Most people of the post-Baby Boomer generation have only seen a “normal” calculator; identical in function from the time they were children to the present day. The thing fits in your pocket and does basic math, some exponents, and simple trigonometry. For certain professions, however, you need more.

When I started engineering school, I saved up my birthday money and bought a Hewlett-Packard 28S scientific calculator. It was a little bit bigger than an iPhone, with a hard plastic flip-open case. Rather than trying to describe it, we have the technology to show you:

For a dork, it was heavenly. Multiple bases, matrix operations, RPN, expanded trig and logarithmic functions, 32 KB (yes, kilobytes) of programmable memory, and best of all, a small screen for graphing equations.

My wife, the accountant, has an old H-P 12C calculator, from her accounting and business classes. I don’t know what any of the buttons do, but it’s almost as imposing as my 28S was in its day. And like my 28S, it was designed to easily allow its operators to find the answers they’re looking for, without having to start from scratch every time.

The push/fold decisions in NLHE, especially those on the bubble of sit-n-go tournaments, have their own math problems to solve, equally as complex as a Laplace transform or an asset depreciation function. What ranges should I be pushing? Should I call a shove with A-Q offsuit when there is a player with 4 BB on the verge of eminent elimination?

The technique for answering these questions is based on a game theory puzzle called a Nash Equilibrium. In a Nash Equilibrium, every player in the game makes decisions based on the assumption that every other player is also making perfect decisions. This predictability takes an important variable out of the decision-making process, and the “correct” strategy becomes easier to determine based on the information we have.

In the case of tournament poker, the pertinent information is:

a) the amount of chips we have in relation to the total number of chips in play, called “chip equity”
b) the amount of money awarded to each place in the payout structure

Mapping the two, to determine the dollar value of each tournament chip you have, is called Independent Chip Modeling (ICM). There are a variety of good specialty ICM calculators online, and they are the lifeblood of a good sit-n-go player. The concepts emphasized by their use apply to spots in any tournament, not just single-table SNGs. If you play NLHE tournaments, and have never used an ICM calculator to analyze your key decisions, you are behind the curve.

The ICM calculator I use is at holdemresources.net/hr/sngs/icmcalculator.html. I like this one because it’s straightforward, and free. You enter the pertinent data into the form – blind & payout structure, and chip stacks for each player. Taking a recent single-table SNG bubble spot from the Hand Advice forum:

PokerStars Game #44920618027: Tournament #278651732, $20+$5+$2 USD Hold’em No Limit – Level VI (100/200) – 2010/06/01 13:49:56 ET
Table ‘278651732 1’ 9-max Seat #5 is the button
Seat 1: Small Blind (970 in chips)
Seat 3: Big Blind (6400 in chips)
Seat 4: Under The Gun (2370 in chips)
Seat 5: The Button (3760 in chips)
Small Blind: posts small blind 100
Big Blind: posts big blind 200
*** HOLE CARDS ***
Dealt to Under The Gun [X X]
Under The Gun: raises 2170 to 2370 and is all-in
The Button: folds
Small Blind: folds
Big Blind: calls 2170
*** FLOP *** [Tc 2s 5h]
*** TURN *** [Tc 2s 5h]
*** RIVER *** [Tc 2s 5h 7h]
*** SHOW DOWN ***
Big Blind: shows [X X]
Under The Gun: shows [X X]

…the ICM calculator data looks like:

When you click “Calculate,” you’re presented with two sets of data. The first is the break-even point, from a differential in chip equity perspective, for the first three player’s shoving ranges. The second is a fully defined set of ranges for shoving, calling and over-calling for each player.

If you’ll indulge me, we’ll take the second part first, and take on the equity-change discussion later. The full set of range data for the hand in question is shown in the following format:

This says Under The Gun, first to act, should shove with the top 11.5% of hands (PU = Push). The button should call (CA = Call) with TT+, AKs. If The Button calls, the Small Blind (very short) should only call (OC = Overcall) with AA, and the Big Blind with QQ+. If The Button folds, the Small Blind should call with the top 5.9% of hands, and the Big Blind overcalls with 99+, AQs+.

If Under the Gun folds, the Button should shove with 55+, any ace, and any Broadway cards. The calling ranges from the blinds are almost identical to those to an Under The Gun shove, as the stacks are similar.

Contrary to what may feel obvious (it did to me), the Small Blind should not be open-shoving Any Two Cards. Top 50% is a wide range of hands, but it’s tighter than I presumed before I ran the calculation. He has little fold equity against an opponent who can afford to call off light.

The criticism people have off Nash ICM calculations is based in its game theory roots: the assumption all players are following perfect strategy. I understand, and agree with the criticism – but not when it comes to the shove/call/over-call ranges. I am being given a guide as to what hands I should play, and what my opponents should. If either one of us don’t, that’s not the calculator’s fault.

Making a play with a hand out of the mathematically correct range is a leak to be fixed, and an ICM calculator is a perfect analysis tool with which to analyze the solution. If you want to be serious about the game, you should spend some time away from the tables, going over hand histories and making sure the calculator agrees with your play. It does have its imperfections…but that’s a topic for next time.

* Read Part 2: Poker Calculator 2 – Imperfect Applications

grapsfan Paul Herzog