Von Neumann and Morgenstern 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. The only properties that matter for this are:
The maximin value v1=max(min(W,X),min(Y,Z)), which represents each player’s best alternative to negotiated agreement (BATNA).
The maximum total utility v2=max(2W,X+Y,2Z).
There are two cases:
The inessential case, v2=2v1. This includes the Abundant Commons with 2W>X+Y. No player has any incentive to negotiate, because the BATNA is Pareto-optimal.
The essential case, v2>2v1. This includes all other games in the OP.
It might seem strange that VNM consider, say, 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, and this is equivalent to threatening to defect.
Von Neumann and Morgenstern 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. The only properties that matter for this are:
The maximin value v1=max(min(W,X),min(Y,Z)), which represents each player’s best alternative to negotiated agreement (BATNA).
The maximum total utility v2=max(2W,X+Y,2Z).
There are two cases:
The inessential case, v2=2v1. This includes the Abundant Commons with 2W>X+Y. No player has any incentive to negotiate, because the BATNA is Pareto-optimal.
The essential case, v2>2v1. This includes all other games in the OP.
It might seem strange that VNM consider, say, 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, and this is equivalent to threatening to defect.