[SumZero]

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Well first, I wonder if there is a "right" answer to this question because "optimal strategy" would essentially be relative to each player's idea of what is and is not a good deal.

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The optimal straties in game theoretic context reffered to as a Nash Equillibrium (NE). A NE is a set of strategies such that player A is reacting optimally (i.e getting the highest payoff) given Bs strategy and player B is reacting optimally given As strategy -i.e. both players are acting optimally given the strategies of the other player. Using this concept of an equillibrium in a finite period game what is fair or what is a good deal doesn not matter.

As was pointed out above there may be reasons to expect typical players to deviate from NE strategies in this sort of game (in the real world).

Passion

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And, iirc, this type of game has been simulated. Where a group of people were supposed to devise rules for how their programs would act as A or B knowing that they'd get to play 100 times as A and 100 times as B against every other program with their goal being to have the most money possible. That is a slight wrinkle on the game because if there were 10 programs then accepting any split is better for you against 8 of the programs not involved in the split but may be worse against the other player you are playing against.