Collusion, 101

Sometimes I feel like I walk around with blinders on. Such is the case with the advantage play method known as collusion. This method simply means that players share information about their hands with each other. The needle always moves towards the player’s side when information can be used to improve a strategic decision. This has long been known to be the case in blackjack. For example, assume there are three players seated at a single-deck game. Right after the shuffle, if the dealer shows an Ace, and no face cards are dealt to any of the three hands, then the insurance wager has a player edge of 6.67%. It turns out that many proprietary games have collusion issues as well.

The vulnerability of a game to collusion is always a function of how much information the players have and how that information constrains the dealer’s possible hands. In assessing this vulnerability, I assume the player’s find a way to share all their cards with each other. Computer-perfect collusion strategy is usually so complex that it cannot be concisely expressed. The task of the AP is to find a simple way of reducing full information to usable information. These simplifications give up a significant part of the edge gained through computer-perfect collusion, but for many games, it’s still good enough.

In what follows, I will be using the Excel function combin(n, k) quite a bit. This function gives the total number of ways that k things can be chosen from a set of n objects. Here is a link to the entry in Wikipedia if you want a refresher.

Here are a few examples of what's possible using collusion.

Three Card Poker. The baseline house edge is 3.37%. A player holding any three cards knows that the dealer has one of combin(49,3) = 18424 hands. Now assume seven players are seated at the table, sharing their hands. Then each player knows 21 cards, leaving 31 cards for the dealer’s three cards. This means that the dealer has one of combin(31,3) = 4495 possible hands. Collusion has allowed the players to exclude 75.60% of the possible dealer hands.

With computer-perfect play, the house still has an edge of 2.32% over the colluding players in Three Card Poker. Collusion moved the house edge by only 1.05% towards the player-side. It simply does not work to gain an edge. By comparison, with hole-card play, the player knows his three cards and one dealer card. That means the dealer has one of combin(48,2) = 1128 hands. With hole-carding, the player has an edge of 3.48% over the house. The more constraints on the dealer’s hand, the more movement of the edge towards the player-side.

Caribbean Stud Poker. The baseline house edge is 5.22%. Each player is dealt five cards. With seven players at the table sharing their cards, a total of 36 cards are known (including the dealer’s up-card). Collusion reduces the number of possible hands for the dealer from combin(48,4) = 194580 to combin(16,4) = 1820. Collusion has allowed the players to exclude 99.06% of the possible dealer hands.

With computer-perfect collusion strategy, the colluding players have a 2.38% edge over the house in Caribbean Stud Poker. This is an improvement of 7.60% over the baseline house edge. A human-feasible collusion strategy (the MAK count, discussed in this post) gives the players a 1.34% edge over the house. This represents a 6.56% improvement over the baseline house edge. The MAK count achieves 86.3% of the gain possible by computer-perfect collusion.

High-Card Flush. Perfect basic strategy has never been quantified, but computer-perfect play gives a house edge of 2.64%. Using a very simple strategy of playing any hand that is T-8-6 or higher and folding all others, the house edge is 2.71%.

In this game, each player is dealt seven cards. With six players at the table sharing their cards, a total of 42 cards are known to the players. Collusion reduces the number of possible hands for the dealer from combin(45,7) = 45379620 to combin(10,7) = 120. Collusion has allowed the players to exclude 99.9989% of the possible dealer hands.This is a tremendous level of constraint on the dealer’s hand and the movement of the edge towards the player-side reflects this.

With computer-perfect collusion strategy, the players have a 7.33% edge over the house in High Card Flush. This represents a gain of 10.04% over the edge obtained by using simple basic strategy.  With a human-feasible strategy (presented in this post), the players can get a 4.04% edge over the house, a 6.75% improvement over the simple basic strategy.

These three examples provide some intuition about the relationship between shared information and the movement of the house edge.
 

• Three Card Poker, seven player computer-perfect collusion. 75.60% of dealer possible hands excluded. The edge moved towards the player-side by 1.05%.

• Caribbean Stud Poker, seven player computer-perfect collusion. 99.06% of dealer possible hands excluded. The edge moved towards the player-side by 7.60%.

• High Card Flush, six player computer-perfect collusion. 99.9989% of dealer possible hands excluded. The edge moved towards the player-side by 10.04%.

When I was at G2E (2013) last week, I opened my eyes wide to collusion for the first time. Most poker games have hole-card problems. It turns out that many have collusion problems as well.

A perfect example of seeing something for the first time is SHFL's game Six-Card Poker. The baseline house edge is 1.27%. In this game, each player is dealt six cards. The dealer is dealt six cards as well, with three of his cards dealt face up and three dealt face down. If there is no information sharing, then the dealer has one of combin(43,3) = 12341 hands. If there are six players sharing their cards, then the dealer has one of combin(13,3) = 286 hands. Collusion has allowed the players to exclude 97.68% of the possible dealer hands. Using the three examples above as intuition, I expected that colluding players could gain a small edge over the house using a human-feasible strategy.  When I looked around, I found this post on Stephen How's blog Discount Gambling, giving a practical strategy for players to gain a 0.43% edge over the house. This game protection issue is why SHFL advises casinos to not allow players to share their cards.

There were two other games that I noticed at G2E (2013) that may have significant collusion issues. These were Lunar Poker and Double Draw Poker. In each game, it is possible for the hand to draw down to the last card.
 

Collusion is not a new form of advantage play. In today’s table game world, it works best on poker-style games where the dealer’s hands can be constrained as much as possible. Of the three main methods, including edge-sorting and hole-carding, collusion is the weakest. But it is also poses a formidable challenge for game protection. Flying under the radar, collusion teams are now crushing many popular games world-wide.