We introduce a new non-zero-sum optimal stopping game with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, while the other one gets nothing. The reward function is different for each player. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. Under the setting driven by a Lévy process with positive jumps, we explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviours of the players’ strategies as well as the quantification of the value of information. Joint work with J.L. Perez (CIMAT) and N. Rodosthenous (Queen Mary, University of London).