There can be no even-money bet. In the US the wheel will have the Eagle and the OO. When they hit everyone loses.
[QUOTE=RTFirefly;21277520The purpose of putting your own money in is to increase your guaranteed winnings by modestly reducing the expected value of your winnings. I don’t see what’s being maximized.[/QUOTE]
Um, the stuff you mentioned at the start of the previous sentence?
But that’s maximized with A=0. No need for a fancy mathematical function for that. Damned if I know what the function was representing. (ETA: Or was intended to represent.)
I think that’s why “even-money” is in quotes in the OP.
Yeah, this A=? etc. stuff baffles me. But nonetheless maximizing your guaranteed net return is a perfectly cromulent goal to some. And for that you have to invest your own money.
My thoughts were: 1. Make the lowest risk bet with the given chip. One of the 1-1 payout ones. 2. Make an equal bet on the opposite of that with your own bet. 3. Bet on 0-00 to cover the last option, with the amount such that if one of those come up you get the same total amount as if you made one of the 1-1 bets.
Reasoning:
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Making a riskier bet with the given chip means having to bet even more of your money in case that one doesn’t win.
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The least amount of money to “cover” that bet but no point of going higher since that increases your potential loss for no gain.
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Same logic, different odds.
Lets say “Alice” had a better idea for “safe” betting. If she bet on one (or more) things with a total less return than my (corrected) idea, if that comes her return is less than mine.
If instead she bet on something with higher total return than mine and the rest at least as high as mine, then she is putting even more money in. Split Alice into two people: Bob and Carol. Bob makes my play, gets my returns. Carol bets the difference between Alice and Bob’s bets. Carol is just a regular punter at the table and she is fated to poor odds. Carol is better off keeping her money and not making any bets. Therefore Alice is better off not making excess bets.
Calculus is not needed.
Even money doesn’t necessarily imply a situation where the true expectation of winning is exactly 50%. It can mean a situation where the implied probability (from the payout the bookie/casino offers) is 50%. In a practical betting context, an “even money bet” would usually mean the latter - and quite naturally so, since the expression taken literally means equal amounts of money at stake, not 50% probability.
Yes. That’s all correct. A represents your “optimal” bet; specifically you bet 0.9A on black and 0.1A on 0/00 (for a total bet of 1.0*A). I’ll address the two cases you complain of: (a) A is negative, (b) A exceeds your bankroll, and then (c) the “normal” case.
(a) Optimal bet is negative. The casino won’t let you make a negative bet, so you bet nothing. If you have more than $400,000 this is your correct strategy. Why concede house vigorish to protect a profit that means little to you? (If the house DID offer the negative bet — in effect allowing you to take the House side — the formula would tell you how much of that House action to assume.)
The idea that you should always take a “guaranteed” profit is easily refuted: If that were a valid idea, everyone would buy U.S. Treasury bonds, never common stocks.
(b) Optimal bet exceeds bankroll. Since you have a “sure thing” (will always win enough to return the loan) you may be able to get someone to stake you for a small fee. (Adjust the formula to reflect that fee.) If you can’t find a staker — so what? I never claimed the formula would produce optimal bets the casino wouldn’t allow you to make! :smack: Instead, I carefully used Bankroll = $40,000 for my small-bankroll example to avoid that issue.
(c) With those outliers out of the way, let’s look at the formula more closely. First of all, this isn’t something I made up; it’s based on ideas introduced by Daniel Bernoulli in the early 18th century (when he was examining things like the St. Petersburg Paradox). Today, the idea of maximizing the weighted geometric mean of total capital is usually called “Kelly Criterion.”
But even without that specific formula, the general principle should be clear, intuitively. A rich person would be foolish to concede the unnecessary vigorish to “protect” a small win. A poor man would be foolish NOT to.
This is exactly the principle on which we should buy insurance, too.
Unless you have knowledge of atypical risk that you are not legally obliged to reveal to the insurer, you should assume that your expected return is negative, since insurance is a profitable business. Self-insure small risks where you could comfortably absorb the loss.
A corollary: the harder a salesperson tries to sell you insurance for small risks, the more you should resist buying it. Extended warranties, for example.
But this isn’t “always” and the “counter”-example is quite incomparable to the situation in the OP.
Not using your own money means that there’s a greater than 50% chance of walking away with nothing. Who in their right mind would invest in stock or some such where the odds of losing all your money are over 50%??? That you might double your money isn’t going to placate anyone other than hard core gamblers.
People routinely avoid risk at a small loss of expected return.
Hypothetically: On your flight to Las Vegas, Brother Bob informs you that Great-Aunt Annie died and left you a $25,000 chip. At the first casino, he tosses it on Red and says “I’m the will’s executor and That’s what I want you to do with your inheritance!” The money is yours, now, but you’re unwilling to call security, especially since he left the documents showing your chip ownership in the hotel room. To avoid making a scene at the busy casino, you let the Red bet ride.
Exactly the same situation now as before, but this time it is your money. Does your plan change?
Which of the following best describes your position.
(A) Mathematical problems like this can be solved with … Mathematics! But septimus’ mathematics is wrong.
(B) Math is useless here; psychology, common sense, etc. are what is needed to find solution.
Just reading the thread, is it really that hard to get a single 0 table in the US? (Whilst I know 00 tables exist, I thought they were just outliers for suckers) Here the single 0 is assumed and the question is "do they also offer “La Partage”? (50% back on even money bets when 0 hits)
Your casinos can afford 1.3% vigorish on even-money roulette action. They don’t need to provide expensive Obamacare for their croupiers, nor security to protect from terrorists like Stephen Paddock, understandably upset when the Mandalay Bay permitted him to take only two dozen guns up to his room.
And while La Partage might be OK, many Americans would get irate if you tried to put their wager En Prison.
Not remotely the same situation. For one thing, Bob hasn’t given me the chip, he’s given it to the casino. I sue Bob for my actual money. Also, how on Earth am I going to get the money down needed to maximize my guaranteed win before the croupier says “no more bets”???
It’s a ridiculous comparison. Far from being “exactly the same”. I mean. come on here …
I did do the Math. Got one thing off which RTFirefly pointed out. Your goal and my goal are not the same. Get used to this!!!
Whatever.
Let me ask RTFirefly: Did I answer your questions? Are you satisfied with my analysis?
You bet a stranger’s $25K chip?
And the ball lands on…the “S”. The “what”?
Popping up all over Las Vegas now are wheels with 39 spots - 1-36, 0, 00, and a third green spot, which can be 000 but is usually some symbol representing the casino. The first such wheel had an “S”, representing “Sands,” which owns the Venetian and Palazzo hotels; in fact, the game is sometimes called “Sands Roulette.”
BTW, none of the payoffs change - for example, a single number still pays 35-1 - so the house edge goes up by 50% over a double-zero wheel.
:smack: I’ll guess many Vegas visitors, especially from rural America, are seeking out the new wheels: “Hey Honey! They’ve got one of the new special wheels; here’s where I want to play. I’m betting on S!”
Instead of just ‘S’ why don’t they credit the new American spirit more directly with — is there room for it? — ‘MAGA’?
Well, no, not at all. From Wikipedia (bolding mine):
Its application to a single bet seems rather specious.
And your application of it to a single bet seems particularly arbitrary. Different people have different tolerances of risk, and different needs or preferences to lock in their gains. I would much rather lock in $22,222 than have an 18/38 chance at $50,000, despite your formula telling me I should do the latter. But your formula is ‘one size fits all’ and has no way of adjusting to different persons’ risk tolerances.
Unless the specific bet is the only bet you make in your life, it is part of a series of bets. 
… and can and should be analyzed the same way.
And it eventually maximizes actual (not just logarithmic) wealth by locally maximizing the expected logarithm of wealth — it looks like that Wiki article needs some editing.
But yes, it is a “one size fits all” formula and different people will have different risk profiles. Think of Kelly’s criterion as being a default profile, with strong mathematical reason to treat it as the default. In my original post I did NOT write
You should choose A to maximize …
I wrote
Kelly will choose A to maximize …
So let’s not quibble about the “one size fits all” claim that I never made Do you at least agree with my case (c) in the follow-up where I argue that, clearly, a sufficiently rich person (say several millions, not just the $400k Kelly threshold) should “hedge” little or none of his forced bet, while a person for whom the forced bet is a substantial fraction of his bankroll should hedge most of it?
Yeah, and you were referring to a person, a fellow Doper who you said would be able to back you up.
What happened to him/her?