Peter McBurney wrote:

Bill --

One difference is that von Neumann/Nash games typically assume a payoff (a reward or loss) to participants upon termination of the game, whereas the abstract games discussed in game semantics usually do not assume this.

Although it seems that three-player generalizations of the latter type of games may need something more nuanced than just "win" or "lose" to ensure that there is not a situation in which a player who cannot himself win may make an arbitrary choice of which other player does so. One obvious solution is to declare a "winner" and "loser" [who has to pay the winner, wash the winner's car, or whatever], and (for instance,in a Nim-type game) to declare that the player due to play immediately after the winner is the loser. It is then not a matter of indifference to any player how the game turns out. However, there are wheels within wheels: for in a game that is not completely trivial [a trivial game would be like Nim with N piles of size 1, in which there are no bad moves] there is the possibility that the player who is due to come in second if everybody plays to maximize their immediate position may choose to "throw" the game, moving to third place and putting the erstwhile loser into the lead. This would be an irrational play on its own, but in combination with a pact for the new leader to throw the game in turn, both conspirators would end up ahead of their original positions. The question now is - is there honor among hustlers? Will the original loser renege? Can the pact be enforced? This lands us fair and square in the middle of the von Neumann/Nash kind of game theory. What if anything this says about generalizations of game sematics I do not know. John H. Conway told me when I was a graduate student that this area was under active investigation by somebody or other, but I haven't heard of anything that came of it. -Robert Dawson