The capacity for the discrete memoryless arbitrarily varying channel (AVC) with cost constraints on the jammer is studied using deterministic list codes under both the maximal and average probability of error criteria. For a cost function l(.) on the state set and constraint Lambda on the jammer, the achievable rates are upper bounded by the random coding capacity Cr(Lambda). For maximal error, the rate R = Cr(Lambda) - epsilon is achievable using list codes with list size O(epsilon^-1). For average error, an integer L_sym(Lambda), called the symmetrizability, is defined. It is shown that any rate below Cr(Lambda) is achievable under average error using list codes of list size L > L_sym. An example is given for a class of discrete additive AVCs.