# Dynamic Signaling Games with Quadratic Criteria under Nash and Stackelberg Equilibria

## TL;DR

This paper analyzes dynamic signaling games with quadratic costs under Nash and Stackelberg equilibria, revealing quantized equilibria in Nash and fully revealing strategies in Stackelberg, with explicit solutions for Gaussian sources.

## Contribution

It provides a comprehensive analysis of multi-stage signaling games with subjective models, deriving explicit policies and conditions for informativeness under both equilibrium concepts.

## Key findings

- Final stage equilibrium is always quantized.
- Stackelberg equilibria are always fully revealing.
- Explicit recursive solutions for Gaussian source encoding.

## 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 (leader-follower) 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 multi-stage 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 non-linear 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.