You don’t have that much leverage as a new grad, so your best bet is counter offers. Keep in mind that you really want to optimize for your long term career, so it makes sense to value an environment in which you can learn a lot and grow rapidly over immediate monetary reward.
You should try but probably won't get anything unless you have an offer letter from elsewhere.
Just try, you have nothing to lose. Just straight up ask for $x base, $x shares etc etc
Multiple offers take 2nd highest to highest and ask them to beat it. Then STFU till they give you more.
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You don’t
Yeah, you basically have no leverage. Honestly, the only way to do it as a new grad would be to get offers from multiple places and be ready to walk if they don't counter.