How does the Lemke-Howson Algorithm choose a vertex? I am trying to get my head around the Lemke-Howson algorithm , in the scenario where it searches for a Nash Equilibrium in a Bimatrix game.
I have seen the Tableau operations, and I can followe the process relatively well.
However I do not understand some of the "higher level" concepts of the algorithm.
In some sources, from what I understand, it is described that the algorithm traverses along edges of the polytope (which represents the game), untill it reaches a solution.
If it is so, I couldn't understand what is the rule for choosing the next vertex which it will travel along. And how the Tableau pivot operations achieve that. And how does 'it know' the Nash Equilibrium is found?
The definition of label, and binding inequality wasnt explained well in the literature that i have read.
 A: See, e.g.,
B. von Stengel (2007), Equilibrium computation for two-player games in strategic and extensive form. Chapter 3, Algorithmic Game Theory, eds. N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, Cambridge Univ. Press, Cambridge, 53-78. 
Available at http://www.maths.lse.ac.uk/personal/stengel/TEXTE/agt-stengel.pdf


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*The first pivot is determined by a free choice of label to drop, which corresponds to choosing a single pure strategy. This dropped label is called the missing label and is missing throughout the computation until it is picked up in the final pivot.

*Subsequent pivots are determined as follows: after a pivot a new label is picked up, either this is the missing label and we are done, or it must be a duplicate label. If we have a duplicate we move away from the other (i.e. not the one we just reached) facet with this duplicate label.
In terms of implementation, the labels we have correspond to the co-basic variables in the current basis (and thus binding nonnegativities, that is, a binding inequality for one of the mixed strategy probabilities being zero, or for the slack associated with a best-response inequality being zero). In every basic feasible solution we visit except the first and last, there are two co-basic variables with the same (duplicate) label. 
