Pierpaolo Battigalli

Dynamic psychological games

Building on recent work on dynamic interactive epistemology, we extend the analysis of extensive-form psychological games (Geneakoplos, Pearce & Stacchetti, Games and Economic Behavior, 1989) to include conditional higher-order beliefs and enlarged domains of payoff functions. The approach allows modeling dynamic psychological effects (such as sequential reciprocity, psychological forward induction, and regret) that are ruled out when epistemic types are identified with hierarchies of initial beliefs. We define a notion of psychological sequential equilibrium, which generalizes the sequential equilibrium notion for traditional games, for which we prove existence under mild assumptions. Our framework also allows us to directly formulate assumptions about ‘dynamic’ rationality and interactive beliefs in order to explore strategic interaction without assuming that players beliefs are coordinated on an equilibrium. In particular, we provide an exploration of (extensive-form) rationalizability in psychological games.

Rationalization and Incomplete Information

We analyze a family of extensive-form solution procedures for games with incomplete information that do not require the specification of an epistemic type space a la Harsanyi, but can accommodate a (commonly known) collection of explicit restrictions D on first-order beliefs. For any fixed D we obtain a solution called D-rationalizability.In static games, D-rationalizability characterizes the set of outcomes (combinations of payoff types and strategies) that may occur in any Bayesian equilibrium model consistent with D; these are precisely the outcomes consistent with common certainty of rationality and of the restrictions D. Hence, our approach to the analysis of incomplete-information games is consistent with Harsanyis, and it may be viewed as capturing the robust implications of Bayesian equilibrium analysis.In dynamic games, D-rationalizability yields a forward-induction refinement of this set of Bayesian equilibrium outcomes. Focusing on the restriction that first-order beliefs be consistent with a given distribution on terminal nodes, we obtain a refinement of self-confirming equilibrium. In signalling games, this refinement coincides with the Iterated Intuitive Criterion.

Buyer power and quality improvements

This paper analyses the sources of buyer power and its effect on sellers’ investment in quality improvements. In our model retailers make take-it-or-leave-it offers to a producer and each of them obtains its marginal contribution to total profits (gross of sunk costs). In turn, this depends on the rivalry between retailers in the bargaining process. Rivalry increases when retailers are less differentiated and when decreasing returns to scale in production are larger. The allocation of total surplus affects the incentives of the producer to invest in product quality, an instance of the hold-up problem. An increase in buyer power not only makes the supplier and consumers worse off, but it may even harm retailers, that obtain a larger share of a smaller surplus. A repeated game argument shows that efficient quality improvements can be supported as an equilibrium outcome if the producer and retailers are involved in a long-term relationship.

Conjectural Equilibria and Rationalizability in a Game with Incomplete Information

Conjectural equilibria and rationalizability are introduced in order to analyze extensive games with incomplete information when the common prior assumption is dropped. We provide a macroeconomic example which clearly illustrates the crucial role of players’ structural and behavioral conjectures: different equilibrium and disequilibrium paths (some with keynesian type inefficiencies) are generated by different expectations patterns. Convergence of rationalizable paths to some conjectural equilibrium may occur, but not necessarily to a full information-rational expectations one.

Rationalizable bidding in first-price auctions

Abstract We analyze the consequences of strategically sophisticated bidding without assuming equilibrium behavior. In particular, we characterize interim rationalizable bids in symmetric first-price auctions with interdependent values and affiliated signals. We show that (1) every nonzero bid below the equilibrium is rationalizable, (2) some bids above the equilibrium are rationalizable, (3) the upper bound on rationalizable bids of a given player is a nondecreasing function of her signal. In the special case of independent signals and quasi-linear valuation functions, (i) the least upper bound on rationalizable bids is concave; hence (ii) rationalizability implies substantial proportional shading for high valuations, but is consistent with negligible proportional shading for low valuations. We argue that our theoretical analysis may shed some light on experimental findings about deviations from the risk-neutral Nash equilibrium.

Costly contracting in a long-term relationship

We examine a model of contracting where parties interact repeatedly and can contract at any point in time, but writing enforceable contracts is costly. A contract can describe contingencies and actions at a more or less detailed level, and the cost of writing a contract is proportional to the amount of detail. We consider both formal (externally enforced) and informal (self-enforcing) contracts. The presence of writing costs has important implications both for the optimal structure of formal contracts, particularly the tradeoff between contingent and spot contracts, and for the interaction between formal and informal contracting. Our model sheds light on these implications and generates a rich set of predictions about the determinants of the optimal mode of contracting.

Interactive Epistemology and Solution Concepts for Games with Asymmetric Information

We use an interactive epistemology framework to provide a systematic analysis of some solution concepts for games with asymmetric information. We characterize solution concepts using expressible epistemic assumptions, represented as events in the canonical space generated by primitive uncertainty about the payoff relevant state, payoff irrelevant information, and actions. In most of the paper, we adopt an interim perspective, which is appropriate to analyze genuine incomplete information. We relate Delta-rationalizability (Battigalli and Siniscalchi, Advances in Theoretical Economics 3, 2003) to interim correlated rationalizability (Dekel, Fudenberg and Morris, Theoretical Economics 2, 2007) and to rationalizability in the interim strategic form. We also consider the ex ante perspective, which is appropriate to analyze asymmetric information about an initial chance move. We prove the equivalence between interim correlated rationalizability and an ex ante notion of correlated rationalizability.

The Logic of Belief Persistence

The interaction bietween knowledge and belief in a temporal context is analyzed. An axiomatic formulation and semantic characterization of the principle of belief persistency implied by the standard conditionalization rule are provided. This principle says that an individual does not change her mind unless new evidence forces her to do so. It is shown that if beliefs are conscious (or stateindependent) and satisfy negative introspection then the principle of persistency of beliefs is characterized by the following axiom schema: the individual believes that 9 at date t if and orilly if she believes at date t that she will believe that 4 at date t+l.

Synchronic Information, Knowledge and Common Knowledge in Extensive Games

Restricting attention to the class of extensive games defined by von Neumann and Morgenstern (1944) with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of ‘‘maximum information’’ we are able to provide an alternative characterization of the meet of the information partitions.

An Epistemic Characterization of Extensive Form Rationalizability

We use an extensive form, universal type space to provide the following epistemic characterisation of extensive form rationalisability. Say that player i strongly believes event E if i is certain of E conditional on each of her information sets consistent with E. Our main contribution is to show that a strategy profile s is extensive form rationalisable if and only if there is a state in which s is played and (0) everybody is rational, (1) everybody strongly believes (0), (2) everybody strongly believes (0) & (1), (3) everybody strongly believes (0) & (1) & (2), .... This result also allows us to provide sufficient epistemic conditions for the backward induction outcome and to relate extensive form rationalisability and conditional common certainty of rationality.