Editorial

Baccarat Math Behind Borgata vs Ivey

John Mehaffey April 16, 2014

There has been a lot of discussion about the Borgata lawsuit alleging that Phil Ivey and a companion used baccarat edge sorting to beat the casino out of $9.6 million in 2012. The lawsuit makes many claims and admits that management accommodated many special requests made by Ivey and his friend.

The most interesting part of the lawsuit might be the portion where Ivey’s winnings are disclosed. According to the lawsuit, Phil Ivey played four session of baccarat in 2012. Here are his results, hours played, and average wager per session:

April 2012 – Ivey won $2,416,000 with an average wager of $25,000 during 16 hours of baccarat.
May 2012 – Ivey won $1,597,400 with an average wager of $36,000 during 56 hours of baccarat.
July 2012 – Ivey won $4,787,700 with an average wager of $89,000 during 17 hours of baccarat.
October 2012 – Ivey won $824,900 with an average wager of $93,800 during 18 hours of baccarat.

Hands Per Hour Played

The lawsuit does not mention how many hands were played. We reached out to Borgata to comment about the average speed of their heads up baccarat games. We did not receive a response.

USPoker asked several experienced baccarat players how many hands would be played in an hour with an automatic shuffler in a normal game. The consensus is that this game would have dealt 100-120 hands per hour. This would have left plenty of time for shuffling, superstitions, and the special requests accommodated by Borgata dealers.

The lawsuit claims that a player that was edge sorting baccarat has a 6.765% player advantage. If we assume that Ivey played 100 hands per hour, none of his sessions reached this level.

Wizard of Odds states a typical baccarat game deals 72 hands an hour, a figure likely attributed to a table with multiple participants since it is used for rating players at an undisclosed Las Vegas casino. For players not familiar with the speed of the game, baccarat and blackjack game speeds are nearly identical. Baccarat is slightly faster as the player has no decision to make once placing a bet.

Ivey’s Reported Return on Play

To accomplish the player return alleged by Borgata, Ivey, who was playing heads up, would have been playing about 25 hands per hour. This would be an extremely slow game, about 35% the speed of a normal baccarat table with multiple players. The game would deal less than one hand every two minutes when there was an automatic shuffler and the player allegedly had a massive advantage, something that should actually encourage fast play.

There is one item mentioned in the lawsuit that suggests that the game did not move this slow. Borgata states:

The process of “edge sorting” all the cards in the decks took more than one shoe.

At 25 hands an hour, it would take over three hours to deal out the estimated 80 hands out of each eight-deck shoe. The play would require at least one more shoe to accomplish the high/low card sorting. A player wanting to burn out an 8-deck shoe quickly could do so in about 20 minutes.

It is possible to skip hands. A player with edge sorting capabilities would have an advantage of up to 5.5% for banker bets if it is the appropriate wager for the next identified card, according to the lawsuit. It is illogical that a player is going to give up that advantage by passing hands when there an edge.

The lawsuit supports the theory that Ivey placed wagers on the banker at what Borgata asserts were opportune times:

[W]hen Ivey bet on “banker,” the first card dealt was significantly more likely to be a strategically unimportant card.

Ivey Results

The results below are stated in the lawsuit filed by Borgata.

April 2012 session

Won $2,416,000
16 hour session
$151,000 hourly rate
$25,000 average bet
+6.04 betting units per hour
6.04% player return at 100 hands per hour
8.89% player return at 72 hands per hour
By far his best session in terms of hourly rate
Shortest session of the four

May 2012

Won $1,597,400
56 hour session
$28,525 hourly rate
$36,000 average bet
+.79 betting units per hour
.79% player return at 100 hands per hour
1.1% player return at 72 hands per hour
Accounted for more than half of all hours played

July 2012

Won $4,787,700
17 hour session
$281,629 hourly rate
$89,000 average bet
+3.16 betting units per hour
+3,16% player return at 100 hands per hour
+4.39% player return at 72 hands per hour

October 2012

Won $824,900
18 hour session
$45,828 hourly rate
$93,800 average bet
+.49 betting units per hour
+.49% player return at 100 hands per hour
+.72% player return at 72 hands per hour
Borgata accuses Ivey of intentionally losing at the end of this session

Assuming 100 hands per hour, Ivey wagered an estimated $562 million over a combined 10,700 hands over 107 hours. This makes the average wager about $52,500. His total win was purportedly $9,626,000. Ivey won a total of 183 betting units. This is about a 1.7 units per hour. At 100 hands per hour, that would be a return of 1.7%. At 72 hands per hour, the return would be 2.3%. Either way, his results are closer to the 1.15% overall house advantage a baccarat game has than the 6.765% player advantage an edge sorter would have.

The lawsuit alleges that Ivey wagered “well below the maximum bet” at the beginning of each session and after the alleged edge sorting was completed, Ivey “bet the maximum amount on every hand”. This accusation implies that Ivey was betting the max when he had the 6.765% player advantage even though his overall return at anything resembling a normal game speed is closer to the actual standard baccarat house edge.

If we simply assume the edge sorting player advantage, and ignore the standard game speed, Ivey theoretically wagered $142.3 million combined across all sessions. It takes time to edge sort the shoe so the actual wager amount is likely somewhat higher.

Let’s assume Ivey did have the advantage stated by Borgata and his results fell exactly in line with the edge sorting advantage. At that rate, here are the number of hands per hour Ivey would have had to play during each visit to have his results:

April 2012 – 89 hands per hour
May 2012 – 12 hands per hour (this session accounted for more than half of all hours)
July 2012 – 47 hands per hour
October 2012 – 7 hands per hour

The average number of theoretical hands per hour across all sessions to post Ivey’s win would be about 25. That is less than one hand every two minutes for a game that requires no player decisions besides placing a bet and has an automatic shuffling machine.

Converting the results into smaller wagers

The win looks so large because of the massive average wager. The sessions start to look more reasonable when we put them into limits that the average person plays when visiting a casino.

The figures below assume the same average hourly rate as Phil Ivey’s Borgata session:

A $10 average bettor would win $17 per hour.
A $25 average bettor would win $42.50 per hour.
A $50 average bettor would win $85 per hour.
A $100 average bettor would win $170 per hour.

How much should Ivey have lost?

The baccarat house edge is about 1.15%. Based on the above figures, we have broken down Ivey’s expected losses.

April 2012

100 hands per hour: -$460,000
72 hands per hour: -$331,200
25 hands per hour: -$115,000

May 2012

100 hands per hour: -$2,318,400
72 hands per hour: -$1,669,248
25 hands per hour: -$579,600

July 2012

100 hands per hour: -$1,739,950
72 hands per hour: -$1,252,764
25 hands per hour: -$434,989

October 2012

100 hands per hour: -$1,941,660
72 hands per hour: -$1,397,995
25 hands per hour: -$485,415

Cumulative Expected Loss

100 hands per hour: -$6,460,010
72 hands per hour: -$4,651,207
25 hands per hour: -$1,615,004

How does someone intentionally lose $2.7 million at baccarat?

Baccarat is purely a game of luck. Borgata alleges that Ivey had edge sorting to his advantage and in his last session he intentionally lost. This is called “cover play” in the world of advantage gambling. At one point during his last session, Borgata states:

During Ivey’s Baccarat play on October 7-8, 2012, he was ahead by as much as $3.5 million. During Ivey’s Baccarat play on October 7-8, 2012 he eventually purportedly won $824,900.

The lawsuit continues:

Upon information and belief, Ivey intentionally lost a portion of his winnings at the end of the October 7-8, 2012 Baccarat session.

Assuming Ivey played 100 hands per hour in 89 hours between the first three sessions, he played 8,900 hands for a total wager handle of $392.9 million. His average wager during this time was $44,146, with a win of $8.8 million. This is a win of 2.24 units per hour, regardless of the number of hands played. Assuming the same results from his alleged first card knowledge, his intentional losses should have mirrored his previous play and advantage.

Ivey’s average bet in the fourth session was $93,800. The maximum wager was $100,000 for this session so there was not much in the way of bet variation.

According to the lawsuit, Ivey appears to have lost $2,676,000 from his peak. Using the 2.24 bets per hour win, and assuming the same average speed as the other sessions, Ivey would lose $210,112 per hour at $93,800 per hand. It would take 12.73 hours to lose $2,676,000 at that rate. The entire session was only 18 hours and he still needed enough time to win $3.5 million.

Ivey could have just bet the tie. That bet at Borgata has a house edge of 4.84%. At $93,800 per hand, Ivey would theoretically lose $454,000 per hour at 100 hands per hour. It would take almost six hours to accomplish this if hitting the expected loss, however, the tie is a high variance bet.

It is also possible that only a portion of a bet went on tie and the rest on player or banker. That would substantially increase the time needed to dump nearly $2.7 million in chips.

There are other possibilities. It is unknown if side bets were available on the table. These tend to carry a high house edge. A bettor could also wager on the player, banker, and tie, all at the same time. This would have a lower edge than simply betting the tie by itself for the max bet. It would also draw scrutiny since a player would be better off just betting the tie by itself and not placing opposing wagers where one side pays a 5% commission.