Simplified Three-Card Poker Game

There are two players: Player 1 and Player 2.

Before the game begins, each player pays $\$1$ into the pot, so the initial pot is

$$\$2.$$

The deck contains exactly three cards:

$$A,\quad K,\quad Q,$$

with ranking

$$A>K>Q.$$

Player 1 receives one card, Player 2 receives one card, and the third card is set aside.

Both players may look at their own card.

Player 1 acts first.

Player 1's action

Player 1 may:
• bet, which means adding exactly $\$1$ to the pot; or
• check, which means adding nothing.

Player 2's action

After seeing Player 1's action, Player 2 may:
• fold, immediately giving the current pot to Player 1; or
• follow, meaning Player 2 matches Player 1's total contribution.

Therefore:
• if Player 1 bet, Player 2 must add another $\$1$ to follow;
• if Player 1 checked, following costs nothing additional because both players have already contributed their $\$1$ ante.

If Player 2 follows, the cards are revealed and the player with the higher card wins the pot.

Subproblem A

Find the optimal strategy for each player.

Subproblem B

Suppose Player 1 bluffs by betting with the Queen with probability

$$p.$$

Suppose Player 2, when holding a King and facing a bet, follows with probability

$$q.$$

Derive equations that make the relevant decisions indifferent, and solve for $p$ and $q$.

Subproblem C

Using the optimal strategies, find Player 1's expected net value per hand.