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[Tables]

(Note: This challenge will require use of simulators, but also a lot of non-simulator related thinking. I also don't know if this challenge has a solution, but I suspect it likely does)

The challenge is simple in concept: Construct a Dominion board with three 'optimal' strategies, such that strategy A will beat strategy B, strat B beats strat C and strat C beats strat A, and there isn't a strategy which beats all three.

Essentially the challenge is two parts: Firstly making the board so these strategies exist, and second making some simulator scripts which show the A>B>C>A strategies. Note that if two of the strategies are very close (e.g. C beats A 45.5-45) then it's not unlikely that A could just be optimised to beat C, and C might not be able to recover that loss, which would make A simply the dominant strategy.

I've done a little thinking about this, and have the following thoughts. None of them will probably be that mind-shattering, though.

To get the RPS effect, you're certainly going to need some kind of player interaction. Although I'm not sure what kind would work.
A good start might be using attacks and counter-type defences - e.g. a player buying Tunnels might not do well against the Council Room player, but would have a good shot against the militia opponent. Or using cards which turn specific opponent attacks into an advantage.
The main difficulty is probably going to be there NOT being an optimal strategy that beats all three. I reckon finding three strategies in a (rather strong) RPS effect might be a good start, though.

[WanderingWinder]

Note that this has been discussed before:
http://forum.dominionstrategy.com/index.php?topic=712.msg9832#msg9832

[DrHades]

Okay, I have an idea: Trader-Ambassador-BMU. Pretty sure this solves the puzzle...

[DrHades]

Another solution and this was harder, because I decided to come up with something without reactions:

Council Room + Militia beats Council Room - Militia is good, it gives a slight advantage.
Council Room beats Library + Militia - Militia is good, but not so much that it compansate horrible call going with Library against Council Room.
Library + Militia beats Council Room + Militia - Both sides have Militia, but it is obvious who founds it more annoying.

I didn't try anything to beat all 3 of them, but I am afraid some kind of Militia+CR+Library might work...

[HiveMindEmulator]

I think this is hardly a "challenge". There are so many such situations involving reactions or defenses. One example related to a recent game I played is:
Wharf > Wharf+Trader > Wharf + Sea Hag > Wharf.
Or similarly:
Young Witch > Jack > Jack + weak bane (herbalist) > Young Witch.

The interesting thing about these "rock paper scissors" scenarios is that they aren't actually like RPS, since you play sequentially. These situations actually give later players an advantage.

[ehunt]

I had an interesting game once with fishing village, talisman, treasure map, remake, wharf, and familiar. It was an RPS type situation and I was very happy to be 2nd player (and, more importantly, to split 3-4 and not 4-3.) The opponent opened remake. I ended up opening fishing village talisman, then rushing for wharves. with talisman and wharf, it's easy to buy and pair up  treasure maps very early and there's no need to remake anything.

if the opponent opens potion, i open remake and by the time he's got a familiar, i've got a thin deck with a wharf or two.

if the opponent opens talisman, i open potion.