confrontations occur: war arises when reality disappoints initial (rational) optimism, and it persist longer when both agents are optimists but reality proves both wrong. Bargaining proposals that are rejected initially might eventually be accepted after several periods of confrontation. We provide an explicit computation of the equilibrium, evaluating the probability of war, and its expected losses as a function of i) the costs of confrontation, ii) the asymmetry of the split imposed under surrender, and iii) the strengths of contenders at attack and defense. Changes in these parameters display non-monotonic effects.

Bargaining one-dimensional policies and the efficiency of super majority rules

We consider negotiations selecting one-dimensional policies. Individuals have single-peaked preferences, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. The existence and uniqueness of stationary subgame perfect equilibrium is established, and its explicit characterization provided. We supply an explicit formula to determine the unique alternative that prevails, as impatience vanishes, for each majority. As an application, we examine the efficiency of majority rules. For symmetric distributions of peaks unanimity is the unanimously preferred majority rule. For asymmetric populations rules maximizing social surplus are characterized.

Simultaneous Nash bargaining with consistent beliefs

We propose and analyze a new solution concept, the R solution, for three-person, transferable utility, cooperative games. In the spirit of the Nash Bargaining Solution, our concept is founded on the predicted outcomes of simultaneous, two-party negotiations that would be the alternative to the grand coalition. These possibly probabilistic predictions are based on consistent beliefs. We analyze the properties of the R solution and compare it with the Shapley value and other concepts. The R solution exists and is unique. It belongs to the bargaining set and to the core whenever the latter is not empty. In fact, when the grand coalition can simply execute one of the three possible bilateral trades, the R solution is the most egalitarian selection of the bargaining set. Finally, we discuss how the R solution changes important conclusions of several well known Industrial Organization models.

Bargaining failures and merger policy

In this paper we study the optimal ex-ante merger policy in a model where merger proposals are the result of strategic bargaining among alternative candidates. We allow for firm asymmetries and, in particular, we emphasize the fact that potential synergies generated by a merger may vary substantially depending on the identity of the participating firms. The model demonstrates that, under some circumstances, relatively inefficient mergers may take place. That is, a particular merger may materialize despite the existence of an alternative merger capable of generating higher social surplus and even higher profits. Such bargaining failures have important implications for the ex-ante optimal merger policy. We show that a more stringent policy than the ex-post optimal reduces the scope of these bargaining failures and raises expected social surplus. We use a bargaining model that is flexible, in the sense that its strategic structure does not place any exogenous restriction on the endogenous likelihood of feasible mergers.

From bargaining solutions to claims rules: a proportional approach

Is it important to negotiate on proportions rather than on numbers? To answer this question, we analyze the behavior of well-known bargaining solutions and the claims rules they induce when they are applied to a "proportionally transformed" bargaining set SP -so-called bargaining-in-proportions set. The idea of applying bargaining solutions to claims problems was already developed in Dagan and Volij (1993). They apply the bargaining solutions over a bargaining set that is the one de ned by the claims and the endowment. A comparison among our results and theirs is provided. Keywords: Bargaining problem, Claims problem, Proportional, Constrained Equal Awards, Constrained Equal Losses, Nash bargaining solution. JEL classi fication: C71, D63, D71.

Parental investments in children: how bargaining and educational homogamy affect time allocation

This study examines parental time investment in their children, distinguishing between developmental and non-developmental care. Our analyses centre on three influential determinants: educational background, marital homogamy, and spouses' relative bargaining power. We find that the emphasis on quality care time is correlated with parents' education, and that marital homogamy reduces couple specialization, but only among the highly educated. In line with earlier research, we identify gendered parental behaviour. The presence of boys is an important condition for fathers' time dedication, but primarly among lower educated fathers. To the extent that parental stimulation is decisive for child outcomes, our findings suggest the persistence of important inequalities. This emerges through our special attention to behavioural differences across the educational distribution among households.

Bargaining, coalitions and competition

We study a decentralized matching model in a large exchange economy,in which trade takes place through non--cooperative bargaining in coalitionsof finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with theWalrasian allocations of the economy. Our method of proof exploits equivalenceresults between the core and Walrasian equilibria. Our model relaxes differentiability and convexity of preferences thereby covering the caseof indivisible goods.