Say there are $2$ players. Each is given one card, and the person with the higher number wins. The first player goes and can either bet 1 dollar or fold (giving the pot to the second player). If the first player bets 1 dollar, the second player can either call by putting 1 dollar in the pot, or fold giving the pot to the first player. If the second player calls, both people show their cards and whichever card is higher wins. Say that you are the first player. Assume you play this game for a while the second player knows your strategy and plays optimally based on it. You are not allowed to change your strategy. Given that the pot starts with $1$ dollar every time, what's the optimal strategy here?
My intuition is that if we have a large card, then we should always call. If we have a "middle" card then we should bluff with a certain percentage. If we have a low card, we should fold. I'm struggling to figure out the probability of how often I should bluff.