Dara O’Kearney: Should you insta-rebuy?

Dara O’Kearney investigates whether you should take the instant rebuy option in online tournaments and explains why you could make big money by saying no

At the start of Collin Moshman’s seminal textbook on sit-and-go strategy, he runs through the maths of what happens when two players in a nine-man sit-and-go get into an all-in coinflip in the first hand. Moshman proves that while the equity of the loser of the flip obviously drops by 100%, the equity of the winner does not increase by 100%. It actually increases by less than 60%, and the remaining equity is divided equally by the players sitting on the sidelines. This example illustrates that it’s a major mistake to get it in flipping in this spot because you are risking 100% to gain (slightly under) 60%. Essentially you’re flipping a coin where you have to pay a tenner if you lose but only get £6 when you win. Trust me on this one: do that enough times and you’ll end up broke.

Every penny counts

This idea is now widely known and sit-and-go grinders will pass on marginal spots early. However, there is one area where similar considerations apply but until recently even good regs were not taking this into account: rebuy tournaments. Consider the classic ‘cubed’ format on PokerStars. In a $10 cubed, you start with 1,500 chips (which costs $11 including registration fee). There is an optional rebuy of a further 1,500 chips for $10 (no reg), and an optional add-on at the end of the rebuy period for $10. Until recently, the accepted consensus among regs was to take the rebuy at the start (thereby effectively buying a double starting stack). Recently however, good regs have increasingly been switching to not taking the rebuy (unless necessary).

Let’s have a look at the maths to see why they think this is a good idea.

If you are playing a $10 cubed and everyone immediately takes the rebuy, then everyone’s equity at the start is $20 (for which they have paid $21). If, however, some of the field don’t take the rebuy, then ICM dictates they gain some equity at the expense of those who do take it. The question is how much equity do they gain? Obviously it won’t be as much as in a sit-and-go where ICM is a bigger factor. So I ran some simulations through an ICM tool using typical field sizes and payout structures.

It turns out that if 90% of the field rebuy, their equity at the start drops about a cent to $19.99 as a result of the 10% who don’t rebuy. Those guys gain $0.09 each. What happens if more people decide not to rebuy? Well, the rebuyers lose significantly more (their equity now drops to $19.89), while those who don’t rebuy gain $0.11 each. If only 10% of the field rebuys, they lose $0.55, while those not rebuying gain about $0.06 each.

Balancing the books

You might say, ‘Oh, these are all tiny edges anyway, who cares?’ Well, a few years ago, very few people did, which is why regs happily went on flicking in the extra buy-in for the extra chips. When your overall edge is 40% or 50%, you’re not going to sweat the 0.1% you lose rebuying, or the 0.9% edge you could gain by not doing so.

However, this is gradually changing for two reasons:

  1. Overall edges are coming down. Many pros are now making a living on very small edges playing high volume. A typical mid-stakes volume grinder might play 20,000 tournaments a year at an average buy-in of $30. If his edge (as measured by his long-term return on investment) is only 5%, then he is making $30,000 a year. If he switches from rebuying (giving up 0.1%) to not rebuying (now gaining 0.9%), he increases his edge by 1%, which translates to earning an extra $6,000 a year. Those cents add up!
  2. As more and more people cotton on to the equity they gain by not rebuying, it becomes an even bigger mistake to go on auto-rebuying. In some spots the difference is over 6%, which is enough to change our high volume mid-stakes grinder from a pro with a 5% edge making $30k a year to a losing player!

Follow the leaders

This is one of those rare occasions in poker where it pays to be a sheep rather than a wolf. Back when everyone was clicking the rebuy button, it made little difference if you did or not. The minor chip equity hit in doing so could easily be compensated for by your superior skill with more chips. The 1,500 extra chips might only have a theoretical value of 1,485 at the start, but most pros would back themselves to turn those 1,485 chips into more than 1,500 in an average hour. But as we move from a scenario where most people were rebuying to one where most don’t, the differential means we are not talking about losing 15 theoretical chips any more, but rather 90. Very few regs would back themselves to turn 1,410 additional chips into 1,500 on average in an hour.

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