Energy system integration can bring several benefits to energy systems, notably to those that are in transition to high shares of renewable energy. Strategies are needed to realize the theoretical benefits of this approach in practice. Therefore, this paper proposes the organization and mathematical formulation of a multi-carrier day-ahead market in which electricity, gas and heat are traded simultaneously. This market set-up is applied to a conceptual test case to identify how - compared to a reference set-up mimicking the current practice - the multi-carrier market is able to unlock the benefits of energy system integration. It is quantitatively shown that the multi-carrier market (1) eliminates the need for forecasts of prices on subsequent markets and the consequences of the related errors, (2) allows to use the flexibility available in one carrier to facilitate the balancing in another, e.g. using the flexibility of a heat system to help balance the electricity system, and (3) enables specific market outcomes, unachievable in a sequential set-up, which increase the optimality of the market outcome. (C) 2018 Elsevier Ltd. All rights reserved.