## Suboptimal Bets, Liquidity Traps & Real Estate

Suppose that two insane billionaires offer you great bets. Billionaire A has a 70% chance of winning (double your bet) and 30% chance of losing (lose your bet). Billionaire B isn’t quite as insane … you have a 2/3rds chance of winning. Being Billionaires, they’ll take whatever stakes you can scrap together.

But A has a catch. It’s a single event. B, on the other hand, is willing to make three tries. Like all Billionaires, they are insanely jealous and suspicious of each other, so if you bet with one, the other won’t deal with you.

The best bet, in terms of expected value, is to wager as much as you can afford to lose on A. But a much better long term strategy is to take that amount, divide it into 3 piles, and place three wagers with B. That’s because your odds of going broke are 30% with A, and a mere 1/27th (under 4%) with B. Of course, your “Ideal” Billionaire would give you some odds (not much, say a 51% of winning) and then take as many bets as you care to place. (Of course, better odds are still better, but you get the point).

The public, by the way, is the ideal billionaire for Las Vegas, State Lotteries, and the like.

But if you generalized the insane billionaires, so that they allowed N bets, you could figure out what the ideal betting strategy would be. You want to grow your bankroll, but the I.B. will go on hot streaks, so you don’t want to bet too much on each bet. But if you bet too little, it takes too many bets (which may take up your time, or risk I.B. getting bored with you and leaving). I believe Poker Theoreticians have solved this problem, assuming you have some Expected Value for your time playing.

Anyway, I was thinking about Acquire, and it occurs to me that you are trying to solve the same basic problem. All the chains will appreciate, but you can only place so many bets at a time. Placing a big bet (3 shares) on company A precludes you from betting big on B. Worse yet, each bet doesn’t resolve instantly, so if your money stays tied up, not only do you have to decide between A and B, you have to worry that either A or B may preclude you from taking bets C, D and E down the road. You’ve run into a liquidity trap — good investments and no money. You lose.

Of course, games aren’t the real world. If I earn a crushing 20% per year safe investment, I don’t lose because someone else squeaked out 22%. But in a game you do. Winner takes all. The net result is that, if your first bet won bigger than everyone elses, you can switch to the slightly lower, but much less swingy return on investment. The worst your early investments failed, the more volatile high-risk/high-reward bets you take later on. If you aren’t winning, betting all of your money hoping for a merger before your next purchase is reasonable.

Of course, in the actual game you probably don’t assign numbers, just a gut feel (“Imperial will fold into American pretty soon and I can take 2nd place, or I could lock up world wide, which will likely be 31+ by the end of the game…”), but the idea holds.

More idly, I was thinking that Acquire had everyone a big winner, but if you assume it represents ~40 years of business (say, 1910 to 1950). You start with $5.5k (I think), and end with ~$50k (if you win), but by the inflation calculator that has a real value of about $20k of starting value, which is a nice steady three and a quarter percent real rate of return!

If billionaire B lets you parlay your winnings if you win your first bet he’s certainly worth more in EV than billionaire A.

The betting structure is also a little more complicated than wagering 1/3 of your starting bankroll each time.

Fred BushApril 8, 2011 at 6:07 pm

Yes, for simplicity I was assuming no parlays, bets had to be placed at the same time.

taogamingApril 8, 2011 at 8:20 pm

If there’s no parlay, then Billionaire A still gives you a better expected value than Billionaire B. No matter how you split up B’s bets, it’s still 70% vs. 67%. You want to assign an enhanced value to not going broke, so you want some kind of utility function. That’s perfectly reasonable, but an expected value won’t get the job done.

In fact, if you CAN parlay your money, your best EV strategy is 3 double or nothing bets with Billionaire B. That gives you about a 30% chance of multiplying your starting capital by 8. Even with a 70% chance of going broke, that still gives you an expected return of 140% profit, which is the best you can do.

Larry LevyApril 8, 2011 at 9:34 pm

I believe that you would take the three smaller bets in an iterated game where the amount you are wagering (cost of 3 shares) exceeds 40% of your bankroll.

Fred BushApril 9, 2011 at 7:50 am

Larry, that’s basically what I meant (without explaining it much). If I have a Million dollars, my utility gained from doubling it is much less than my utility lost from losing it all. Even if I only bet 1/2 a million, my utility upside is less than the downside. In the real world, that counts.

In a multiplayer game which each person “places bets”, the utility function is quite different. There’s not much difference between ending with my current value or no value … I probably lose either way, because at least one opponent will place a winning bet (or bets). (I may enjoy the loss more with current value, but we’ll ignore that).

taogamingApril 9, 2011 at 9:41 am

I was about to make the same point as the previous comment (less utility for increasing dollars than decreasing dollars) but since it has already been made I’ll point out a rarely considered exception.

Sometimes the utility can be greater for the increasing dollar. For example:

You need $200,000 to pay a loan shark (or have an operation) but only have $100,000. Being broke or having any amount less than $200,000 means you will die. You don’t much care if you end up with $0 or $190,000.

DavidApril 10, 2011 at 10:56 pm