Friday, February 11, 2011

Optimal usage of Full Tilt Points, featuring more expected utility analysis of MTTs

With online poker rooms putting more of a focus on promotions and loyalty programs that involve player loyalty points, it's becoming more important for the bottom line of a serious players to optimize his or her points usage.

For the tl;dr crowd, THIS is the table you should probably be using:

Best Table: Accounting For Tax and Risk Aversion
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.510.202985.4%
$5k Bonus11000000$5,000.00$1,350.00$2,535.840.230597.0%
$209+$7 Super Turbo47500280$209.00$49.41$110.040.233098.0%
$200+$16 FTOPS43500640$200.00$42.66$89.210.208287.6%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$102.530.2377100.0%

We will now go through the underlying assumptions and the steps of building up to it.

What Most People Do

While there are a variety of tournament tickets available in the Full Tilt Store, for small-stakes players, the tournament options for lower buyins are not as good as those at the $216 level. We'll restrict our attention to these $216 tournaments and account for a case of extra risk aversion through smaller wealth for small-stakes players. We'll look at three different tournaments:
  • $209+$7 Super Turbo — A 6-handed single-table tournament (STT) that fires quickly and takes only a few minutes to play, thanks to a 10BB starting stack... a great setting for the assumption that we're a break-even player!
  • $200+$16 FTOPS — A generic NL multi-table tournament (MTT), with broad appeal and a large field. Number of entrants (6607) and prize structure is based off of FTOPS XIX Event #1.
  • $207+$9 FTOPS Super Turbo — A Rush Super Turbo MTT, with niche appeal and limited opportunities to exercise skill. On the plus side, it has a smaller field and a very short running time and thus is an appealing way to roll through points quickly. Number of entrants (1726) and prize structure is based off of FTOPS XIX Event #12.

It's not too hard to look at the offerings in the Full Tilt store and calculate the pure EV per point of each item.

Naïve Table #1: No Rakeback
ItemCost in pointsValue¢/point% of best
Carte Blanche250$1.000.484.1%
$5k Bonus1100000$5,000.000.45454545595.5%
$209+$7 Super Turbo47500$209.000.4492.5%
$200+$16 FTOPS43500$200.000.45977011596.6%
$207+$9 FTOPS Super Turbo43500$207.000.475862069100.0%

The above table is for players without rakeback. Most quick analyses also factor in the effects of rakeback:

Naïve Table #2: 27% Rakeback
ItemCost in pointsBase ValueLost RBTotal Value¢/point% of best
Carte Blanche250$1.00$0.27$0.730.292078.2%
$5k Bonus1100000$5,000.00$1,350.00$3,650.000.331888.9%
$209+$7 Super Turbo47500$209.00$49.41$159.590.336090.0%
$200+$16 FTOPS43500$200.00$42.66$157.340.361796.9%
$207+$9 FTOPS Super Turbo43500$207.00$44.55$162.450.3734100.0%

The final column, "% of best", lets us know how much value we're losing if we went with an option other than the best option.

For example, if we have rakeback and we can break even at tournaments, under the incomplete methods of this table, we should be saving our points and waiting for the FTOPS to come around. If we instead want to roll through our points faster, we're getting only 90% of our maximum value by playing the Super Turbos. If we instead want a sure thing, we're getting only 88.9% of our maximum value by taking the bonuses. If we don't want to spend time clearing the bonus and instead go with Carte Blanche or electronics, we're getting 78.2% of our maximum value.

Carte Blanche and the $5k (and other, worse-value) cash bonuses are available only to Black Card members; players with less than Black Card are limited to the tournaments or to electronics, the value of which all seem to be worse than the Carte Blanche rate (in the 70-90% range, checking a few against Amazon.com). Luckily, Full Tilt has chosen their direct points-to-cash Carte Blanche rate to be awful compared to the bonuses. While clearing a bonus does take time, we will ignore that for the purposes of this article. So we can look at the Carte Blanche rate as an upper bound to the points-to-cash value for non-Black Card members, and the $5k bonus rate as an upper bound (some people might need to choose smaller bonuses at worse rates, depending on their volume) to the points-to-cash value for Black Card members.

So, rakeback or not, the basic value hierarchy seems to go:

Carte Blanche & Electronics < $5k Cash Bonus < Tournament Tickets < FTOPS Tournament Tickets But, as we've seen with the WSOP, even a break-even tournament player loses value in tournaments due to taxes and risk aversion, particularly in large-field tournaments with high payouts for the top places.

Will adjusting to certainty equivalents make the fixed-value cash bonus preferable to the tournaments?

Will the high risk in FTOPS events versus 6-handed $209+$7 Super Turbos make us prefer the latter?



If You're A Tournament Player

First, we should notice that, if we are tournament players and would be playing these particular tournaments anyway, the tickets are obviously the best value. We need not consider the utility effects; we were going to spend $216 cash to enter that STT or MTT, so the ticket is worth exactly $216 (minus rakeback adjustments) to us. We're done.

If you play tournaments anyway, obviously just get the tournament tickets.

For The Rest Of Us

From here on out, we make these assumptions:
  • We don't regularly play tournaments, but we are willing to in order to roll through our points. We'll ignore the opportunity cost of our time spent playing the tournament.
  • We are an approximately break-even player in any tournament we choose to play, so we will finish in each place with equal probability.
  • Therefore our expected value (NOT expected utility!) of each tournament is equal to the value of the tournament after the rake. For example, the expected value of the $209+$7 Super Turbo is $209, not $216.
  • We have rakeback and are the highest tier of Black Card, receiving a 4x points multiplier on points we earn (like all elements of the Black Card program, the points multiplier turns out to have a very negligible effect on the numbers we'll get).
If we're going to be playing a tournament that we wouldn't have played anyway, we need to adjust the above table to account for two benefits of playing this extra tournament: points and rakeback that we gain from paying the rake on the tournament.

Less Naïve Table: Accounting For Points+Rake From Tournaments
ItemCost in pointsPoints You GetBase ValueLost RBTotal Value¢/point% of best
Carte Blanche2500$1.00$0.27$0.730.292077.5%
$5k Bonus11000000$5,000.00$1,350.00$3,650.000.331888.1%
$209+$7 Super Turbo47500280$209.00$49.41$159.590.338089.8%
$200+$16 FTOPS43500640$200.00$42.66$157.340.367197.5%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$162.450.3766100.0%

The effects are pretty minimal. Much more important are the effects of taxes and risk aversion.

Certainty Equivalents Under Tax and Risk Aversion

In the style of our WSOP analysis, we will start by making similar assumptions on our wealth, income, and risk tolerance:
  • Our preferences for different levels of wealth and risk are governed by our usual utility function (isoelastic with ρ=0.8). We are subject to both U.S. Federal and New Jersey State income tax. again, NJ has higher income tax than most other states, but the effect on the certainty equivalents we produce is not large.
  • Our net worth is $80,000.
  • Our income for the year aside from our points usage is $40,000. (We assume that we will almost certainly be winning poker players for the year, but since the loss associated with entering a tournament and losing is simply the rakeback hit, we don't really need the distinction between poker and non-poker income this time.)
We also make a few other small assumptions about the tournaments:
  • FTOPS events will run with no overlay. If there ends up being an overlay, we get more EV than our base buyin amount, but we also get a rakeback hit (I think). I don't think this happens very often in NL events, historically.
  • FTOPS events actually have extra value that we're ignoring because there are bounties for eliminating a red pro. This probably adds somewhere around 1% to our EV for an FTOPS event if 1% of the field are red pros. We'll ignore it here rather than try to model it.
With these assumptions, we calculate the certainty equivalents for each of the tournaments under consideration:

Best Table: Accounting For Tax and Risk Aversion
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.510.202985.4%
$5k Bonus11000000$5,000.00$1,350.00$2,535.840.230597.0%
$209+$7 Super Turbo47500280$209.00$49.41$110.040.233098.0%
$200+$16 FTOPS43500640$200.00$42.66$89.210.208287.6%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$102.530.2377100.0%

Observe that:
  • Now that we're accounting for tax, the cents per point is lower across the board, but some options suffer more than others.
  • The relative value of the riskless options has risen, due to the discounting of the values of the tournaments from taxes and risk aversion.
  • The low-rake FTOPS event is still the best, but now the Super Turbo STT has overtaken the $200+$16 FTOPS. Keep in mind that the "variance" is much lower in a 6-person tournament than a tournament with over 1,000 players, and accordingly, the reduction to the certainty equivalent is lessened.
  • The difference in value between the three best options ($5k bonus, Super Turbo, and FTOPS Super Turbo) is now quite small. This average player should probably just go with whichever option is convenient.
Other cases
Let's look at the numbers for a few other cases of wealth and income. To model a small-stakes grinder who would be out of his comfort zone playing $216 tournaments, let's look at a case of very small wealth and income:

Small-Stakes Grinder ($5k net worth, $2k annual income)
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.650.258788.0%
$5k Bonus11000000$5,000.00$1,350.00$3,233.900.2940100.0%
$209+$7 Super Turbo47500280$209.00$49.41$137.000.290198.7%
$200+$16 FTOPS43500640$200.00$42.66$59.570.139047.3%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$70.810.164155.8%
The variance-free bonus has become the best option, though the small-stakes player will have trouble clearing it quickly. Surprisingly, even someone with this small of a net worth is pretty close to indifferent between the guaranteed cash and the 6-handed $216 tournament. The large-field, high-risk FTOPS have become very unfavorable, however. A player wealthier than our default player has lesser tax consequences and less risk aversion to the higher "variance" of the FTOPS events:

Affluent Pro ($500k net worth, $100k annual income)
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.480.191678.9%
$5k Bonus11000000$5,000.00$1,350.00$2,395.500.217889.6%
$209+$7 Super Turbo47500280$209.00$49.41$104.710.221791.3%
$200+$16 FTOPS43500640$200.00$42.66$96.370.224892.5%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$104.820.2430100.0%
The wealthier player should probably save up his points for the FTOPS.

Conclusions
A non-tournament player of average wealth/income ($80k/$40k) is close to indifferent between spending points on $5k bonuses, $216 single-table tournaments, and $216 FTOPS events.

A player with less wealth/income should probably choose the $5k bonus or the $216 single-table tournament.

A player with more wealth/income should probably save his points up for the FTOPS.

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