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Reward functions encode desired behavior in multi-agent Markov decisionprocesses, but onlookers may learn reward functions by observing agents, whichcan reveal sensitive information. Therefore, in this paper we introduce andcompare two methods for privatizing reward functions in policy synthesis formulti-agent Markov decision processes. Reward functions are privatized usingdifferential privacy, a statistical framework for protecting sensitive data.Both methods we develop rely on the Gaussian mechanism, which is a method ofrandomization we use to perturb (i) each agent's individual reward function or(ii) the joint reward function shared by all agents. We prove that both ofthese methods are differentially private and compare the abilities of each toprovide accurate reward values for policy synthesis. We then develop analgorithm for the numerical computation of the performance loss due to privacyon a case-by-case basis. We also exactly compute the computational complexityof this algorithm in terms of system parameters and show that it is inherentlytractable. Numerical simulations are performed on a gridworld example and inwaypoint guidance of an autonomous vehicle, and both examples show that privacyinduces only negligible performance losses in practice.

Differentially Private Reward Functions for Multi-Agent Markov Decision Processes