Nash Equilibrium?

OK, so I'm out of my depth, by a lot, just by writing that title.  I saw "A Beautiful Mind" but that doesn't make me an expert on Nash equilibrium or game theory.  Somebody who really knows that stuff can call me out on my posing.  But...

In my last post on stacking of tennis lineups (here), I said once we dump the "strength order" rule, I prefer having coaches specify lineups to the alternative I proposed earlier of random draws for match ups (here).  I think that both rules would result in the same outcome, the random match ups.

Let's say we make a rule that each coach writes down his line up on a line up card and the coaches exchange line up cards before the matches begin, just like baseball coaches exchange line ups with each other and the umpire at home plate before a baseball game begins.  No changes are allowed once the lineups are exchanged.  What order do you place your players?

If the opposing team has placed its players in a predictable order in previous matches, you would place your players in positions that give you the best chance to win.  That's obvious.  But would it make sense for any coach to play a predictable line up in previous matches?  Nope.  So what line up is not predictable?  Only a random one.  Any non-random method of setting the line up opens up the possibility of an opponent figuring out your method and selecting match ups that favor the opponent, not you.

Now it is possible to set traps and so forth, deception being a tried and true way to gain an edge.  A team could use one line up in all its easy matches and then pull the switcheroo for a later match against a big rival.  That could work.  Once.

So the "submitting line ups simultaneously without changes" rule leads to the random pairing of matches.  That's fine with me.

What other rules for revealing line ups might lead to different strategies?

Visiting team reveals position number one first.  Home team submits a match for number one and then reveals number two position.  The visitors submit their number two player and reveal number three, etc.  That might be fun.   I haven't experimented with that one.

A friend and I just tried a rule where one team submits their order of players and the other team gets to match.  But to make it more fair, the first team is allowed some number of switches.  We used three singles and four doubles and allowed one singles and one doubles switch.  That was a hoot.  For NCAA format of three doubles and six singles, a rule with one doubles switch (team or player?) and two singles switches might be fun to try.  This rule really puts the coach, and players if the coach wants their input, to work.  This one could be darn exciting.

Any way, those are my latest thoughts on this.  We need to get a ground swell of coaches and players to get rid of the conflict-inducing "strength order" rule we now have.  That conflict-inducing aspect of the current rule probably needs a post of its own.