Scientific American did this once, or something quite similar. The “right answer” is that you each randomly pick a number from 1 to 20. Whoever gets the number 1 asks for $9,999,999,999. The others all ask for the full $10,000,000,000.
This maximizes everyone’s expected profit.
ETA: S.A. suggested that this answer can be deduced by everyone, and does not require collusion. Without collusion, everyone draws a random number independently.
The only nash equilibrium is $.01. So yeah, I’d ask for that.
And your co-workers, who each thought “$10 billion split 20 ways is still $500 million each, so we all come out way ahead.” murder you the next day…
I ask for $9,999,998,427. Close enough to $10 billion, and hopefully avoids some “there’s no single smallest number, so no-one wins it.” crappy loophole.
It’s funny that so many of you are going so high. I can’t imagine millions ever paying out, even with only two participants, unless you allow some kind of collaboration.
With 20 random people I’d probably go with $100. Maybe even $50. Even going that low, my expectation of winning is $0, but I’m gambling on the small chance that I’m surrounded by people who are bigger gamblers than me. (By the way, if there was a smaller pool of people, I’d probably go higher. But even faced with a single person, I’m not sure I’d go higher than $1,000.)
As Joker says, the only real equilibrium here is $0.01, but I’m willing to lose out on that much to try for something higher. 0.01 isn’t worth the effort to play the game.
I’d just take a guess and submit a relatively low request. Let’s say $9500.
It’s low enough that I figure I have a reasonable shot at being the lowest bid. But it’s high enough that I’d notice the money.
Frankly, $100 is a pathetic amount to win in a game like this. Screw that, go big or go home. I’m going millions, easy. Tell me I have a 1% chance of never having to step foot in an office again, or a 50% chance of winning Dinner & a Movie, I’m not going for D&M.
So here’s the bids so far:
$10,000,000,000
$10,000,000,000
$10,000,000,000
$9,999,999,999
$9,999,998,427
$20,000,000
$4,000,000
$2,000,000
$899,999
$800,000
$500,000
$99,000
$9,500
$3,000
$2,000
100
.01
It’s not like the billionaire is going to pull a name out of a hat. You either have the lowest bid or you lose. You go big and get nothing. Dracoi plays it safe and gets a hundred dollars. Winning a hundred dollars might seem pathetic to you but it beats zero.
That’s probably why I’m accountant. 
But I assess the odds differently than you do. I don’t even think $100 has a 50/50 chance of winning, it’s just the lowest amount that I’d consider worth bidding. If I lose to a lower amount, I really haven’t lost anything.
If I had to assign odds, I’d give my $100 a 20% chance. I’d give $1000 a 1% chance. By the time we’re talking a million, I’d say the odds are about a million against it. Of course, I can get 0.01 with absolute certainty. So expected winning are $20, $10, $1 and $0.01. $20 expected winnings sounds good to me.
If you’re assigning 1% to a million, then it would be reasonable for you to bet that, since your expected winnings would be 10,000… but I think that’s a rosy assessment. In any group of 20 people, I’d expect there to be at least one pessimist who’s willing to take pocket change over nothing. (And I think Nemo’s summation proves my thinking. It only takes one of us to screw up everything.)
$30k
Would certainly help me out and would feel like I “won” something, and would have a reasonable chance of being the lowest bid.
$799,000.
As I recall, it was a little different (or maybe it was a different, but related contest) - SA would give a million (or billion - some large number anyway) dollars divided by the sum of all the numbers sent in, to the person who provided the largest number. Nobody won, since the numbers sent in were so astronomical, than the “prize” was way, way smaller than a cent.
Here’s an article on the subject Platonia dilemma - Wikipedia
Nothing in the OP prevents me from going to co-workers and drawing up an agreement–I’d write it in Word and get everyone to sign it and witness everyone else signing it. One of us asks for $9,999,999,999.99, and the rest of us ask for $10,000,000,000. The one that asks for the lower amount agrees to pay all of us equal shares.
The OP states that the rich gamester randomly approaches 20 people and asks for individual responses. I think it’s fair to say that an immediate response is needed and that collusion is not allowed. The spirit of the question would require secret and immediate responses.
42, of course. Now, should that be cents or dollars???
Rule Clarification:
No collaboration at all: The billionaire pulls you aside and tells you to give her a number; you don’t know if you’re the 1st or the 20th person she’s asked, or have any idea who the other 19 are.
If exact lowest happerns, she’ll pay all tied people an equal share or give nothing depending on how she feels. She certainly won’t give everybody who tied the full amount.
It wasn’t the largest number that won. The winning number was randomly chosen. The idea was that you could submit as many “tickets” as you wanted. You didn’t even have to write them all out - you could just say “I’m submitting a trillion tickets”. Submitting a large number increased the chance that one of your numbers would be drawn. But as you note, the large number of entries reduced the prize to virtual zero.
I’ll take $50,000. Enough to fix up my 1891-crate-of-a-house. I’m 64; my spouse is 72. We don’t have that much time to live in this house (and it doesn’t have to be THAT “quite perfect”) to get to enjoy it.
OTOH … I’ll ask for $250,000 more … a fabulous circuitous trip around the world while the house is being fixed up would be – sorta nice. 
Umm, that’s not how it works.
You’re right. In practice, though, people tried to create outrageously large numbers, so that a random selection would make them almost certain to win, making the game in effect a contest in which the biggest number won.