Incomplete Contracts with Asymmetric Information: Exclusive v. Optional Remedies

Law and economics scholars have always had a strong interest in contract remedies. Perhaps the most explored issue in contract law has been the desirability of various contract remedies, such as expectation damages, specific performance, or liquidated damages, to name the most common. Scholars have been debating for years, from various perspectives, the comparative advantage of these remedies. Yet, most scholars have assumed that each of these remedies is exclusive, and their work has compared a single remedy contract to another single remedy contract. Interestingly, an analysis that assumes these remedies are optional (or cumulative) has not yet been explored, in spite of the fact that contract law provides the non-breaching party with a variety of optional remedies to choose from in case of a breach, and in spite of the fact that parties themselves write contracts which provide such an option. In this paper we attempt to start filling in this gap by studying the relationship between these remedies. Specifically, we study the conditions at which a contract that grants the non-breaching party an option to choose from optional remedies is superior to an exclusive remedy contract. We show that under conditions of double-sided uncertainty and asymmetric information between a seller (who might breach) and a buyer (who never breaches) the interaction of the parties\u27 distributions should determine whether a contract provides for exclusive or optional remedies. Specifically, if the buyer\u27s conditional expected valuation is larger than the seller\u27s conditional expected valuation (in both cases - conditional that their expected valuation is above the buyer\u27s mean valuation), then a contract which provides the buyer an option to choose between liquidated damages or specific performance (or actual damages) is superior. Our analysis in this paper informs transactional lawyers of the relevant economic factors they should consider when deciding the optimal composition of remedies in a given context. Moreover, our analysis is relevant for courts that interpret contracts because it will help them to better understand whether rational parties would have agreed that a particular remedy would be an exclusive remedy or an optional remedy when the language of the contract is ambiguous. Lastly, our analysis provides yet another economic rationale for why courts should enforce parties\u27 liquidated damages clauses even if it seems ex-post over, or under, compensatory. We present a model which shows when parties will agree on a non-exclusive liquidated damages clause. Under such a contract the parties stipulate ex-ante that the buyer will have the option to choose upon breach whether she prefers an optional remedy, such as actual damages or specific performance, to the pre-determined liquidated damages. We focus on the ex-ante design of the contract in light of the new information that the parties anticipate they will gain after they draft the contract. Therefore, we assume that no renegotiation or investments are involved. We demonstrate the optimal way to design contract clauses which takes advantage of the information that the seller and the buyer receive between the time they enter into the contract and the time of the actual breach. We further suggest that parties indeed use such clauses and that courts honor them. After laying out the basic model we provide some extensions to it. As is well known, an exclusive liquidated damages contract is equivalent to granting the seller a call option to breach and pay, where the exercise price is equal to the amount of the agreed liquidated damages. What is perhaps less known is that a non-exclusive, or optional, contract, where the buyer can choose performance, is equivalent to giving the buyer a consecutive call option with the same exercise price. Yet, the consecutive call option to the buyer does not have to have the same exercise price but can rather have a higher one. We call this new contract a two-price contract and show that it is even more efficient than the basic contract we have explored before. Next, we introduce more rounds of sequential options and show that while the regular ex-ante contract can achieve on average about 4 Indeed, in an environment of asymmetric information renegotiation costs are high. More on this below. 90% of the first-best allocative efficiency, an n-rounds contract approaches the first best, as n goes to infinity. We show numerically that within just 4 rounds, 96% of the allocative efficiency can be achieved. Section two describes the legal background against which we have designed our model. Section three surveys the literature that evaluates contract remedies from an economic perspective. Section four presents a simple model with two-sided incomplete information and with a liquidated damages clause. In section four we compare the performance of a regime with optional remedies with a regime of exclusive remedy and then determine the conditions at which each regime should be applied. Section five discusses some interesting extensions meant to approach the first-best allocative efficiency. The appendix provides a more rigorous mathematical demonstration of the model

Measurements of the branching fractions for B→K∗γB \to K^{*}\gamma decays at Belle II

This paper reports a study of $B \to K^{*}\gamma$ decays using $62.8\pm 0.6$ fb$^{-1}$ of data collected during 2019--2020 by the Belle II experiment at the SuperKEKB $e^{+}e^{-}$ asymmetric-energy collider, corresponding to $(68.2 \pm 0.8) \times 10^6$ $B\overline{B}$ events. We find $454 \pm 28$, $50 \pm 10$, $169 \pm 18$, and $160 \pm 17$ signal events in the decay modes $B^{0} \to K^{*0}[K^{+}\pi^{-}]\gamma$, $B^{0} \to K^{*0}[K^0_{\rm S}\pi^{0}]\gamma$, $B^{+} \to K^{*+}[K^{+}\pi^{0}]\gamma$, and $B^{+} \to K^{*+}[K^{+}\pi^{0}]\gamma$, respectively. The uncertainties quoted for the signal yield are statistical only. We report the branching fractions of these decays: $$\mathcal{B} [B^{0} \to K^{*0}[K^{+}\pi^{-}]\gamma] = (4.5 \pm 0.3 \pm 0.2) \times 10^{-5},$$ $$\mathcal{B} [B^{0} \to K^{*0}[K^0_{\rm S}\pi^{0}]\gamma] = (4.4 \pm 0.9 \pm 0.6) \times 10^{-5},$$ $$\mathcal{B} [B^{+} \to K^{*+}[K^{+}\pi^{0}]\gamma] = (5.0 \pm 0.5 \pm 0.4)\times 10^{-5},\text{ and}$$ $$\mathcal{B} [B^{+} \to K^{*+}[K^0_{\rm S}\pi^{+}]\gamma] = (5.4 \pm 0.6 \pm 0.4) \times 10^{-5},$$ where the first uncertainty is statistical, and the second is systematic. The results are consistent with world-average values

Measurement of the integrated luminosity of the Phase 2 data of the Belle II experiment

From April to July 2018, a data sample at the peak energy of the γ(4S) resonance was collected with the Belle II detector at the SuperKEKB electron-positron collider. This is the first data sample of the Belle II experiment. Using Bhabha and digamma events, we measure the integrated luminosity of the data sample to be (496.3 ± 0.3 ± 3.0) pb-1, where the first uncertainty is statistical and the second is systematic. This work provides a basis for future luminosity measurements at Belle II

Measurement of the branching fraction for the decay B→K∗(892)ℓ+ℓ−B \to K^{\ast}(892)\ell^+\ell^- at Belle II

We report a measurement of the branching fraction of $B \to K^{\ast}(892)\ell^+\ell^-$ decays, where $\ell^+\ell^- = \mu^+\mu^-$ or $e^+e^-$, using electron-positron collisions recorded at an energy at or near the $\Upsilon(4S)$ mass and corresponding to an integrated luminosity of $189$ fb$^{-1}$. The data was collected during 2019--2021 by the Belle II experiment at the SuperKEKB $e^{+}e^{-}$ asymmetric-energy collider. We reconstruct $K^{\ast}(892)$ candidates in the $K^+\pi^-$, $K_{S}^{0}\pi^+$, and $K^+\pi^0$ final states. The signal yields with statistical uncertainties are $22\pm 6$, $18 \pm 6$, and $38 \pm 9$ for the decays $B \to K^{\ast}(892)\mu^+\mu^-$, $B \to K^{\ast}(892)e^+e^-$, and $B \to K^{\ast}(892)\ell^+\ell^-$, respectively. We measure the branching fractions of these decays for the entire range of the dilepton mass, excluding the very low mass region to suppress the $B \to K^{\ast}(892)\gamma(\to e^+e^-)$ background and regions compatible with decays of charmonium resonances, to be \begin{equation} {\cal B}(B \to K^{\ast}(892)\mu^+\mu^-) = (1.19 \pm 0.31 ^{+0.08}_{-0.07}) \times 10^{-6}, {\cal B}(B \to K^{\ast}(892)e^+e^-) = (1.42 \pm 0.48 \pm 0.09)\times 10^{-6}, {\cal B}(B \to K^{\ast}(892)\ell^+\ell^-) = (1.25 \pm 0.30 ^{+0.08}_{-0.07}) \times 10^{-6}, \end{equation} where the first and second uncertainties are statistical and systematic, respectively. These results, limited by sample size, are the first measurements of $B \to K^{\ast}(892)\ell^+\ell^-$ branching fractions from the Belle II experiment

Advances in quantum metrology

The statistical error in any estimation can be reduced by repeating the measurement and averaging the results. The central limit theorem implies that the reduction is proportional to the square root of the number of repetitions. Quantum metrology is the use of quantum techniques such as entanglement to yield higher statistical precision than purely classical approaches. In this Review, we analyse some of the most promising recent developments of this research field and point out some of the new experiments. We then look at one of the major new trends of the field: analyses of the effects of noise and experimental imperfections

Determination of ∣Vub∣|V_{ub}| from untagged B0→π−ℓ+νℓB^0\to\pi^- \ell^+ \nu_{\ell} decays using 2019-2021 Belle II data

We present an analysis of the charmless semileptonic decay $B^0\to\pi^- \ell^+ \nu_{\ell}$, where $\ell = e, \mu$, from 198.0 million pairs of $B\bar{B}$ mesons recorded by the Belle II detector at the SuperKEKB electron-positron collider. The decay is reconstructed without identifying the partner $B$ meson. The partial branching fractions are measured independently for $B^0\to\pi^- e^+ \nu_{e}$ and $B^0\to\pi^- \mu^+ \nu_{\mu}$ as functions of $q^{2}$ (momentum transfer squared), using 3896 $B^0\to\pi^- e^+ \nu_{e}$ and 5466 $B^0\to\pi^- \mu^+ \nu_{\mu}$ decays. The total branching fraction is found to be $(1.426 \pm 0.056 \pm 0.125) \times 10^{-4}$ for $B^0\to\pi^- \ell^+ \nu_{\ell}$ decays, where the uncertainties are statistical and systematic, respectively. By fitting the measured partial branching fractions as functions of $q^{2}$, together with constraints on the nonperturbative hadronic contribution from lattice QCD calculations, the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ub}$, $(3.55 \pm 0.12 \pm 0.13 \pm 0.17) \times 10^{-3}$, is extracted. Here, the first uncertainty is statistical, the second is systematic and the third is theoretical

Angular analysis of B+→ρ+ρ0B^+ \to \rho^+\rho^0 decays reconstructed in 2019, 2020, and 2021 Belle II data

We report on a Belle II measurement of the branching fraction ($\mathcal{B}$), longitudinal polarization fraction ($f_L$), and CP asymmetry ($\mathcal{A}_{CP}$) of $B^+\to \rho^+\rho^0$ decays. We reconstruct $B^+\to \rho^+(\to \pi^+\pi^0(\to \gamma\gamma))\rho^0(\to \pi^+\pi^-)$ decays in a sample of SuperKEKB electron-positron collisions collected by the Belle II experiment in 2019, 2020, and 2021 at the $\Upsilon$(4S) resonance and corresponding to 190 fb$^{-1}$ of integrated luminosity. We fit the distributions of the difference between expected and observed $B$ candidate energy, continuum-suppression discriminant, dipion masses, and decay angles of the selected samples, to determine a signal yield of $345 \pm 31$ events. The signal yields are corrected for efficiencies determined from simulation and control data samples to obtain $\mathcal{B}(B^+ \to \rho^+\rho^0) = [23.2^{+\ 2.2}_{-\ 2.1} (\rm stat) \pm 2.7 (\rm syst)]\times 10^{-6}$,$f_L = 0.943 ^{+\ 0.035}_{-\ 0.033} (\rm stat)\pm 0.027(\rm syst)$, and$\mathcal{A}_{CP}=-0.069 \pm 0.068(\rm stat) \pm 0.060 (\rm syst)$. The results agree with previous measurements. This is the first measurement of$\mathcal{A}_{CP}$in$B^+\to \rho^+\rho^0$ decays reported by Belle II

Measurement of the branching fractions and CPCP asymmetries of B+→π+π0B^+ \rightarrow \pi^+ \pi^0 and B+→K+π0B^+ \rightarrow K^+ \pi^0 decays in 2019-2021 Belle II data

We determine the branching fractions ${\mathcal{B}}$ and $CP$ asymmetries ${\mathcal{A}_{{\it CP}}}$ of the decays $B^+ \rightarrow \pi^+ \pi^0$ and $B^+ \rightarrow K^+ \pi^0$. The results are based on a data set containing 198 million bottom-antibottom meson pairs corresponding to an integrated luminosity of $190\;\text{fb}^{-1}$ recorded by the Belle II detector in energy-asymmetric electron-positron collisions at the $\Upsilon (4S)$ resonance. We measure ${\mathcal{B}(B^+ \rightarrow \pi^+ \pi^0) = (6.12 \pm 0.53 \pm 0.53)\times 10^{-6}}$, ${\mathcal{B}(B^+ \rightarrow K^+ \pi^0) = (14.30 \pm 0.69 \pm 0.79)\times 10^{-6}}$, ${\mathcal{A}_{{\it CP}}(B^+ \rightarrow \pi^+ \pi^0) = -0.085 \pm 0.085 \pm 0.019}$, and ${\mathcal{A}_{{\it CP}}(B^+ \rightarrow K^+ \pi^0) = 0.014 \pm 0.047 \pm 0.010}$, where the first uncertainties are statistical and the second are systematic. These results improve a previous Belle II measurement and agree with the world averages