Via kottke.org, an article in New Scientist on the mathematics of gambling.  One bit concerns arbitrage in online sports wagering.

Let’s say, for example, you want to bet on one of the highlights of the British sporting calendar, the annual university boat race between old rivals Oxford and Cambridge. One bookie is offering 3 to 1 on Cambridge to win and 1 to 4 on Oxford. But a second bookie disagrees and has Cambridge evens (1 to 1) and Oxford at 1 to 2.

Each bookie has looked after his own back, ensuring that it is impossible for you to bet on both Oxford and Cambridge with him and make a profit regardless of the result. However, if you spread your bets between the two bookies, it is possible to guarantee success (see diagram, for details). Having done the calculations, you place £37.50 on Cambridge with bookie 1 and £100 on Oxford with bookie 2. Whatever the result you make a profit of £12.50.

I can verify that arbitrage opportunites abound.  In my research with Toomas Hinnosaar on sorophilia, we investigated an explanation involving betting.  In the process we discovered that the many online bookmakers often quote very different betting lines for basketball games.

How could bookmakers open themselves up to arbitrage and still stay in business?  Here is one possible story.  First note that, as mentioned in the quote above, no one bookmaker is subject to a sure losing bet.  The arbitrage involves placing bets at two different bookies.

Now imagine you are one of two bookmakers setting the point spread on a Clippers-Lakers game and your rival bookie has just set a spread of Lakers by 5 points.  Suppose you think that is too low and that a better guess at the spread is Lakers by 8 points.  What spread do you set?

Lakers by 6.  You create an arbitrage opportunity.  Gamblers can place two bets and create a sure thing:  with you they take the Clippers and the points.  With your rival they bet on the Lakers to cover.  You will win as long as the Lakers win by at least 7 points, which is favorable odds for you (remember you think that Lakers by 8 is the right line.)  Your rival loses as long as the the Lakers win by at least 6 points, which is unfavorable odds for your rival.  You come away with (what you believe to be) a winning bet and you stick your rival with a losing bet.

Now this begs the question of why your rival stuck his neck out and posted his line early.  The reason is that he gets something in return: he gets all the business from gamblers wishing to place bets early.  Put differently, when you decided to wait you were trading off the loss of some business during the time his line is active and yours is not versus the gain from exploiting him if he sets (what appears to you to be) a bad line.

Since both of you have the option of playing either the “post early” or “wait and see” strategy, in equilibrium you must both be indifferent so the costs and benefits exactly offset.

Of course, with online bookmaking the time intervals we are talking about (the time only one line is active before you respond, and the time it takes him to adjust to your response, closing the gap) will be small, so the arbitrage opportunities will be fleeting.  (As acknowledged in the New Scientist article.)