  We analyse the typical structure of games in terms of the connectivity
properties of their best-response graphs. Our central result shows that almost
every game that is 'generic' (without indifferences) and has a pure Nash
equilibrium and a 'large' number of players is connected, meaning that every
action profile that is not a pure Nash equilibrium can reach every pure Nash
equilibrium via best-response paths. This has important implications for
dynamics in games. In particular, we show that there are simple, uncoupled,
adaptive dynamics for which period-by-period play converges almost surely to a
pure Nash equilibrium in almost every large generic game that has one (which
contrasts with the known fact that there is no such dynamic that leads almost
surely to a pure Nash equilibrium in every generic game that has one). We build
on recent results in probabilistic combinatorics for our characterisation of
game connectivity.
