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Information-Theoretic Approach to Strategic Communication as a Hierarchical Game

Emrah Akyol, Cedric Langbort, Tamer Basar

arXiv:1510.00764v2cs.ITcs.GTmath.ITmath.OC

TL;DR

Mismatched objectives make strategic communication distinct from conventional information-theoretic settings. The paper characterizes strategic-compression limits and Q-G equilibrium strategies, finding linear optimality in several settings while identifying narrower conditions when the receiver has side information.

  • Problem

    Information-theoretic results for mismatched transmitter and receiver objectives are relatively scarce, and strategic communication raises questions about compression and equilibrium structure.

    Source: §I · p. 1

  • Method

    The paper characterizes strategic-compression limits and studies linear-strategy optimality in Q-G communication and decentralized-control settings.

    Source: §I · p. 2, F. Decentralized Stochastic Control… · p. 5

  • Results

    The paper characterizes strategic-compression limits and finds linear strategies optimal in Q-G simple and noisy equilibria, with receiver-side-information optimality conditional in noisy cases.

    Source: §I · p. 2, §VI · p. 10

  • Takeaways & Limitations

    The paper identifies necessary and sufficient conditions for linear optimality in general strategic games and notes possible extensions to multidimensional, networked, and non-Gaussian settings.

    Source: §VI · p. 10

Abstract

from arXiv · show

This paper analyzes the information disclosure problems originated in economics through the lens of information theory. Such problems are radically different from the conventional communication paradigms in information theory since they involve different objectives for the encoder and the decoder, which are aware of this mismatch and act accordingly. This leads, in our setting, to a hierarchical communication game, where the transmitter announces an encoding strategy with full commitment, and its distortion measure depends on a private information sequence whose realization is available at the transmitter. The receiver decides on its decoding strategy that minimizes its own distortion based on the announced encoding map and the statistics. Three problem settings are considered, focusing on the quadratic distortion measures, and jointly Gaussian source and private information: compression, communication, and the simple equilibrium conditions without any compression or communication. The equilibrium strategies and associated costs are characterized. The analysis is then extended to the receiver side information setting and the major changes in the structure of optimal strategies are identified. Finally, an extension of the results to the broader context of decentralized stochastic control is presented.

I. INTRODUCTION

The paper frames strategic communication as an information-theoretic problem with mismatched objectives and a committed transmitter, then introduces its Q-G and strategic-compression contributions alongside economic communication models.

I. INTRODUCTION

  • The model treats the transmitter as a committed leader whose policy the receiver knows and responds to, enabling Shannon-theoretic analysis of compression and communication limits.The commitment is to a policy based on statistics rather than the realization.Source: §I · p. 1
  • The paper characterizes strategic-compression limits and establishes linear-strategy optimality for Q-G communication and simple-equilibrium settings, while also examining receiver side information and decentralized control.For simple equilibrium, the paper reports uniquely optimal linear policies.Source: §I · pp. 1–2

A. Notation

  • The notation distinguishes the source, private information, and reconstruction alphabets, and uses S for Borel-measurable, square-integrable real functions.The paper uses natural logarithms and denotes Gaussian densities by their mean and variance.Source: A. Notation · p. 2

B. Overview of Communication Games in Economics

  • In cheap talk, the transmitter chooses its mapping after observing the source, and equilibrium mappings are non-injective quantizers, with quantization arising from mismatched objectives.The transmitter is not committed to its mapping before seeing the source realization.Source: 1) Strategic Information Transmission… · p. 2

2) Overview of Information Disclosure Model (Bayesian Persuasion):

The overview places the model in a committed transmitter–receiver game; in the noiseless quadratic-Gaussian setting, linear encoding and decoding form the essentially unique optimal pair.

2) Overview of Information Disclosure Model (Bayesian Persuasion):

  • The transmitter commits to an encoding map first, and the receiver optimizes its map knowing that commitment.The transmitter anticipates the receiver's response when choosing its encoding map.Source: 2) Overview of Information Disclosure… · p. 2

C. Setting-1: Simple (Noiseless) Equilibrium

1) Functional Representation Lemma:

The section presents a functional representation of W using a variable independent of Y, then states a maximal-correlation result with a unique linear optimizer.

1) Functional Representation Lemma:

2) Maximal Correlation Coefficient:

D. Setting-2: Strategic Compression

The section defines strategic compression through rate-and-distortion triples, then gives Gaussian test-channel optimality results and a receiver-side-information qualification.

D. Setting-2: Strategic Compression

  • Strategic compression maps source and private-information sequences to an index, from which the decoder produces a source reconstruction.The achievable objects are triples of rate and transmitter and receiver distortions.Source: D. Setting-2: Strategic Compression · p. 4

E. Setting-3: Gaussian Test Channel and Equilibrium with Noisy Channel

  • The noisy communication setup sends a Gaussian source through an additive Gaussian channel and gives both agents the squared-error reconstruction objective, with a transmitter power constraint.Source: E. Setting-3: Gaussian Test Channel… · p. 4
  • Allowing n-letter Shannon strategies gives the same solution as zero-delay strategies for the Gaussian test channel.The passage attributes this result to Goblick.Source: E. Setting-3: Gaussian Test Channel… · p. 4
  • For the Gaussian test channel without receiver side information, single-letter mappings are essentially unique Shannon-sense optimal encoding and decoding mappings.Source: E. Setting-3: Gaussian Test Channel… · p. 4
  • With receiver side information, linear strategies are no longer optimal even among zero-delay strategies.In the strategic side-information setting, a similar optimality result can hold depending on problem parameters.Source: E. Setting-3: Gaussian Test Channel… · p. 5

F. Decentralized Stochastic Control Problems

The paper connects linear optimality in Gaussian communication to the form of the quadratic objective, then extends the analysis to strategic settings.

F. Decentralized Stochastic Control Problems

A. Equilibrium Conditions

The noiseless quadratic-Gaussian equilibrium uses linear encoding and decoding strategies, with the encoder balancing its objective against the receiver’s response. The equilibrium’s behavior varies with the correlation between source and private information.

A. Equilibrium Conditions

  • In the noiseless Q-G setting, the essentially unique equilibrium strategies are linear: the encoder sends X + αθ and the decoder uses κY.The equilibrium coefficients α and κ depend on the problem parameters.Source: A. Equilibrium Conditions · p. 5
  • The receiver’s optimal decoder is the conditional expectation E{X|Y}, reducing the encoder’s problem to choosing its encoding map.The analysis also characterizes the encoder’s objective through the joint distribution of the reconstruction error and private information.Source: A. Equilibrium Conditions · p. 5
  • At equilibrium, the encoder uses a Gaussian test-channel form Y = X + αθ + T, with independent Gaussian noise T, and determines α and σ_T² by optimizing its objective.The analysis establishes this form by showing that the Gaussian reconstruction error minimizes the relevant objective.Source: A. Equilibrium Conditions · p. 6
  • The equilibrium coefficient satisfies |α| < 1 for all problem parameters, so the encoder does not directly transmit X + θ.The paper interprets this as the transmitter never “flat out” lying at equilibrium.Source: A. Equilibrium Conditions · p. 6

B. Compression

The paper characterizes strategic compression through a rate-distortion formulation and specializes the equilibrium to the quadratic-Gaussian case. There, compression can be treated as operating on an effective source, though this simplification does not carry over to receiver side information.

B. Compression

  • Strategic compression has a single-letter rate-distortion characterization, and in the Q-G setting its equilibrium points are achieved uniquely by jointly Gaussian (X, Y, θ).The rate-distortion region is characterized as a convex hull over feasible decoder functions and conditional distributions.Source: B. Compression · p. 6
  • In strategic Q-G compression, the forward test channel is Y = X + β*θ + S, where β* is independent of the allowed rate and equals the coefficient α from the simple equilibrium.S is independent of X and θ.Source: B. Compression · p. 7
  • Strategic Q-G compression therefore reduces to compressing the effective source X + αθ, allowing standard encoding codes to be used on that source.The paper notes that this simplification does not carry over to settings with receiver side information.Source: B. Compression · p. 7

C. Equilibrium with Noisy Channel

For the noisy-channel Q-G equilibrium, the paper tests whether single-letter strategies remain optimal in the Shannon sense. It reports Shannon-sense optimality for all power levels and an encoding strategy that is independent of channel-noise variance.

C. Equilibrium with Noisy Channel

  • The analysis asks whether single-letter strategies retain Shannon-sense optimality when the Q-G channel is noisy.The proof invokes data processing and a lower bound on distortion for source-channel coding schemes.Source: C. Equilibrium with Noisy Channel · p. 7
  • The equilibrium strategies in the noisy-channel Q-G problem are Shannon-sense optimal for all power levels.The section establishes the result by matching inner and outer bounds.Source: C. Equilibrium with Noisy Channel · pp. 7–8
  • The equilibrium encoding map does not depend on channel-noise variance and uses the non-strategic strategy on the effective source X + αθ.The effective-source interpretation parallels the one given for strategic compression.Source: C. Equilibrium with Noisy Channel · p. 8

IV. IMPACT OF RECEIVER SIDE INFORMATION

With receiver side information, the paper characterizes equilibrium strategies and compression in the quadratic-Gaussian setting, then identifies how SI changes rate-constrained and noisy-channel optimality. Receiver SI can help estimation and reduce rate, but strategic misalignment can eliminate the rate-reduction benefit.

IV. IMPACT OF RECEIVER SIDE INFORMATION

  • In the noiseless quadratic-Gaussian setting with receiver SI, equilibrium mappings are linear: the encoder uses g(X, θ) = X + αSIθ and the decoder uses h(Y, W) = bY + cW.The coefficients can be explicitly computed, though the computation is omitted.Source: §IV · p. 8

E and DRSI

  • Without a constraint on Y, receiver SI at the transmitter does not improve either agent’s distortion; an average-power constraint can make W useful to both.The proof states that transmitter SI does not affect distortions; the unconstrained rate expression differs only by a Markov-chain constraint that is always satisfied for jointly Gaussian variables.Source: E and DRSI · pp. 8–9
  • Receiver SI provides estimation benefit and can reduce rate through I(Y; W), but strategic misalignment can make Y independent of W and eliminate that rate-reduction benefit.The rate-reduction benefit is specific to the information-theoretic setting.Source: E and DRSI · p. 8
  • For quadratic-Gaussian compression with receiver SI, the forward test channel includes a rate-dependent coefficient β(R), and its independent noise variance also depends on R.The achievable rate-distortion region is characterized as a convex hull of feasible rate-distortion triplets.Source: E and DRSI · pp. 8–9
  • With channel noise and receiver SI, single-letter linear strategies are Shannon-sense optimal only under the matching condition ρX,W = −ρθ,W β(R).The condition requires the rate-distortion test channel to match the communication channel; the matching rate implies Y and W are independent.Source: E and DRSI · p. 9
  • Linear mappings may remain optimal among single-letter strategies without that condition, while being strictly suboptimal among n-letter strategies.The compression coefficient with SI depends on the allowed rate, unlike the coefficient without SI.Source: E and DRSI · p. 9

V. DECENTRALIZED STOCHASTIC CONTROL PROBLEMS

The paper extends its strategic communication analysis to decentralized stochastic control problems. It gives a necessary-and-sufficient condition for essentially unique linear strategies in Problem 4 and develops a proof that addresses cross terms in the objectives.

V. DECENTRALIZED STOCHASTIC CONTROL PROBLEMS

  • Problem 4 has essentially unique linear mappings g(X) = c(X + αθ) and h(Y) = dY if and only if the objective functions contain no product terms in U and X̂.The theorem extends the main result of to the strategic setting.Source: §V · p. 10
  • The proof cannot directly reuse Theorem 7 because Cauchy-Schwarz does not upper-bound the objective’s cross term when a linear encoder need not be proportional to a1X + a2θ.The proof instead reformulates the objectives and optimizes the encoder under fixed receiver distortion.Source: §V · p. 10
  • For fixed receiver distortion, a jointly Gaussian (Y, X, θ) minimizes transmitter distortion, leading to a linear or affine encoding strategy.The argument uses fixed power and distortion constraints to establish joint Gaussianity of the variables.Source: §V · p. 10

VI. DISCUSSION

The discussion summarizes linear-strategy results across the paper’s strategic communication settings and identifies their conditions. It highlights the specific matched case required for noisy channels with receiver side information and points to broader extensions.

VI. DISCUSSION

  • The quadratic-Gaussian equilibrium has unique linear optimal strategies, established using probability tools including the functional representation lemma and maximal correlation.The paper identifies these as information-theoretic tools used in the derivation.Source: §VI · p. 10
  • The noisy quadratic-Gaussian equilibrium also admits linear optimal strategies, with a proof using data processing, rate-distortion, and channel capacity.Source: §VI · p. 10
  • With receiver SI, linear strategies are optimal in the noiseless quadratic-Gaussian setting, but in the noisy setting only for a specific match of channel noise, allowed power, and joint statistics.The paper contrasts this with the non-strategic analogue, which does not admit linear optimal strategies.Source: §VI · p. 10
  • The paper identifies necessary and sufficient conditions for linear optimality in general strategic games and proposes multidimensional, networked, non-Gaussian, and privacy-constrained extensions.The discussion cites preliminary privacy-constraint work.Source: §VI · p. 10
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