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The following is the first part of an excerpt from our next book, The Intelligent Poker Player by Philip Newall. We expect to have the book for sale by either late March or early April 2011. The second part of this excerpt will appear in next month’s magazine.

Information Hiding

Backwards induction is the method for solving finite sequential games of perfect information such as chess, go, or tic-tac-toe. Starting from the opening move, you draw a game tree — a diagram that represents each possible move a player can make as a single branch. The tree grows with each move until it eventually covers each possible course of play. Starting at the end branches, you prune the tree by deleting each sub-optimal decision in every round. When you finally work back to the opening round, the only branches left will represent the Nash equilibrium.

This method can be easily used to solve a simple perfect information game like tic-tac-toe. Although it can in principle be used for a more complicated games, chess and go are both too complex for today’s computers to solve via this method.

See the box below for the backwards induction solution to the sequentially played Prisoners’ Dilemma game:

 

Figure I: Prisoner’s Dilemma

Backwards Induction Solution

 

Backwards induction cannot be used to solve poker since it’s a game of incomplete information, (each player’s holecards are known only to him), meaning you are uncertain about which branch of the game tree the current play is on. But the more information about your hand you reveal, the easier it becomes for your opponent to pick his most profitable play since he can eliminate certain branches from consideration.

The classic example of revealing information is through tells — some mannerism that indicates the strength of your hand.¹ This is less of a factor in online play, although if your speed of decision is correlated with your hand strength, it can reveal information to an observant opponent.

Your choice of action can also reveal information. Suppose a freeroll qualifier overbets the pot, moving all-in preflop during the first level in the Main Event at the World Series of Poker. Busting out of the first level of this prestigious tournament would be a massive disappointment for this player, so using this information wisely, we can surmise that he must be holding aces — the only hand he wouldn’t be afraid to make this play.

As we can see, information hiding is a concept that crops up in all poker variants. The idea is that it’s beneficial to play parts of our range in a consistent manner during early decision points. We do this in order to conceal the strength or weakness of our hand until later decision points where the bets and pots can be larger and thus more meaningful.

As a broad illustration of the concept, suppose we put many more bets into the pot on the opening round with our strong hands than with our weak hands. This is doubly disadvantageous since our strong hands lose the element of surprise, and furthermore, our weak hands are inviting our opponents to put pressure on us during the later streets because they are no longer provided cover by our better hands. Therefore, early on, both sets of hands benefit from being played in a consistent manner.

This concept is most vital for limit games where the cost of giving our hand away on the opening round is seldom worth the value of getting an extra bet in relation to the size of the pot. In big bet games, particularly with low stack to pot ratios, the concept is less relevant, although, in deep stack play it once again becomes a major factor.

The earliest articulation of this concept that I’ve found is in Nesmith C. Ankeny’s Poker Strategy, originally published in 1981. This book presents a game theoretic approach to five card draw, jacks or better. On the opening round, players are allowed to open with a bet only if they hold a pair of jacks or better, otherwise they must pass. (Once a player opens, you must either fold, call, or raise; if all players pass, then the next hand is dealt.)

Ankeny’s information hiding rule is very simple: never draw two cards. Suppose, in an eight handed game, the under the gun player opens. Due to the rules of the game, we can say that he must hold at least a pair of jacks. However, due to his early position, his minimum opening hand is probably a pair of aces. We raise from middle position, the opener calls, and we enter the draw heads up, at which point we draw two cards. What is our range of hands? We must hold a made hand since drawing two cards to a straight or flush is suicidal, we cannot have two pair, and one pair is not strong enough to raise since our opponent’s hand is at least a pair of aces. Simple hand reading concludes that our only possible hand is three of a kind. In a situation with wider ranges, we might possibly want to draw two cards with a pair and a high kicker. However, these are the only hands that will be motivated to draw two cards.

Ankeny’s solution is to only ever draw three, one, or zero cards. Drawing three cards admits to holding only a pair before the draw — implying a relatively weak distribution of hands after the draw. Drawing one card results in a wider pre-draw distribution of straight and flush draws (without a pair of jacks or better, these hands are unable to open the pot), two pair, three of a kind, and the unlikely four of a kind. This is a nicely balanced distribution of hands that is able to make a range of value bets and bluffs after the draw.

Note that the cost to three of a kind of drawing only one card is minimal. The chances of improving to a full house or better is only 1.9 percent less when drawing one card rather than two,^{2 3} and the benefit of hiding the hand’s strength in a wider distribution of hands far outweighs any reduction in post-draw expected hand strength.

Another early expression of the concept was in David Sklansky’s The Theory of Poker. In “The Cost of Giving Your Hand Away” he says:

If opponents know exactly what you have, they will never make a mistake except on very close mathematical decisions. The more your play gives away what you have, the less likely it is that your opponents will make a mistake. Yet you want them to make mistakes. Creating mistakes is, in a sense, the whole objective of the game.⁴