An official website of the United States government
Abstract Many-body dynamical models in which Boltzmann statistics can be derived directly from the underlying dynamical laws without invoking the fundamental postulates of statistical mechanics are scarce. Interestingly, one such model is found in econophysics and in chemistry classrooms: the money game, in which players exchange money randomly in a process that resembles elastic intermolecular collisions in a gas, giving rise to the Boltzmann distribution of money owned by each player. Although this model offers a pedagogical example that demonstrates the origins of Boltzmann statistics, such demonstrations usually rely on computer simulations. In fact, a proof of the exponential steady-state distribution in this model has only become available in recent years. Here, we study this random money/energy exchange model and its extensions using a simple mean-field-type approach that examines the properties of the one-dimensional random walk performed by one of its participants. We give a simple derivation of the Boltzmann steady-state distribution in this model. Breaking the time-reversal symmetry of the game by modifying its rules results in non-Boltzmann steady-state statistics. In particular, introducing ‘unfair’ exchange rules in which a poorer player is more likely to give money to a richer player than to receive money from that richer player, results in an analytically provable Pareto-type power-law distribution of the money in the limit where the number of players is infinite, with a finite fraction of players in the ‘ground state’ (i.e. with zero money). For a finite number of players, however, the game may give rise to a bimodal distribution of money and to bistable dynamics, in which a participant’s wealth jumps between poor and rich states. The latter corresponds to a scenario where the player accumulates nearly all the available money in the game. The time evolution of a player’s wealth in this case can be thought of as a ‘chemical reaction’, where a transition between ‘reactants’ (rich state) and ‘products’ (poor state) involves crossing a large free energy barrier. We thus analyze the trajectories generated from the game using ideas from the theory of transition paths and highlight non-Markovian effects in the barrier crossing dynamics.
more »
« less
This content will become publicly available on March 1, 2026
Hallmarks of deception in asset-exchange models
We investigate the transient and steady-state dynamics of the Bennati-Dragulescu-Yakovenko money game in the presence of probabilistic cheaters, who can misrepresent their financial status by claiming to have no money. We derive the steady-state wealth distribution per player analytically, and we show how the presence of hidden cheaters can be inferred from the relative variance of wealth per player. In scenarios with a finite number of cheaters amidst an infinite pool of honest players, we identify a critical probability of cheating at which the total wealth owned by the cheaters experiences a second-order discontinuity. Below this point, the transition probability to lose money is larger than the probability to gain; conversely, above this point, the direction is reversed. We further establish a threshold cheating probability at which cheaters collectively possess half of the total wealth in the game. Lastly, we provide bounds on the rate at which both cheaters and honest players can gain or lose wealth, contributing to a deeper understanding of deception in asset-exchange models.
more »
« less
- Award ID(s):
- 2400424
- PAR ID:
- 10623749
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review Research
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2643-1564
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”more » « less
-
Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than 1/2, and that our objective is to determine this best player. Under the requirement that the policy employed guarantees a correct choice with a probability of at least some specified value, we look for a policy that has a relatively small expected number of games played before decision. We consider this problem both under the assumption that the best player wins each game with a probability of at least some specified value >1/2, and under a Bayesian assumption that the probability that player i wins a game against player j is its value divided by the sum of the values, where the values are the unknown values of n independent and identically distributed exponential random variables. In the former case, we propose a policy where chosen pairs play a match that ends when one of them has had a specified number of wins more than the other; in the latter case, we propose a Thompson sampling type rule.more » « less
-
Robust Adaptive Secure Secret Sharing (RASSS) is a protocol for reconstructing secrets and information in distributed computing systems even in the presence of a large number of untrusted participants. Since the original Shamir's Secret Sharing scheme, there have been efforts to secure the technique against dishonest shareholders. Early on, researchers determined that the Reed-Solomon encoding property of the Shamir's share distribution equation and its decoding algorithm could tolerate cheaters up to one third of the total shareholders. However, if the number of cheaters grows beyond the error correcting capability (distance) of the Reed-Solomon codes, the reconstruction of the secret is hindered. Untrusted participants or cheaters could hide in the decoding procedure, or even frame up the honest parties. In this paper, we solve this challenge and propose a secure protocol that is no longer constrained by the limitations of the Reed-Solomon codes. As long as there are a minimum number of honest shareholders, the RASSS protocol is able to identify the cheaters and retrieve the correct secret or information in a distributed system with a probability close to 1 with less than 60% of hardware overhead. Furthermore, the adaptive nature of the protocol enables considerable hardware and timing resource savings and makes RASSS highly practical.more » « less
-
Guruswami, Venkatesan (Ed.)Traditional fraud detection is often based on finding statistical anomalies in data sets and transaction histories. A sophisticated fraudster, aware of the exact kinds of tests being deployed, might be difficult or impossible to catch. We are interested in paradigms for fraud detection that are provably robust against any adversary, no matter how sophisticated. In other words, the detection strategy should rely on signals in the data that are inherent in the goals the adversary is trying to achieve. Specifically, we consider a fraud detection game centered on a random walk on a graph. We assume this random walk is implemented by having a player at each vertex, who can be honest or not. In particular, when the random walk reaches a vertex owned by an honest player, it proceeds to a uniformly random neighbor at the next timestep. However, when the random walk reaches a dishonest player, it instead proceeds to an arbitrary neighbor chosen by an omniscient Adversary. The game is played between the Adversary and a Referee who sees the trajectory of the random walk. At any point during the random walk, if the Referee determines that a {specific} vertex is controlled by a dishonest player, the Referee accuses that player, and therefore wins the game. The Referee is allowed to make the occasional incorrect accusation, but must follow a policy that makes such mistakes with small probability of error. The goal of the adversary is to make the cover time large, ideally infinite, i.e., the walk should never reach at least one vertex. We consider the following basic question: how much can the omniscient Adversary delay the cover time without getting caught? Our main result is a tight upper bound on this delay factor. We also discuss possible applications of our results to settings such as Rotor Walks, Leader Election, and Sybil Defense.more » « less
