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IEEE Transactions on Automatic Control

Almost-Bayesian Quadratic Persuasion

Olivier Massicot, Cédric Langbort

Published Jan 6, 2025• 1 citation
Publisher

Abstract

In this article, we relax the Bayesianity assumption in the now-traditional model of Bayesian Persuasion introduced by Kamenica & Gentzkow. Unlike preexisting approaches—which have tackled the possibility of the receiver (Bob) being non-Bayesian by considering that his thought process is not Bayesian yet known to the sender (Alice), possibly up to a parameter—we let Alice merely assume that Bob behaves ‘almost like’ a Bayesian agent, in some sense, without resorting to any specific model. Under this assumption, we study Alice's strategy when both utilities are quadratic and the prior is isotropic. We show that, contrary to the Bayesian case, Alice's optimal response may not be linear anymore. This fact is unfortunate as linear policies remain the only ones for which the induced belief distribution is known. What is more, evaluating linear policies proves difficult except in particular cases, let alone finding an optimal one. Nonetheless, we derive bounds that prove linear policies are near-optimal and allow Alice to compute a near-optimal linear policy numerically. With this solution in hand, we show that Alice shares less information with Bob as he departs more from Bayesianity, much to his detriment.

DOI
10.1109/tac.2025.3526521

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In this article, we relax the Bayesianity assumption in the now-traditional model of Bayesian Persuasion introduced by Kamenica & Gentzkow. Unlike preexisting approaches—which have tackled the possibility of the receiver (Bob) being non-Bayesian by considering that his thought process is not Bayesian yet known to the sender (Alice), possibly up to a parameter—we let Alice merely assume that Bob behaves ‘almost like’ a Bayesian agent, in some sense, without resorting to any specific model. Under this assumption, we study Alice's strategy when both utilities are quadratic and the prior is isotropic. We show that, contrary to the Bayesian case, Alice's optimal response may not be linear anymore. This fact is unfortunate as linear policies remain the only ones for which the induced belief distribution is known. What is more, evaluating linear policies proves difficult except in particular cases, let alone finding an optimal one. Nonetheless, we derive bounds that prove linear policies are near-optimal and allow Alice to compute a near-optimal linear policy numerically. With this solution in hand, we show that Alice shares less information with Bob as he departs more from Bayesianity, much to his detriment.
In this article, we address Bayesian persuasion between a sender and a receiver with state-dependent quadratic cost measures for general classes of distributions. The receiver seeks to make mean-square-error estimate of a state based on a signal sent by the sender while the sender signals strategically in order to control the receiver’s estimate in a certain way. Such a scheme could model, e.g., deception and privacy, problems in multiagent systems. Existing solution concepts are not viable since here the receiver has continuous action space. We show that for finite state spaces, optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. We then establish its strong duality to a copositive program. To exemplify the effectiveness of this equivalence result, we adopt sequential polyhedral approximation of completely positive cones and analyze its performance numerically. We also quantify the approximation error for a quantized version of a continuous distribution and show that a semi-definite program relaxation of the equivalent problem could be a benchmark lower bound for the sender’s cost for large state spaces.
We investigate a privacy-signaling game problem in which a sender with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The sender aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework with quadratic costs where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. We formulate the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the random variables is observed at the sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a linear solution which is explicitly characterized in the paper. We provide explicit conditions on when the optimal solutions, or equilibrium solutions in the Nash setup, are informative or noninformative.
We introduce a privacy-signaling game problem in which a transmitter with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The transmitter aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. We formulate the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the parameters is observed at the sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a linear solution which is explicitly characterized in the paper. We provide explicit conditions on when the optimal solutions, or equilibrium solutions in the Nash setup, are informative or noninformative.
In this paper we investigate the potential for persuasion arising from the quantum indeterminacy of a decision-maker's beliefs, a feature that has been proposed as a formal expression of well-known cognitive limitations. We focus on a situation where an agent called Sender only has few opportunities to influence the decision-maker called Receiver. We do not address the full persuasion problem but restrict attention to a simpler one that we call targeting, i.e. inducing a specific belief state. The analysis is developed within the frame of a n-dimensional Hilbert space model. We find that when the prior is known, Sender can induce a targeted belief with a probability of at least 1/n when using two sequential measurements. This figure climbs to 1/2 when both the target and the belief are known pure states. A main insight from the analysis is that a well-designed strategy of distraction can be used as a first step to confuse Receiver. We thus find that distraction rather than the provision of relevant arguments is an effective means to achieve persuasion. We provide an example from political decision-making.
Secure key distribution among two remote parties is impossible when both are classical, unless some unproven computation-complexity assumptions are made, such as the difficulty of factorizing large numbers. On the other hand, a secure key distribution is possible when both parties are quantum. What is possible when only one party (Alice) is quantum, yet the other (Bob) has only classical capabilities? We present a protocol with this constraint and prove its robustness against attacks: we prove that any attempt of an adversary to obtain information necessarily induces some errors that the legitimate users could notice.
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are kkk senders, Alice1_11​, …\ldots…, Alicek_kk​, and a single receiver, Bob. Each Alicei_ii​ has a data vector Wi∈FmiW_i \in \mathbb{F}^{m_i}Wi​∈Fmi​, where F\mathbb{F}F is a finite field. Bob wants to compute the linear combination Y=V1W1+V2W2+⋯+VkWk∈FmY = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^mY=V1​W1​+V2​W2​+⋯+Vk​Wk​∈Fm, where ViV_iVi​ is an m×mim \times m_im×mi​ matrix over F\mathbb{F}F. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information WiW_iWi​, i=1,…,ki=1,\ldots,ki=1,…,k, into their local subsystems and transmit them to Bob so that he can recover YYY through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.