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We study countably infinite Markov decision processes (MDPs) with real-valuedtransition rewards. Every infinite run induces the following sequences ofpayoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2.Total payoff (the sequence of the sums of all rewards so far), and 3. Meanpayoff. For each payoff type, the objective is to maximize the probability thatthe $\liminf$ is non-negative. We establish the complete picture of thestrategy complexity of these objectives, i.e., how much memory is necessary andsufficient for $\varepsilon$-optimal (resp. optimal) strategies. Some cases canbe won with memoryless deterministic strategies, while others require a stepcounter, a reward counter, or both.

Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs