CFR on Kuhn poker: bluff rate, best-reply gain and game value after 1,000 to 100,000 iterations
Our own run, 8 November 2026 (UTC), for the article "How poker AIs learn to bluff". Counterfactual regret minimization (CFR) with chance sampling on Kuhn poker (three cards, one bet; player 1 acts first), seed 20261108, reading off the average strategy over all iterations. Best-reply gain (exploitability) is how much a best reply to each seat gains over the game value, summed over both seats, in bets per hand: it is 0 for a perfect strategy.
What we found
- After 1,000 iterations the strategy bets the Jack 0.211 of the time and the King 0.511; a best reply gains 0.0354 bets per hand.
- After 10,000 iterations: Jack 0.166, King 0.581; a best reply gains 0.0111 bets per hand.
- After 100,000 iterations: Jack 0.158, King 0.497; a best reply gains 0.0082 bets per hand. So the strategy is close to unexploitable, not solved.
- After 100,000 iterations the King bets 3.14 times as often as the Jack (the textbook ratio is 3), and the Queen calls a bet after checking 0.506 of the time against the textbook's Jack rate plus 1/3, 0.492.
- The Queen bets first only 0.00004 of the time after 100,000 iterations: she checks.
- The first player's game value, two ways: exact enumeration of the 100,000-iteration strategy gives -0.0554 bets per hand, and a simulation of 2,000,000 hands gives -0.0541.
- The textbook value is -1/18, which is -0.0556 bets per hand; computed in exact fractions it is -1/18 for any Jack-bet rate from 0 to 1/3 (rates 0, 1/6 and 1/3 checked).
| Self-play iterations | Jack bets | King bets | Best-reply gain (bets per hand) | Exact value to player 1 |
|---|---|---|---|---|
| 1,000 | 0.211 | 0.511 | 0.0354 | -0.0541 |
| 10,000 | 0.166 | 0.581 | 0.0111 | -0.0553 |
| 100,000 | 0.158 | 0.497 | 0.0082 | -0.0554 |
What this cannot tell you: one seed, and Kuhn poker is the smallest poker; the equilibrium is a family (any Jack-bet rate from 0 to 1/3), so the rate CFR lands on depends on the run.