Re: AA
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Its not a value bet in the traditional sense. He is not planning on being ahead 55% of the time that he is called. Its just that he is more of an underdog when checks and calls than when he bets and is called.
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Being more of an underdog doesn't matter in this case given the pot size. If she never bets AA it's a check-fold. But if she bets AA 50% of the time it's a check-call (earns .75BB if we're wondering if she has AA or KK), which is better than a bet, which loses 1BB the vast majority of the time here. If she bets AA 40% of the time she has AA (even accounting for only one combo of AA) we break even on our 1BB investment. In order for our river bet to be breakeven she will have to have JJ or AK some portion of the time, given how often we will be bet-folding or bet-chopping. Given how this hand has played out, especially preflop and on the flop, that doesn't seem nearly likely enough for a bet to have more EV than a check-call.
IMO the decision is somewhere between check-calling and check-folding, but betting seems particularly spewy.
Rob
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