Poker with three cards · Problem 75

A deck has three cards: a jack, a queen and a king (king highest). Two players each put 1 dollar in the pot and each gets one card; the third card is not used. Each player sees only their own card.

Player 1 may check or bet 1 dollar.
If player 1 bets, player 2 may call (add 1 dollar, then the higher card takes the pot) or fold (player 1 takes the pot).
If player 1 checks, player 2 may check (the higher card takes the pot) or bet 1 dollar. After that bet, player 1 may call (higher card takes the pot) or fold (player 2 takes the pot).

So a hand is won or lost by 1 dollar, or by 2 dollars when a bet was called. Both players play optimally and may randomise their choices.

What is player 1's expected gain per hand?

Idea. Strip away the choices that are never right, guess an equilibrium in which each player's mixing makes the other indifferent, check that no one can do better by deviating, and then compute the value by going through the six possible deals.

Step 1: the obvious choices. Facing a bet, the jack always loses a showdown, so it folds; the king always wins one, so it calls. Player 2 with the king bets after a check. Player 2 with the queen checks after a check: a bet would be called only by the king. Player 1 with the queen checks: a bet would be called only by the king and folded by the jack, which is worse than checking.

Step 2: a candidate equilibrium.

Player 1 always checks. Facing a bet, they fold the jack, call with the queen with probability $\frac{1}{3}$ , and call with the king.
Player 2, after a check, bets with the king, checks with the queen, and bets (bluffs) with the jack with probability $\frac{1}{3}$ . Facing a bet, they fold the jack, call with the queen with probability $\frac{1}{3}$ , and call with the king.

Step 3: nobody gains by deviating.

Player 1 with the queen, facing a bet. Player 2 holds the jack or the king, each with probability $\frac{1}{2}$ , and bets always with the king, a third of the time with the jack. So given the bet, the jack has probability $\frac{1/3}{1 + 1/3} = \frac{1}{4}$ . Calling gives $\frac{1}{4} \cdot 2 - \frac{3}{4} \cdot 2 = - 1$ , the same as folding. Indifferent.
Player 2 with the jack, after a check. Checking loses 1. Bluffing loses 2 against the king; against the queen it wins 1 two thirds of the time and loses 2 one third of the time, so 0 on average. Overall $\frac{1}{2} (- 2) + \frac{1}{2} \cdot 0 = - 1$ . Indifferent.
Player 1 betting instead of checking. With the king, betting wins $1$ against the jack (fold) and $\frac{1}{3} \cdot 2 + \frac{2}{3} \cdot 1 = \frac{4}{3}$ against the queen: $\frac{7}{6}$ on average. Checking wins $\frac{1}{3} \cdot 2 + \frac{2}{3} \cdot 1 = \frac{4}{3}$ against the jack and $1$ against the queen: also $\frac{7}{6}$ . With the jack, betting averages $\frac{1}{2} \cdot 0 + \frac{1}{2} (- 2) = - 1$ , and checking loses 1 for sure. Indifferent in both cases, so checking is a best reply.

Step 4: the value. Each of the six deals has probability $\frac{1}{6}$ . Player 1's average gain in each (player 1's card first):

J against Q. Both check and the queen wins: $- 1$ .
J against K. Check, bet, the jack folds: $- 1$ .
Q against J. The jack bluffs a third of the time; then the queen calls a third of the time, so player 1 gains $\frac{1}{3} \cdot 2 - \frac{2}{3} \cdot 1 = 0$ . Otherwise both check and the queen wins 1. Average: $\frac{2}{3}$ .
Q against K. Check, bet; the queen calls a third of the time: $- \frac{1}{3} \cdot 2 - \frac{2}{3} \cdot 1 = - \frac{4}{3}$ .
K against J. The jack bluffs a third of the time and the king calls; otherwise the king wins the showdown: $\frac{1}{3} \cdot 2 + \frac{2}{3} \cdot 1 = \frac{4}{3}$ .
K against Q. Both check and the king wins: $1$ .

These add up to $- \frac{1}{3}$ , so the value for player 1 is $\frac{1}{6} \times (- \frac{1}{3}) = - \frac{1}{18} .$

Remarks. Player 1 has a whole family of equilibria: bluff the jack with any probability $α \leq \frac{1}{3}$ , bet the king with probability $3 α$ , and call with the queen with probability $α + \frac{1}{3}$ . All give the same value $- \frac{1}{18}$ : acting second is worth about 5.6 cents per hand.