Understanding the interdependence between autonomous and human-operated vehicles remains an ongoing challenge, with significant implications for the safety and feasibility of autonomous driving.This interdependence arises from inherent interactions among road users.Thus, it is crucial for Autonomous Vehicles (AVs) to understand and analyze the intentions of human-driven vehicles, and to display behavior comprehensible to other traffic participants.To this end, this paper presents GTP-UDRIVE, a unified game-theoretic trajectory planner and decision-maker considering a mixed-traffic environment. Our model considers the intentions of other vehicles in the decision-making process and provides the AV with a human-like trajectory, based on the clothoid interpolation technique.% This study investigates a solver based on Particle Swarm Optimization (PSO) that quickly converges to an optimal decision.Among highly interactive traffic scenarios, the intersection crossing is particularly challenging. Hence, we choose to demonstrate the feasibility and effectiveness of our method in real traffic conditions, using an experimental autonomous vehicle at an unsignalized intersection. Testing results reveal that our approach is suitable for 1) Making decisions and generating trajectories simultaneously. 2) Describing the vehicle's trajectory as a piecewise clothoid and enforcing geometric constraints. 3) Reducing search space dimensionality for the trajectory optimization problem.