Optimal Pre-flop All-in Strategy for 3 or More Players

Introduction

In a previous post, we looked at optimal pre-flop all-in strategies in heads-up situations, but this time we’ll look at pre-flop all-in (push or fold) strategies in situations with three or more players. A well-known textbook approach is the strategy using a table called the Power Number, and we will first provide an overview of this approach. Next, I will explain the problems with power numbers and explain an optimal preflop all-in strategy using counterfactual regret optimization (CFR). CFR is not necessarily the optimal strategy when there are three or more players, but it is expected to be close to the Nash equilibrium if everyone adopts the same strategy, so there is some merit in presenting it.

Power Number

The power number is calculated by comparing the ratio of the effective stack to the value below, called the M value, with the value in the table below, and going all-in if the latter value is larger. The M value is calculated using

　　M value = 1.5 + Ante × Number of players.

This is a number that represents the amount of chips that will be lost if all positions are folded in one rotation. The power number can be calculated using

　　PN = Effective stack ÷ M value × number of players making decisions after the player.

Your effective stack is basically your own stack, but if you are the chip leader, it is the stack of the player with the largest stack who will make a decision after you.

 The numbers to compare with the power number are given for each hand in the table below.

	A	K	Q	J	T	9	8	7	6	5	4	3	2
A	∞	∞	∞	∞	∞	50	37	32	28	31	27	26	24
K	∞	∞	∞	75	66	44	17	15	14	13	11	10	9
Q	∞	48	∞	75	58	38	16	11	10	8	8	8	8
J	50	31	26	∞	58	39	21	12	7	7	7	7	5
T	36	19	17	22	∞	43	26	15	10	9	7	5	4
9	27	12	9	9	11	∞	31	17	10	9	3	3	–
8	24	10	8	8	10	10	66	19	15	10	5	–	–
7	22	9	6	5	6	7	10	58	15	10	9	–	–
6	18	9	6	4	3	3	4	7	51	11	10	4	–
5	21	9	6	4	–	–	–	–	–	44	11	8	–
4	18	8	5	3	–	–	–	–	–	–	39	6	–
3	16	8	5	3	–	–	–	–	–	–	–	33	–
2	13	7	4	3	–	–	–	–	–	–	–	–	28

Compare your power number with the value in this table, and if PN < the value in the table, you go all-in.

 The following points should be noted:

1. The recommended stack size is 20bb or less.
2. If there is a player ahead of you who is all-in, you will need to choose your hand more tightly than this.
3. This is basically a strategy to maximize expected value, and towards the end of a tournament you will need to be more selective about your hand.

Problems 1 and 2 can be solved by normal calculations, so below we will use CFR to calculate the optimal strategies for three and four players.

Optimal Pre-flop All-in Strategy with CFR

The optimal preflop all-in strategy for a three-player game can be calculated using the same algorithm as for heads-up. However, since there are six possible states \(\small s\) when determining each player’s strategy \(\small \pi(a, s)\), the expected regret in each state must be calculated separately. Specifically, there are six:

1. Position is BTN
2. Position is SB, BTN goes all-in
3. Position is SB, BTN folds
4. Position is BB, BTN and SB go all-in
5. Position is BB, BTN goes all-in and SB folds
6. Position is BB, BTN folds and SB goes all-in

The number of hand charts that must be calculated increases, which makes the convergence of the simulation worse. Because the calculation process is very heavy, I have included a link so that you can see only the calculation results.

 Similarly, we calculated the optimal strategy for four players. The calculations took more than a week, but some parts did not appear to converge, so I have manually corrected some parts based on my own intuition, so please use this as a reference. When there are four players, there are 14 possible strategic situations:

1. Position is CO
2. Position is BTN, CO goes all-in
3. Position is BTN, CO folds
4. Position is SB, CO and BTN go all-in
5. Position is SB, CO goes all-in and BTN folds
6. Position is SB, CO folds and BTN goes all-in
7. Position is SB, CO and BTN fold
8. Position is BB, CO,BTN and SB go all-in
9. Position is BB, CO and BTN go all-in, SB folds
10. Position is BB, CO and SB go all-in, BTN folds
11. Position is BB, BTN and SB go all-in, CO folds
12. Position is BB, CO goes all-in, BTN and SB fold
13. Position is BB, BTN goes all-in, CO and SB fold
14. Position is BB, SB goes all-in, CO and BTN fold

As with the three players case, I will provide a link so that you can see only the calculation results.

Conclusion

In reality, when there are three or more players, the all-in strategy may not be optimal due to CFR, but on average it is considered a reasonable strategy. Also, if there are three or more players who go all-in, even if you have a strong hand, it will inevitably feel like a game of luck, so it may be better to use it carefully if there are players who go all-in repeatedly.

 The calculations can be expanded to include more than five players, but there are too many tables and it would be impractical to memorize them all. The calculations also have poor convergence, so the calculations may have to run for more than a week, which is something the author does not want to do. So what happens when there are five or more players? It may be surprising, but when you use the Preference Order Model, even when there are five or more players, it feels like they are going all-in with a fairly realistic hand. Perhaps when there are five or more players, Push or Fold is a game of luck, so it comes down to deciding whether to go with a strong hand, and there may not be much need to consider whether it is optimal. In the future, I may only adopt the strategies calculated by CFR for NPC strategies in mobile games I develop when there are up to four players. For now, I’ve modified the All-in or Fold game so that it can be played Game Theory Optimally with up to four players.