This article is the follow-up to Great minds: 10 players who excel at both chess and poker. We are going to look at 5 key concepts in poker and chess. These concepts are not always easy to understand but become exciting when you make the effort to explore them.
Variance is the reason why Magnus Carlsen doesn’t win every tournament he plays in. It can be described as the measure of the “glorious uncertainty of sport” (which competitive chess and poker both are!).
In chess, the difference between the Elo ratings of the two players gives the expected score according to this table:
For instance, a player rated 2193 playing against a 2000 will, on average, score 0.75 points (win = 1 point and draw = 0.5 points).
This table is also used to calculate a performance: if you score 7.5/10 against an average rating of 2000, your performance will be 2193.
The table doesn’t give the draw percentage. You could score 75% by winning 75% and losing 25% of the games or by winning 50% and drawing 50%.
The draw percentage depends on many factors like the level (top players draw more), the style (So draws more than Nepomniachtchi), the time control (the draw percentage increases with the time control), or the color of the better player (there are more draws when the stronger player has Black).
Based on François Labelle’s work, here is what we could expect from a 2800 player:
As we see, there is variance in chess, but its impact is limited and, soon enough, the better player will prevail.
In poker, variance plays a much more central role.
Without getting into too many details, let’s say that if you play poker for $100 ($1 big blind), after 100 hands, you will be considered as
- good if you win, on average, $5
- very good if you win, on average, $10
- insanely good if you win, on average, $20
As you have probably guessed, “on average” implies your win rate is subject to variance. As in chess, variance will depend on many factors like the format (tournaments have greater variance than cash games), the variant (Omaha has greater variance than Holdem), the number of people you play against (in cash games, fewer people means greater variance), or your style of play.
The following table gives the probability of losing money after a certain number of hands.
As we see, variance has a deep impact in poker and even though the better player will eventually prevail, the long-term is much, much longer than in chess.
Perfect information vs. imperfect information
Unlike chess, poker has hidden information (the opponent’s hole cards). Game theorists call chess a game of perfect information and poker a game of imperfect information.
This difference has a funny implication:
In chess, we usually want to have the move (except when in zugzwang). As both players have all the information, there is no giving-away when making a move.
In poker, we usually want to act last because we’ll have more information than our opponents had when they acted.
Perfect play, pure strategy, and mixed strategy
A perfect chess player could always play the same move every time they encounter the same situation. It seems self-evident, but as we are about to see, it’s not true in poker.
In chess there are theoretically only 3 possible evaluations of a position:
- White is winning
- The position is drawn
- Black is winning
NB: The traditional evaluations like += or +/- and the computer’s evaluation in centipawns like +0.22 are only useful in practice, between imperfect players.
According to game theory, playing chess perfectly simply means playing a move that doesn’t change the evaluation of the position.
If several moves do not change the evaluation of the position, you could choose one of them randomly or always chose the same move every time you encounter this position. Doing the latter would be following a pure strategy, one where randomness is not involved.
To understand perfect play in poker, you should first understand perfect play in rock paper scissors. Let’s say you play an AI which analyses its opponent’s history, tries to detect patterns, and then guesses his or her next move. (In case you are wondering, yes, such AI does exist).
If you only play one round, it doesn’t matter what you choose: playing rock, paper or scissors will in each case be a perfect move.
You play rock and so does the AI. It’s a draw. Now if you keep on playing only rock, the AI will adjust by playing only paper and win every round.
In rock paper scissors, perfect play is to choose each of the 3 possibilities with the same probability of 1/3. This is what we call a mixed strategy because you have to assign a probability of being played to more than one possible action.
In poker, perfect play in a given situation is almost always a mixed strategy. It could, for instance, be to raise a certain amount 80% of the time, to raise another amount 15% of the time, and to call 5% of the time.
A perfect poker player will not always play the same move every time they encounter the same situation!
Playing perfectly vs. playing well
This position is drawn and every legal move leads to a draw. So, if you are playing against a perfect player (like someone using tablebases) and if you can also play perfectly, it doesn’t matter what you play here, as the game will for sure end in a draw. In that sense, 1.Kb6, 1.Na3, and 1.Ra2 are all perfect moves.
But between humans, 1.Kb6 is clearly the only good move, with more or less winning chances, depending on the level of the players. And 1.Ra2?? is an even bigger blunder than 1.Na3??.
We just saw that a perfect move according to game theory might not be a good move according to common evaluation. That said, it seems impossible to consider good a move that changes the evaluation, like a losing move in a drawn position. To sum up: in chess, good moves are always perfect, but perfect moves are not always good.
We are about to see that in poker, while the perfect strategy is always good, the best strategy is seldom perfect!
Let’s imagine your opponent doesn’t try to catch your bluffs as often as they should. The best strategy would then be to bluff more. But by doing so, you expose yourself to a simple counter-strategy: your opponent could call you more often. But by doing so they expose themselves to a simple counter-strategy: never bluffing and betting only with good hands. But by doing so… you got it.
To play perfectly, you have, among other things, to bluff and to call just the right amount in every situation, what poker players refer to as having balanced ranges or playing according to the Game Theoretical Optimum.
If you play GTO, your opponent can do whatever he wants, playing many hands or just a few, bluffing often or rarely, making big or little bets, and in the long run you will never lose money. In fact, if your opponent doesn’t play GTO himself, you will win money.
But if your opponent doesn’t play perfectly (and no one does), you would make more money by playing an exploitive – yet exploitable – strategy like bluffing more if he doesn’t call you as often as he should.
To sum up, in poker the best strategy against a non-perfect player is always to play another non-perfect strategy!
Two-player game vs. Multiplayer game
In a 2-player game like chess or heads-up poker, you are always happy to see your opponent make a mistake. As surprising as it sounds, in a multiplayer game, a mistake by one of your opponents can be detrimental to you.
The notion of perfect play in a multiplayer game is only valid if every player acts in their own best interest. But players could group together into a coalition to make seemingly bad plays, that will be detrimental to both them and you, but beneficial to one of their allies.
Let’s imagine you play against 5 weaker opponents. If each of them tries to act in their own best interest, as well as they can, you will outplay them. But if they form a coalition, you will probably get crushed, because they will make intentional “mistakes” that are beneficial to their allies.
It goes without saying that coalitions are considered cheating and forbidden. Still, an opponent may make an unintentional mistake that will be detrimental to both him and you and beneficial to a third player.
We’ll be organising casual poker tournaments with GM Jan Gustafsson, GM Laurent Fressinet and others in collaboration with partypoker soon. If you’d like to receive further information, send an email to [email protected] or leave a comment below!