Assuming you have 44% equity vs someones overall open shoving range in end game poker, at what stack depth is calling from the big blind better than folding and where is the break even point?
S = effective stack
SB = 0.5 BB
BB = 1.0 BB
P0 = (1.0 + 0.5) = 1.5 BB
villain open jams from SB
J = (S - 0.5)
P1 = 1.5 + (S - 0.5) = (S + 1.0) BB
You need to call (S - 1.0) BB
to buy your equity share of
P2 = (S + 1.0) + (S - 1.0) = 2S BB
Let your equity in P2 be 'e'
EV(call) = e * P2 - (S - 1.0)
The breakeven point is where EV(call) = 0
We know e = 44% = 0.44
then you can solve for S
0 = 0.44 * 2 * S - (S - 1.0)
0 = -0.12 * S + 1.0
S = 8.3 BB
Therefore when S = 8.3 BB you are indifferent between calling or folding to an open jam range against which you have 44% equity.
Therefore @ S < 8.3 BB calling a jam with 44% equity is better than folding.
Also FYI: some reference points for the required equity vs open jams for different S values in hyper HUSNGs:
S
e
25
48%
24
48%
23
48%
22
48%
21
48%
20
48%
19
47%
18
47%
17
47%
16
47%
15
47%
14
46%
13
46%
12
46%
11
45%
10
45%
9
44%
8
44%
7
43%
6
42%
5
40%
4
38%
3
33%
Note that as effective stacks decrease, ceteris paribus you can call open jams with a wider range because your equity in the blinds becomes more valuable relative to your remaining stack.
S = effective stack
SB = 0.5 BB
BB = 1.0 BB
P0 = (1.0 + 0.5) = 1.5 BB
villain open jams from SB
J = (S - 0.5)
P1 = 1.5 + (S - 0.5) = (S + 1.0) BB
You need to call (S - 1.0) BB
to buy your equity share of
P2 = (S + 1.0) + (S - 1.0) = 2S BB
Let your equity in P2 be 'e'
EV(call) = e * P2 - (S - 1.0)
The breakeven point is where EV(call) = 0
We know e = 44% = 0.44
then you can solve for S
0 = 0.44 * 2 * S - (S - 1.0)
0 = -0.12 * S + 1.0
S = 8.3 BB
Therefore when S = 8.3 BB you are indifferent between calling or folding to an open jam range against which you have 44% equity.
Therefore @ S < 8.3 BB calling a jam with 44% equity is better than folding.
Also FYI: some reference points for the required equity vs open jams for different S values in hyper HUSNGs:
S
e
25
48%
24
48%
23
48%
22
48%
21
48%
20
48%
19
47%
18
47%
17
47%
16
47%
15
47%
14
46%
13
46%
12
46%
11
45%
10
45%
9
44%
8
44%
7
43%
6
42%
5
40%
4
38%
3
33%
Note that as effective stacks decrease, ceteris paribus you can call open jams with a wider range because your equity in the blinds becomes more valuable relative to your remaining stack.