The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are 1-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely non-simple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a non-simple cycle present in a heteroclinic network of the Rock-Scissors-Paper game. 3 See Keller [10] and Mohd Roslan and Ashwin [18] for other contexts. 4 Part of our method is an adaptation of some of the techniques of Podvigina [19], concerning simple cycles of type Z, to obtain results in a far more general context. 5 The form of a basic transition matrix is given in (7).

The Rock-Scissors-Paper game has been studied to account for cyclic behaviour under various game dynamics. We use a two-person parametrised version of this game to illustrate how cyclic behaviour is a dominant feature of the replicator dynamics. The cyclic behaviour is observed near a heteroclinic cycle, in a heteroclinic network, with two nodes such that, at each node, players alternate in winning and losing. This cycle is shown to be as stable as possible for a wide range of parameter values. The parameters are related to the players' payoff when a tie occurs.JEL codes: C72, C73, C02

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