


PART
II
By Andrew MacDonald and William R. Eadington. 
Optimal Utilization: A science
and an art.
Many gaming executives might mistakenly believe that optimal utilization on a gaming table
is a simple function of getting as many people as possible to crowd around a single gaming
table and wager as much as possible. At first glance, this might appear to be the most
efficient utilization of floor space and labor. This logic appeals even more in times when
managers are being asked to trim their operating costs. This can occur when management is
under shortterm profit pressure and is looking wherever it can to enhance overall
performance.
The major costs in most casino operations are gaming tax, labor and complimentary benefits
to players. These often make up around 80% of direct operating costs in a casino's table
game operation and, as such, come into focus in times of economic downturn when greater
efficiencies are being sought. As gaming tax is generally unavoidable as a percentage of win,
and player complimentaries  when properly administered  are a marketing cost that helps
drive business, it is often direct gaming labor that receives the greatest scrutiny. It is the
"most variable" of the variable costs.
However, though it may seem counterintuitive, optimal table game utilization is not
necessarily a function of having every available position at every open gaming table occupied
by customers. Indeed, that could quite possibly be a long way from the optimal outcome; the
labor cost savings might be more than offset by foregone table game winnings. Targeting
100% gaming position utilization may be the best way to reduce labor costs, but it may also
sacrifice the highest possible yield in profit from a given set of customers. This might be
paralleled to a farmer maximizing the yield per tractor in harvesting his fields, but leaving
much of the crop to rot because the few tractors in use could not get to some of the farmland.
Determining optimal utilization per table can be approached as a scientific question. This
analysis attempts to do so by applying a relatively straightforward mathematical equation
which can be estimated by carefully understanding and modeling certain important
components of the mechanics of table game play.
An appropriate table games analysis needs to take into account time and motion aspects of
table gaming processes, including the elements that control the speed at which dealers
conduct the game. In the game of Baccarat, as it is played in Macau or Las Vegas for
example, the important elements are:
• the time the dealer takes to deal the cards;
• the time taken by the players of the Player hand and Dealer hand to expose their
results;
• the time taken by the dealer to take and pay individual wagers; and
• the time allowed for players to make subsequent wagers.
Issues like fills, credits, cash buyins, color changes, supervisor acknowledgements and
dealer changes all have some impact on the time and execution requirements at the table.
However, these are usually considerably less significant and perhaps less sensitive to
variations in the number of players at the game over any reasonable time frame. For our
purposes, we are going to concentrate on the number of players as the key variable in seeking
out optimal table allocations under varying player demand conditions.
Using empirical observations to monitor, segregate, and estimate the elements of the
mechanics of the game under different playing conditions, especially with respect to the
number of players at the table, will provide the basis for finding closer to true optimal
solutions with respect to the table games labor question. Such data would permit baseline
calculations to estimate the average speed of the game under varying circumstances, and
would allow analysts to determine how changes in the number of players affect the number of
decisions per player and for the entire table over any given time period.
The game of Blackjack can be used to illustrate this issue and how it might be addressed.
Consider a situation where six $100 players, each planning on betting one box per round at
$100 per hand, step onto the casino floor. Management's alternatives at one extreme could be
to cater to the players by having them all play at a single table with one dealer and one
supervisor. At the other extreme, management could offer each player a private table with
their own dealer and supervisor. Suppose that the applicable gaming tax rate is 20% and the
labor cost per gaming table per operating hour is $40. Which scenario provides the better
return?
To be most efficient and maintain operating costs at the lowest level, management might opt
for the single table scenario. If that is the case, then operating costs would be $40 rather than
the alternative $240, and this is clearly more costefficient.
But is this solution the more profitable for the casino? Suppose that empirical observations
yielded the following findings for the casino's blackjack games. The casino's dealers can
deal a total of 350 hands per hour on average (including their own hand) and this number of
hands is more or less independent of the number of players at the table. House rules are such
that a perfect Basic Strategy player plays to a House Advantage of 0.5% of handle. The
average $100 player that the casino attracts plays a strategy about 1.0% inferior to the Basic
Strategy, creating a 1.5% average House Advantage for the casino.
Under these assumptions, the single table will generate 300 player hands per hour, with a
total handle of $30,000 per hour, and an expected win of $450. After labor costs are
computed, the contribution to income from the single table is $410.
On the other hand, the second alternative of one player per table would permit each player to
make 175 wagers per hour, resulting in a handle per table of $17,500 and an expected win per
table for the casino of $262.50. For the entire six tables, the expected win would be $1,575;
after labor costs of $40 per table per hour are subtracted, the contribution to income for the
casino from this option is $1,335 per hour, more than three times better than the one table
alternative. In addition, players may be tempted to play more than one box. This increases the
yield by increasing the proportion of the 350 of dealt hands which go to the player. As will
always be the case with taxes on win at a fixed percentage, the aftertax results will not
change the optimal solution, and the onetableperplayer option remains more than three
times better than the onetable option. (If gaming taxes are accrued as a percentage of
gaming win, it will not change the optimal number of players per table, but it will increase
the required minimum average wager to make the game show positive income contribution.)
This type of simple example is often cited to demonstrate that optimal utilization is not the
same as 100% table occupancy. In the above scenario, optimal utilization occurs with sixbox
Blackjack tables when there would be a single player per table. (Given the assumptions,
using three tables with two players each would result in hourly expected winnings per table
of $350 and a contribution per table of $310, for a total of $930, a third less profitable than
the onetableperplayer option.)
Based on an algebraic model developed for the values assumed in the above situation, any
average wager of $23 or more would call for one player per table. If the average wager was
between $15 and $22, then the optimal number of players per table would be two; between
$11 and $14 average wager, the optimal number of players per table is three; and at between
$9 and $10 per average wager, the optimal number of players would be six. When the
average wager per player drops below $9, the table cannot earn enough to cover its labor
costs at $40 per hour.
What needs to be calculated for each game type and for each average wager level is the
number of players at which profit per player is maximized. To do this for Blackjack requires
estimates of how the game's speed of play and the number of decisions delivered to each
player changes as more players join the game. For each average wager size, it is then
conceptually possible to calculate the optimal utilization for that game, taking into account
gaming tax rates and labor costs.
For Blackjack, it is also necessary to assess the relative average skill for each classification of
player by average wager size. For example, it is conceivable that low limit players (with $10
average wagers) play with an average House Advantage (player disadvantage) of 2.0%
whereas higher limit players (with $100 average wagers) might play with an average player
disadvantage of 1.0%. To estimate the skill levels of different player categories, hours of
observations with appropriate sampling strategies would be required to determine the average
skill levels of players (based on their play strategy deviations from Basic Strategy). Existing
software in the casino's surveillance department, such as the Blackjack tracking tool
"Bloodhound," can calculate the casino House Advantage against any player, based on actual
player decisions. Such a tool could facilitate determining player skill by players grouped
into average wagering level categories.
Calculations for the game of Blackjack will be distinct for each market depending on tax
rates, labor costs, player skills and game rules. These factors, along with dealer speed and
procedural efficiencies, will affect the values of the parameters in the model. It is then a
relatively straightforward task to apply the logic of the model to determine an optimal
utilization rate as a function of average wager size.
Of course, players are not always so accommodating to be easily classified and segregated
with respect to average wager size and skill level, so consideration needs to be given to the
social aspects of table games. However, it is in the casino operator's interests to consider
how best to maximize returns in light of such realities. One simple solution might be to offer
some higher limit Blackjack tables with a lesser number of playing stations than the standard
six. In the same vein, the casino might change the size and shape of higher limit tables and
offer more luxurious and comfortable seating for each player.
Yet another solution might be to offer a dynamic (electronic) table with betting areas (boxes)
that could be activated on demand. This could be achieved by LED's embedded into the
tables to delineate betting areas. The number of boxes could then be altered dynamically.
When games are initially opened, it may make sense to have only three to five betting spots
activated per table. Betting areas could also be activated in circumstances where total demand
exceeds supply at some average betting level, or where an individual higher value player
might want to play multiple boxes. Management can then get closer to optimal table
utilization either by considered alteration as playing conditions change. Alternatively,
technology might allow the number of spots at the table to change as a matter of
electronically observed playing patterns and carefully written algorithms.
Harking back to the example of six $100 players and a choice of operating either six
Blackjack tables or one, it was shown that it was more effective to operate six individual
tables as long as the players chose to play on a single table each. However, what if the players
instead chose to all play together on one table? In the example, this situation would be suboptimal
not only for the reasons found in the analysis, but also because the five inactive
tables would incur an additional operating cost in aggregate of $200 per hour. Thus, part of
the management challenge is to encourage the right kinds of players to spread out while
discouraging lower betting and more highly skilled players to cluster together. To some
extent, this can be influenced by opening or closing tables at various minimum betting limits.
Similar calculations to the blackjack example could also be made for any other table game.
Each game should be analyzed by estimating its particular mechanical characteristics and
direct cost components. Such estimates should also reflect the relative change in game speed
that occurs when additional players are added to a single gaming table. For example, in
Macau Baccarat, there will be substantial differences for squeeze or no squeeze games and
for commission versus no commission variants. Within the Macau gaming market, baccarat is
far more relevant to consider  compared to blackjack  because of the high preference for
the game amongst players.
To illustrate the situation, and apply a slight variation of the model for Macau Baccarat, the
following assumptions might be made;
Gaming Tax = 40%
Labor cost = MOP$100
House advantage = 1.35%
Decisions per hour = 55 decreasing by 3 for each additional player (to 37 then decreasing
at a lower rate as player numbers increase).
Squeeze game with standard payout structure.
Average Bet 
Optimal Number of
Players 
MOP$ 
100 
6 
MOP$ 
200 
5 
MOP$ 
300 
4 
MOP$ 
500 
3 
MOP$ 
1,000 
2 
MOP$ 
5,000 
1 
This would suggest that if game speed declines in the manner assumed  as more players are
playing on the same table  then, as with Blackjack, it is better to spread players out across a
greater number of tables. Depending on the tax rate and labor costs per hour, the minimum
average wager to make the game profitable is increased by the slower play with multiple
players. For such games, this suggests that rituals that slow the game, such as squeezing the
cards, might need to be eliminated for relatively low limit games in the interest of efficiency.
As with the earlier discussion, this also suggests that some higher limit Baccarat tables in
Macau should be redesigned to cater to fewer players per table; smaller tables with more
comfortable and luxurious chairs and better player services might produce greater returns.
What is also evident is that one needs to incorporate a number of important variables in order
to approach the optimal utilization of tables in a casino.
One more area of consideration is the existing structures and customs in a particular gaming
market. For baccarat tables in Macau for example, almost all tables currently have nine
boxes, even though optimal utilization  using the principles developed in this analysis 
might suggest optimal numbers of players varying from one to nine players per table,
depending on average wager size and speed of the game. This may be difficult to manage,
partly because of typical player behavior in Macau. Players tend to bet together against the
house and seek situations where they feel they are on a run  or a hot streak  and cluster
around a single table betting sometimes threedeep, so that a ninebox table might have 27
wagers with back betting, all riding on the same outcome. While it may be optimal to
eliminate that situation from a profit maximization perspective, it may not be in keeping with
the motives or desires of the players, and thus any attempt to do so must be balanced
carefully with how players might react.
One lesson that comes from this exercise is the importance of segregation of gaming tables
by the average wager size. If the optimal number of players for high limit wagers is one per
table and for low limit wagers is six per table, there is no circumstance where the casino
should permit low limit players from slowing the rate of play at high limit tables. Where one
draws the line really depends on the crossover points on the "optimal number of players per
table" for a given set of parameter values, and it might be hard to get to that actual result. It is
also crucial for accurate management reporting and analysis to not aggregate and average
data from tables with different price points.
All of these complications noted, one can still conclude that management's manipulation of
the size, shape and congregation patterns around higher limit tables and its control over the
average number of players per table at various wagering levels should allow the casino to
increase profitability. That is better than allowing a lot of potential wagering to go
unharvested by being too cautious in controlling the casino floor's labor costs.






