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When bidders bid on complex objects, they might be unaware of characteristics effecting their valuations. We assume that each buyer's valuation is a sum of independent random variables, one for each characteristic. When a bidder is unaware of a characteristic, he omits the random variable from the sum. We study the seller's decision to raise bidders' awareness of characteristics before a second-price auction with entry fees. Optimal entry fees capture an additional unawareness rent due to unaware bidders misperceiving their probability of winning and the price to be paid upon winning. When raising a bidder's individual awareness of a characteristic with positive expected value, the seller faces a trade-off between positive effects on the expected first order statistic and unawareness rents of remaining unaware bidders on one hand and the loss of the unawareness rent from the newly aware bidder on the other. We present characterization results on raising public awareness together with no versus full information. We discuss the winner's curse due to unawareness of characteristics., Comment: 29 pages

Nambu-determinant brackets on $R^d\ni x=(x^1,...,x^d)$, $\{f,g\}_d(x)=\rho(x) \det(\partial(f,g,a_1,...,a_{d-2})/\partial(x^1,...,x^d))$, with $a_i\in C^\infty(R^d)$ and $\rho\partial_x\in\mathfrak{X}^d(R^d)$, are a class of Poisson structures with (non)linear coefficients, e.g., polynomials of arbitrarily high degree. With good cocycles in the graph complex, Kontsevich associated universal -- for all Poisson bivectors $P$ on affine $R^d_{aff}$ -- elements $\dot{P}=Q^\gamma(P)\in H^2_{P}(R^d_{aff})$ in the Lichnerowicz-Poisson second cohomology groups; we note that known graph cocycles $\gamma$ preserve the Nambu-Poisson class $\{P(\rho,a)\}$, and we express, directly from $\gamma$, the evolution $\dot{\rho}$,$\dot{a}$ that induces $\dot{P}$. Over all $d\geq2$ at once, there is no universal mechanism for the bivector cocycles $Q^\gamma_d$ to be trivial, $Q^\gamma_d=[\![P,\vec{X}^\gamma_d(P)]\!]$, w.r.t. vector fields defined uniformly for all dimensions $d$ by the same graph formula. While over $R^2$, the graph flows $\dot{P} = Q^{\gamma_i}_{2D}(P(\rho))$ for $\gamma\in\{\gamma_3,\gamma_5,\gamma_7,...\}$ are trivialized by vector fields $\vec{X}^{\gamma_i}_{2D}=(dx\wedge dy)^{-1}d_{dR}(Ham^{\gamma_i}(P))$ of peculiar shape, we detect that in $d\geq3$, the 1-vectors from 2D, now with $P(\rho,a_1,...,a_{d-2})$ inside, do not solve the problems $Q^{\gamma_i}_{d\geq3}=[\![P,{\vec{X}^{\gamma_i}_{d\geq3}}(P(\rho,a))]\!]$, yet they do yield good Ansatz where we find solutions $\vec{X}^{\gamma_i}_{d=3,4}(P(\rho,a))$. In the study of the step $d\mapsto d+1$, by adapting the Kontsevich graph calculus to the Nambu-Poisson class of brackets, we discover more identities for the Jacobian determinants within $P(\rho,a)$, i.e. for multivector-valued $GL(d)$-invariants on $R^d_{aff}$., Comment: Plenary talk given by AVK at the international conference on Integrable Systems and Quantum Symmetries (ISQS28) held on 1-5 July 2024 at CVUT Prague, Czech Republic; 13 pages, 1 table; followed by papers II. (arXiv:2409.12555 [math.QA]) and III. (arXiv:2409.15932 [math.QA]) by the same authors

Kontsevich constructed a map between `good' graph cocycles $\gamma$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz--Poisson cohomology. For the tetrahedral graph cocycle $\gamma_3$ and for the class of Nambu-determinant Poisson bivectors $P$ over $\mathbb{R}^2$, $\mathbb{R}^3$ and $\mathbb{R}^4$, we know the fact of trivialization, $\dot{P}=[[ P, \vec{X}^{\gamma_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ are unique modulo Hamiltonian vector fields $\vec{X}_{H}=d_P(H)= [[ P, H]]$, where $d_P$ is the Lichnerowicz--Poisson differential and where the Hamiltonians $H$ are also represented by Kontsevich (micro-)graphs. However, we find that the choice of Kontsevich (micro-)graphs to represent the aforementioned multivectors is not unique.

Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu--Poisson brackets on $\mathbb{R}^d$. Within Kontsevich's graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu--Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for $d = 2$ (1996). In 2020, Buring and the third author found that the deformation is trivial for $d = 3$. Building on these discoveries, we now establish that the deformation is trivial for $d = 4$.

We consider a decision maker who is unaware of objects to be sampled and thus cannot form beliefs about the occurrence of particular objects. Ex ante she can form beliefs about the occurrence of novelty and the frequencies of yet to be known objects. Conditional on any sampled objects, she can also form beliefs about the next object being novel or being one of the previously sampled objects. We characterize behaviorally such beliefs under subjective expected utility. In doing so, we relate "reverse" Bayesianism, a central property in the literature on decision making under growing awareness, with exchangeable random partitions, the central property in the literature on the discovery of species problem and mutations in statistics, combinatorial probability theory, and population genetics. Partition exchangeable beliefs do not necessarily satisfy "reverse" Bayesianism. Yet, the most prominent models of exchangeable random partitions, the model by De Morgan (1838), the one parameter model of Ewens (1972), and the two parameter model of Pitman (1995) and Zabell (1997), do satisfy "reverse" Bayesianism., Comment: 34 pages, 4 figures

Level-k thinking has been widely applied as a solution concept for games in normal form in behavioral and experimental game theory. We consider level-k thinking in games in extensive form. Player's may learn about levels of opponents' thinking during the play of the game because some information sets may be inconsistent with certain levels. In particular, for any information set reached, a level-k player attaches the maximum level-l thinking for l < k to her opponents consistent with the information set. We compare our notion of strong level-k thinking with other solution concepts such as level-k thinking in the associated normal form, strong rationalizability, Delta-rationalizability, iterated admissibility, backward rationalizability, backward level-k thinking, and backward induction. We use strong level-k thinking to reanalyze data from some prior experiments in the literature., Comment: 51 pages, 10 figures

Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic awareness correspondence and accessibility relations modeling implicit knowledge. They are a flexible model of unawareness, and best interpreted from a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and Schipper (2006, 2008) (HMS) model awareness by a lattice of state spaces and explicit knowledge via possibility correspondences. Sublattices thereof can be interpreted as subjective views of agents. Open questions include (1) how implicit knowledge can be defined in HMS structures, and (2) in which way FH structures can be extended to model the agents' subjective views. In this paper, we address (1) by defining implicit knowledge such that it is consistent with explicit knowledge in HMS models. We also introduce a variant of HMS models that instead of explicit knowledge, takes implicit knowledge and awareness as primitives. Further, we address (2) by introducing a category of FH models that are modally equivalent relative to sublanguages and can be interpreted as agents' subjective views depending on their awareness. These constructions allow us to show an equivalence between HMS and FH models. As a corollary, we obtain soundness and completeness of HMS models with respect to the Logic of Propositional Awareness, based on a language featuring both implicit and explicit knowledge., Comment: 46 pages. arXiv admin note: substantial text overlap with arXiv:2307.05041 author note: This is the full version of the arXiv:2307.05041