Yesterday at a live game I raised all-in vs another players C-bet.

His decision depends on whether I would "run it twice".

If yes, he calls, if no, he folds.

Hand was 1/2/5/10, hero open 40, 3bet 110, hero call 110.

Flop has £228 in middle, Cbet is £160, my raise all in is £550.

So what is higher EV?

Take the £270 risk free 100% of time

Or run it twice vs his 6 outs / 24% (1 pair vs 2 overcards) to potentially win £660 (56%) or chop which I consider a loss of £270 (About 50%) or lose £660 (6.7%) outright

Run it twice does not affect EV. This is just a pot odds vs. equity problem.

I don't know how you know his equity unless he showed you his cards, but if he has 24% equity, he needs at least about 3:1 pot odds to call. He's calling £390 to win £938. He's only getting 938:390 = 2.4:1 pot odds, so you want him to call.

Where are you getting 270 from?

I’m not able to follow your dollar (pound) numbers or your % numbers. But since EV is the same for run it once and run it twice, the decision can be made quite easily by just considering run it once.

If the pot is Pot prior to your bet of Bet and villain’s call of Call, you should agree to run it twice if

P(Win)*(Pot + Call) + P(Chop)*(Pot/2) - P(Lose)*Bet > Pot

The probabilities must add to 100%.

For this unique situation, I don’t think the benefit of the reduced variance of RIT is a factor.

That is the amount V has bet pre and flop, ignoring my call (110) and dead money (8)

Maybe the nature of possible chop or lost has confused me, after he made a direct offer of 100% of the pot if I say no to RIT.

I have not seen his cards, but his equity is between "8% under pair" to "33% up n down", assuming I would not be able to fold a better hand.

Why would you ignore either one of those? This is important for you to understand if you want to solve these types of problems.

He needs 29.4% equity to call (it's £390 to call, pot is £938. 390/(938+390)=0.29367). If his equity is 24%, you want him to call.

Another way of looking at it, if he calls (and has 24% equity), you win a £1328 pot 76% of the time. 0.76*1328= £1009.28. If he folds, you win £938 100% of the time. £1009.28>£938 so you'd rather he calls.

This assumes you somehow know for a fact his equity is 24%. Running it once or twice is irrelevant. The equation for running it twice would be (0.76*664)+(0.76*664)=1009.28. £1009.28>£938 so it's the same thing as above.

The conclusion is correct but I question the EV analysis. You only considered hero's win amount with a call. What about his possible loss?

Prior to hero's raise of 550, the pot was 228 + 160 = 388. If villain calls, adding 390 to hero's potential winnings, hero EV for doing RIT is

EV = 0.76 *778 - 0.24*550 = 459.

This is greater than the 388 pot hero wins if villain folds.

The equation for running it twice is not correct. In its most general comparative EV form it would be

EV = P(W1)*[P(W2|W1)*Pot + P(L2|W1)*Pot/2] + P(L1)*[P(W2|L1)*Pot /2

where Wi and Li are win and lose run i, respectively.

Without knowing the card specifics I don't see how this can be solved, though it's easy to show the EVs for one or two runs are equal.

I'm not sure I understand the point you're arguing. We're not considering hero's win amount, we're considering his equity in the pot (currently his 'stack' is zero, so the only relevant factor is his equity in two different outcomes - namely 100% of £938 (the current pot he would win which raise his stack from 0 to 938) or 76% of £1328 (the equity in the pot if called, this raises his stack from 0 to £1009 on average). I'm not sure what else is relevant (assuming the question is "do I want villain to call or fold").

Okay, I get your point that the decision is after hero bet so there is no further loss. I was using as the the decision point his raise under the premise of the RIT question coming up.