Dynamic signaling games with quadratic criteria under Nash and Stackelberg equilibria

Abstract

This paper considers dynamic (multi-stage) signaling games involving an encoder and a decoder who have subjective models on the cost functions. We consider both Nash (simultaneous-move) and Stackelberg (leaderfollower) equilibria of dynamic signaling games under quadratic criteria. For the multi-stage scalar cheap talk, we show that the final stage equilibrium is always quantized and under further conditions the equilibria for all time stages must be quantized. In contrast, the Stackelberg equilibria are always fully revealing. In the multistage signaling game where the transmission of a Gauss-Markov source over a memoryless Gaussian channel is considered, affine policies constitute an invariant subspace under best response maps for Nash equilibria; whereas the Stackelberg equilibria always admit linear policies for scalar sources but such policies may be nonlinear for multi-dimensional sources. We obtain an explicit recursion for optimal linear encoding policies for multi-dimensional sources, and derive conditions under which Stackelberg equilibria are informative. * (decoder), who generates his M-valued optimal decision U upon receiving X. The policies of the encoder and decoder are assumed to be deterministic; i.e., x = γ e (m) and u = γ d (x) = γ d (γ e (m)). Let c e (m, u) and c d (m, u) denote the cost functions of the encoder and the decoder, respectively, when the action u is taken for the corresponding message m. Then, given the encoding and decoding policies, the encoder's induced expected cost is J e γ e , γ d = E [c e (m, u)], whereas, the decoder's induced expected cost is J d γ e , γ d = E c d (m, u) . If the transmitted signal x is also an explicit part of the cost functions c e and/or c d , then the communication between the players is not costless and the formulation turns into a signaling game problem. Such problems are studied under the tools and concepts provided by game theory since the goals are not aligned. Although the encoder and decoder act sequentially in the game as described above, how and when the decisions are made and the nature of the commitments to the announced policies significantly affect the analysis of the equilibrium structure. Here, two different types of equilibria are investigated: the Nash equilibrium, in which the encoder and the decoder make simultaneous decisions, and the Stackelberg equilibrium, in which the encoder and the decoder make sequential decisions where the encoder is the leader and the decoder is the follower 1 . In this paper, the terms Nash game and the simultaneous-move game 2 will be used interchangeably, and similarly, the Stackelberg game and the leader-follower game will be used interchangeably.In the simultaneous-move game, the encoder and the decoder announce their policies at the same time, and a pair of policies (γ * ,e , γ * ,d ) is said to be a Nash equilibrium [4] if where Γ e and Γ d are the sets of all deterministic (and Borel measurable) functions from M to X and from X to M, respectively. As observed from the definition (1), under the Na...

Hypothesis Testing Under Subjective Priors and Costs as a Signaling Game

“…However, in some situations, security measures may not be observable for the attacker; therefore, a simultaneous-move game is preferred to model such situations; i.e., the Nash equilibrium analysis is needed [28]. These two concepts may have equilibria that are quite distinct: As discussed in [25,29], in the Nash equilibrium case, building on [17], equilibrium properties possess different characteristics as compared to team problems; whereas for the Stackelberg case, the leader agent is restricted to be committed to his announced policy, which leads to similarities with team problem setups [26,30]. However, in the context of binary signaling, we will see that the distinction is not as sharp as it is in the case of quadratic signaling games [25,29].…”

On the Number of Bins in Equilibria for Signaling Games

“…Furthermore, in a recent work, where we generalized signaling games and cheap talk problems to dynamic (multistage) setups, a crucial property that allowed the generalization was the assumption that the number of bins for each stage equilibrium, conditioned on the past actions, is uniformly bounded [3,Theorem 2.4]. In view of this, showing that the number of bins is finite would be a useful technical result.…”

“…, N (Note that m 0 = 0 and m N = +∞ for an exponential source, whereas m 0 = −∞ and m N = +∞ for a Gaussian source). By [2, Theorem 3.2], 1 We note that, unlike Crawford and Sobel's simultaneous Nash equilibrium formulation, if one considers a Stackelberg formulation (see [4, p.133] for a definition), then the problem would reduce to a classical communication problem since the encoder would be committed a priori and the equilibrium would not be quantized; i.e., there exist affine equilibria [2], [3], [5]- [7]. Recently, the problem of strategical coordination has been considered in [8]; specifically, the information design and a point-to-point strategic sourcechannel coding problems (originated from the Bayesian persuasion game [9]) between an encoder and a decoder with non-aligned utility functions have been investigated under the Stackelberg equilibrium.…”

“…Cheap talk and signaling game problems find applications in networked control systems when a communication channel/network is present among competitive and non-cooperative decision makers [4], [10]. Also, there have been a number of related results in the economics and control literature in addition to the seminal work by Crawford and Sobel, which are reviewed in [2], [3] (see [11] for an extensive survey).…”

Strategic Communication with Decoder Side Information

“…Communication between autonomous devices that have distinct objectives is under study. This problem, referred to as the strategic communication problem, is at the crossroads of different disciplines such as Control Theory [1], [2], Computer Science [3] and Information Theory [4], [7], [8], [9], [10], [11], [12], where it was introduced by Akyol et al in [5], [6].…”