I. Introduction

Piggybacking on Tommy Angelo's concept of reciprocality, I have studied the concept of poker insurance to see where I can make better decisions than my foes.

As many are aware, some underground games provide insurance. For example, 12 outs with 1 to come is 2:1. You can lock up $200 for every $300 in the pot against that 12 outer. In a $350 pot, the player with the 12 outer has the option to take the $100 and give up $200 and play for the rest ($50). If the player with the draw declines the insurance and decides to play for the entire pot, the house will lay the insurance.

Insurance comes with a premium of course. 12 outs is really 32:12, not 24:12 with 1 to come and two hands exposed. You give up some of your edge when you insure. The benefits of paying this "premium" include: preventing tilt when they suck out, keeping you in a good game if you are playing on your last buy-in, or locking up a win if you get involved in a big pot right before you planned to leave. Basically, insurance reduces your variance at a price.

Of course, dealing it twice (for half the pot each time) also reduces variance at no lost EV.

Some players deal it once when they are ahead no matter what. They also tend not to take the insurance when behind. Other players love to deal twice when ahead or behind. Many will pay the insurance premium when ahead, and not take the insurance when behind. Some insure their hands to the felt. Others insure only what they put into the pot.

The goal of this post is to enable you to recognize what the premium is so you can make an informed decision whether to insure your hand if ahead, or take the insurance if offered when behind.

II. The Cost of Insurance With 1 Card to Come

Perfect reciprocality, of course, would mean running it once or twice when ahead, and (almost) always taking the insurance when offered when behind. This preserves your full EV when ahead, and allows you to capture some EV gain from a bad set of circumstances when behind.

But what if you tilt? What if you are on your last buy in a juicy game? What if you found yourself in a big pot right before you were preparing to leave? Here's the scoop on how to manipulate insurance in your favor versus the unthinking crowd.

First, the price. Here is a typical insurance chart, with the house edge in percentage, with 1 card to come:

1 out - 20:1 - 2.5%
2 outs - 12:1 - 3.1%
3 outs - 8:1 - 4.3%
4 outs - 6:1 - 5.2%
5 outs - 5:1 - 5.3%
6 outs - 4:1 - 6.4%
7 outs - 3.5:1 - 6.3%
8 outs - 3:1 - 6.8%
9 outs - 2.75:1 - 6.2%
10 outs - 2.5:1 - 5.8%
11 outs - 2.25:1 - 5.8%
12 outs - 2:1 - 6.1%
13 outs - 1.9:1 - 4.9%
14 outs - 1.8:1 - 3.9%
15 outs - 1.7:1 - 2.9%
16 outs - 1.6:1 - 2.1%
17 outs - 1.5:1 - 1.4%
18 outs - 1.4:1 - 0.8%
19 outs - 1.3:1 - 0.3%
20 outs - 1.2:1 - 0.0%
21 outs - 1.1:1 - -0.1%
22 outs - EVEN - 0%

III. Observations

The first thing we can conclude is that insurance is cheapest when insuring 1-3 outs or 14 outs or more. From 4-13 outs, you will be paying at least 5%. So, generally speaking, when you are ahead, be more willing to run it twice instead of taking insurance with 4-13 outs with 1 to come. If you do insure at 4-13 outs, only insure what you have put in the pot. Don't insure to the felt. The less you insure, the less premium you pay compared to insuring the whole pot. On the other hand, with 1 out, you can insure to the felt and not give up very much in relation to the tilt factor that might occur if you went buck naked and they hit.

If you are behind, conversely, take that insurance when they want to insure those 4-13 outers. If you have a 1 outer, however, you don't lose much when you decline the insurance and go for the big score.

Another thing is that insurance gets very cheap at 15-20 outs, and actually is in the player in the lead's favor at 21 outs. Dealing it twice is not that much of a better deal than insurance for reducing variance at, for example, 18 outs. Further, if you have the 18 outer, taking the insurance is not that great a deal.

IV. The Effect of Tie Outs

Finally, the last thing to consider is tie outs. Insurance is off in case of a tie. Thus, tie outs reduce the house edge. For example, with 8 outs insurance is 24:8 while the true odds are 36:8. With 6 tie outs, however, the true odds become 30:8, since insurance is off when one of the 6 tie cards hit. Thus, 6 tie outs reduce the house edge in half on an 8 outer. From 6.8% to 3.4%. Which gets us into 2 outer territory in terms of house edge (3.1%).

The more outs the player drawing has, the fewer tie outs are needed to severely cut the house edge. For example, on a 4 outer the insurance is 24:4. True odds are 40:4. You'd need 8 tie outs to cut the house edge in half.

However, on a 12 outer, the insurance is 24:12 and the true odds are 32:12. So you only need 4 tie outs to cut the house edge in half.

Finally, the discussion above assumes two players all in. If three or more players are all in on the hand, then every other hand thats exposed has the same effect of 2 tie outs in terms of cutting the house edge.

Bottom line, when there are tie outs, you should be more willing to insure your hands, and less eager to take the insurance if behind since the juice is cut.

V. Two Card Insurance

With 2 cards to come, the house doubles the outs for insurance purposes. 5 outs twice becomes 10 outs on the chart. This overstates the chance of the draw hitting. For example, we know the true formula for figuring a flush draw is not 9/45 + 9/44 (40.45%) but rather 1-(36/45*35/44) or 35%.

As a result, two-card insurance is generally very expensive. The only insurance lay where the house edge is less than 5% is 1 out twice for 12:1, which is 3.3% juice. The house edge can reach as high as 8% with 2 to come.

Plus, there are many variables at play. There can be backdoor draws (which would need to be reinsured on the turn). Tie outs are not fully defined on the flop.

Clearly, if you can turn your foe dead, you never want to insure both the turn and the river. Why pay for 2 cards of insurance when its possible you won't need it?

When you can turn your foe dead, it's much better to offer to deal 1 turn and 2 rivers if you want to reduce variance. This way, you win the whole pot if you turn him dead, and if he draws out on the turn, you have 2 chances at a redraw.

So, only consider 2 card insurance if: you cannot turn your foe dead, and there are tie outs buffering the big house edge.

Conversely, if you are behind and are offered two-card insurance, you should almost always take it. Especially if he can turn you dead. The only way you should even consider declining is if there are a ton of tie outs.

Finally, some houses allow for 1-card insurance on the turn. This is a particularly bad deal, especially if you cannot turn them dead. Take one pair vs a flush draw. 90K pot. One card insurance is 2.75:1 or 66K to 24K. 9 times you win 66K. 36 times you have to reinsure the full pot at 66-24, guaranteeing you 66K from which you have to pay back 24K from the turn blank. So, 36 times you net 42K. 9*66K + 36*42K = 46.8K net on average. However, if you insured both cards, thats 1.4:1 or 52.5K to 37.5K in a 90K pot. So, taking two-card insurance versus one at a time saves you 52.5K - 46.8K or +5.7K. This is true because while two card insurance is more expensive than 1 card, it's not twice as expensive.

VI. Conclusion

I hope this post has opened your eyes to how much insurance decisions are costing you, either when you insure the best hand or decline the insurance when you are drawing. I hope that the discussion of tie outs and turning your foe dead will help you make better insurance decisions.