On the other hand, if for some reason player A already knew that for all players it was good to purchase the auction prize for 9 or less and bad to purchase it for 10 or more, then yes player A should immediately bid 9.
You assume again perfect rationality and perfect information. Human beings are not perfect so player A could gamble, bid 9 and hope that everybody else undervalues the asset.
It also depends on how granular the currency is. If you could e.g. bid 9,01 there is little cost in not gambling and not immediately bidding 9.
1. In the case of cursed bottle, the correct amount is either the minimum amount or not to take cursed bottle. So in fact it is very possible to compute the exact amount.
I wanna see these computations. Seriously, I know nothing about solving stochastic games and your knowledge about it could probably help a lot in creating decent Dominion bots.
I hope that the irony comes through. No human being calculates the optimum amount for Mountain pass or Bottle Imp while playing the game, not even experts in stochastic game theory. We use heuristics, rules of thumb.
2. This does not come into play with cursed bottle. Paying more for the bottle imp is both a higher cost and a lower reward. On the other hand, if you had to pay more than the previous time, it could work very nicely.
Huh? Bottle Imp doesn't have two upsides, it has an upside and a downside. If another player takes Bottle Imp away at a moment which is good for you because the costs of -13VPs outweighs the benefit of auto-Wishes each turn you are happy that he does it.
And, this is crucial, the other player might not behave irrationally at all! His deck could differ a lot from yours so for him the benefits of auto-Wishes outweigh the -13VPs.
3. Again, this doesn't apply to cursed bottle because of how you have to underbid. Like, am I going to buy bottle imp for 6 hoping that my opponent will spend 5 on it? That's nonsense; they are losing less than me, and regardless they can still just buy it for min amount.
Ccreating an option for you to get rid of -13VPs is always good. Creating an option for you to get rid of auto-Wish is always bad. You ignore this basic trade-off of the card the entire time.
In a situation in which you have played Cursed Bottle, have 6 Coins and want to buy an Ironmonger (because that is the b best card right now) you have 2 Coins left. There isn't any difference cost-wise between paying 0, 1 or 2 for Bottle Imp so your choice depends on how direly you want to hang onto it and how much you value the option that somebody might snatch it away from you. The best option is Kingdom-, time-, deck- and player-dependent.
Holunder:
I think that you are trolling, sorry if you aren't, I won't respond to you anymore.
You do implicitly assume stuff which is not always automatically the case in a real game of Dominion: that decks are symmetric, that the evaluation of the strength of a card is equal among players and constant over time, that nobody makes mistakes and so on. I might disagree too harshly with these assumptions but I have no interest in trolling you; this is about the game, not the players.