This note presents two results. First, it shows that under mild conditions, a decision problem is quasi-concave if the set of optimal actions is convex under every belief. Second, it shows that if a decision problem is quasi-concave, then it satisfies the local single crossing property after relabeling the states.
In a strand of the literature, it is assumed that the common prior has full support; that is, every type of every player is assigned positive probability. Morris (1991,1994) established an epistemological-behavioral duality characterisation of the common prior with full support, showing that a finite type space admits such a prior if and only if it contains no acceptable bet. This result forms the basis of the present paper. The paper makes three contributions: (1) The characterisation of Morris (1991,Morris1994) is extended to infinite type spaces. (2) The extension is robust: it does not depend on whether the infinite model applies countably additive or purely additive probabilities as beliefs. (3) The analysis implies that the notion of a real common prior-understood as a single probability distribution or a set of probability distributions-is not necessarily meaningful.
We set out to solve a dual puzzle regarding reproductive strategies: The "Ancient vs. Modern" Puzzle (why pre-modern elites adopted a "Survival" strategy while modern elites adopt an "Anxiety" strategy) and the "Class Divide" Puzzle (why modern involution manifests as a U-shaped fertility pattern). We develop a unified computational framework (DP + Monte Carlo) that introduces Cognitive Heterogeneity across classes. Our Hybrid Model (M-H) posits that the poor act as "Rational Survivors" (M1 utility, Reality parameters), while the middle/rich act as "Biased Strivers" (M4b utility, Belief parameters). Our simulations yield three core findings. First, we confirm that the "Survival" strategy is objectively rational whenever risk exceeds a low threshold ($σ> 0.45$). Given that real-world risk is massive ($σ_{Real} \approx 4.9$), the modern "Quality" strategy is objectively fragile. Second, the trap for the Middle/Rich ($B \ge 200$) is driven by a "Two-Stage Belief Error": they are first "baited" by a Causal Error (underestimating risk) to enter the status game, and then "trapped" by a Marginal Error (underestimating returns) which triggers a stop in fertility. Third, the U-shape is driven by the cognitive divide. The Poor escape the trap by retaining a "Rational Survival" strategy in the face of real high risk. Conversely, the Aspirational Middle Class ($HC \approx 12, B \ge 200$) is uniquely trapped by their Biased Beliefs. Their high competence raises their dynastic reference point ($R$) to a level where, under perceived low returns, restricting fertility to $N=1$ becomes the only rational choice within their biased belief system.
To generalize complementarities for games, we introduce some conditions weaker than quasisupermodularity and the single crossing property. We prove that the Nash equilibria of a game satisfying these conditions form a nonempty complete lattice. This is a purely order-theoretic generalization of Zhou's theorem.
We give two generalizations of the Zhou fixed point theorem. They weaken the subcompleteness condition of values, and relax the ascending condition of the correspondence. As an application, we derive a generalization of Topkis's theorem on the existence and order structure of the set of Nash equilibria of supermodular games.
We critique the formulation of Arrow's no-dictator condition to show that it does not correspond to the accepted informal/intuitive interpretation. This has implications for the theorem's scope of applicability.
We present a new proof for the existence of a Nash equilibrium, which involves no fixed point theorem. The self-contained proof consists of two parts. The first part introduces the notions of root function and pre-equilibrium. The second part shows the existence of pre-equilibria and Nash equilibria.
We study Nash implementation by stochastic mechanisms, and provide a surprisingly simple full characterization, which is in sharp contrast to the classical, albeit complicated, full characterization in Moore and Repullo (1990).
We use a combinatorial approach to compute the number of non-isomorphic choices on four elements that can be explained by models of bounded rationality.
These lecture notes accompany a one-semester graduate course on information and learning in economic theory. Topics include common knowledge, Bayesian updating, monotone-likelihood ratio properties, affiliation, the Blackwell order, cost of information, learning and merging of beliefs, model uncertainty, model misspecification, and information design.
Two type structures are hierarchy-equivalent if they induce the same set of hierarchies of beliefs. This note shows that the behavioral implications of "cautious rationality and common cautious belief in cautious rationality" (Catonini and De Vito 2021) do not vary across hierarchy-equivalent type structures.
We consider an observational learning model with exogenous public payoff shock. We show that confounded learning doesn't arise for almost all private signals and almost all shocks, even if players have sufficiently divergent preferences.
We present mathematical definitions for rights structures, government and non-attenuation in a generalized N-person game (Debreu abstract economy), thus providing a formal basis for property rights theory.