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Register a new userDisclosure Games with Large Evidence Spaces
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Shaofei Jiang
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We study a disclosure game with a large evidence space. There is an unknown binary state. A sender observes a sequence of binary signals about the state and discloses a left truncation of the sequence to a receiver in order to convince him that the state is good. We focus on truth-leaning equilibria (cf. Hart et al. (2017)), where the sender discloses truthfully when doing so is optimal, and the receiver takes off-path disclosure at face value. In equilibrium, seemingly sub-optimal truncations are disclosed, and the disclosure contains the longest truncation that yields the maximal difference between the number of good and bad signals. We also study a general framework of disclosure games which is compatible with large evidence spaces, a wide range of disclosure technologies, and finitely many states. We characterize the unique equilibrium value function of the sender and propose a method to construct equilibria for a broad class of games.
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Evidence games study situations where a sender persuades a receiver by selectively disclosing hard evidence about an unknown state of the world. Evidence games often have multiple equilibria. Hart et al. (2017) propose to focus on truth-leaning equilibria, i.e., perfect Bayesian equilibria where the sender prefers disclosing truthfully when indifferent, and the receiver takes off-path disclosure at face value. They show that a truth-leaning equilibrium is an equilibrium of a perturbed game where the sender has an infinitesimal reward for truth-telling. We show that, when the receivers action space is finite, truth-leaning equilibrium may fail to exist, and it is not equivalent to equilibrium of the perturbed game. To restore existence, we introduce a disturbed game with a small uncertainty about the receivers payoff. A purifiable equilibrium is a truth-leaning equilibrium in an infinitesimally disturbed game. It exists and features a simple characterization. A truth-leaning equilibrium that is also purifiable is an equilibrium of the perturbed game.
This paper proposes a new equilibrium concept robust perfect equilibrium for non-cooperative games with a continuum of players, incorporating three types of perturbations. Such an equilibrium is shown to exist (in symmetric mixed strategies and in pure strategies) and satisfy the important properties of admissibility, aggregate robustness, and ex post robust perfection. These properties strengthen relevant equilibrium results in an extensive literature on strategic interactions among a large number of agents. Illustrative applications to congestion games and potential games are presented. In the particular case of a congestion game with strictly increasing cost functions, we show that there is a unique symmetric robust perfect equilibrium.
This paper considers incentives to provide goods that are partially excludable along social links. Individuals face a capacity constraint in that, conditional upon providing, they may nominate only a subset of neighbours as co-beneficiaries. Our model has two typically incompatible ingredients: (i) a graphical game (individuals decide how much of the good to provide), and (ii) graph formation (individuals decide which subset of neighbours to nominate as co-beneficiaries). For any capacity constraints and any graph, we show the existence of specialised pure strategy Nash equilibria - those in which some individuals (the Drivers, D) contribute while the remaining individuals (the Passengers, P) free ride. The proof is constructive and corresponds to showing, for a given capacity, the existence of a new kind of spanning bipartite subgraph, a DP-subgraph, with partite sets D and P. We consider how the number of Drivers in equilibrium changes as the capacity constraints are relaxed and show a weak monotonicity result. Finally, we introduce dynamics and show that only specialised equilibria are stable against individuals unilaterally changing their provision level.
We study full implementation with evidence in an environment with bounded utilities. We show that a social choice function is Nash implementable in a direct revelation mechanism if and only if it satisfies the measurability condition proposed by Ben-Porath and Lipman (2012). Building on a novel classification of lies according to their refutability with evidence, the mechanism requires only two agents, accounts for mixed-strategy equilibria and accommodates evidentiary costs. While monetary transfers are used, they are off the equilibrium and can be balanced with three or more agents. In a richer model of evidence due to Kartik and Tercieux (2012a), we also establish pure-strategy implementation with two or more agents in a direct revelation mechanism. We also obtain a necessary and sufficient condition on the evidence structure for renegotiation-proof bilateral contracts, based on the classification of lies.
We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call $varepsilon$-estimated equilibrium and prove its existence for any positive $varepsilon$. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk--Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that paper the existence of pure-strategy Nash equilibria has been proved for any nonatomic game with a continuum of players, endowed with an atomless countably additive probability. But, requiring Borel measurability of strategy profiles may impose some limitation on players choices and introduce an exogenous dependence among players actions, which clashes with the nature of noncooperative game theory. Our suggested solution is to consider every subset of players as measurable. This leads to a nontrivial purely finitely additive component which might prevent the existence of equilibria and requires a novel mathematical approach to prove the existence of $varepsilon$-equilibria.
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