Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Advanced Fabrication Method for the Preparation of MOF Thin Films: Liquid-Phase Epitaxy Approach Meets Spin Coating Method

Here, we report a new and advanced method for the fabrication of highly oriented/polycrystalline metal–organic framework (MOF) thin films. Building on the attractive features of the liquid-phase epitaxy (LPE) approach, a facile spin coating method was implemented to generate MOF thin films in a high-throughput fashion. Advantageously, this approach offers a great prospective to cost-effectively construct thin-films with a significantly shortened preparation time and a lessened chemicals and solvents consumption, as compared to the conventional LPE-process. Certainly, this new spin-coating approach has been implemented successfully to construct various MOF thin films, ranging in thickness from a few micrometers down to the nanometer scale, spanning 2-D and 3-D benchmark MOF materials including Cu₂(bdc)₂·<i>x</i>H₂O, Zn₂(bdc)₂·<i>x</i>H₂O, HKUST-1, and ZIF-8. This method was appraised and proved effective on a variety of substrates comprising functionalized gold, silicon, glass, porous stainless steel, and aluminum oxide. The facile, high-throughput and cost-effective nature of this approach, coupled with the successful thin film growth and substrate versatility, represents the next generation of methods for MOF thin film fabrication. Therefore, paving the way for these unique MOF materials to address a wide range of challenges in the areas of sensing devices and membrane technology

Supramolecular Isomers of Metal–Organic Frameworks Derived from a Partially Flexible Ligand with Distinct Binding Motifs

Three novel metal–organic frameworks (MOFs) were isolated upon reacting heterofunctional ligand 4-(pyrimidin-5-yl)­benzoic acid (4,5-pmbc) with mixed valence Cu­(I,II) under solvothermal conditions. X-ray crystal structural analysis reveals that the first compound is a layered structure composed of one type of inorganic building block, dinuclear paddlewheel [Cu₂(O₂C−)₄], which is linked through 4,5-pmbc ligands. The two other supramolecular isomers are composed of the same Cu­(II) dinuclear paddlewheel and a dinuclear Cu₂I₂ cluster, which are linked via the 4,5-pmbc linkers to yield two different 3-periodic frameworks with underlying topologies related to <b>lvt</b> and <b>nbo</b>. The observed structural diversity in these structures is due to the distinct coordination modes of the two coordinating moieties (the carboxylate group on the phenyl ring and the N-donor atoms from the pyrimidine moiety)

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

A Family of Porous Lonsdaleite‑e Networks Obtained through Pillaring of Decorated Kagomé Lattice Sheets

A new and versatile class of metal–organic materials (MOMs) with augmented lonsdaleite-e (<b>lon-e</b>-a) topology is presented herein. This family of <b>lon-e</b> nets are built by pillaring of hexagonal two-dimensional kagomé (<b>kag</b>) lattices constructed from well-known [Zn₂(CO₂R)₄] paddlewheel molecular building blocks (MBBs) connected by 1,3-benzenedicarboxylate (bdc^2–) linkers. The pillars are [Cr₃(μ₃-O)­(RCO₂)]₆ trigonal prismatic primary MBBs decorated by six pyridyl moieties (tp-PMBB-1). The three-fold symmetry (<i>D</i>_3<i>h</i>) of tp-PMBB-1 is complementary with the alternating orientation of the axial sites of the paddlewheel MBBs and enables triple cross-linking of the <b>kag</b> layers by each pillar. These MOMs represent the first examples of axial-to-axial pillared undulating <b>kag</b> layers, and they are readily fine-tuned because the bdc^2– moieties can be varied at their 5-position without changing the overall structure. This <b>lon-e</b> platform possesses functionalized hexagonal channels since the <b>kag</b> lattices are necessarily eclipsed. The effects of the substituent at the 5-positions of the bdc^2– linkers upon gas adsorption, particularly the heats of adsorption of carbon dioxide and methane, were studied

A Family of Porous Lonsdaleite‑e Networks Obtained through Pillaring of Decorated Kagomé Lattice Sheets

A new and versatile class of metal–organic materials (MOMs) with augmented lonsdaleite-e (<b>lon-e</b>-a) topology is presented herein. This family of <b>lon-e</b> nets are built by pillaring of hexagonal two-dimensional kagomé (<b>kag</b>) lattices constructed from well-known [Zn₂(CO₂R)₄] paddlewheel molecular building blocks (MBBs) connected by 1,3-benzenedicarboxylate (bdc^2–) linkers. The pillars are [Cr₃(μ₃-O)­(RCO₂)]₆ trigonal prismatic primary MBBs decorated by six pyridyl moieties (tp-PMBB-1). The three-fold symmetry (<i>D</i>_3<i>h</i>) of tp-PMBB-1 is complementary with the alternating orientation of the axial sites of the paddlewheel MBBs and enables triple cross-linking of the <b>kag</b> layers by each pillar. These MOMs represent the first examples of axial-to-axial pillared undulating <b>kag</b> layers, and they are readily fine-tuned because the bdc^2– moieties can be varied at their 5-position without changing the overall structure. This <b>lon-e</b> platform possesses functionalized hexagonal channels since the <b>kag</b> lattices are necessarily eclipsed. The effects of the substituent at the 5-positions of the bdc^2– linkers upon gas adsorption, particularly the heats of adsorption of carbon dioxide and methane, were studied

A Family of Porous Lonsdaleite‑e Networks Obtained through Pillaring of Decorated Kagomé Lattice Sheets

A new and versatile class of metal–organic materials (MOMs) with augmented lonsdaleite-e (<b>lon-e</b>-a) topology is presented herein. This family of <b>lon-e</b> nets are built by pillaring of hexagonal two-dimensional kagomé (<b>kag</b>) lattices constructed from well-known [Zn₂(CO₂R)₄] paddlewheel molecular building blocks (MBBs) connected by 1,3-benzenedicarboxylate (bdc^2–) linkers. The pillars are [Cr₃(μ₃-O)­(RCO₂)]₆ trigonal prismatic primary MBBs decorated by six pyridyl moieties (tp-PMBB-1). The three-fold symmetry (<i>D</i>_3<i>h</i>) of tp-PMBB-1 is complementary with the alternating orientation of the axial sites of the paddlewheel MBBs and enables triple cross-linking of the <b>kag</b> layers by each pillar. These MOMs represent the first examples of axial-to-axial pillared undulating <b>kag</b> layers, and they are readily fine-tuned because the bdc^2– moieties can be varied at their 5-position without changing the overall structure. This <b>lon-e</b> platform possesses functionalized hexagonal channels since the <b>kag</b> lattices are necessarily eclipsed. The effects of the substituent at the 5-positions of the bdc^2– linkers upon gas adsorption, particularly the heats of adsorption of carbon dioxide and methane, were studied

A Family of Porous Lonsdaleite‑e Networks Obtained through Pillaring of Decorated Kagomé Lattice Sheets

A new and versatile class of metal–organic materials (MOMs) with augmented lonsdaleite-e (<b>lon-e</b>-a) topology is presented herein. This family of <b>lon-e</b> nets are built by pillaring of hexagonal two-dimensional kagomé (<b>kag</b>) lattices constructed from well-known [Zn₂(CO₂R)₄] paddlewheel molecular building blocks (MBBs) connected by 1,3-benzenedicarboxylate (bdc^2–) linkers. The pillars are [Cr₃(μ₃-O)­(RCO₂)]₆ trigonal prismatic primary MBBs decorated by six pyridyl moieties (tp-PMBB-1). The three-fold symmetry (<i>D</i>_3<i>h</i>) of tp-PMBB-1 is complementary with the alternating orientation of the axial sites of the paddlewheel MBBs and enables triple cross-linking of the <b>kag</b> layers by each pillar. These MOMs represent the first examples of axial-to-axial pillared undulating <b>kag</b> layers, and they are readily fine-tuned because the bdc^2– moieties can be varied at their 5-position without changing the overall structure. This <b>lon-e</b> platform possesses functionalized hexagonal channels since the <b>kag</b> lattices are necessarily eclipsed. The effects of the substituent at the 5-positions of the bdc^2– linkers upon gas adsorption, particularly the heats of adsorption of carbon dioxide and methane, were studied