Commitment and Cyclic Strategies in Multi-Objective Games

Multi-objective games present a natural framework for studying strategic interactions between rational individuals concerned with more than one objective. We explore both the impact of commitment on the equilibria as well as the learning behaviour of agents in such games. It is well known that in single-objective normal-form games, committing to a future strategy can never be worse than the utility from any Nash equilibrium. We show that this property does not hold in multi-objective games. On the other hand, we are able to construct games in which commitment is beneficial for both players, highlighting the nuances that commitment introduces. Furthermore, we observe that commitment can sometimes induce the same joint-action distribution as a cyclic Nash equilibrium and show that such cyclic Nash equilibria may exist even when no Nash equilibrium exists. We evaluate these characteristics in a learning setting, to explore whether this behaviour can be expected in applications as well. We find that all proposed theorems can clearly be observed from the learning dynamics. In addition, we observe that commitment can learn the same joint-action distribution as in a cyclic Nash equilibrium, but that it is not guaranteed when there are multiple best-responses for the follower.