# Game Theory Optimal

A couple of commenters scoffed at the conclusion to my recent PokerStars recap of Martin Jacobson’s all in hands, where I suggested that Jorryt van Hoof—who had open-raised before two players he had covered 5:1 both went all in—should have considered calling the all ins.

To recap, this was hand 224, where William Tonking was eliminated by eventual champion Martin Jacobson. There were only four players remaining. Van Hoof was the chip leader with 101.625 million, Felix Stephensen was in second at 57.2 million, Jacobson had 21.5 million, and Tonking was the shortest, with 20.15 million. Blinds were 500,000/1,000,000 with a 150,000 ante. Stephensen was big blind, Jacobson was small blind. Van Hoof opened under the gun for 2.2 million, Tonking went all in from the cutoff, and Jacobson followed suit. Stephensen folded his big blind. The pot holds 42.45 million.

Van Hoof opened with just a Q7, which, according to the naysayers, had horrible equity for a call against two shoves. My reasoning was that in this particular situation, the possibility of knocking out two players, ensuring a \$1.35 million pay jump from third-place money to second-place money, and increasing the chance of the \$6.2 million jump to first place, made the chance worthwhile. In fact, had he called, van Hoof would have won the hand with a queen hitting the river.

Let’s take a look at some possibilities.

Odds and Numbers

Before the hand began, Tonking had 20BB with an M-ratio of 9.6. Jacobson had 21BB and 10.2M. And van Hoof had more than 101BB with M of nearly 50. Van Hoof has to call 19.15 million to close pre-flop action, getting 2.21:1 in the 43 million chip pot.

If we assign the tightest Sklansky hand ranges that include the deuces Tonking shoved with (#7, top 30.9% of hands) and the tens that Jacobson re-shoved with (#2, top 4.4%), van Hoof’s hand has about 22% equity pre-flop. That puts it behind both of the other players’ ranges. Card odds are 3.5:1.

From a conventional calculation of whether to call, this is definitely a fold. That is, in fact, the option van Hoof takes immediately after Jacobson’s re-shove.

What Happened

The way the hand played out was van Hoof remaining a substantial chip leader, with 99.275 million chips, Stephensen in second place with 56.05 million, Jacobson in third with 45.150 million, and Tonking out of the game in fourth place.

The Call

If van Hoof had called, there would have been a pot of 62.1 million, consisting of a main pot with 59.4 million and a side pot between van Hoof and Jacobson of 2.7 million.

Possibilities

1. Tonking wins main pot; Jacobson wins side pot. If Tonking’s deuces hit a set or straight and Jacobson’s tens held up otherwise, Tonking would have ended the hand with 61.6 million and Jacobson would have been sucking air with 2.7 million, with van Hoof at 80.125 million after the call.
2. Tonking wins main pot; van Hoof wins side pot. Tonking’s deuces hit set/straight, and a queen pairs van Hoof’s top card. Jacobson is eliminated, Tonking at 61.6, van Hoof at 80.125.
3. Jacobson beats Tonking. Jacobson has 64.8 with both main and side pots, Tonking eliminated, van Hoof with 80.125.
4. Van Hoof wins. Tonking and Jacobson eliminated. Van Hoof has 144.925 million heads up with Stephenson.

Payouts

Payouts for the top four spots in the November 9 this year had a steep jump between first and second because of the \$10 million guarantee. Here’s how it was actually distributed.

PlacePlayerPrize
1Martin Jacobson\$10,000,000
2Felix Stephensen\$5,145,968
3Jorryt van Hoof\$3,806,402
4William Tonking\$2,848,333

ICM

These tables show how ICM affects the value of each stack under the various scenarios.

Actual PayoutsPreflop ICMActual Result ICMMain: Tonking, Side: Jacobson (ICM)
Jacobson, \$10 millionvan Hoof, \$7.34 million (101.6M chips)van Hoof, \$7.32 million (99.28M)van Hoof, \$6.72 million (80.13M)
Stephensen, \$5.15 millionStephensen, \$5.96 million (57.2M)Stephensen, \$6.01 million (56.05M)Tonking, \$6.12 million (61.6M)
van Hoof, \$3.8 millionJacobson, \$4.29 million (21.5M)Jacobson, \$5.61 million (45.15M)Stephensen, \$5.93 million (56.05M)
Tonking, \$2.85 millionTonking, \$4,21 million (20.15M)Tonking, \$2.85 millionJacobson, \$3.03 million (2.7M)
Actual PayoutsMain: Tonking, Side: van Hoof (ICM)Main and Side: Jacobson (ICM)Main and Side: van Hoof (ICM)
Jacobson, \$10 millionvan Hoof, \$6.83 million (82.8M)van Hoof, \$6.42 million (80.13M)van Hoof, \$8.64 million (144.4M)
Stephensen, \$5.15 millionTonking, \$6.16 million (61.6M)Jacobson, \$6.25 million (64.3M)Stephensen, \$6.5 million (56.05M)
van Hoof, \$3.8 millionStephensen, \$5.97 million (56.05M)Stephensen, \$5.96 million (56.05M)Jacobson, \$3.8 million
Tonking, \$2.85 millionJacobson, \$2.85 millionTonking, \$2.85 millionTonking, \$2.85 million

Van Hoof’s fold and the elimination of Tonking in 4th resulted a loss of only \$20,000 of equity according to the ICM calculations.

Against the ranges assigned to Jacobson and Tonking, the most likely result of a call—a win by Jacobson—was approximately 52% (56% using the actual cards dealt), resulting in a loss of \$920K.

A Tonking-range win was likely to happen about 26% of the time (only 17% with his actual hand) for a loss of \$620K if Jacobson took the side pot (70% of the time Tonking won, or 18% overall) and \$510K if van Hoof beat Jacobson for the 2.7 million chips.

The 22% chance that van Hoof knocked out both of the other players results in a \$1.3 million increase in equity.

If we run the numbers for the call we get:

-0.52(\$920,000)-0.18(\$620,000)-0.08(\$510,000)+0.22(1,300,000) =
-\$478,400-\$111,600-\$40,800+\$286,000 =
-\$344,800

While that represents a more than 17-fold increase in equity loss over what an ICM calculation of the fold represents, and many would argue that a fold was therefore the optimal move, \$350,000 is only about 25% of the difference between third-place money and second-place money, and less than 6% of the difference between the third-and first-place payouts.

Considering that van Hoof would have remained the chip leader by more than 15 million chips in any scenario, that he could never finish worse than third no matter what the result, and that there was no scenario in which he finishes worse than third place, I still maintain a call would have been the optimal move.