We consider the following belief free solution concepts for games with incomplete information: (i) incomplete information rationalizability, (ii) incomplete information correlated equilibrium and (iii) ex post equilibrium. We present epistemic foundations for these solution concepts and establish relationships between them. The properties of these solution concepts are further developed in supermodular games and potential games.
Keywords: Correlated equilibrium, Rationalizability, Ex post equilibrium, Belief free, Types, Payo types, Belief types, Supermodular games, Potential games
We consider truthful implementation of the socially efficient allocation in a dynamic private value environment in which agents receive private information over time. We propose a suitable generalization of the Vickrey-Clarke-Groves mechanism, based on the marginal contribution of each agent. In the marginal contribution mechanism, the ex post incentive and ex post participations constraints are satisfied for all agents after all histories. It is the unique mechanism satisfying ex post incentive, ex post participation and efficient exit conditions.
We develop the marginal contribution mechanism in detail for a sequential auction of a single object in which each bidders learn over time her true valuation of the object. We show that a modified second price auction leads to truthtelling.
We consider an auction environment with interdependent values. Each bidder can learn her payoff type through costly information acquisition. We contrast the socially optimal decision to acquire information with the equilibrium solution in which each agent has to privately bear the cost of information acquisition.
In the context of the generalized Vickrey-Clarke-Groves mechanism, we establish that the equilibrium level exceeds the socially optimal level of information with positive interdependence. The individual decisions to acquire information are strategic substitutes. The difference between the equilibrium and the efficient level of information acquisition is increasing in the interdependence of the bidders’ valuations and decreasing in the number of informed bidders.
We consider an auction environment with interdependent values. Each bidder can learn her payoff type through costly information acquisition. We contrast the socially optimal decision to acquire information with the equilibrium solution in which each agent has to privately bear the cost of information acquisition.
In the context of the generalized Vickrey-Clarke-Groves mechanism, we establish that the equilibrium level exceeds the socially optimal level of information with positive interdependence. The individual decisions to acquire information are strategic substitutes. The difference between the equilibrium and the efficient level of information acquisition is increasing in the interdependence of the bidders’ valuations and decreasing in the number of informed bidders.
JEL Classification: C72, C73, D43, D83
Keywords: Vickrey-Clarke-Groves mechanism, Information acquisition, Strategic substitutes, Informational efficiency
We consider truthful implementation of the socially efficient allocation in a dynamic private value environment in which agents receive private information over time. We propose a suitable generalization of the Vickrey-Clarke-Groves mechanism, based on the marginal contribution of each agent. In the marginal contribution mechanism, the ex post incentive and ex post participations constraints are satisfied for all agents after all histories. It is the unique mechanism satisfying ex post incentive, ex post participation and efficient exit conditions.
We develop the marginal contribution mechanism in detail for a sequential auction of a single object in which each bidders learn over time her true valuation of the object. We show that a modified second price auction leads to truthtelling.
Keywords: Vickrey-Clarke-Groves mechanism, Pivot mechanism, Ex post equilibrium, Marginal contribution, Multi-armed bandit, Bayesian learning
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function: this definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function: this definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two payoff types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function. This definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable payoff types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two payoff types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function. This definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable payoff types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function: this definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents.
A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function: this definition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and sufficient for robust virtual implementation.
We consider an single object auction environment with interdependent valuations and a generalized Vickrey–Clark–Groves allocation mechanism that allocates the object almost efficiently in a strict ex post equilibrium. If there is a significant amount of interdependence, there are multiple rationalizable outcomes of this direct mechanism and any other mechanism that allocates the object almost efficiently. This is true whether the agents know about each others’ payoff types or not.
We consider an ascending price dynamic version of the generalized VCG mechanism. When there is complete information among the agents of their payoff types, we show that the almost efficient allocation is the unique backward induction (i.e., extensive form rationalizable) outcome of the auction, even when there are multiple rationalizable outcomes in the static version. This example illustrates the role that open auctions may play in obtaining efficient allocations by reducing strategic uncertainty.
We consider an single object auction environment with interdependent valuations and a generalized Vickrey–Clark–Groves allocation mechanism that allocates the object almost efficiently in a strict ex post equilibrium. If there is a significant amount of interdependence, there are multiple rationalizable outcomes of this direct mechanism and any other mechanism that allocates the object almost efficiently. This is true whether the agents know about each others’ payoff types or not.
We consider an ascending price dynamic version of the generalized VCG mechanism. When there is complete information among the agents of their payoff types, we show that the almost efficient allocation is the unique backward induction (i.e., extensive form rationalizable) outcome of the auction, even when there are multiple rationalizable outcomes in the static version. This example illustrates the role that open auctions may play in obtaining efficient allocations by reducing strategic uncertainty.
We consider the truthful implementation of the socially efficient allocation in a dynamic private value environment in which agents receive private information over time. We show that a suitable generalization of the Vickrey-Clark-Groves mechanism, based on the marginal contribution of each agent, leads to truthtelling in every period.
A leading example of a dynamic allocation model is the sequential auction of a single good in which the current winner of the object receives additional information about her valuation. We show that a modified sequential second price auction in which only the current winner makes a positive payment leads to truthtelling. In general allocation problems, the marginal contribution mechanism continues to induce truthtelling in every period but may now include positive transfers for many agents.