Von Neumann and Morgenstern [LW · GW] also classify the two-player games, but they get only two games, up to equivalence. The reason is they assume the players get to negotiate beforehand. For them the only properties that matter are:

• The maximin value $v_1=max(min(W,X),min(Y,Z\left)\right)$, which represents each player's best alternative to negotiated agreement (BATNA).

• The maximum total utility $v_2=max(2W,X+Y,2Z)$.

There are two cases:

1. The inessential case, $v_2=2\ast v_1$. This includes the Abundant Commons with $2W>X+Y$. No player has any incentive to negotiate, because the BATNA is Pareto-optimal.

2. The essential case, $v_2>2\ast v_1$. This includes all other games in the OP.

It might seem strange that VNM consider Cake Eating to be equivalent to Prisoner's Dilemma. But in the VNM framework, Player 1 can threaten not to eat cake in order to extract a side payment from Player 2, just and this is the same as threatening to defect.