How to choose between multiple Nash equilibria and solution concepts? Please help!  I don't remember what happened last night but I woke up this morning in a prison cell with nothing but my pile of game theory lecture notes. The guard came in and says they have another rational person in the cell next door, and we both have to play a one-shot nonzero-sum matrix game without communicating.    (The utilities include various prison-themed events like early release, food, being beaten with sticks etc.)
Using the algorithms from these books, I've worked though the game matrix and found it has several Nash equilibria.   Some of them have various 'solution concept' properties like being trembling hand perfect or evolutionarily stable. Some have higher overall social utilities and some have more egalitarian outcomes.
My question then is: which of these equilibria should I choose to actually play?
My pile of lecture notes don't seem to say anything very helpful about this, instead they all have vague statements like 'solution concepts might explain some choices seen in the world'.   I don't care about explaining what is seen in the world, I want to know the optimal action to take right now.
In particular -- some of the N equilibria here are mixed strategies and some of the others are pure.   If no-one gives me any better advice, I'm going to default to picking one of the strategies completely at random (ie. with 1/N probability).   However if I do this, I'm effective creating and using a new mixed strategy with new action weights, which wasn't in the original set, and therefore isn't optimal any more?   Can anyone explain if this is a good idea or not, and why?   And why are all the authors of these lecture notes avoiding these questions?
 A: Most of game theory concerns interacting agents: what is optimal for you to do depends on what your opponent does (and vice versa). Thus, most of  game theory focuses on equilibria, interpreted as profiles of strategies were all agents are playing optimally given how their opponents are playing.
There are classes of games where it is possible to recommend an essentially unique strategy; e.g., zero-sum games. It is often possible to pinpoint strategies that one should not play; e.g., avoid strictly dominated strategies. But in general examples (including Battle of Sexes), there is no overarching principle when you try to solve the problem of what to do in isolation.
Finally, if you are really interested in a theory that selects a unique strategy profile for every game, you should read J. C. Harsanyi and R. Selten. A General Theory of Equilibrium Selection in Games. Cambridge, MA: The MIT Press, 1988. Beware: their advice is going to work in general only if it is common knowledge that you and your opponents have read the same book, understood its contents, and decided to follow its recommendation.  
A: There exist a several solution concepts, for this situation, but don´t exist any "universal", valid for all types of games. Operation is limited to situations where everyone is guided by the same principle.
Simple examples of principles.


*

*If unique NE from set of NE is Pareto dominant, use it (that mean,
that this equilibrium gives every player at least as much as the
other equilibrium)



*If unique NE from NE set is efficient, use it. (NE, which maximizes a sum of payoffs)

*Use a symmetric NE, if exist (especially, for a symmetric game really exist, but no necessarry pure)

*If you play repeatedly, watch where the NE game converges and use it first



But these concepts are not universal . For example, a Chicken game:
$$ \left(\begin{array}{cc}
2/2 & 5/0\\
0/5 & -10/-10
\end{array}\right)$$ 
The simple 2x2 symmetric game has a 2 efficent equilibria $(5,0)$ and $(0,5)$ and one (worse) symmetric mixed equilibrium. But the goal of the game is to force the opponents to accept the equilibrium $ (5,0) $, a vice versa.
Appendix: (Reply to comment, but contains a picture)
Experimentally one has proved, that the final used equilibrium  is not predictable. From example, an experimental game of 7 players, from Camerer-Behavioral game theory. Experiments in strategic interaction, page  13-15, has a many of NE, but exactly two symmetric: 3 with payoff 60 for every player and 12 with payoff 112 for every player. 
Experimental results with repeated game and accross groups are here Decision between 12 and 3 is unpredictable! Despite the fact that 12 dominates 3. 
Note: In this game, payoff to player i has been a function of strategy of i and median strategy. Use 12 against median 3 leeds to payoff -100 
