Some measures have been proposed, see
The basic idea from the first paper is to estimate
skill = (potential learning effect) / (potential learning effect + potential random effect)
which gives skill as a number between 0 and 1. Alas, these effects are analytically computable only for "easy" games. For a one-player game, the above equation comes down to
skill = (Gm - G0) / (Gu - G0)
where the G's are the expected net gains of three players
'0': a beginner who plays the game in the naive way of somebody who has
just mastered the rules of the game.
'm': a real average player who can be thought to represent the vast majority
'u': a virtual average player whom we tell in advance (i.e. before he has
to decide) the outcome of the chance elements.
As an example they calculate for American Roulette: Gu = 35 and Gm = -1/74, the latter corresponding a "simple" play (e.g. rouge/noir,
pair/impair). The value for G0 is actually a matter of debate, even for this game. If the beginner goes for a simple strategy, then the skill is 0 obviously. However if G0 is for a non-simple strategy (e.g. plein, cheval, carre), then G0 is -1/37 (i.e. worse average loss.) So with the latter assumption, there is a minor potential for learning, so skill is 0.0004. I have to say I'm a little miffed that they use French terminology for the American Roulette; alas they source they cite for further details is in Dutch.
For Blackjack they derive from a computer simulation that Gm = 0.11, Gu = 27, and take G0 = -0.057 for a "mimic the dealer" strategy, and from that obttain a skill of 0.006.
For games where players compete directly and strategies like sandbagging or bluffing matter (these are the only games called multi-player games in game theory by the way), the second paper has a more sensible approach in that it considers players potentially changing strategy a source of randomness. They use the same skill formula as above (except that they calll the three types of players beginner, optimal and fictive player). The difference in their approach is that
the expected gains for player i as an optimal player
are given by his expected gains in the Nash
equilibrium of the related two-person zero-sum
game against the coalition of the other players
and for the "fictive" player they also assume that he knows the the outcome of the
randomization process of his opponents.
Alas there aren't any examples that are intersting but simple enough to relate in detail here. They compute for a simplified version of drawpoker a skill of 0.22.
Both papers emphasise however that the exact skill value depends on the definition/assumption of beginners’ behavior.
An experimental approach is needed for more complex games of practical interest, e.g.
Those players identified a priori as being highly skilled achieved an average return on investment of over 30 percent, compared to a -15 percent for all other players. This large gap in returns is strong evidence in support of the idea that poker is a game of skill.