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In modern communication systems, different users have different requirements for quality of service (QoS). In this work, QoS refers to the average codeword error probability experienced by the users in the network. Although several practical schemes (collectively referred to as unequal error protection schemes) have been studied in the literature and are implemented in existing systems, the corresponding performance limits have not been studied in an information-theoretic framework. In this paper, an information-theoretic framework is considered to study communication systems which provide heterogeneous reliabilities for the users. This is done by defining individual probabilities of error for the users in the network and obtaining the fundamental tradeoffs of the corresponding error exponents. In particular, we quantify the reliability tradeoff by introducing the notion of error exponent region (EER), which specifies the set of error exponent vectors that are simultaneously achievable by the users for a fixed vector of users' rates. We show the existence of a tradeoff among the users' error exponents by deriving inner and outer bounds for the EER. Using this framework, a system can be realized, which can provide a tradeoff of reliabilities among the users for a fixed vector of users' rates. This adds a completely new dimension to the performance tradeoff in such networks, which is unique to multiterminal communication systems, and is beyond what is given by the conventional performance-versus-rate tradeoff in single-user systems. Although this is a very general concept and can be applied to any multiterminal communication system, in this paper we consider Gaussian broadcast and multiple-access channels (MACs).