*If you use a point-earning credit card and pay $3.95 for a $500 reload card, what is your cost per point? The answer may surprise you (it surprised me). *

If you find a store that allows you to buy reload cards with a credit card (see “Reader reload research“), you can essentially buy points for less than a penny each. Right? The math is simple: cost = $3.95. points earned (with a 1X card) = 504. Cost per point = .78 cents. It’s simple.

Not so fast…

I recently held an email conversation about this with the “Traveling Mileonaire” where he said:

In my mind it is always 2 cpm or so

If I go to CVS and buy 500$ VR and pay 4$ load to Bluebird I have a choice of 2 cards to do it with:

A) My SPG [where] I get 504 SPG=605 AA miles (or US etc etc) for 395c = 0.63 cpm

B) I use my 2% cash back Amex / Visa. I get 10$ cash back and spend 4$ for it (250% return). I am ahead by 6$

So by using option A, I am not only spending 4$ for fees, I am also losing 10$ I would have got back from the cash back card. So 504 SPG – 4$ = 10$-4$ or, 504 SPG = 10$ or 1000 SPG = 20$ = 2 cpm

So even when you go and buy VR at CVS, you are still paying 2 cpm in opportunity cost.

Traveling Mileonaire’s viewpoint is directly aligned with how I think about everyday credit card spend. In my 2011 post “The Cost of Credit Card Points” I essentially argued the same thing. I said that when buying things with a credit card you always have the choice of using a 2% cash back card instead. So, if you earn only 1 point per dollar with your credit card, you are essentially buying those points for 2 cents each since that is the amount of cash you are forfeiting. 2 cents per dollar is the opportunity cost of using a point earning credit card.

Despite my prior posts on the subject, I wasn’t convinced that this was the right way to think about reload cards. I replied with this:

I’m not sure that opportunity cost is the right measure here. If you go to a store and buy a box of cereal for $4, that’s the cost regardless of whether you could have spent your time making money mowing lawns instead. So, when I go to the store and buy [a reload card] for the purpose of buying points cheaply, I think that $3.95 is the cost regardless of what I could have done instead. I know this goes directly against things I’ve written in the past but there seems to be a difference (at least to me) when manufacturing spend vs. doing regular spend. Let me try to give an example (I’m sort of thinking “aloud” here):

If I use a credit card to buy dinner, then my goal is to get dinner. The choice of credit card is a choice of what type of rebate I want. If, on the other hand, my goal is to buy points, then I don’t really have any choice but to use the credit card that offers the points I want. So, I think you can choose to buy points at less than a penny each or to buy pennies (cash) at a cost of less than a penny each.

I’m still struggling with how best to think about all of this, but the above is my current position… I think 🙂

In my response, I essentially argued that the purpose of your purchase makes a difference to the cost. This is, in hindsight, complete nonsense. At the end of the day, your bank account and point balances will be the same regardless of the purpose of your purchase. Still, its hard to shake the thought that my 504 points cost only $3.95 so of course I’ve paid only .78 cents each. What’s the right answer? Is the cost .78 cents or 2 cents per point? That’s a big difference!

### The Monty Hall problem

The situation described above reminds me of the famous Monty Hall problem where the answer is counterintuitive. Wikipedia has the details:

[The Monty Hall problem] became famous as a question from a reader’s letter quoted in Marilyn vos Savant‘s “Ask Marilyn” column in

Parademagazine in 1990 (vos Savant 1990a):

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?Vos Savant’s response was that the contestant should switch to the other door. (vos Savant 1990a)

The argument relies on assumptions, explicit in extended solution descriptions given by Selvin (1975a) and by vos Savant (1991a), that the host always opens a different door from the door chosen by the player and always reveals a goat by this action—because he knows where the car is hidden. Leonard Mlodinow stated: “The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated.” (Mlodinow 2008)

Contestants who switch have a 2/3 chance of winning the car, while contestants who stick have only a 1/3 chance. One way to see this is to notice that there is a 2/3 chance that the initial choice of the player is a door hiding a goat. When that is the case, the host is forced to open the other goat door, and the remaining closed door hides the car. “Switching” only fails to give the car when the player had initially picked the door hiding the car, which only happens one third of the time.

I expect that some readers will argue that the answer presented above is wrong, but it’s not. Before any decisions are made, the probability of the car being behind any given door is exactly 1 out of 3. *After* a door is selected and the host shows a different door to have a goat, the probability of the car being behind the remaining door (not the one you picked) is exactly 2 out of 3. Trust me (or write angry comments, if you prefer).

The important take away from above, is that the right answer changes once action is taken and extra information is revealed.

### Back to reload cards

The comparison of the cost of points via reload cards to the Monty Hall problem is imperfect, but I do think there is some similarity. After lots of thought about it, here is (I think) the counterintuitive solution to Frequent Miler’s Reload Card problem:

- When deciding between doing nothing at all or buying a reload card with a credit card that earns 1 point per dollar, then the cost
*is*.78 cents per point. *After deciding*to buy a reload card, you can choose to stick with your 1X point earning card or switch to a 2% cash back card. If you stick with the 1X card, then that decision costs you 2 cents per point.

The interesting thing about this solution is to note that the costs are *not additive*. That is, it doesn’t cost a total of 2.78 cents per point to use a 1X card. It either costs .78 cents or 2 cents per point depending upon how you think about it.

### The solution, in practice

There are two important decisions to make regarding reload cards: 1. Should you buy them at all? and 2. Which credit card should you use?

With the first question, you need to decide first whether it is worth your time and likely frustration when things don’t go as easily as expected. If this is your hobby and so don’t mind spending the time, then I think it is reasonable to use the .78 cents calculation as a rough way to decide whether or not to do this at all.

Once you’re committed to buying reload cards, it makes sense to decide which credit card is best to use. In the post “Questioning Delta loyalty” I showed how I compared my spend with Delta credit cards to the same spend with 2% cash back cards and decided ultimately that I would get more travel value from the Delta cards. Thanks to high spend bonuses with the Delta cards, I earn approximately 1.5 miles per dollar. I determined that, for me, it was worth giving up 2 cents per dollar to get 1.5 Delta miles per dollar.

### Conclusion

When deciding whether or not to buy reload cards to earn points, the cost per point is easy to calculate. Once the decision has been made, though, it is important to consider the opportunity cost of choosing one credit card over another. For example, buying Starwood points for .78 cents each is a great deal, but is it a *better* deal than earning a profit by using a 2% cash back card? For you, the answer may be yes, but keep in mind that *that* answer is costing you more than .78 cents per point.

Another argument for the VR’s is that I can apply for more credit cards because now I have a way to meet the minimum spend. I used to apply for 1-2 cc’s every 6 months due to this issue. Recently, I just applied and received 5 cc’s with a total minimum spend of $15k in 3-6 months. So I value the VR’s a bit more since it allows me to get the 50k bonuses from the cc’s.

If you thought about everything in terms of opportunity cost, there would be very little you could afford.

Opportunity cost is useful as a comparison tool only, not as a means of account.

It is true that you could be getting 2% cashback instead of 1 point, but the value of 1 point may very well be greater than 2 cents. Also, getting $6 doesn’t get me out in the morning, getting a first class trip for free after x number of trips to CVS does.

Thanks for the post, as always.

I think this opens up the discussion on what your return is on those points. Yes, you’re paying .78 cents for the points, but if you make an outsized return, then you’re ultimately benefiting. Of course, that brings about the discussion of fair market value for a point vs. what a point redemption should be counted as (e.g. if I use my SPG points for a $10k SQ Suites ticket, are those points really worth, say, 7 cents a point, or because I would never dream of spending $10k on that same ticket, are they worth less?).

Ultimately, I think from a hobbyist standpoint, $10 will always be $10, but, part of the fun, is that a collection of .78 cent points, can lead to outsized returns.

Another monkey –

Signup bonuses. If I’m using VRs to fulfill bonus spend, The $3000 I spend is suddenly worth, say, 50k AA miles instead of 3024 of them. My cpm drops to .04.

Or, my value for the purchase jumps to $800 (assuming 1.6 cpm value) as opposed to the $60.48 I would have made with a 2% card.

If you’re constantly thinking about how much greener the grass could have been, you’ll never be happy.

While I dont disagree with FMs article. I have to say sometimes you can get lost in discussions about how much it costs to earn. So I agree with Anon, I have a card that earns 4% and another earning 6.33% and I still from time to time spend on other cards like Ink or Sapphire just to earn UR pts.

I don’t have a cash back credit card so this dilemma does not affect me. 😉

to beat the dead horse……

If you are using manufactured spend solely as a perpetual point machine notwithstanding CC sign up bonuses then the conversation is a valid one. The CC sign up bonuses are the major selling point to why we are all here. Sure I could go try the UR mall and generate 11 UR per dollar and make a small profit but that is not a guaranteed return and my capital outlay is considerably higher. The reload is a guaranteed point return on a cash loss.

Why wouldn’t one use the Club Carlson Biz card for 5x points?

The thing with opportuniy costs is that you have to be consistent. The cash card also has an opportunity cost (ie. the best return you could have gotten elsewhere).

Don’t act like the 2% is without cost if you are doing it this way.

As said before, be sure to factor in all benefits of using the non-cash credit card (be it initial spend or other level that gets you something – bonus points, status, companion ticket or whatever).

The benfits of the best card are then an opportuniy cost of the cash reawrd card.

You are basically asking, Do you value starpoints at or above 2cpm? If you do, then buy VR with your starwood card. If not, buy VR with a 2% cashback card. Which would also be true on any purchase whatsoever. If you don’t value starpoints above 2cpm why would you have the card outside of the sign up bonus (or starwood hotel purchases)?

Enjoyed the article but you are way overthinking a simple question about the value of starpoints.

My question is how to dispose of reload cards. My wife and I both have a Bluebird, but there do not seem to be other good options. Any input?

I think you missed one key concept of opportunity cost.

It only applies if you are facing a consumption limit.

If you can buy unlimited reload cards (or as many as you want to consume) then there is no opportunity cost.

But, if you are like me, and have 1 BB, then I am limited to $5,0000 in reload cards.

For what it is worth, I buy reload cards with my cash back card unless there is a bonus I am chasing.

I normally find your articles well thought out, but this just seems confusing, and wrong.

You are contorting what opportunity costs are (they aren’t an actual cost that you can tack on as if it were a fee)! It is only a tool for considering two alternatives, when you have limited resources. The choice is EITHER 504 points for your $3.90 OR $10 for your $3.90.

PLUS, if you want to even try to equate these, you have to equate the currencies. 504 points is worth a minimum of .01 cpp and up to .02 or more … making it possibly worth more than $10. The $10 on the other hand wouldn’t buy you 504 points directly from any carrier!

I think this one issue has got your brain jumbled. Very uncharacteristic.

This is very simple from a manufactured spending standpoint. Rule 1, You want a gOod relationship with the bank. Don’t spend too much on one card and spread out your spending. Thus all point wil cost 0.79 cents. The opportunity cost you’re describing does not exist. You’re making sure that the opportunity to use the CC is still there in the future.

I love game theory and you have me scratching my head on the Monte Hall problem. “After a door is selected and the host shows a different door to have a goat, the probability of the car being behind the remaining door (not the one you picked) is exactly 2 out of 3.”

Why isn’t it 1 out of 2? Of the remaining doors, one has a goat and one has a car. Or is the underlying assumption that if the player was to choose the car with the first choice, that the game would be over immediately. That’s the only unstated assumption I can think of that would make the probabilities work out as stated.

Too bad you can’t get vanilla @ OD anymore, this conversation would never have happened.

I would first equate 504 SPG points at 2.2 cents a point.. so that would equal $11.08 – $3.95, which equals $7.13 profit.

Using my Barclaycard arrival 2X on all purchases, I would then equate 1108 Barclaycard points subtract $3.95, which also equals $7.13 profit.

I value both these options the same. Therefore, there is no opportunity cost when comparing these two options. However, if you value SPG points less than 2.2 cents per point, there would be an opportunity cost.

What do you think?

Once the door is opened, the game is new. The new probabilities are 1/2 and 1/2.

The misuse of the concept of opportunity cost here is becoming pretty rampant (including in the OP).

Opportunity cost is not relevant as a system of *account* because we don’t account for the rest of our economic lives in that way. When you buy a $500 TV, do you think about the cost of that TV as being $500 plus the loss associated with no longer having $500 to buy a front row ticket at your favorite band’s concert? No, you don’t. You may think about these factors in a comparative sense in your decision to buy the TV (how much would you enjoy the Rollin Stones compared with the TV?) but you will never consider yourself as having spent more than $500.

If we don’t do our personal accounting this way in other aspects of our lives, why should we when dealing with points and miles?

what bosstom said, with emphasis on the fact that you are neglecting the different values of the two types of currencies you are receiving.

@Matt B, @Dan…The Wikipedia link that FM provided has all of the details about the Monty Hall problem and why the optimal strategy is to switch rooms. This is a classic problem that is discussed in any course that covers conditional probability.

Also for those not getting the Monty Hall problem: the probability doesn’t change, but your information does. Think about it like this: with 3 doors, any door your choose with have a probability of being correct of 1/3, implying that the remaining unchosen doors collectively have a 2/3 probability of holding the prize (1 – 1/3 = 2/3). But, after choosing, you then learn that the probability of one of the unchosen doors having the prize is 0, which in turn implies that the remaining unchosen door must have a probability of 2/3 (2/3 – 0 = 2/3). So you double your chance of being correct by switching doors.

That said, not sure why this was included as its not relevant to the discussion of opportunity costs.

@Matt B, @Dan…If you still don’t believe the explanation, play it yourself a bunch of times with one of the many simulations available online such as this one: http://math.ucsd.edu/~crypto/Monty/monty.html

Wow, lots of comments, so I’ll try to address them via the theme rather than one by one…

Along the lines of the signup bonus argument, you are also not accounting for spend on cards that have a calendar year bonus. I just earned the TT ticket for the Chase BA card from VRs, but ALSO got the Avios points. Not sure how to value that but it’s certainly >0.

You will lose your mind once you go down the opportunity cost rabbit hole.

There is simply no end to the different ways that you could look at the value. Do 22,000 UR points (For a high-end Hyatt room) equal the cost incurred to earn the points(maybe free if it was spending you were going to do anyway), the cost to manufacture the spend (22 x 7.90 = $173.80), the cash value of the points (22,000 x $.01 = $220), the travel value (in the UR Mall) of the points (22,000 x $.0125 = $275), the potential value of the points when converted to United Miles (22,000 x $.02(maybe) = $440), or some other transfer where the points could be worth 4-6 cents (amtrak, British Airways)? …and that’s just UR points.

When it’s all said and done, you’ll be fit for a padded cell.

@dan — There is a 1/3 chance that the car is behind the curtain you picked and 2/3 chance it is behind one of the others. That is still true after one goat is exposed. Therefore, after one goat is exposed, there is a 1/3 chance that the car is behind the curtain you originally picked and 2/3 chance it is behind the remaining curtain.

@DBest — The TT ticket is worth about 5 cents in my book. The taxes and fees are a huge rip-off.

The Monty Hall problem: The way I think of it is like this: when you first pick a door, the probability of it having a car is 1/3. That doesn’t change. The probability of it being in either of the two other doors is 2/3. Monty Hall purposely opens one door that he knows does not have the car. That doesn’t change the fact that the probability of the car being in those 2 doors is 2/3, it just narrows down for you exactly which door has the 2/3 chance.

If that doesn’t convince you, then try modeling this in Excel. Create a table with 3 columns (representing doors) and three rows. The rows represent the possible outcomes and look like this:

car sheep sheep

sheep car sheep

sheep sheep car

Now pick any column as if that’s the door you picked. Notice that only 1 of 3 possibilities is a winner if you keep that original choice. Now imagine that you pick a column, but then once another column is shown to be sheep, you pick the other column. Notice that you will win 2 out of 3 times.

To follow on to FM’s Excel suggestion, use one of the many online simulations available after a quick search for “Monty Hall problem simulation”

Great post FrequentMiler. I distinctly remember studying the goat and car prize in statistics class in college. I think it was Applied Probability course or something very similar to it. Ah the good times.

Opportunity cost: All I’m saying is that you have a choice between points or cash back. Regardless of whether you agree that opportunity cost is a good way of thinking about things here, at the end of the day you’ll either end up with a bunch of points or a bunch of cash. I think it is important to decide which you prefer. Even if your opportunity is unlimited (which it isn’t), you are still making a choice and the choice will leave you with more cash or more points.

The value of points: Of course it is critical to consider the value of points. That’s the point! I never said that SPG points aren’t worth 2 cents each or more, that’s up to each person to decide for themselves and within their circumstances. The question is: when deciding whether to buy a reload card with an SPG card, do you ask yourself whether SPG points are worth more than .78 cents each? Or, do you ask yourself whether the points will each give you more than 2 cents value? I used the think the former, but now I think the latter.

.

Here is maybe a better way to think about it than what I presented in the post:

.

The acquisition cost of SPG points is .78 cents each, but the decision of whether to acquire SPG points or cash depends on which rebate will give you more value. If you pick SPG over 2% cash, then you are assuming you will get more than 2% value from your points.

Sign-up bonuses (and other bonuses): If you are working towards a sign-up bonus (or other type of spend bonus), then the math changes because you no longer compare to 1X. For example, if you’ll get 30K points for 5K of spend, then you would end up with 35K points (30K bonus plus the 5K points you would normally get). In this case, you are earning 7X for your purchases up to $5K. Most point systems will easily give you better value at 7X than 2% cash (although, with some hotel programs that’s not necessarily the case).

Dan, the game is only new if they randomly re-position the goat after the door was opened. Only then would it be a 50-50 chance.

I get the idea for everyday spend but I don’t think it applies, at least not for me, to reloads. I can make more money I can’t in any other way buy points so cheaply. If I only had the 2% cash back card I would not bother with reloads to save $6. But with the manufactured spend I can buy points cheaply for redemption’s that I would not pay for or save me tons of cash. This summer I had a redemption of $.02 per Carlson point on a busy summer weekend. The rate was $550+ for 2 nights. Had I had to pay OOP I would not have gone. Buying those points on reloads would cost $44.24 (3.95/2,500*28,000). That’s payback you can’t get with a 2% cash back card.

Now days between myself and my wife it is all about clearing bonuses. Without reloads this would not be possible. Our 4 new cards require $12,000 in spending in 6 months. Without reloads there is no way this would be possible. But with them it is easy. The points earned on clearing bonuses destroys the 2% argument. For me I always want more points so I can fly up front and stay at luxury hotels. Or even dumpy hotels on an expensive weekend in a tourist town. Last month I spent 2 nights at the Park Hyatt Tokyo and next month I’m flying Buisness Class to Europe for Oktoberfest. Without the points getting me the flight and hotels for nearly free there is no way I could/would afford it. Points give me things no amount of cash-back can. That’s why I spend the time hunting for Vanilla Reloads every month.

Has anyone ever watched “Deal or No Deal”? If so, is it to the contestant’s advantage to switch briefcases before the final opening? My family has disagreed on this for years.

You have to answer the question of what are you going to do with your miles before you answer the 2 cent question. As I am only using them for Hyatt international premium awards and First Class awards the value is dramatically more than 2 cents……..so for me that exercise is a no brainer………..a more difficult decision is whether to chase Hilton diamond with manufactured spend verses building UR points……

@FM

Right, so a useful comparison (make sure you know what you are giving up), but not accurate at all to say that your cost of acquiring points is 2cpm.

I think your blog is one of the best out there for being the most technical (and I’d almost even call your approach empirical), but I think the fewer economics theories involved, the better (this is a good heuristic for life in general as well) 🙂

Here is another way to look at the door scenario.

If you pick Door 1, you have a 2/3 chance of being wrong, then a 100% chance of being right if you switch. (Overall probability of switching to win = 2/3 X 1/1 = 2/3).

If you pick Door 1, you have a 1/3 chance of being right, then a 100% chance of being wrong if you switch. (Overall probability of switching to lose = 1/3 X 1/1 = 1/3).

Math gets so confusing. Thats why salesmen use it all the time. If I am buying a reload card then I am buying points. There should not be an opportunity to buy cash. (I have cash and want points) It’s different then if you are buying dinner or a TV.

I do love math, and the Monty Hall problem is always so fun.

The somewhat “brute force” approach, for those who still don’t get it:

Suppose the car is behind door #1, keeping in mind we could repeat this for either of the other two doors, so the result would be the same.

If you pick door #1, Monty reveals 2 or 3. If you switch, you lose. If you stay, you win.

If you pick door #2, Monty reveals 3. If you switch, you win. If you stay, you lose.

If you pick door #3, Monty reveals 2. If you switch, you win. If you stay, you lose.

To sum up, in two of the three above cases, switching wins, and staying loses. In one of the three cases, switching loses and staying wins. Thus, your probability of winning if you switch is 2/3 and if you stay is 1/3.

FM, being a west coaster I’m a bit late to the discussion. In the Monty Hall scenario, wouldn’t your odds be 1 out of 2 in the 2nd round? The door with the goat is now removed from the equation – you have 2 doors and one has a car.

For me, it is quite simple when you think of a hierarchy:

Do I need to manufacture spend to meet a bonus:

If yes, then that buy the VR on that card, always the best.

Do I need points for an award right now (or planned near future) to achieve an award?

If yes, then even at 1X that is the best option.

It is only after these that I address:

Right now, for my $3.95 do I want to generate a rebate of 504 points (.01 to .02+ cpp value) or $10.

Only at this time, if I value the points at less than .02cpp, would I choose the $10.

@Matt – I was typing when your post went up…thank you for squaring me on the 2/3.

Putting this in the context of manufactured spend and VR cards is irrelevant. It’s actually a simple comparison between SPG and a 2% type card. If you would choose a 2% card for VR, you should choose 2% card for all spending (non bonus categories of course).

A simpler article would be: Is SPG card better than a 2% card for non bonused spending.

Part of what makes this difficult to understand is the difference between acquisition cost and redemption value. Those are two different things for points, but they’re identical for money.

What you’re doing here is comparing the $.79 you’re *paying out* for an SPG point (acquisition cost) to the 2% cash you’re *getting back* (redemption value). This is much easier to explain if you stop thinking of cash back as “cash back” and think of it as “buying pennies” instead. When you do, you find that you’re buying pennies for $.0039 each.

Long story short: The math works out exactly the same because “pennies” just happen to be worth 1 cent each, but it’s important to keep acquisition and redemption values straight when you’re comparing the values of different points because “pennies” may not always be on one side of the equation.

Agree with Bosstom. Some comments. Feel free to chime in.

1. This is only relevant IF you would otherwise go and try to earn net $12 for each $1000 spent (i am not including cents here to preserve my mind for more useful things) multiple times a month.

2. Hence, the comparison should not be based on 2cpm. The question you ask yourself is: What is more valuable: spending 0.078cpm to get 1 SPG or 0.012 cents in cash. In another words, if I spend $10K in VR cards each month on a 2% cash back card, I would get $120 net cash in my account. If I used my SPG card, I would spend $80 to get 10,000 points. The key is, you are doing one OR the other (and that’s only relavant [back to #1] if you would even consider spending this much time to collect a meager $120/month). So, yes, while in a true sense, the cost of using a SPG card is the $78 + the lost opportunity to have earned $120, the truth of the matter is that you wouldn’t have done BOTH. Your assumption of 2cpm is very theoretical.

The real question in my mind is that do you value 0 points + $120 cash back (on $10k spend) OR 10,000 points for $78. The way I like to think is that can you get more than 0.012cpm out of your SPG. If you can, then the latter is a better option. Its

There’s a very simple answer to the dilemma. Why not both? (Okay, that’s a question, but cue the cheering crowd anyway)

I have an amount of spend that I’m comfortable manufacturing on any given card and I allocate my VR purchases accordingly. Some go onto a 2% card, some go onto Club Carlson, and some go onto a *wood or equivalent. Putting all that spend onto any one card raises the potential for scrutiny that I just don’t want.

On the goat problem in case anyone is interested a good way, for me anyways, to break it down is to make a matrix

GGC

GCG

CGG

If you look at column 1 you see your 1/3 chance. Now in each row eliminate from each row 1 G excluding column 1 and you will see your 2/3 chance that the other door has the car. Definitely counter intuitive. I am sure there is a fancy math proof out there for it.

As to the reload issue I just prefer to just analyze which card will provide me with the most value. Your big risk with points cards is devaluation before you can use the points.

I would love some help in understanding the Monty Hall question. There are 2 facts that I see: 1) our first choice has no bearing on the outcome since we are being forced to make another choice after the host has eliminated one door, and 2) the host has to pick a door with a goat. The probability of picking a goat or car is then completely independent of the first choice. The entire exercise of picking a door in the beginning is pointless. If we had not been made to pick one of the three, and the host is still required to remove one of the goat doors, then we are still left with an independent choice between 2 doors; one with a goat and one with a car.

Good one Matt. I like your setup.

The conceptually challenging part of the Monty Hall dilema is that our mind intuitively wants to set up a new scenario with only two doors. This would only be true is the car is randomly re-positioned after the door is exposed – THEN it is a 50-50 chance and that is where our minds want to go. It was my first reaction when I read the dilemma.

A more intuitive example might be: You pick a card from a 52 card deck – you have a 1 in 52 chance of guessing the card. If you pick out another card, what are the odds of guessing the second card? Well, since one card is removed, there are only 51 left in the deck, so you have a 1 in 51 chance of guessing. This is a simpler and more intuitive scenario since it is a whole new game for the second card, yet in the Monty Hall case it is not.

I think in terms of what I am using the points or cash back for.

For example: 90,000 SPG = 110,000 AA miles (80,000 SPG = 100,000 AA + 10,000 SPG = 10,000 AA).

or $1800 on Arrival card.

To use on Cathay from North America to SE Asia in business would cost $6000+ in May 2014. To me the SPG point is giving me more value than the cash back, even factoring in the ~25,000 miles I would have earned using the cash (I don’t have status or using special route bonus).

Ben, if I understand your point correctly. . .

Your scenario is: what if there was no initial choice of doors? What if the host just eliminated a door with a goat and then you choose? In that scenario it would be a 50-50 chance. The key in your scenario is that the host would ALWAYS have a choice of two doors to eliminate.

This is very different than the Monty Hall scenario. The Monty Hall scenario is counter-intuitive: you win by initially picking the wrong door. Because, if you initially pick the wrong door, you have the host by the balls: the host has only ONE door he can eliminate and he WILL reveal the correct door to you.

There is a 2/3rd chance of initially choosing the wrong door. So, 2/3rds of the time, the host finds himself in a position where he must reveal the correct door to you.

Another viewpoint:

Let’s say that you choose your door (out of 3). Then, without showing what’s behind any of the doors, Monty says you can stick with your first choice or you can have both of the two other doors. I think most everyone would then take the two doors collectively.

This is pretty spooky. I just did a post awhile back , saying pretty much the same thing, although not quite so eloquently.Is mine a Frequent Miler for dummies blog?

If you do go for the miles, you had better at least be getting redemption value of .02 or better long-term, or you’re just fooling yourself. Most people (read- people who don’t read blogs like this) are going to be better off with the straight 2% rebate.

The thing about the monty hall problem that no one ever talks about and is so vital to understanding is this:

you must imagine you ARE the game show host. if you just say that the game show host “knows” it doesn’t convey to people what they’d understand if they WERE monty hall.

as monty hall, you put up the curtains, goats, and car.

your contestant chose one of the doors. you know whether it’s car or goat. regardless, you must reveal a door without the car. it cannot be the one your contestant chose. so you’re restricted in choosing the “other” goat curtain. if your contestant had chosen car, you can choose either curtain. if your contestant chose goat, you’re restricted to choosing the only goat door left. therefore it’s not the same as a “new” game at all. the door that opened was NOT random.

I agree with Ed on acquisition cost and redemption value. I’ve always looked at points this way. The circular discussion around value of mile or point doesn’t help me as points are arbitrage.

When I calc my cost of points for mfg spend, I only look at out of pocket costs. That’s why I don’t manufacture spend unless there is a decent spread between what I pay for points and what I get in return (award).

Most people are constrained by liquidating the VRs, but if the constraint is instead on how much you can feasibly manufacture/month on one credit card, then the dilemma (opportunity cost) goes away. For instance, I assume a max of 5k in VRs on each card per month. Assuming i can unload both, that’s $60 (10x$6 profit), AND 5000 points (at cost of $39). But if bluebird constrains me to 5k per month though, then I must choose. I do believe I extract more value than 2 cpm out of starwood, but it doesn’t scale. Why? Because another constraint is my ability to use the points (or sell them). Beyond 1 or 2 international trips a year I am restricted by the amount of vacation I have. Since manufacturing actually doesn’t produce that quantity (60k SPG/year assuming 5k/mo), it’s a bit of a non-issue. Cashback scales farther, but SPG points have more marginal value (to transfer to an account with nearly enough miles for an award).

I don’t really understand why there is an opportunity cost issue when buying reload cards.

Just look at this example:

When you want to buy a $500 reload card, if you have a 1% cashback card and a 2% cashback card, which one should you use?

Definitely 2% card, because it generates higher value.

The “value” matters, especailly when BB has an annual $60K reload cap.

This also applies to points card.

When you try to compare a points card to a cashback card, you have to know how much “value” they generate.

For example:

When you want to buy a $500 reload card, if your points card can generate a value (after minus $3.95) higher than 503.95*2%-3.95=$6.129, just use that card;

otherwise, use your 2% cashback card.

Only you know the “point value” is, because people redeem different stuff.

Do not care how much money you paid for one single point – care about the value one single point can generate/redeem.

62 comments on this already…

You are driving me SANE !!!

I disagree that there is 2/3 winning chance to pick car in monty hall problem. since the total chance to pick up car and goats is 100%, if the chance for picking up car is 2/3, then the chance to picking the remaining goat would be 1/3. do you agree with this result? Also let’s think the same monty hall problem differently. 1 car and 2 goats behind 3 doors each. however, instead of asking the contestant to choose the car, let the contestant to pick a goat. So at the very beginning, there is 2/3 of chance to get the goat. Now after the contestant pick, the host open one of the two left doors and remove the car, at this moment, what is the chance for the left door being goat? 50/50, 1/3 or 2/3? should the contestant switch his pick or continue to hold the first pick?

Ack, you really shouldn’t have posted the Monty Hall problem in the same post. Now you have people who don’t understand statistics as confused as those who can’t see they are buying SPG for 2 cents (minimum). I recall discussing this topic with you ~3 months ago and concluding a 2% card was probably the best proxy for measuring opportunity cost because using a 2% card isn’t a capped or limited opportunity (unlike juicing spend with AMEX GCs and higher X cards).

@jesse, if there is 1 car and 2 goats, and you remove the car, your chances of getting the remaining goat are smidge better than 2/3 whether you switch your pick or not.

Setting that aside, I tend to agree that you need to look at the opportunity cost of purchasing VRs using a cash back card. Haven’t thought this completely through, but if you imagine you could buy points directly for 1 cent per point, looking only at the acquisition cost of getting starpoints by buying VRs using an SPG card, you’d think buying VRs with an SPG card would be a good deal. $4 would get you 504 starpoints. Whereas that $4 would only purchase 400 starpoints. But if you could buy them for 1 cent per point, you’d actually be better off using the cashback card and purchasing the starpoints with the cashback you received. It’s because the acquisition cost of starpoints through other channels is so high, e.g., directly purchasing from starwood, that this is confusing people. It’s still a good deal to acquire them by using VRs if you value them above 2 cents per point. If you only valued them at 1 cent per point, you wouldn’t want to use VRs to acquire them even though the acquisition cost is .78 cents per point.

Folks, the easiest way to validate the Monty Hall problem is play the game yourself (do this before posting a question)! Have a friend hide a penny under 1 of 3 cups. Guess which cup has the penny, then ask the friend to show you which of the other 2 cups doesn’t have the penny. You will see very quickly that 2/3 of the time the penny will be under the cup they didn’t show you.

It seems to me that you cannot assign the value of forgone cash back, because you did not forgo it.

First, you chose to manufacture points, not cash.

Second, manufacturing points does not stop you from also manufacturing cash back.

You’re really just begging the question of what is better to manufacture.

I find it interesting to see that different explanations for the Monty Hall problem appeal to different people. That is, the explanation that makes the solution most clear to me, might not be the one that works best for you. I think that’s true also for the acquisition cost vs opportunity cost discussion. Everyone has their own way of thinking of it that makes sense to them. Nothing wrong with that.

It’s amazing to watch the Monty Hall problem reaction. It is the same reaction as 1994. Many people could not accept the 2/3 answer from statisticians. Some schools actually conducted Monty Hall experiments to test the answer.

The answer is absolutely 2/3. Try the experiment yourself if you don’t believe.

Perhaps this may help:

There is a car behind one of three doors. You are given the option of choosing one door or choosing two doors. Which option would you take? Of course, you would want to choose two doors since this doubles your chance of winning. So, you choose two doors. You KNOW that one of those doors has no car. The host, knowing the doors, reveals your door with no car. Who Cares?! You already knew one of your doors had no car.

That is what Monty Hall boils down to: you get to choose one door or two doors. Monty revealing a loser door is just a distraction so you don’t realize you would get two doors.

After the door is opened revealing a goat, the odds now change. The odds with 2 remaining closed doors is now exactly 50/50. It no longer matters what the original odds were with 3 unknown doors. After opening one door it is now only relevant that there are 2 unknown doors, 50/50.

When the Monty Hall problem was published in 1990, over 90% of the public disagreed with the 2/3 answer. The author received thousands of letters that she was wrong (including over 1,000 letters from people with PhDs). Mathematicians constructed computer simulations and, sure enough, the answer is 2/3. There was one famous mathematician who refused to accept the 2/3 until he watched the computer simulation.

Eric: your rational is by far the most common. It was my initial reasoning as well. Yet, incorrect. The 2/3 solution is counter-intuitive and difficult to grasp.

OK, last thought tonight on Monty Hall:

For those of you who still believe in the 50-50 answer, explain to me how I can lose in the following scenario.

1) Everyone agrees that there is a 2/3 chance of initially choosing the wrong door. If you do not believe this, do not continue.

2) I am committed to “switching”. I will pick a door and I WILL switch.

3) I initially pick a door and I have a 2/3 chance of being wrong. I will switch after a loser door is revealed.

4) If I initially choose a wrong door (2/3 chance), explain to me how I lose.

If you are a switcher and initially choose the wrong door (2/3 chance) then you will ALWAYS win.

Hopefully a few “lightbulbs” went off out there!

“After a door is selected and the host shows a different door to have a goat, the probability of the car being behind the remaining door (not the one you picked) is exactly 2 out of 3.”

So, it’s not a goat — it’s Schrodinger’s cat behind the other door!

Oh, wow, has there been a response to this question…..and I know the original version of the issue was 1 or 2 percent type cards. But for me it is about 3 things:

1 – clearing the minimum spending thresholds for new card bonuses quickly and easily.

2 – topping off one account or another

3 – nailing tons of point on Club Carlson system and then using the points for redeeming at expensive hotels all over the world, as well as SPG card, and Barclays card at 2%. One needs free hotels when one travels.

I’ve been thinking about this and this seems to be over thinking the straight forward answer of where do you want to go and how tripped out do you want your trip? I think even though we all love the miles chasing game we have an end game in mind when we schlep to get those reload cards……….I just returned from a trip to France where ITA says the cost per mile on the FC seat I got from BA was $1.05 per mile……combined with the 5 free nights at hilton and the 5 free nights at Marriott I felt that I got dramatically more than 2 cents per mile. And it was for seats that I NEVER would have bought……….FM you are the best analyst in this game………..don’t take your eye off the prize……..

If you continue to believe the odds are 50/50, please come see me. We’ll run the game. I will wager $100 against your $90. We can repeat the wager until you are convinced.

I agree exactly with FM’s reasoning. I travel 8 months of the year and can only manufacture gift card spend when in the USA, so my limit is $20K per year on BB and $12K via AP. I calculate the opportunity cost on every spending decision I make.

The problem with the Monte Hall problem is that it never existed. If you ever watched the show,Monte did not give the contestant the choice to switch doors on the final big door prize. You picked your door and you were stuck with it. Same with the new Let’s Make a Deal. Use a different metaphor for that example of chance.

Anyone suggesting that VRs cost 2cpp has their head where the sun don’t shine.

I go down the opportunity cost rabbit hole when I’m dealing with limited points resources. For example: let’s say that someone buys Visa gift cards with a PIN for $4.95. That’s about 1cpp at $500.

I can use liquidate that card on Amazon Payments for free. Or I can take it to Walmart and buy a money order for 70¢. Because my Amazon Payment usage is limited, I can move that $500 spending to another non-PIN card, all it costs me is an additional 70¢ (and trip to WalMart).

Btw, the crux of the Monty Hall problem is that you want your first choice to be wrong (a goat). If there are 2 goats and 1 car, you have a 2/3 chance of having it wrong. By showing where the other goat is, you are given a “free pass” to change to the car.

If you don’t swap, then you are dependent on having chosen the car first — 1/3 chance.

@ Matt in response 43 – thanks, this explanation made it click for me.

FrequentMiler – based on the number of responses, perhaps you should throw out a brain teaser every few weeks to stretch everyone’s brains beyond the world of miles and points 🙂

A lot of incorrect reasoning in these comments, I think somewhat motivated by the idea that earning free travel is fun while earning a modest cash profit is less exciting.

Opportunity cost is indeed important, but the opportunity cost is different for different people. The way this would be put in a microeconomics 101 class is: rational people think at the margin. That means you aren’t concerned with the value of the best redemption you ever got with SPG (or whatever program), you are concerned with the value of the redemption another 504 points would go towards. Because if you didn’t buy that VR, you would still have gotten that best redemption ever (or even the average redemption), just not the marginal one that you would have to give up. (This is why, as FM has pointed out, UR points are so valuable – they can be transferred to top off an account where they will have very high marginal value.) Likewise whether FM’s scenario #1 (cost = 0.79 cents) or scenario #2 (cost = 2 cents) is correct for you depends on your situation. What are YOU giving up by buying that VR on your SPG Amex? For some people it is 2 cents, for others it is 0.79 cents.

If you aren’t manufacturing spending at all, you can put your spending (ignoring issues like merchants that don’t accept Amex) on whatever CC you like without fear of shutdown. So CC spending ability is not a limited resource and the cost of $1 of spending on SPG Amex is at least 2 cents (maybe higher if you have the JCB Marukai card, buy Amex GC, etc.) because you could have put that $1 of spending on another card. FM is implicitly applying this principle when he says 2 cents should be used rather than a higher value because the 2% is unlimited and unrestricted, but you can buy $5k worth of Amex GC every 14 days on a 2% card through TCB. Some people might not manufacture that much spending anyway, so Amex GC might be effectively unlimited for them, making the true cost of points higher than 2 cents.

Now let’s consider someone who only manufactures a little bit of spend, say $5k/mo of VR they can load to BB. Unless they have a low CL they can still probably put that 5k in addition to their normal spending for the month on one card, especially if they make multiple payments during a cycle. So we are still in FM’s scenario #2 where the cost is 2+ cents per point. If you value SPG points at 1.5 cents each, you might say you are still ahead if you buy VR and pay 0.79 cents. True, but you could have been more ahead if you had used the Priceline Visa instead. If you think that’s not very worth it to use the 2% card for such a small profit, well you were willing to do it for a smaller profit (unless the SPG points are worth more than 2 cents to you, and they might be).

But now suppose you are a heavier hitter, I won’t give any specific numbers because how much you have to do to fall into this category is highly variable but suppose you manufacture enough spending so that you have to spread it around multiple credit cards in order to not get shut down. Then, you might already be doing as much as you feel comfortable on the 2% card (and it’s wise to go for the long run benefits by playing it safe). So, your cost might only be the 0.79 cents if the SPG Amex was the only card you felt comfortable putting that extra $503.95 of spend on. This is the only way FM’s scenario #1 is correct.

On the other hand, as I pointed out, the value of the first SPG point earned from CC spending is higher than the value of the last. If you are earning sign-up bonuses or getting a category bonus buying VR on a miles/points card, then it’s probably better to earn that bonus than get the 2% CB. FM has discussed the proper way to value points before but it depends on what YOU would buy with them (at the margin!). As with valuing miles and points, this is a situation where you have to think for yourself about your own circumstances.

Lemma,

awesome analysis. FM should repost using this.

What this article (and many posters) misses is that when you are getting 2% cash back that represents ~ 1% profit that is actually taxable.

Normally cash back is not taxable since in most cases it is like a rebate on a regular purchase but not in this case. It will be extremely easy for IRS to show that you are making $6.05 profit on each VR purchase so you should consider your tax bracket. And I am willing to bet that people engaging in manufactured spending are much more likely to get audited than general public. So for me personally 1.25 AA points wins easily not even considering that 30,000 spending on SPG card gives you gold status.

xcg001: Theoretically, if profits from manufactured spending are taxable, then the points are taxable at their fair market value. The fact that it’s difficult to measure their value doesn’t stop Citibank et al. from sending a 1099 in non-CC situations where they give miles (bonus miles for opening a bank account). It seems to me like either it’s taxable or not regardless of whether it’s cash or points, like in situations where you would be sent a 1099. I’m not a CPA nor a lawyer so consult one about your taxes.

Lemma: It is not true that points should be taxable as they don’t have intrinsic value – most points expire and subject to devaluation as they are technically not owned by you but by the operator of the program. In fact ALL frequent program (miles, hotel points) state EXPLICITLY that points have no value. And in fact Citibank stopped issuing 1099 after being criticized for that.

Cash and other financial derivatives on the other had has nominal value and there is no way to argue that it is not a profit. May be you can wiggle your way as a “income from hobby” but not for amounts that most people manufacture spend. If it is worth getting the cash back (>$1000 per year) there is no way to argue that this is non taxable profit.

I’m a math teacher and tried to explain the Monty Hall problem to a group of sophomores my first year teaching. It was a disaster.

I think an example that might help is to try to pick the Ace of Spades (the car) from a 52 card deck. Once you’ve made your pick, the all knowing host reveals 50 of the remaining 51 cards and none of those are the ace of spades. You’re left with your card and the card leftover by the host’s actions. Do you want to switch?

Of course. It’s not 50-50. You only had a 1/52 chance of being right originally.

I also like the explanation to ignore what the host does and consider choosing between 1 door/card and the other two doors/51 cards.

Thanks, everyone, for contributing your thoughts, opinions, and suggestions for how to think about this stuff. I love it!

Cole, I like your example. If you pick one card from a 52 card deck, then at least 50 of the 51 remaining cards will not be the Ace of Spades. The dealer, knowing the cards, just shows you those 50 cards.

Monty Hall initially seems so simple and it is amazing how strong the human impulse is to think 50-50. I’m surprised casinos haven’t used this to their advantage.

Consider a casino game based on Monty Hall (but where a player cannot switch). This leverages the player’s false belief that their odds are improving as the game progresses.

There a 10 doors. Behind one door is a magic ball, the other 9 doors are empty. The player makes an initial guess of which door has the magic ball. One by one, the remaining empty doors are opened. As each one is opened, the player gets more and more excited s/he could win – its looking better and better for me! Once there are two doors left, the player is given the option to bet on their door with 5:1 odds. The player thinks There are only two doors left, there is a 50-50 chance the magic ball is behind my door – and I get 5:1 odds?!! Heck yea! I’ll bet the farm on that! The player actually has a 1:10 chance and the casino would make a fortune.