Thursday, March 31, 2005

Craps, Odds, and Expected Returns

I’ll be in Vegas this weekend and have been brushing up on some gambling strategies. One great resource that I like to use is the Wizard of Odds. It gives you the best play on all the basic casino games and allows you to play some of them for free.

But one strategy that has always struck me as intuitively wrong deals with taking the fullest odds allowed in craps. The wizard's craps guide, like other strategies I’ve seen, suggest taking the fullest odds possible after a pass line bet. The basic idea is that the house has an 1.41% on the pass line bet while there is no house edge on the odds bet. So every extra dollar on the odds bet reduces the average house edge:

House Edge – Pass Line Bet With Odds
Odds Edge
1X .848%
2X .606%
3X .471%
5X .326%
10X .184%
20X .099%
100X .021%

As the table shows this strategy seems to make sense. You can reduce the house edge all the way to .021% with 100X odds! But hold on. What’s the expected return on each of these bets? I’ll use a pass line bet of $100 (I wish!) to demonstrate:

Expected Return – Pass Line Bet With Odds
Odds Expected Return
1X -$1.41
2X -$1.41
3X -$1.41
5X -$1.41
10X -$1.41
20X -$1.41
100X -$1.41

As you can see, even though the house edge is reduced, the expected return remains unchanged. However, you now have a lot more cash on the table.

You can look at it another way to see the logic doesn’t hold. What if they weren’t "even money" odds? What if the odds bet actually had a .50% house edge. Well, again, every extra dollar bet on the odds would reduce the overall house edge. But every extra dollar is still a losing proposition and I don’t think many people would suggest betting 100X odds.

To make one more illustrative point, let’s take a car company that loses $5 dollars for every car they make. Because of this loss, they decide to make trucks that only lose $1 per vehicle. Now they can reduce the average loss per vehicle to $3! Does this sound like a good plan? Of course not. What should drive the investment decision is the marginal profit/loss not the average profit and loss.

Now of course the odds bet is still among the best you will find in the casino and anyone who gambles will want to take advantage of it. But the decision of how much to put down on the odds bet is completely independent of the house edge on the pass line bet. The pass line bet is a sunk cost and there is nothing you can do to change it. The marginal return of each bet should be the only factor in the decision making process, and of course the marginal return on the odds bet is zero.

So if your not a high roller don’t worry about only taking 1X or 2X odds on the pass line bet. There is not a valid economic reason to bet more. You will have the same expected return of the bettor who puts down 3x odds or 10x odds.