# The Freshness Game: Timely Communications in the Presence of an Adversary

## TL;DR

This paper analyzes the strategic interaction between a base station and an adversary in a wireless status update system, exploring optimal jamming strategies, equilibrium points, and system performance under different diversity conditions.

## Contribution

It introduces a game-theoretic framework for analyzing timely communications under adversarial jamming, providing optimal strategies and equilibrium analysis for both diversity and non-diversity models.

## Key findings

- Optimal adversarial jamming strategy in non-diversity model is to target the least scheduled user.
- Uniform scheduling and sub-carrier selection are near-optimal in the diversity model for large T.
- Nash equilibrium exists in the diversity model, but not in the non-diversity model.

## Abstract

We consider a communication system where a base station (BS) transmits update packets to $N$ users, one user at a time, over a wireless channel. We investigate the age of this status updating system with an adversary that jams the update packets in the downlink. We consider two system models: with diversity and without diversity. In the model without diversity, we show that if the BS schedules the users with a stationary randomized policy, then the optimal choice for the adversary is to block the user which has the lowest probability of getting scheduled by the BS, at the middle of the time horizon, consecutively for $\alpha T$ time slots. In the model with diversity, we show that for large $T$, the uniform user scheduling algorithm together with the uniform sub-carrier choosing algorithm is $\frac{2 N_{sub}}{N_{sub}-1}$ optimal. Next, we investigate the game theoretic equilibrium points of this status updating system. For the model without diversity, we show that a Nash equilibrium does not exist, however, a Stackelberg equilibrium exists when the scheduling algorithm of the BS acts as the leader and the adversary acts as the follower. For the model with diversity, we show that a Nash equilibrium exists and identify the Nash equilibrium. Finally, we extend the model without diversity to the case where the BS can serve multiple users and the adversary can jam multiple users, at a time.