Bargaining models with asymmetric agents

Resumo

The classic models in game theory can be classified into two categories. The models of the cooperative game theory are defined by a set of agents and a characteristic function (benefit or cost). The models of the non-cooperative game theory are defined by a set of agents and information structure which contains a set of actions for each agent, strategies, payoffs and the sequence of movements in the interaction of the game. In the present dissertation, models from the two categories are studied and contain asymmetric agents, with asymmetry defined from two perspectives. In some cases, asymmetry is by nature, where we consider a unique agent with different characteristics from the rest, e.g. a unique tenant and several lessors. In some other cases, asymmetry is by temporal component, where some agents can act and/or develop simultaneously and others sequentially, e.g. the activities of a complex or large project. From both perspectives, these characteristics play an important role when we define the models of the present study. In the first case, we consider as example the acquisition of land for mineral exploitation or extraction by a mining firm in Peru and for settlement of a military base by the Ministry of Defence in Pontevedra - Spain. In these cases we can see a negotiation process between a firm or institution, and a set of owners or communities. These situations are studied as a conflict of interests problem from the cooperative and non-cooperative viewpoints. In the second case, we consider as example the cost that is generated by a delay of the activities in complex or large projects, where some activities can develop simultaneously while other sequentially. This situation is studied as a cost assignment problem between the responsible agents. Due to the huge costs on the one hand by the land conflicts (Dufwenberg et al., 2016) and, on the other hand, the delays of large projects, we consider as a relevant issue to study both situations from a game theory viewpoint. The contribution of this dissertation is on the one hand an axiomatic approach and on the other hand non-cooperative. This dissertation is organized in five chapters as follows: Chapters 2, 3 and 4 consider land rental problems where there are several communities that can act as lessors and a single tenant who does not necessary need all the available land, while Chapter 5 considers delay of activities problems. Chapter 2 is based on Valencia-Toledo and Vidal-Puga (2016a). We propose a non-cooperative game theory viewpoint. In this case, we consider that the tenant should negotiate sequentially with each lessor for the available land. In each stage, we apply the Nash bargaining solution (Nash, 1950). Our results imply that, when all land is necessary, a fixed price per unit is more favourable for the tenant than a lessor-dependent price. Furthermore, a lessor is better off with a lessor-dependent price only when negotiating first. For the tenant, lessors’ merging is relevant with lessor-dependent price but not with fixed price. Chapter 3 is based on the article Valencia-Toledo and Vidal-Puga (2015). We propose a cooperative game theory viewpoint. We establish that a rule should determine which communities become lessors, how much land they rent and at which price. According to this, our first result is a complete characterization of the family of rules that satisfy two properties: “land monotonicity” in the sense that an increase in the amount of available land benefits to every agent involved, and “non-manipulability under land reassignment” in the sense that the land reassignment does neither benefit nor harm the interest of the agents involved. We also define two parametric subfamilies. The first one is characterized by adding a property of “weighted standard for two”, that says the way to share the benefit of cooperation in case we have only one lessor and one tenant. The second one is characterized by adding two properties: “consistency” in the sense that leaving of some agents do not affect the share rule, and “continuity” in the sense that a small change in the initial problem does not mean big changes in the proposed assignment. Chapter 4 is based on Valencia-Toledo and Vidal-Puga (2016b). We study self-duality in land rental problems where there are several lessors and a single tenant. We assume that the available land which is owned by the lessors is larger than the rented land. We propose the property of self-duality based on the definition given by Aumann and Maschler (1985) in bankruptcy problems (O’Neill, 1982). In land rental problems, this is equivalent to share the rented land and remaining not rented land. We provide a complete characterization of the family of rules that satisfy self-duality. We obtain that the amount of land should be shared proportionally. We characterize the family of rules that satisfy “self-duality” and “price independence”. We also characterize a unique rule that satisfies “self-duality”, “price independence” and “standard for two”, which says that the benefit of cooperation should be shared equally in case there are only one tenant and one lessor. Finally, Chapter 5 is based on Bergantiños et al. (2016). We study a tool used to schedule and coordinate activities in a complex project, which is commonly called as PERT (Program Evaluation Review Technique). In assigning the cost of a potential delay, we characterize an unique rule which is based on the Shapley value (Shapley, 1953) as the unique that satisfies “consistency” and other desirable properties such as: “anonymity” says that the cost sharing does not depend on the name of the activities; “monotonicity”, which says that if the delay of an activity increases, the quantity that it should pay must not decrease; “scale invariance”, which says that the change of unit of delay does not affect the assigned cost; “standard for two”, which says that if there are only two activities, the cost must share in standard way; and “independence of irrelevant delay”, which says that if the delay of an activity does not affect the total cost of the project, then the agent responsible for this activity pay nothing.