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Leveraging Noisy Observations in Zero-Sum Games

Emmanouil M Athanasakos, Samir M Perlaza

TL;DR

This work extends zero-sum game theory to settings where the follower observes a noisy version of the leader’s committed action through a channel $P_{ ilde{A}_2|A_2}$. It introduces the extended game $\mathscr{G}(u, P_{ ilde{A}_2|A_2})$, proves equilibrium existence for any channel, and derives sufficient conditions for almost-sure uniqueness with follower best responses concentrating on single actions. Key contributions include a characterization of best responses via $\mathrm{BR}_1$, an almost-surely unique equilibrium under mild assumptions, and a Gaussian-channel instance illustrating how observation noise shifts equilibria between Nash, Stackelberg, and pure strategies. The results illuminate the impact of data acquisition noise on strategic leverage in leader-follower interactions and have implications for adversarial ML and decision-making under imperfect monitoring, especially when channel densities with respect to the Lebesgue measure are relevant.

Abstract

This paper studies an instance of zero-sum games in which one player (the leader) commits to its opponent (the follower) to choose its actions by sampling a given probability measure (strategy). The actions of the leader are observed by the follower as the output of an arbitrary channel. In response to that, the follower chooses its action based on its current information, that is, the leader's commitment and the corresponding noisy observation of its action. Within this context, the equilibrium of the game with noisy action observability is shown to always exist and the necessary conditions for its uniqueness are identified. Interestingly, the noisy observations have important impact on the cardinality of the follower's set of best responses. Under particular conditions, such a set of best responses is proved to be a singleton almost surely. The proposed model captures any channel noise with a density with respect to the Lebesgue measure. As an example, the case in which the channel is described by a Gaussian probability measure is investigated.

Leveraging Noisy Observations in Zero-Sum Games

TL;DR

This work extends zero-sum game theory to settings where the follower observes a noisy version of the leader’s committed action through a channel . It introduces the extended game , proves equilibrium existence for any channel, and derives sufficient conditions for almost-sure uniqueness with follower best responses concentrating on single actions. Key contributions include a characterization of best responses via , an almost-surely unique equilibrium under mild assumptions, and a Gaussian-channel instance illustrating how observation noise shifts equilibria between Nash, Stackelberg, and pure strategies. The results illuminate the impact of data acquisition noise on strategic leverage in leader-follower interactions and have implications for adversarial ML and decision-making under imperfect monitoring, especially when channel densities with respect to the Lebesgue measure are relevant.

Abstract

This paper studies an instance of zero-sum games in which one player (the leader) commits to its opponent (the follower) to choose its actions by sampling a given probability measure (strategy). The actions of the leader are observed by the follower as the output of an arbitrary channel. In response to that, the follower chooses its action based on its current information, that is, the leader's commitment and the corresponding noisy observation of its action. Within this context, the equilibrium of the game with noisy action observability is shown to always exist and the necessary conditions for its uniqueness are identified. Interestingly, the noisy observations have important impact on the cardinality of the follower's set of best responses. Under particular conditions, such a set of best responses is proved to be a singleton almost surely. The proposed model captures any channel noise with a density with respect to the Lebesgue measure. As an example, the case in which the channel is described by a Gaussian probability measure is investigated.
Paper Structure (7 sections, 6 theorems, 1 equation, 1 figure)

This paper contains 7 sections, 6 theorems, 1 equation, 1 figure.

Table of Contents

  1. introduction
  2. Preliminaries and Notation
  3. Game Formulation
  4. Equilibrium
  5. Existing Special Cases of Game $\mathscr{G}\left(u, P_{\tilde{A}_2 | A_2}\right)$
  6. main results
  7. Example: Gaussian channel

Key Result

Lemma 3.1

Given a commitment $P \in \Delta\left( \mathcal{A}_2 \right)$ and a channel output $\tilde{b} \in \tilde{\mathcal{A}_2}$, the set of best responses of Player $1$ is where

Figures (1)

  • Figure 1: Plots of $\hat{v}$ in \ref{['Eq_Best_PLayer2']} and $\hat{u}$ in \ref{['Eq_fun_u_hat']} as a function of the commitment $P_{A_2}$. The channel is the AWGN defined in \ref{['Eq_Gaussian_measures_def1']} with variance $\sigma^2_1=\sigma^2_2=\sigma^2$; and $-a_{2,1} =a_{2,2}=10^2$.

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 4.1