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Isaac M. Sonin

Publications and source records attributed to Isaac M. Sonin.

8 recordsLinked to original sources
arXiv2026

The Test and Find Model

We introduce the Test and Find (TF) problem, where a decision maker (DM) faces the following situation: $k$ identical objects are randomly allocated to $n$ distinct boxes (sites) according to some distribution $π$, with no more than one object to a box. The DM tests all of the boxes. However, the tests are imperfect: they can give false positive or false negative results. DM has $m$ tags, $1\leq m\leq n$, and, after testing all boxes, she can place a tag on any box that she thinks has a hidden object. She is rewarded $c_i$ for a correct guess and penalized $d_i$ for a wrong guess in box $i$. DM knows all of the parameters of the model and her goal is to maximize the expected reward. We give an explicit solution to this problem. We then turn to the symmetric case, for which we derive more computationally efficient results. We also consider several extensions of the TF model and give detailed solutions. One of these extensions is to the realistic case where $k$, the number of objects, is unknown and random.

math.PR↗
arXiv2026

The Game of Marginal Utilities

We study a noncooperative resource-allocation game in which $m$ players distribute fixed resources among $n$ projects and the payoff of player $j$ is given by \[ F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}, \] where $a_i$ and $b_i$ are project parameters, while $x_i^j$ is the amount of resources player $j$ allocates to project $i$. This specification combines diminishing returns with congestion generated by competitors. We prove that the game has a unique Nash equilibrium and characterize it by an equimarginal principle. We show that, after the projects are ordered by $a_i/b_i$, each player invests in an initial segment of projects, and these segments are nested across players, so the equilibrium decomposes into consecutive activity zones; players with larger resources invest weakly more in every project. In the fully active regime, where every player invests in every project, we reduce the equilibrium to a single scalar nonlinear equation for the aggregate marginal-utility rate; all individual marginal rates, investments, and payoffs then follow from explicit formulas. We also provide a projected marginal-utility algorithm with global linear convergence under an explicit step-size condition, together with a structure-exploiting \textit{Block Pandora} algorithm that reconstructs and certifies the equilibrium conditional on a proposed nested cutoff structure.

cs.GT↗
arXiv2023

Symmetric Case of Locks, Bombs and Testing Model

We present a Defense/Attack resource allocation model, where Defender has some number of ``locks" to protect $n$ vulnerable boxes (sites), and Attacker is trying to destroy these boxes, having $m$ ``bombs" that can be placed into the boxes. Similar models were studied in game theory - (Colonel) Blotto games, but our model has a feature absent in the previous literature. Attacker tests the vulnerability of all sites before allocating her resources, and these tests are not perfect, i.e., a test can be positive for a box without a lock and negative for a box with a lock. We describe the optimal strategies for a special case of a general Locks-Bombs-Testing (LBT) model when all boxes are identical and the Defender has a fixed number of locks.

cs.GT↗
arXiv2020

Banks as Tanks: A Continuous-Time Model of Financial Clearing

We present a simple continuous-time model of clearing in financial networks. Financial firms are represented as "tanks" filled with fluid (money), flowing in and out. Once "pipes" connecting "tanks" are open, the system reaches the clearing payment vector in finite time. This approach provides a simple recursive solution to a classical static model of financial clearing in bankruptcy, and suggests a practical payment mechanism. With sufficient resources, a system of mutual obligations can be restructured into an equivalent system that has a cascade structure: there is a group of banks that paid off their debts, another group that owes money only to banks in the first group, and so on. Technically, we use the machinery of Markov chains to analyze evolution of a deterministic dynamical system.

econ.GN↗
arXiv2019

Bayesian Game of Locks, Bombs and Testing

We consider the game with discrete units of resources for protection and destruction of some sites. In our model, Defender (DF) has locks and Attacker (AT) has bombs to allocate among sites, trying to destroy these sites. One or more bombs can be placed into the same site. A site is destroyed if at least one explosion occurs. A lock is a protection device which, placed in a site, prevents its destruction with any number of bombs in it. The important feature of the model is that AT can and will test every site, trying to find sites without locks. This testing is not perfect: a test of a site may have a positive result, even if there is no lock at the site, or a negative result, even if there is a lock. The probabilities of correct identification of both types (sensitivity and specificity) are known to both players. In the Bayesian setting we assume that the probability distribution of the locks is known to AT, although the actual positions of the locks are not. After the locks are allocated, AT receives the test results, and distributes the bombs among the sites, trying to maximize the expected sum of values of all destroyed sites. The goal of DF is to select a prior distribution of locks to minimize this loss. Some special cases of this game are completely solved, and some partial results are obtained for the general model.

math.PR↗
arXiv2018

Some Nontrivial Properties of a Formula for Compound Interest

We analyze the classical model of compound interest with a constant per-period payment and interest rate. We examine the outstanding balance function as well as the periodic payment function and show that the outstanding balance function is not generally concave in the interest rate, but instead may be initially convex on its domain and then concave.

econ.GN↗
arXiv2012

Independent Events in a Simple Random Experiment and the Meaning of Independence

We count the number and patterns of pairs and tuples of independent events in a simple random experiment: first a fair coin is flipped and then a fair die is tossed. The first number, equal to 888,888, suggest that there are some open questions about the structure of independence even in a finite sample space. We discuss briefly these questions and possible approaches to answer them.

math.PR↗