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A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graphGn,1/2typically admits a covering of a constant fraction of its edges by edge‐disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: forandA1,…,Atchosen uniformly and independently from thek‐subsets of {1,…,n}, what can one say aboutOur main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.
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Optimal stopping under model ambiguity: A time‐consistent equilibrium approach
Abstract An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account howambiguity‐aversean agent is. This inclusion of ambiguity attitude, via an‐maxmin nonlinear expectation, renders the stopping problem time‐inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one‐dimensional diffusion with drift and volatility uncertainty, we show that any initial stopping policy will converge to an equilibrium through a fixed‐point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst‐case (or best‐case) analysis. In a concrete example of real options valuation under model ambiguity, all equilibrium stopping policies, as well as thebestone among them, are fully characterized under appropriate conditions. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity‐averse, the more eager to stop—so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.
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- Award ID(s):
- 1715439
- PAR ID:
- 10450864
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Mathematical Finance
- Volume:
- 31
- Issue:
- 3
- ISSN:
- 0960-1627
- Format(s):
- Medium: X Size: p. 979-1012
- Size(s):
- p. 979-1012
- Sponsoring Org:
- National Science Foundation
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