So you're arguing that, taking RoR into account, Ivey shouldn't have played Beal in the first place, so therefore now that he was playing him running it twice/reducing the variance would be meaningless? Sounds pretty illogical to me...

It's illogical because you're not getting it as well.

The initial discussion was that the reason why Ivey doesn't run it twice is because his RoR is lower relative to his opponent's, and if he was to play against Bill Gates for much bigger pots, he would run it twice because his RoR is higher in such case.

I disagree and argued that if the reason why Ivey doesn't run it twice is because of RoR, then he probably wouldn't have played against Beal, who is capable of increasing Ivey's RoR exponentially. I also do not think it matters much that the game was LHE, because the argument is about RoR.

Oh. I think you're probably right. Pretty sure Ivey only runs it once, because he is oldschool and he wants maximum pain. Perhaps also because he feels he can deal with the swings better than his opponents.

This is not very relevant. Risking the Kelly fraction maximises the expected long term return when a bet is repeated many times. Risking up to twice the Kelly fraction has positive long term return. But in this case, a single event is under discussion. The most appropriate quantitative measure would be the expectation of the utility function for the person considering the wager. If the utility function is concave, running it twice will increase the expected utility and will be preferable, if the utility function is convex, running it twice will reduced the expected utility and will be not preferable. A typical concave utility function would be where a chance of doubling your net worth would not make up for an equal chance of losing your entire net worth.

There was an interesting true story a few years back where some guy cashed his entire net worth (around $100,000), walked into a Las Vegas casino (in his boxers) and put the whole amount on a roulette bet on red. After doubling up, he declined the croupier's offer to play again, indicating that his utility function had suddenly gone from being convex enough to not worry about a 5% house edge, to a function that was not convex enough to justify a second bet.

challenge accepted sir, lol

Me neither when it comes to general independent chip model calculations, but the simple idea is the non-linear relationship between chips and expectations in a tournament, compared to the linear relationship between chips and money in a cash game. This sometimes means it is optimal to be a risk-seeker rather than risk-aversive or risk-neutral (as in a cash game). [Sometimes the opposite can also happen, with it being best to be risk-averse for a while]

The clearest example I can think of is where you are the shortest stack in the last 4 of a tournament which pays 3 people and so short stacked that you will be eliminated by the antes very soon. You have a chance to go all-in for a 50-50 shot with the player in 3rd place (who has much less than twice your chips) just before you get anted out. In a cash game taking the 50-50 shot would increase your variance without increasing your expectation and you could pass. Here it is extremely profitable to take the shot since it gives you a good chance of third prize rather than 100% chance of nothing. [Actually worse than a 50-50 shot would be good enough, if in the next hand you would get forced all-in in the blinds, with 2 big stacks looking for a chance to knock you out]